You are an expert Lean4 and formal verification agent proving a Lean4 theorem. Your output **must** strictly adhere to the specified format.

### Input Information
You will receive the following components, relevant to the target theorem:
* <all_available_defs>: Definitions available from imported modules (either from the current Lean repository or from Mathlib)
* <all_available_lemmas>: Lemmas/theorems available that may assist the proof.
* <local_ctx>...</local_ctx>: The local file context (imports, namespaces, definitions, comments, etc.) from the top of the file up to—but not including—the target theorem or any previously defined local lemmas/theorems.
* <target_theorem>...</target_theorem>: The target theorem statement, ending with `:=`.

**Notes**
1. Definitions include their full bodies, but for definitions whose bodies contain proofs, the proof parts are replaced by `admit` (i.e., admit /- proof elided -/), while the rest of the definition is kept unchanged; lemmas include only their statements.
2. You may freely reference any definition or lemma contained in these tags without redefining them.
3. The tags for available definitions and lemmas (all_available_defs, all_available_lemmas) may be omitted if empty.

### Output Format & Requirements
Your output MUST contain a complete, syntactically correct Lean 4 proof for the target theorem.

**Strict Output Structure:**
<reasoning> 
...
</reasoning> 

<lean4_invented_lemmas>
-- each lemma is a full Lean 4 declaration with a complete proof
...
</lean4_invented_lemmas>

<lean4_proof>
-- Lean 4 proof body of the target theorem
...
</lean4_proof>

**Requirements:**
1. <reasoning> and <lean4_invented_lemmas> are OPTIONAL. Include them only if they aid proof construction.
2. <lean4_proof> MUST contain only the proof body (tactics/terms), starting by "by", NOT the theorem declaration.
3. <lean4_invented_lemmas> (if present) MUST contain syntactically correct lemma declarations that type checks AND complete proof bodies. Defining a lemma using `axiom` is NOT allowed.
4. <reasoning> contains high-level proof planning.
5. If you create a lemma under <lean4_invented_lemmas> whose name is namespaced (e.g., X.Y.Z), you MUST always refer to it using its fully qualified name. Do NOT rely on shortened names. For example, write rw [X.Y.Z], not rw [Z] for invented lemmas.
6. All Lean 4 code MUST be syntactically correct. No `sorry`, `admit`, or non-Lean text. Comments allowed using `--` or `/- ... -/`.
7. Follow the tags and structure EXACTLY. No extra text before, between, or after the sections.
8. Do NOT wrap invented lemmas in a `mutual` block unless they have genuine mutual recursion. Declare each lemma independently.

### Examples are provided here (example input are under "User Prompt", and example assistant response are under "Agent Response"):
Example 1 ------------------- 
User Prompt:
<all_available_defs>
structure BitVec (w : Nat) where
  /-- Construct a `BitVec w` from a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  ofFin ::

  /-- Interpret a bitvector as a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  toFin : Fin (hPow 2 w)
</all_available_defs>

<all_available_lemmas>
theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) : (l₁ ++ l₂)[i] = l₁[i]
theorem getElem_eq_testBit_toNat (x : BitVec w) (i : Nat) (h : i < w) :
  x[i] = x.toNat.testBit i := rfl
@[simp, grind =] theorem toNat_cons (b : Bool) (x : BitVec w) :
    (cons b x).toNat = (b.toNat <<< w) ||| x.toNat
@[simp, grind =] theorem getLsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false
theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsbD i
</all_available_lemmas>

<local_ctx>
import Mathlib.Computability.NFA
import Mathlib.Data.FinEnum
import Mathlib.Data.Rel
import Mathlib.Data.Vector.Basic
import Blase.AutoStructs.ForLean

open Set
open Mathlib
open SetRel

-- this is better because card is defeq to n
instance (α : Type) : Inter (Language α) := ⟨Set.inter⟩
instance (α : Type) : Union (Language α) := ⟨Set.union⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width.
-/
structure BitVecs (n : Nat) where
  w : Nat
  bvs : List.Vector (BitVec w) n

def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=
  { w := bvs.w, bvs := h ▸ bvs.bvs }

abbrev BitVecs.empty : BitVecs n := ⟨0, List.Vector.replicate n .nil⟩
abbrev BitVecs.singleton {w : Nat} (bv : BitVec w) : BitVecs 1 := ⟨w, bv ::ᵥ .nil⟩
abbrev BitVecs.pair {w : Nat} (bv1 bv2 : BitVec w) : BitVecs 2 := ⟨w, bv1 ::ᵥ bv2 ::ᵥ .nil⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width, encoded as a list of
bit vectors of width `n`. The width of the encoded bit vectors is the length of
the list.
-/
abbrev BitVecs' (n : Nat) := List (BitVec n)

@[simps]
def dec (bvs' : BitVecs' n) : BitVecs n where
  w := bvs'.length
  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k

@[simp]
</local_ctx>

<target_theorem>
lemma dec_snoc n (bvs' : BitVecs' n) (a : BitVec n) :
  dec (bvs' ++ [a]) =
    { w := bvs'.length + 1
      bvs := List.Vector.ofFn fun k =>
        BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k) } :=
</target_theorem>

Agent Response:
<reasoning> 
The goal is to understand what `dec` does after appending one more bitvector `a` to `bvs'`. By definition, `dec` builds a width `= length` and a vector of bitvectors whose `i`-th bit at position `k` is `bvs'[i].getLsbD k`. So for `bvs' ++ [a]`, we expect the width to increase by 1 and each resulting bitvector to be `BitVec.cons` of the new bit coming from `a`.

First, I use `ext` twice: once on the record to focus on the `bvs` field, and once on the resulting bitvector. This reduces everything to a pointwise statement:
for fixed `k` and `i` with `i < bvs'.length + 1`,
the bit `(bvs' ++ [a])[i].getLsbD k` should match the bit at index `i` of `BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k)`.

At this point I need a clean description of indexing into `BitVec.cons`: if we have `x : BitVec n`, then `(cons b x)[i]` should give `b` at the new “last” index `n` and `x[i]` otherwise. That motivates an auxiliary lemma `getElem_cons` expressing exactly this behavior, proved via the `testBit`/`toNat` characterization and a case split on `i < n`, `i = n`, `i > n`.

Using `getElem_cons`, the RHS becomes an `if` on whether `i = bvs'.length`. So I split into cases:
- If `i = bvs'.length`, then `(bvs' ++ [a])[i]` is the appended `a`, so both sides reduce to `a.getLsbD k`.
- If `i ≠ bvs'.length`, the bound `i < bvs'.length + 1` forces `i < bvs'.length`. Then `(bvs' ++ [a])[i] = bvs'[i]` by `List.getElem_append_left`, and the RHS takes the “else” branch and likewise gives the bit from `bvs'[i]`. In both subcases, `simp` (plus the definition of `dec`) finishes.

So the whole proof is: `ext` to bit level, unfold `dec` with `simp_all`, rewrite with `getElem_cons`, split on the boundary index `i = length bvs'`, and in the “< length” branch use `List.getElem_append_left` and bounds to line up both sides.
</reasoning> 

<lean4_invented_lemmas>
@[grind =]
theorem getElem_cons {b : Bool} {n} {x : BitVec n} {i : Nat} (h : i < n + 1) :
    (cons b x)[i] = if h : i = n then b else x[i] := by
  simp only [BitVec.getElem_eq_testBit_toNat, BitVec.toNat_cons, Nat.testBit_or]
  rw [Nat.testBit_shiftLeft]
  rcases Nat.lt_trichotomy i n with i_lt_n | i_eq_n | n_lt_i
  · have p1 : ¬(n ≤ i) := by omega
    have p2 : i ≠ n := by omega
    simp [p1, p2]
  · simp only [i_eq_n, ge_iff_le, Nat.le_refl, decide_true, Nat.sub_self, Nat.testBit_zero,
    Bool.true_and, BitVec.testBit_toNat, BitVec.getLsbD_of_ge, Bool.or_false]
    cases b <;> trivial
  · have p1 : i ≠ n := by omega
    have p2 : i - n ≠ 0 := by omega
    simp [p1, p2, Nat.testBit_bool_toNat]
</lean4_invented_lemmas>

<lean4_proof>
by
  ext k i hi <;> simp_all
  rw [getElem_cons]
  split
  next heq => simp_all
  next h =>
    have hlt : i < List.length bvs' := by simp at hi; omega
    rw [List.getElem_append_left hlt]
    simp
</lean4_proof>

### Your task:
<all_available_defs>
BitVec.xor in Init.Data.BitVec.Basic
Bool in Init.Prelude
DecidableEq in Init.Prelude
Repr in Init.Data.Repr
List in Init.Prelude
Finset in Mathlib.Data.Finset.Defs
Sum in Init.Core
Sum.elim in Init.Data.Sum.Basic
Polynomial.X in Mathlib.Algebra.Polynomial.Basic
ZMod in Mathlib.Data.ZMod.Defs
Sum.inl in Init.Core
Sum.inr in Init.Core
HOr in Init.Prelude
HOr.hOr in Init.Prelude
OrOp in Init.Prelude
AndOp in Init.Prelude
HAnd in Init.Prelude
HAnd.hAnd in Init.Prelude

noncomputable def f : (ZMod q)[X] := X^(2^n) + 1

def revertBvHyps (g : MVarId) : MetaM MVarId := do
  let (_, g) ← g.revert (← g.getNondepPropHyps)
  return g

def trunc_shl_7_i8_i64_before := [llvm|
{
^0(%arg37 : i64):
  %0 = llvm.mlir.constant(7) : i64
  %1 = llvm.shl %arg37, %0 : i64
  %2 = llvm.trunc %1 : i64 to i8
  "llvm.return"(%2) : (i8) -> ()
}
]

Id.run in Init.Control.Id

def test_after := [llvm|
{
^0(%arg0 : i26):
  "llvm.return"(%arg0) : (i26) -> ()
}
]

def p0_scalar_after := [llvm|
{
^0(%arg14 : i32, %arg15 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.add %arg14, %0 : i32
  %2 = llvm.add %1, %arg15 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def trunc_sel_smaller_zext_before := [llvm|
{
^0(%arg69 : i64, %arg70 : i1):
  %0 = llvm.mlir.constant(42 : i16) : i16
  %1 = llvm.trunc %arg69 : i64 to i16
  %2 = "llvm.select"(%arg70, %1, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i16, i16) -> i16
  %3 = llvm.zext %2 : i16 to i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def xor_icmp_true_unsigned_before := [llvm|
{
^0(%arg7 : i32):
  %0 = llvm.mlir.constant(5 : i32) : i32
  %1 = llvm.mlir.constant(6 : i32) : i32
  %2 = llvm.icmp "ugt" %arg7, %0 : i32
  %3 = llvm.icmp "ult" %arg7, %1 : i32
  %4 = llvm.xor %2, %3 : i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def mul9_low_after := [llvm|
{
^0(%arg8 : i9, %arg9 : i9):
  %0 = llvm.mlir.constant(15 : i9) : i9
  %1 = llvm.mlir.constant(4 : i9) : i9
  %2 = llvm.and %arg8, %0 : i9
  %3 = llvm.lshr %arg8, %1 : i9
  %4 = llvm.and %arg9, %0 : i9
  %5 = llvm.lshr %arg9, %1 : i9
  %6 = llvm.mul %5, %2 overflow<nuw> : i9
  %7 = llvm.mul %4, %3 overflow<nuw> : i9
  %8 = llvm.mul %4, %2 overflow<nsw,nuw> : i9
  %9 = llvm.add %6, %7 : i9
  %10 = llvm.shl %9, %1 : i9
  %11 = llvm.add %10, %8 : i9
  "llvm.return"(%11) : (i9) -> ()
}
]

def no_masks_with_logical_or2_before := [llvm|
{
^0(%arg7 : i32, %arg8 : i32, %arg9 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.mlir.constant(63 : i32) : i32
  %2 = llvm.mlir.constant(true) : i1
  %3 = llvm.icmp "ne" %arg7, %0 : i32
  %4 = llvm.icmp "ne" %arg8, %1 : i32
  %5 = "llvm.select"(%3, %2, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %6 = llvm.icmp "ne" %arg9, %0 : i32
  %7 = llvm.or %5, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def test_sub_nuw__nsw_is_safe_before := [llvm|
{
^0(%arg22 : i32):
  %0 = llvm.mlir.constant(-2147483648 : i32) : i32
  %1 = llvm.mlir.constant(-2147483647 : i32) : i32
  %2 = llvm.mlir.constant(-1 : i32) : i32
  %3 = llvm.or %arg22, %0 : i32
  %4 = llvm.icmp "eq" %3, %1 : i32
  %5 = llvm.sub %0, %3 overflow<nuw> : i32
  %6 = "llvm.select"(%4, %2, %5) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def add_undemanded_low_bits_before := [llvm|
{
^0(%arg1 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(1616 : i32) : i32
  %2 = llvm.mlir.constant(4 : i32) : i32
  %3 = llvm.or %arg1, %0 : i32
  %4 = llvm.add %3, %1 : i32
  %5 = llvm.lshr %4, %2 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

IO.FS.Handle.mk in Init.System.IO

def a_true_implies_b_true2_comm_after := [llvm|
{
^0(%arg27 : i8, %arg28 : i1, %arg29 : i1):
  %0 = llvm.mlir.constant(20 : i8) : i8
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.icmp "ugt" %arg27, %0 : i8
  %3 = "llvm.select"(%2, %arg28, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def not_and_and_not_commute1_before := [llvm|
{
^0(%arg561 : i32, %arg562 : i32, %arg563 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.xor %arg562, %0 : i32
  %2 = llvm.xor %arg563, %0 : i32
  %3 = llvm.and %1, %arg561 : i32
  %4 = llvm.and %3, %2 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]
    (new_arity : Type)        
    (q_arity : arity → Type)  
    (vars : ∀ (a : arity), q_arity a → new_arity)
    
    
    (q : ∀ (a : arity), FSM (q_arity a)) : 
    FSM new_arity :=
  { α := p.α ⊕ (Σ a, (q a).α),
    i := by admit /- proof elided -/

def icmp_sle9_after := [llvm|
{
^0(%arg4 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "ne" %arg4, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

propext in Init.Core

def foo_after := [llvm|
{
^0(%arg0 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.sub %0, %arg0 : i32
  %3 = llvm.and %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def not_xor_to_or_not1_after := [llvm|
{
^0(%arg39 : i3, %arg40 : i3, %arg41 : i3):
  %0 = llvm.mlir.constant(-1 : i3) : i3
  %1 = llvm.or %arg40, %arg41 : i3
  %2 = llvm.and %arg39, %arg41 : i3
  %3 = llvm.xor %1, %0 : i3
  %4 = llvm.or %2, %3 : i3
  "llvm.return"(%4) : (i3) -> ()
}
]

def src_srem_shl_demand_max_mask_hit_demand_before := [llvm|
{
^0(%arg18 : i32):
  %0 = llvm.mlir.constant(4 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.mlir.constant(-4 : i32) : i32
  %3 = llvm.srem %arg18, %0 : i32
  %4 = llvm.shl %3, %1 : i32
  %5 = llvm.and %4, %2 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def printPrefix (x : BitStream) : Nat → String
  | 0   => "0b"
  | n+1 =>
    let h := if x.head then "1" else "0"
    let t := x.tail.printPrefix n
    t ++ h

def lshrult_02_09_after := [llvm|
{
^0(%arg342 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def main3b_before := [llvm|
{
^0(%arg153 : i32):
  %0 = llvm.mlir.constant(7 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(16 : i32) : i32
  %3 = llvm.mlir.constant(1 : i32) : i32
  %4 = llvm.and %arg153, %0 : i32
  %5 = llvm.icmp "eq" %4, %1 : i32
  %6 = llvm.and %arg153, %2 : i32
  %7 = llvm.icmp "ne" %6, %2 : i32
  %8 = llvm.and %5, %7 : i1
  %9 = "llvm.select"(%8, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%9) : (i32) -> ()
}
]

def test1_after := [llvm|
{
^0(%arg14 : i32, %arg15 : i32):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def ashrslt_03_13_after := [llvm|
{
^0(%arg224 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

@[simp_denote]
def icmp_eq_riscv_32_pseudo := [LV| {
  ^entry (%lhs: i32, %rhs: i32):
    %lhsr = "builtin.unrealized_conversion_cast"(%lhs) : (i32) -> (!riscv.reg)
    %rhsr = "builtin.unrealized_conversion_cast"(%rhs) : (i32) -> (!riscv.reg)
    %0 = xor    %lhsr, %rhsr : !riscv.reg
    %1 = seqz    %0 : !riscv.reg
    %2 = "builtin.unrealized_conversion_cast"(%1) : (!riscv.reg) -> (i1)
    llvm.return %2 : i1
  }]

def zext_nneg_sext_sle_op0_narrow_before := [llvm|
{
^0(%arg50 : i8, %arg51 : i16):
  %0 = llvm.zext nneg %arg50 : i8 to i32
  %1 = llvm.sext %arg51 : i16 to i32
  %2 = llvm.icmp "sle" %0, %1 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

def test4_after := [llvm|
{
^0(%arg193 : i32, %arg194 : i32):
  %0 = llvm.icmp "slt" %arg193, %arg194 : i32
  %1 = llvm.zext %0 : i1 to i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def shl_add_before := [llvm|
{
^0(%arg26 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.shl %0, %arg26 : i32
  %3 = llvm.add %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def mul_of_sdiv_fail_ub_before := [llvm|
{
^0(%arg1 : i8):
  %0 = llvm.mlir.constant(6 : i8) : i8
  %1 = llvm.mlir.constant(-6 : i8) : i8
  %2 = llvm.sdiv exact %arg1, %0 : i8
  %3 = llvm.mul %2, %1 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def positive_biggerashr_shlnuwnsw_ashrexact_before := [llvm|
{
^0(%arg25 : i8):
  %0 = llvm.mlir.constant(6 : i8) : i8
  %1 = llvm.mlir.constant(3 : i8) : i8
  %2 = llvm.ashr exact %arg25, %0 : i8
  %3 = llvm.shl %2, %1 overflow<nsw,nuw> : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def PR38139_after := [llvm|
{
^0(%arg0 : i8):
  %0 = llvm.mlir.constant(-64 : i8) : i8
  %1 = llvm.icmp "ult" %arg0, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def or_eq_notY_eq_0_before := [llvm|
{
^0(%arg45 : i8, %arg46 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.xor %arg46, %0 : i8
  %2 = llvm.or %arg45, %1 : i8
  %3 = llvm.icmp "eq" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def udiv_exact_eq_1_before := [llvm|
{
^0(%arg10 : i8, %arg11 : i8):
  %0 = llvm.mlir.constant(1 : i8) : i8
  %1 = llvm.udiv exact %arg10, %arg11 : i8
  %2 = llvm.icmp "ne" %1, %0 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

@[simp_denote]
def add_hi_and_lo_negone_llvm_i64 := [LV| {
    ^entry (%arg: i64):
    %1 = llvm.mlir.constant (-1) : i64
    %0 = llvm.add %arg, %1 overflow<nsw> : i64
    llvm.return %0 : i64
  }]

def slt_or_not_max_logical_before := [llvm|
{
^0(%arg60 : i8, %arg61 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "slt" %arg60, %arg61 : i8
  %3 = llvm.icmp "ne" %arg60, %0 : i8
  %4 = "llvm.select"(%2, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def bools_logical_commute0_before := [llvm|
{
^0(%arg108 : i1, %arg109 : i1, %arg110 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.xor %arg110, %0 : i1
  %3 = "llvm.select"(%2, %arg108, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %4 = "llvm.select"(%arg110, %arg109, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %5 = "llvm.select"(%3, %0, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def PR42691_3_logical_after := [llvm|
{
^0(%arg338 : i32):
  %0 = llvm.mlir.constant(-2147483647 : i32) : i32
  %1 = llvm.icmp "ult" %arg338, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def test_shl_nsw__none_are_safe_after := [llvm|
{
^0(%arg66 : i32, %arg67 : i64):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.mlir.constant(-8 : i32) : i32
  %2 = llvm.shl %arg66, %0 : i32
  %3 = llvm.and %2, %1 : i32
  %4 = llvm.zext nneg %3 : i32 to i64
  %5 = llvm.ashr %arg67, %4 : i64
  "llvm.return"(%5) : (i64) -> ()
}
]

def sub_ashr_or_i8_after := [llvm|
{
^0(%arg39 : i8, %arg40 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.icmp "slt" %arg40, %arg39 : i8
  %2 = "llvm.select"(%1, %0, %arg39) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  "llvm.return"(%2) : (i8) -> ()
}
]

def test11f_before := [llvm|
{
^0(%arg91 : i32, %arg92 : i32, %arg93 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.mul %arg92, %arg93 : i32
  %2 = llvm.xor %1, %arg91 : i32
  %3 = llvm.xor %arg91, %0 : i32
  %4 = llvm.xor %1, %3 : i32
  %5 = llvm.and %4, %2 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def and_two_ranges_to_mask_and_range_no_add_on_one_range_before := [llvm|
{
^0(%arg249 : i16):
  %0 = llvm.mlir.constant(12 : i16) : i16
  %1 = llvm.mlir.constant(16 : i16) : i16
  %2 = llvm.mlir.constant(28 : i16) : i16
  %3 = llvm.icmp "uge" %arg249, %0 : i16
  %4 = llvm.icmp "ult" %arg249, %1 : i16
  %5 = llvm.icmp "uge" %arg249, %2 : i16
  %6 = llvm.or %4, %5 : i1
  %7 = llvm.and %3, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def llvm_zext_lower_riscv_16_to_32 : LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 16)] :=
  {lhs:= zext_llvm_16_to_32, rhs:= zext_riscv_16_to_32}

def masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_3_logical_after := [llvm|
{
^0(%arg21 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(8 : i32) : i32
  %2 = llvm.and %arg21, %0 : i32
  %3 = llvm.icmp "ne" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def p_before := [llvm|
{
^0(%arg60 : i32, %arg61 : i32, %arg62 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg60, %arg62 : i32
  %2 = llvm.xor %arg62, %0 : i32
  %3 = llvm.and %2, %arg61 : i32
  %4 = llvm.or %1, %3 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def lshr_16_add_zext_basic_multiuse_before := [llvm|
{
^0(%arg31 : i16, %arg32 : i16):
  %0 = llvm.mlir.constant(16 : i32) : i32
  %1 = llvm.zext %arg31 : i16 to i32
  %2 = llvm.zext %arg32 : i16 to i32
  %3 = llvm.add %1, %2 : i32
  %4 = llvm.lshr %3, %0 : i32
  %5 = llvm.or %4, %1 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def sub_ashr_or_i32_nuw_nsw_before := [llvm|
{
^0(%arg30 : i32, %arg31 : i32):
  %0 = llvm.mlir.constant(31 : i32) : i32
  %1 = llvm.sub %arg31, %arg30 overflow<nsw,nuw> : i32
  %2 = llvm.ashr %1, %0 : i32
  %3 = llvm.or %2, %arg30 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def ashr_ule_noexact_before := [llvm|
{
^0(%arg158 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.mlir.constant(10 : i8) : i8
  %2 = llvm.ashr %arg158, %0 : i8
  %3 = llvm.icmp "ule" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def icmp_power2_and_icmp_shifted_mask_2147483648_805306368_after := [llvm|
{
^0(%arg41 : i32):
  %0 = llvm.mlir.constant(805306368 : i32) : i32
  %1 = llvm.icmp "ult" %arg41, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def lshrugt_03_11_after := [llvm|
{
^0(%arg372 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def is_ascii_alphabetic_inverted_after := [llvm|
{
^0(%arg247 : i32):
  %0 = llvm.mlir.constant(-33 : i32) : i32
  %1 = llvm.mlir.constant(-91 : i32) : i32
  %2 = llvm.mlir.constant(-26 : i32) : i32
  %3 = llvm.and %arg247, %0 : i32
  %4 = llvm.add %3, %1 : i32
  %5 = llvm.icmp "ult" %4, %2 : i32
  "llvm.return"(%5) : (i1) -> ()
}
]

def ashrslt_03_06_exact_before := [llvm|
{
^0(%arg15 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(6 : i4) : i4
  %2 = llvm.ashr exact %arg15, %0 : i4
  %3 = llvm.icmp "slt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def sub_const_const_sub_before := [llvm|
{
^0(%arg35 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.sub %arg35, %0 : i32
  %3 = llvm.sub %1, %2 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def ashrsgt_02_14_exact_before := [llvm|
{
^0(%arg71 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(-2 : i4) : i4
  %2 = llvm.ashr exact %arg71, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def substitute_constant_and_eq_eq_before := [llvm|
{
^0(%arg322 : i8, %arg323 : i8):
  %0 = llvm.mlir.constant(42 : i8) : i8
  %1 = llvm.icmp "eq" %arg322, %0 : i8
  %2 = llvm.icmp "eq" %arg322, %arg323 : i8
  %3 = llvm.and %1, %2 : i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def narrow_sext_xor_after := [llvm|
{
^0(%arg48 : i16, %arg49 : i32):
  %0 = llvm.trunc %arg49 : i32 to i16
  %1 = llvm.xor %arg48, %0 : i16
  "llvm.return"(%1) : (i16) -> ()
}
]

def irc_constants_APlusC1MinusC2_1_0 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (1) : i64
      %c2 = llvm.mlir.constant (0) : i64
      %0 = llvm.add %a, %c1 : i64
      %1 = llvm.sub %0, %c2 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (1) : i64
      %c2 = llvm.mlir.constant (0) : i64
      %0 = llvm.sub %c1, %c2 : i64
      %1 = llvm.add %a, %0 : i64
      llvm.return %1 : i64
  }]

def add_nsw_poison_constant_fold_after := [llvm|
{
^0(%arg2 : i1, %arg3 : i8):
  %0 = llvm.mlir.constant(64 : i8) : i8
  %1 = llvm.mlir.constant(-127 : i8) : i8
  %2 = llvm.add %arg3, %0 overflow<nsw> : i8
  %3 = "llvm.select"(%arg2, %2, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def trunc_sel_equal_sext_before := [llvm|
{
^0(%arg77 : i32, %arg78 : i1):
  %0 = llvm.mlir.constant(42 : i16) : i16
  %1 = llvm.trunc %arg77 : i32 to i16
  %2 = "llvm.select"(%arg78, %1, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i16, i16) -> i16
  %3 = llvm.sext %2 : i16 to i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def test5_after := [llvm|
{
^0(%arg224 : i32, %arg225 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg224, %arg225 : i32
  %2 = llvm.xor %1, %0 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

EquivBEq in Init.Data.BEq

def test47_before := [llvm|
{
^0(%arg139 : i8):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.sext %arg139 : i8 to i32
  %2 = llvm.or %1, %0 : i32
  %3 = llvm.zext %2 : i32 to i64
  "llvm.return"(%3) : (i64) -> ()
}
]

def trunc_shl_lshr_var_before := [llvm|
{
^0(%arg33 : i64, %arg34 : i64):
  %0 = llvm.mlir.constant(2) : i64
  %1 = llvm.lshr %arg33, %arg34 : i64
  %2 = llvm.shl %1, %0 : i64
  %3 = llvm.trunc %2 : i64 to i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=
  ⟨_, NFA.autSignedCmp cmp⟩

def xor_ugt_2_before := [llvm|
{
^0(%arg13 : i8, %arg14 : i8, %arg15 : i8):
  %0 = llvm.mlir.constant(63 : i8) : i8
  %1 = llvm.mlir.constant(64 : i8) : i8
  %2 = llvm.add %arg13, %arg15 : i8
  %3 = llvm.and %arg14, %0 : i8
  %4 = llvm.or %3, %1 : i8
  %5 = llvm.xor %2, %4 : i8
  %6 = llvm.icmp "ugt" %2, %5 : i8
  "llvm.return"(%6) : (i1) -> ()
}
]

def test_after := [llvm|
{
^0(%arg0 : i64):
  %0 = llvm.mlir.constant(0 : i32) : i32
  "llvm.return"(%0) : (i32) -> ()
}
]

def not_and_and_or_no_or_commute1_and_before := [llvm|
{
^0(%arg160 : i32, %arg161 : i32, %arg162 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.or %arg161, %arg160 : i32
  %2 = llvm.xor %1, %0 : i32
  %3 = llvm.xor %arg160, %0 : i32
  %4 = llvm.and %arg162, %arg161 : i32
  %5 = llvm.and %4, %3 : i32
  %6 = llvm.or %5, %2 : i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def runParser (parser : @ParseFun ParseOutput) (fileName : String) : IO (Option ParseOutput) := do
   
  let packagePaths : List String :=
    match (← IO.getEnv "LEAN_PATH") with
    | .none => []
    | .some colonSeparatedPaths => colonSeparatedPaths.splitOn ":"
  if packagePaths.isEmpty then
    throw <| IO.userError ("Expected `LEAN_PATH` environment variable to be set. " ++
    " Are you running via `lake exec opt`?")
  initSearchPath (← Lean.findSysroot) packagePaths
  let modules : Array Import := #[⟨`LeanMLIR.MLIRSyntax.EDSL, false, false, false⟩]
  let env ← importModules (loadExts := true)  modules {}
  let filePath := System.mkFilePath [fileName]
  if !(← isFile filePath) then
    throw <| IO.userError s!"File {fileName} does not exist"
  parseFile env parser filePath

def urem? {w : Nat} (x y : BitVec w) : IntW w :=
  if y = 0 then
    .poison
  else
    .value <| x % y

def mkWidthFSM (wcard : Nat) (tcard : Nat) (bcard : Nat) (ncard icard : Nat) (pcard : Nat) (w : Nondep.WidthExpr) :
    (NatFSM wcard tcard bcard ncard icard pcard w) :=
  match w with
  | .const nat => {
      toFsm := (FSM.trueUptoExcluding nat).map Fin.elim0
    }
  | .var wnat =>
    if h : wnat < wcard then
      { toFsm :=
        composeUnaryAux FSM.scanAnd (FSM.var' (StateSpace.widthVar ⟨wnat, h⟩))
        
      }
    else
      { toFsm := FSM.zero' } 
  | .min v w =>
      { toFsm :=
        (mkWidthFSM wcard tcard bcard ncard icard pcard v).toFsm &&& (mkWidthFSM wcard tcard bcard ncard icard pcard w).toFsm
      }
  | .max v w =>
        { toFsm :=
        (mkWidthFSM wcard tcard bcard ncard icard pcard v).toFsm ||| (mkWidthFSM wcard tcard bcard ncard icard pcard w).toFsm
      }
  | .addK v k =>
    { toFsm :=
        composeUnaryAux (FSM.repeatN true k)  (mkWidthFSM wcard tcard bcard ncard icard pcard v).toFsm
    }
  | .kadd k v =>
    { toFsm :=
        composeUnaryAux (FSM.repeatN true k)  (mkWidthFSM wcard tcard bcard ncard icard pcard v).toFsm
    }

def ashrsgt_02_05_before := [llvm|
{
^0(%arg296 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(5 : i4) : i4
  %2 = llvm.ashr %arg296, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def test35_with_trunc_after := [llvm|
{
^0(%arg48 : i64):
  %0 = llvm.mlir.constant(2147483648) : i64
  %1 = llvm.mlir.constant(0) : i64
  %2 = llvm.mlir.constant(60 : i32) : i32
  %3 = llvm.mlir.constant(100 : i32) : i32
  %4 = llvm.and %arg48, %0 : i64
  %5 = llvm.icmp "eq" %4, %1 : i64
  %6 = "llvm.select"(%5, %2, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def select_icmp_ne_0_and_8_or_1073741824_after := [llvm|
{
^0(%arg189 : i8, %arg190 : i32):
  %0 = llvm.mlir.constant(8 : i8) : i8
  %1 = llvm.mlir.constant(0 : i8) : i8
  %2 = llvm.mlir.constant(1073741824 : i32) : i32
  %3 = llvm.and %arg189, %0 : i8
  %4 = llvm.icmp "eq" %3, %1 : i8
  %5 = llvm.or %arg190, %2 : i32
  %6 = "llvm.select"(%4, %5, %arg190) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def select_sub_icmp_bad_before := [llvm|
{
^0(%arg71 : i32, %arg72 : i32, %arg73 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "eq" %arg71, %0 : i32
  %2 = llvm.sub %arg71, %arg73 : i32
  %3 = "llvm.select"(%1, %2, %arg72) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def shl_lshr_pow2_not_const_case2_after := [llvm|
{
^0(%arg71 : i16):
  %0 = llvm.mlir.constant(2 : i16) : i16
  %1 = llvm.mlir.constant(0 : i16) : i16
  %2 = llvm.mlir.constant(8 : i16) : i16
  %3 = llvm.icmp "eq" %arg71, %0 : i16
  %4 = "llvm.select"(%3, %1, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i16, i16) -> i16
  "llvm.return"(%4) : (i16) -> ()
}
]

def or_or_and_pat8_after := [llvm|
{
^0(%arg46 : i8, %arg47 : i8, %arg48 : i8, %arg49 : i8):
  %0 = llvm.or %arg49, %arg46 : i8
  %1 = llvm.and %0, %arg47 : i8
  %2 = llvm.or %1, %arg48 : i8
  "llvm.return"(%2) : (i8) -> ()
}
]

def c_1_0_0_before := [llvm|
{
^0(%arg20 : i4, %arg21 : i4):
  %0 = llvm.mlir.constant(-2 : i4) : i4
  %1 = llvm.xor %arg21, %arg20 : i4
  %2 = llvm.and %1, %0 : i4
  %3 = llvm.xor %2, %arg21 : i4
  "llvm.return"(%3) : (i4) -> ()
}
]

@[simps]
def lift (M: NFA (BitVec n1) σ) (f : Fin n1 → Fin n2) : NFA (BitVec n2) σ where
  start := M.start
  accept := M.accept
  step q a := M.step q (a.transport f)

def in_constant_14_vary_after := [llvm|
{
^0(%arg22 : i4, %arg23 : i4):
  %0 = llvm.mlir.constant(-2 : i4) : i4
  %1 = llvm.and %arg22, %0 : i4
  "llvm.return"(%1) : (i4) -> ()
}
]

def f_var0_before := [llvm|
{
^0(%arg52 : i32, %arg53 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.and %arg52, %arg53 : i32
  %3 = llvm.icmp "eq" %2, %0 : i32
  %4 = llvm.lshr %arg52, %1 : i32
  %5 = llvm.and %4, %1 : i32
  %6 = "llvm.select"(%3, %5, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def test_nuw_nsw_and_signed_pred_after := [llvm|
{
^0(%arg54 : i64):
  %0 = llvm.mlir.constant(7) : i64
  %1 = llvm.icmp "ugt" %arg54, %0 : i64
  "llvm.return"(%1) : (i1) -> ()
}
]

List.sum in Init.Data.List.Basic

def test9_after := [llvm|
{
^0(%arg298 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "slt" %arg298, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def andcond.010.inv.inner.cond.in.inner.sel_before := [llvm|
{
^0(%arg65 : i1, %arg66 : i1, %arg67 : i1, %arg68 : i1, %arg69 : i1):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.mlir.constant(true) : i1
  %2 = "llvm.select"(%arg65, %arg66, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %3 = llvm.xor %arg65, %1 : i1
  %4 = "llvm.select"(%3, %arg68, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %5 = "llvm.select"(%2, %arg69, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def shlMax (a : BitVec v) (b : BitVec w) : BitVec (max v w) :=
   a.signExtend (max v w) <<< b.signExtend (max v w)

def ripple_nsw3_after := [llvm|
{
^0(%arg19 : i16, %arg20 : i16):
  %0 = llvm.mlir.constant(-21845 : i16) : i16
  %1 = llvm.mlir.constant(21843 : i16) : i16
  %2 = llvm.and %arg20, %0 : i16
  %3 = llvm.and %arg19, %1 : i16
  %4 = llvm.add %2, %3 overflow<nsw,nuw> : i16
  "llvm.return"(%4) : (i16) -> ()
}
]

Ideal.span in Mathlib.RingTheory.Ideal.Span

def positive_with_aggressive_icmp_before := [llvm|
{
^0(%arg56 : i32):
  %0 = llvm.mlir.constant(128 : i32) : i32
  %1 = llvm.mlir.constant(256 : i32) : i32
  %2 = llvm.mlir.constant(512 : i32) : i32
  %3 = llvm.icmp "ult" %arg56, %0 : i32
  %4 = llvm.add %arg56, %1 : i32
  %5 = llvm.icmp "ult" %4, %2 : i32
  %6 = llvm.and %3, %5 : i1
  "llvm.return"(%6) : (i1) -> ()
}
]

def test2_after := [llvm|
{
^0(%arg6 : i32, %arg7 : i32, %arg8 : i32):
  %0 = llvm.xor %arg7, %arg6 : i32
  "llvm.return"(%0) : (i32) -> ()
}
]

def test36_before := [llvm|
{
^0(%arg158 : i32):
  %0 = llvm.mlir.constant(31 : i32) : i32
  %1 = llvm.mlir.constant(0 : i8) : i8
  %2 = llvm.lshr %arg158, %0 : i32
  %3 = llvm.trunc %2 : i32 to i8
  %4 = llvm.icmp "eq" %3, %1 : i8
  "llvm.return"(%4) : (i1) -> ()
}
]

def p1_scalar_urem_by_nonconst_after := [llvm|
{
^0(%arg18 : i32, %arg19 : i32):
  %0 = llvm.mlir.constant(128 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.and %arg18, %0 : i32
  %3 = llvm.icmp "eq" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def ripple_nsw1_after := [llvm|
{
^0(%arg23 : i16, %arg24 : i16):
  %0 = llvm.mlir.constant(1 : i16) : i16
  %1 = llvm.mlir.constant(-16385 : i16) : i16
  %2 = llvm.and %arg24, %0 : i16
  %3 = llvm.and %arg23, %1 : i16
  %4 = llvm.add %2, %3 overflow<nsw,nuw> : i16
  "llvm.return"(%4) : (i16) -> ()
}
]

def ashr_pow2_ult_after := [llvm|
{
^0(%arg59 : i8):
  %0 = llvm.mlir.constant(-128 : i8) : i8
  %1 = llvm.mlir.constant(-96 : i8) : i8
  %2 = llvm.ashr exact %0, %arg59 : i8
  %3 = llvm.icmp "ult" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def and_or_not_not_commute6_before := [llvm|
{
^0(%arg357 : i32, %arg358 : i32, %arg359 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg358 : i32
  %3 = llvm.and %2, %arg357 : i32
  %4 = llvm.xor %3, %1 : i32
  %5 = llvm.and %arg359, %arg357 : i32
  %6 = llvm.xor %5, %1 : i32
  %7 = llvm.or %2, %6 : i32
  %8 = llvm.and %7, %4 : i32
  "llvm.return"(%8) : (i32) -> ()
}
]

def zero_sign_bit_before := [llvm|
{
^0(%arg31 : i16):
  %0 = llvm.mlir.constant(32767 : i16) : i16
  %1 = llvm.mlir.constant(512 : i16) : i16
  %2 = llvm.and %arg31, %0 : i16
  %3 = llvm.add %2, %1 : i16
  "llvm.return"(%3) : (i16) -> ()
}
]

def ashrslt_03_03_exact_after := [llvm|
{
^0(%arg18 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def positive_biggerashr_shlnsw_before := [llvm|
{
^0(%arg50 : i8):
  %0 = llvm.mlir.constant(6 : i8) : i8
  %1 = llvm.mlir.constant(3 : i8) : i8
  %2 = llvm.ashr %arg50, %0 : i8
  %3 = llvm.shl %2, %1 overflow<nsw> : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def test42_before := [llvm|
{
^0(%arg208 : i32, %arg209 : i32, %arg210 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.mul %arg209, %arg210 : i32
  %2 = llvm.or %arg208, %1 : i32
  %3 = llvm.xor %arg208, %0 : i32
  %4 = llvm.xor %3, %1 : i32
  %5 = llvm.and %4, %2 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def andn_or_cmp_1_logical_after := [llvm|
{
^0(%arg173 : i37, %arg174 : i37, %arg175 : i37):
  %0 = llvm.mlir.constant(42 : i37) : i37
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.icmp "sle" %arg173, %arg174 : i37
  %3 = llvm.icmp "ugt" %arg175, %0 : i37
  %4 = "llvm.select"(%2, %3, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def rgn {Γ : Ctxt _} (k : Nat) (input : Var Γ .nat) (body : Com [ExTy.nat] ExTy.nat) :
    Expr Γ .nat :=
  Expr.mk
    (op := .runK k)
    (ty_eq := rfl)
    (eff_le := EffectKind.pure_le _)
    (args := .cons input .nil)
    (regArgs := HVector.cons (body.castPureToEff _) HVector.nil)

def test22_logical_before := [llvm|
{
^0(%arg5 : i32, %arg6 : i32):
  %0 = llvm.mlir.constant(100663295 : i32) : i32
  %1 = llvm.mlir.constant(268435456 : i32) : i32
  %2 = llvm.mlir.constant(7 : i32) : i32
  %3 = llvm.mlir.constant(-1 : i32) : i32
  %4 = llvm.mlir.constant(true) : i1
  %5 = llvm.and %arg5, %0 : i32
  %6 = llvm.icmp "ult" %5, %1 : i32
  %7 = llvm.and %arg6, %2 : i32
  %8 = llvm.icmp "sgt" %7, %3 : i32
  %9 = "llvm.select"(%6, %4, %8) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%9) : (i1) -> ()
}
]

def and_or3_commuted_before := [llvm|
{
^0(%arg164 : i1, %arg165 : i1, %arg166 : i32, %arg167 : i32):
  %0 = llvm.icmp "eq" %arg166, %arg167 : i32
  %1 = llvm.and %0, %arg165 : i1
  %2 = "llvm.select"(%1, %arg164, %arg165) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%2) : (i1) -> ()
}
]

def and_or_not_not_before := [llvm|
{
^0(%arg375 : i32, %arg376 : i32, %arg377 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg376, %arg375 : i32
  %2 = llvm.xor %1, %0 : i32
  %3 = llvm.and %arg375, %arg377 : i32
  %4 = llvm.xor %3, %0 : i32
  %5 = llvm.or %4, %arg376 : i32
  %6 = llvm.and %5, %2 : i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def srem2_ashr_mask_before := [llvm|
{
^0(%arg6 : i32):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.mlir.constant(31 : i32) : i32
  %2 = llvm.srem %arg6, %0 : i32
  %3 = llvm.ashr %2, %1 : i32
  %4 = llvm.and %3, %0 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def elabDefSignatureFor (dialect : Term) (alts : TSyntax ``matchAltsSig) : CommandElabM Unit := do
  let msg := (return m!"{exceptEmoji ·} Defining operation signature for {dialect} dialect")
  withTraceNode `LeanMLIR.Elab msg (collapsed := false) <| do
    let alts := getMatchAlts alts
    let alts ← do
      alts.mapM fun view => do
        trace[LeanMLIR.Elab] m!"Parsing match alternative with\n\
          \tpatterns: {view.patterns}\n\
          \tsignature: {view.rhs}"
        let rhs ← parseSignature view.rhs
        return { view with rhs }

    |

structure Tensor2d' (α : Type) where
  dim₀ : Index
  dim₁ : Index
  mat : Matrix (Fin dim₀) (Fin dim₁) α

def trunc_shl_zext_64_before := [llvm|
{
^0(%arg0 : i64):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.trunc %arg0 : i64 to i8
  %2 = llvm.shl %1, %0 : i8
  %3 = llvm.zext %2 : i8 to i64
  "llvm.return"(%3) : (i64) -> ()
}
]

def productsList : List (List α) -> List (List α)
| [] => [[]] 
| (xs::xss) => Id.run do
  let mut out := []
  let xss' := productsList xss 
  for x in xs do  
    out := out.append (xss'.map (fun xs => x :: xs))
  return out

def src_srem_shl_demand_min_signbit_after := [llvm|
{
^0(%arg23 : i32):
  %0 = llvm.mlir.constant(1073741823 : i32) : i32
  %1 = llvm.mlir.constant(-2147483648 : i32) : i32
  %2 = llvm.srem %arg23, %0 : i32
  %3 = llvm.and %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

@[simp_denote]
def trunc_llvm_64_to_32_nuw := [LV| {
  ^entry (%lhs: i64):
    %0 = llvm.trunc %lhs overflow<nuw> : i64 to i32
    llvm.return %0 : i32
  }]

def pr40493_neg3_after := [llvm|
{
^0(%arg15 : i32):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.mlir.constant(4 : i32) : i32
  %2 = llvm.shl %arg15, %0 : i32
  %3 = llvm.and %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def ule_swap_or_not_min_logical_after := [llvm|
{
^0(%arg262 : i8, %arg263 : i8):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def product (final? : Prop → Prop → Prop) (M N : NFA' n) : NFA' n where
  σ := _
  M := M.M.product final? N.M

def uge_swap_and_max_logical_after := [llvm|
{
^0(%arg228 : i8, %arg229 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.icmp "eq" %arg228, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def test_mul_nuw__all_are_safe_before := [llvm|
{
^0(%arg14 : i32):
  %0 = llvm.mlir.constant(255 : i32) : i32
  %1 = llvm.mlir.constant(17 : i32) : i32
  %2 = llvm.mlir.constant(9 : i32) : i32
  %3 = llvm.mlir.constant(153 : i32) : i32
  %4 = llvm.and %arg14, %0 : i32
  %5 = llvm.icmp "eq" %4, %1 : i32
  %6 = llvm.mul %4, %2 overflow<nuw> : i32
  %7 = "llvm.select"(%5, %3, %6) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def or_and_shifts1_after := [llvm|
{
^0(%arg14 : i32):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.mlir.constant(8 : i32) : i32
  %2 = llvm.mlir.constant(5 : i32) : i32
  %3 = llvm.mlir.constant(32 : i32) : i32
  %4 = llvm.shl %arg14, %0 : i32
  %5 = llvm.and %4, %1 : i32
  %6 = llvm.shl %arg14, %2 : i32
  %7 = llvm.and %6, %3 : i32
  %8 = llvm.or disjoint %5, %7 : i32
  "llvm.return"(%8) : (i32) -> ()
}
]

def demorgan_nand_before := [llvm|
{
^0(%arg38 : i8, %arg39 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.xor %arg38, %0 : i8
  %2 = llvm.and %1, %arg39 : i8
  %3 = llvm.xor %2, %0 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

Lean.pp.explicit in Lean.PrettyPrinter.Delaborator.Options

def src4_before := [llvm|
{
^0(%arg6 : i32, %arg7 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg6, %arg7 : i32
  %2 = llvm.or %arg7, %arg6 : i32
  %3 = llvm.xor %2, %0 : i32
  %4 = llvm.add %1, %3 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def ashr_and_after := [llvm|
{
^0(%arg17 : i8):
  %0 = llvm.mlir.constant(-64 : i8) : i8
  %1 = llvm.ashr %0, %arg17 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def fold_xor_zext_icmp_after := [llvm|
{
^0(%arg91 : i64, %arg92 : i64, %arg93 : i64):
  %0 = llvm.icmp "sgt" %arg91, %arg92 : i64
  %1 = llvm.icmp "slt" %arg91, %arg93 : i64
  %2 = llvm.xor %0, %1 : i1
  %3 = llvm.zext %2 : i1 to i8
  "llvm.return"(%3) : (i8) -> ()
}
]

Function.RightInverse in Init.Data.Function

def original_sdiv_srem := [LV| {
  ^entry (%a: i64, %b: i64):
    %0= llvm.sdiv exact %a, %b : i64
    %1 = llvm.srem %a, %b : i64
    llvm.return %1 : i64
  }]

def test6a_after := [llvm|
{
^0(%arg1 : i1):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def and_sext_multiuse_before := [llvm|
{
^0(%arg0 : i32, %arg1 : i32, %arg2 : i32, %arg3 : i32):
  %0 = llvm.icmp "sgt" %arg0, %arg1 : i32
  %1 = llvm.sext %0 : i1 to i32
  %2 = llvm.and %1, %arg2 : i32
  %3 = llvm.and %1, %arg3 : i32
  %4 = llvm.add %2, %3 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def t6_after := [llvm|
{
^0(%arg7 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "slt" %arg7, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

@[simp, reducible]
instance : DialectDenote (LLVMPlusRiscV) where
  denote
  | .llvm llvmOp, args, .nil => do
      let xs ← DialectDenote.denote llvmOp (llvmArgsFromHybrid args) .nil
      return xs.map' Ty.llvm (fun t x => x)
  | .riscv (riscvOp), args, .nil => do
      let xs ← DialectDenote.denote riscvOp (riscvArgsFromHybrid args) .nil
      return xs.map' Ty.riscv (fun t x => x)
  | .castRiscv _ , elemToCast, _ =>
    let toCast : BitVec 64 :=
      elemToCast.getN 0 (by admit /- proof elided -/
      )
    [castriscvToLLVM toCast]ₕ
  | .castLLVM _,
    (elemToCast : HVector TyDenote.toType [Ty.llvm (.bitvec _)]), _ =>
    let toCast : PoisonOr (BitVec _) :=
      elemToCast.getN 0 (by admit /- proof elided -/
      )
    [castLLVMToriscv toCast]ₕ

def test29_after := [llvm|
{
^0(%arg173 : i1):
  %0 = llvm.mlir.constant(915 : i32) : i32
  %1 = llvm.mlir.constant(113 : i32) : i32
  %2 = "llvm.select"(%arg173, %0, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def sext_zext_apint1_before := [llvm|
{
^0(%arg86 : i77):
  %0 = llvm.zext %arg86 : i77 to i533
  %1 = llvm.sext %0 : i533 to i1024
  "llvm.return"(%1) : (i1024) -> ()
}
]

def HVector.vars {l : List d.Ty} (T : HVector (Var Γ) l) : VarSet Γ :=
  T.foldl (fun _ s a => insert ⟨_, a⟩ s) ∅

def f_before := [llvm|
{
^0(%arg0 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.or %arg0, %0 : i32
  %3 = llvm.srem %2, %0 : i32
  %4 = llvm.icmp "ne" %3, %1 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def src3_before := [llvm|
{
^0(%arg8 : i32, %arg9 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg9, %arg8 : i32
  %2 = llvm.xor %arg8, %0 : i32
  %3 = llvm.xor %arg9, %0 : i32
  %4 = llvm.and %2, %3 : i32
  %5 = llvm.add %1, %4 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def t2_commutative_before := [llvm|
{
^0(%arg20 : i8, %arg21 : i8):
  %0 = llvm.sub %arg20, %arg21 : i8
  %1 = llvm.icmp "uge" %arg20, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def test_with_1_before := [llvm|
{
^0(%arg317 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.shl %0, %arg317 : i32
  %2 = llvm.and %1, %0 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def test_imply_not2_after := [llvm|
{
^0(%arg3 : i32, %arg4 : i1):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ne" %arg3, %0 : i32
  %3 = "llvm.select"(%2, %1, %arg4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def ne_nuw_rem_zero_before := [llvm|
{
^0(%arg153 : i8):
  %0 = llvm.mlir.constant(5 : i8) : i8
  %1 = llvm.mlir.constant(-126 : i8) : i8
  %2 = llvm.mul %arg153, %0 overflow<nuw> : i8
  %3 = llvm.icmp "ne" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def t1_otherbitwidth_before := [llvm|
{
^0(%arg11 : i16):
  %0 = llvm.mlir.constant(15 : i16) : i16
  %1 = llvm.shl %arg11, %0 : i16
  %2 = llvm.ashr %1, %0 : i16
  "llvm.return"(%2) : (i16) -> ()
}
]

def lshrult_01_12_exact_before := [llvm|
{
^0(%arg137 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(-4 : i4) : i4
  %2 = llvm.lshr exact %arg137, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def t15_before := [llvm|
{
^0(%arg198 : i32, %arg199 : i32):
  %0 = llvm.shl %arg198, %arg199 overflow<nuw> : i32
  %1 = llvm.udiv %0, %arg198 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def irc_constants_AMinusC1PlusC2_neg2_0 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (-2) : i64
      %c2 = llvm.mlir.constant (0) : i64
      %0 = llvm.sub %a, %c1 : i64
      %1 = llvm.add %0, %c2 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (-2) : i64
      %c2 = llvm.mlir.constant (0) : i64
      %0 = llvm.sub %c2, %c1 : i64
      %1 = llvm.add %a, %0 : i64
      llvm.return %1 : i64
  }]

def trunc_shl_16_i32_i64_after := [llvm|
{
^0(%arg49 : i64):
  %0 = llvm.mlir.constant(16 : i32) : i32
  %1 = llvm.trunc %arg49 : i64 to i32
  %2 = llvm.shl %1, %0 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def t10_almost_highest_bit_before := [llvm|
{
^0(%arg22 : i32, %arg23 : i64, %arg24 : i32):
  %0 = llvm.mlir.constant(64 : i32) : i32
  %1 = llvm.mlir.constant(-2 : i32) : i32
  %2 = llvm.mlir.constant(0 : i32) : i32
  %3 = llvm.sub %0, %arg24 : i32
  %4 = llvm.shl %arg22, %3 : i32
  %5 = llvm.add %arg24, %1 : i32
  %6 = llvm.zext %5 : i32 to i64
  %7 = llvm.lshr %arg23, %6 : i64
  %8 = llvm.trunc %7 : i64 to i32
  %9 = llvm.and %4, %8 : i32
  %10 = llvm.icmp "ne" %9, %2 : i32
  "llvm.return"(%10) : (i1) -> ()
}
]

class HasUB (m : Type u → Type v) where
   
  throwUB {α : Type u} : m α

def llvm_const_lower_riscv_li13 : LLVMPeepholeRewriteRefine 64 [] :=
  {lhs := constLLVM13, rhs:= liRiscv13,
   correct := by admit /- proof elided -/

def select_icmp_ne_0_and_4096_xor_32_before := [llvm|
{
^0(%arg207 : i32, %arg208 : i32):
  %0 = llvm.mlir.constant(4096 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(32 : i32) : i32
  %3 = llvm.and %arg207, %0 : i32
  %4 = llvm.icmp "ne" %1, %3 : i32
  %5 = llvm.xor %arg208, %2 : i32
  %6 = "llvm.select"(%4, %arg208, %5) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def reduce_xor_common_op_commute0_before := [llvm|
{
^0(%arg10 : i4, %arg11 : i4, %arg12 : i4):
  %0 = llvm.xor %arg10, %arg11 : i4
  %1 = llvm.xor %0, %arg12 : i4
  %2 = llvm.or %1, %arg10 : i4
  "llvm.return"(%2) : (i4) -> ()
}
]

def neg_mask_before := [llvm|
{
^0(%arg1 : i32, %arg2 : i16):
  %0 = llvm.mlir.constant(15 : i16) : i16
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.sext %arg2 : i16 to i32
  %3 = llvm.sub %arg1, %2 overflow<nsw> : i32
  %4 = llvm.lshr %arg2, %0 : i16
  %5 = llvm.zext %4 : i16 to i32
  %6 = llvm.sub %1, %5 overflow<nsw> : i32
  %7 = llvm.and %3, %6 : i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def ashrslt_01_00_exact_after := [llvm|
{
^0(%arg53 : i4):
  %0 = llvm.mlir.constant(0 : i4) : i4
  %1 = llvm.icmp "slt" %arg53, %0 : i4
  "llvm.return"(%1) : (i1) -> ()
}
]

def different_size_zext_zext_eq_after := [llvm|
{
^0(%arg22 : i4, %arg23 : i7):
  %0 = llvm.zext %arg22 : i4 to i7
  %1 = llvm.icmp "eq" %arg23, %0 : i7
  "llvm.return"(%1) : (i1) -> ()
}
]

def or_xor_commute4_before := [llvm|
{
^0(%arg160 : i32, %arg161 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.udiv %0, %arg160 : i32
  %2 = llvm.udiv %0, %arg161 : i32
  %3 = llvm.or %1, %2 : i32
  %4 = llvm.xor %2, %3 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def main6c_logical_before := [llvm|
{
^0(%arg89 : i32):
  %0 = llvm.mlir.constant(7 : i32) : i32
  %1 = llvm.mlir.constant(3 : i32) : i32
  %2 = llvm.mlir.constant(48 : i32) : i32
  %3 = llvm.mlir.constant(16 : i32) : i32
  %4 = llvm.mlir.constant(true) : i1
  %5 = llvm.mlir.constant(0 : i32) : i32
  %6 = llvm.mlir.constant(1 : i32) : i32
  %7 = llvm.and %arg89, %0 : i32
  %8 = llvm.icmp "ne" %7, %1 : i32
  %9 = llvm.and %arg89, %2 : i32
  %10 = llvm.icmp "ne" %9, %3 : i32
  %11 = "llvm.select"(%8, %4, %10) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %12 = "llvm.select"(%11, %5, %6) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%12) : (i32) -> ()
}
]

@[simp_denote]
def udiv_riscv_32: Com  LLVMPlusRiscV ⟨[.llvm (.bitvec 32), .llvm (.bitvec 32)]⟩
  .pure (.llvm (.bitvec 32)) := [LV| {
  ^entry (%reg1: i32, %reg2: i32):
    %0 = "builtin.unrealized_conversion_cast"(%reg1) : (i32) -> (!i64)
    %1 = "builtin.unrealized_conversion_cast"(%reg2) : (i32) -> (!i64)
    %2 = divu %0, %1 : !i64
    %3 = "builtin.unrealized_conversion_cast"(%2) : (!i64) -> (i32)
    llvm.return %3 : i32
  }]

def scalar_i32_lshr_and_negC_eq_X_is_constant1_after := [llvm|
{
^0(%arg9 : i32):
  %0 = llvm.mlir.constant(12345 : i32) : i32
  %1 = llvm.mlir.constant(8 : i32) : i32
  %2 = llvm.lshr %0, %arg9 : i32
  %3 = llvm.icmp "ult" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def test15c_after := [llvm|
{
^0(%arg25 : i32):
  %0 = llvm.mlir.constant(16 : i32) : i32
  %1 = llvm.and %arg25, %0 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def ashr_ugt_3_before := [llvm|
{
^0(%arg111 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(3 : i4) : i4
  %2 = llvm.ashr %arg111, %0 : i4
  %3 = llvm.icmp "ugt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def test4_after := [llvm|
{
^0(%arg1 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.icmp "ne" %arg1, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def eq_umin1_before := [llvm|
{
^0(%arg60 : i32, %arg61 : i32):
  %0 = llvm.icmp "ult" %arg60, %arg61 : i32
  %1 = "llvm.select"(%0, %arg60, %arg61) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %2 = llvm.icmp "eq" %1, %arg60 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

def icmp_ne_and_pow2_lshr_pow2_before := [llvm|
{
^0(%arg41 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(4 : i32) : i32
  %2 = llvm.mlir.constant(0 : i32) : i32
  %3 = llvm.lshr %0, %arg41 : i32
  %4 = llvm.and %3, %1 : i32
  %5 = llvm.icmp "eq" %4, %2 : i32
  %6 = llvm.zext %5 : i1 to i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def test_shl_nuw__nuw_is_safe_after := [llvm|
{
^0(%arg80 : i32, %arg81 : i64):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.mlir.constant(-8 : i32) : i32
  %2 = llvm.shl %arg80, %0 : i32
  %3 = llvm.and %2, %1 : i32
  %4 = llvm.zext nneg %3 : i32 to i64
  %5 = llvm.ashr %arg81, %4 : i64
  "llvm.return"(%5) : (i64) -> ()
}
]

def elimEmpty {α : Sort u} (i : Inputs Empty i) : α :=
  i.input.elim

def ashr_ule_noexact_before := [llvm|
{
^0(%arg158 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.mlir.constant(10 : i8) : i8
  %2 = llvm.ashr %arg158, %0 : i8
  %3 = llvm.icmp "ule" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def sltiu_test: LLVMPeepholeRewriteRefine 16 [Ty.llvm (.bitvec 16)] where
  lhs := sltiu_llvm
  rhs := sltiu_riscv

def src_x_and_nmask_sge_before := [llvm|
{
^0(%arg17 : i8, %arg18 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.shl %0, %arg18 : i8
  %2 = llvm.and %arg17, %1 : i8
  %3 = llvm.icmp "sge" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def t2_after := [llvm|
{
^0(%arg220 : i8, %arg221 : i8):
  %0 = llvm.mlir.constant(42 : i8) : i8
  %1 = llvm.shl %0, %arg221 : i8
  %2 = llvm.add %1, %arg220 : i8
  "llvm.return"(%2) : (i8) -> ()
}
]

def scalar_i32_lshr_and_signbit_eq_X_is_constant1_after := [llvm|
{
^0(%arg5 : i32):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def test1_before := [llvm|
{
^0(%arg3 : i33):
  %0 = llvm.mlir.constant(4096 : i33) : i33
  %1 = llvm.urem %arg3, %0 : i33
  "llvm.return"(%1) : (i33) -> ()
}
]

def main4b_after := [llvm|
{
^0(%arg132 : i32):
  %0 = llvm.mlir.constant(23 : i32) : i32
  %1 = llvm.and %arg132, %0 : i32
  %2 = llvm.icmp "ne" %1, %0 : i32
  %3 = llvm.zext %2 : i1 to i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def not_signbit_after := [llvm|
{
^0(%arg22 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.icmp "sgt" %arg22, %0 : i8
  %2 = llvm.zext %1 : i1 to i8
  "llvm.return"(%2) : (i8) -> ()
}
]

def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=
  BitVec.ofFn fun i => bv.getLsbD (f i)

def masked_icmps_mask_notallzeros_bmask_mixed_negated_6_after := [llvm|
{
^0(%arg58 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(8 : i32) : i32
  %2 = llvm.and %arg58, %0 : i32
  %3 = llvm.icmp "ne" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def foo_before := [llvm|
{
^0(%arg0 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(7 : i32) : i32
  %2 = llvm.or %0, %1 : i32
  %3 = llvm.and %arg0, %2 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def add_smin_simplify_after := [llvm|
{
^0(%arg34 : i32):
  %0 = llvm.mlir.constant(-3 : i32) : i32
  %1 = llvm.add %arg34, %0 overflow<nsw> : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def narrow_sext_xor_before := [llvm|
{
^0(%arg48 : i16, %arg49 : i32):
  %0 = llvm.sext %arg48 : i16 to i32
  %1 = llvm.xor %0, %arg49 : i32
  %2 = llvm.trunc %1 : i32 to i16
  "llvm.return"(%2) : (i16) -> ()
}
]

def add_nsw_mul_nsw_before := [llvm|
{
^0(%arg33 : i16):
  %0 = llvm.add %arg33, %arg33 overflow<nsw> : i16
  %1 = llvm.add %0, %arg33 overflow<nsw> : i16
  "llvm.return"(%1) : (i16) -> ()
}
]

def main4e_like_logical_before := [llvm|
{
^0(%arg125 : i32, %arg126 : i32, %arg127 : i32):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(1 : i32) : i32
  %3 = llvm.and %arg125, %arg126 : i32
  %4 = llvm.icmp "eq" %3, %arg126 : i32
  %5 = llvm.and %arg125, %arg127 : i32
  %6 = llvm.icmp "eq" %5, %arg127 : i32
  %7 = "llvm.select"(%4, %6, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %8 = "llvm.select"(%7, %1, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%8) : (i32) -> ()
}
]

def xor_icmp_invalid_range_after := [llvm|
{
^0(%arg3 : i8):
  %0 = llvm.mlir.constant(-5 : i8) : i8
  %1 = llvm.mlir.constant(0 : i8) : i8
  %2 = llvm.and %arg3, %0 : i8
  %3 = llvm.icmp "ne" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

abbrev Width φ := ConcreteOrMVar Nat φ

def ashrslt_02_10_exact_before := [llvm|
{
^0(%arg27 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(-6 : i4) : i4
  %2 = llvm.ashr exact %arg27, %0 : i4
  %3 = llvm.icmp "slt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def positive_samevar_shlnsw_ashrexact_before := [llvm|
{
^0(%arg37 : i8, %arg38 : i8):
  %0 = llvm.ashr exact %arg37, %arg38 : i8
  %1 = llvm.shl %0, %arg38 overflow<nsw> : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def sge_swap_or_max_after := [llvm|
{
^0(%arg182 : i8, %arg183 : i8):
  %0 = llvm.icmp "sle" %arg183, %arg182 : i8
  "llvm.return"(%0) : (i1) -> ()
}
]

def not_or_or_and_no_and_before := [llvm|
{
^0(%arg121 : i32, %arg122 : i32, %arg123 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg122, %arg121 : i32
  %2 = llvm.xor %1, %0 : i32
  %3 = llvm.xor %arg121, %0 : i32
  %4 = llvm.or %3, %arg122 : i32
  %5 = llvm.or %4, %arg123 : i32
  %6 = llvm.and %5, %2 : i32
  "llvm.return"(%6) : (i32) -> ()
}
]

@[simp_denote]
def icmp_ule_llvm_32 : Com LLVMPlusRiscV ⟨[.llvm (.bitvec 32), .llvm (.bitvec 32)]⟩
  .pure (.llvm (.bitvec 1)) := [LV| {
  ^entry (%lhs: i32, %rhs: i32):
    %1 = llvm.icmp.ule %lhs, %rhs : i32
    llvm.return %1 : i1
  }]

@[simp_denote]
def llvm_shl_lower_riscv_32: LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 32)] where
  lhs := shl_llvm_32
  rhs := shl_riscv_32

def fold_eq_lhs_fail_eq_nonzero_after := [llvm|
{
^0(%arg21 : i8, %arg22 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.mlir.constant(1 : i8) : i8
  %2 = llvm.shl %0, %arg21 overflow<nsw> : i8
  %3 = llvm.and %2, %arg22 : i8
  %4 = llvm.icmp "eq" %3, %1 : i8
  "llvm.return"(%4) : (i1) -> ()
}
]

def masked_or_allzeroes_notoptimised_logical_after := [llvm|
{
^0(%arg116 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(39 : i32) : i32
  %3 = llvm.and %arg116, %0 : i32
  %4 = llvm.icmp "eq" %3, %1 : i32
  %5 = llvm.and %arg116, %2 : i32
  %6 = llvm.icmp "eq" %5, %1 : i32
  %7 = llvm.or %4, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

@[simp]
def WidthExpr.ofDep_kadd {wcard : Nat} {v : MultiWidth.WidthExpr wcard} {k : Nat} :
    (WidthExpr.ofDep (MultiWidth.WidthExpr.kadd k v)) =
    (.kadd k (.ofDep v)) := rfl

def sext_xor_sub_3_before := [llvm|
{
^0(%arg22 : i64, %arg23 : i1):
  %0 = llvm.sext %arg23 : i1 to i64
  %1 = llvm.xor %0, %arg22 : i64
  %2 = llvm.sub %0, %1 : i64
  "llvm.return"(%2) : (i64) -> ()
}
]

def test1_before := [llvm|
{
^0(%arg14 : i32, %arg15 : i32):
  %0 = llvm.icmp "slt" %arg14, %arg15 : i32
  %1 = llvm.zext %0 : i1 to i32
  %2 = llvm.icmp "sgt" %arg14, %arg15 : i32
  %3 = llvm.zext %2 : i1 to i32
  %4 = llvm.and %1, %3 : i32
  %5 = llvm.trunc %4 : i32 to i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def icmp_and_ashr_multiuse_logical_after := [llvm|
{
^0(%arg4 : i32):
  %0 = llvm.mlir.constant(240 : i32) : i32
  %1 = llvm.mlir.constant(224 : i32) : i32
  %2 = llvm.mlir.constant(496 : i32) : i32
  %3 = llvm.mlir.constant(432 : i32) : i32
  %4 = llvm.and %arg4, %0 : i32
  %5 = llvm.icmp "ne" %4, %1 : i32
  %6 = llvm.and %arg4, %2 : i32
  %7 = llvm.icmp "ne" %6, %3 : i32
  %8 = llvm.and %5, %7 : i1
  "llvm.return"(%8) : (i1) -> ()
}
]

def positive_sameconst_lshrexact_after := [llvm|
{
^0(%arg41 : i8):
  "llvm.return"(%arg41) : (i8) -> ()
}
]

@[simp_llvm_option]
def not {w : Nat} (x : IntW w) : IntW w := do
  let x' ← x
  not? x'

def or_xor_tree_0011_before := [llvm|
{
^0(%arg36 : i32, %arg37 : i32, %arg38 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mul %arg36, %0 : i32
  %2 = llvm.mul %arg37, %0 : i32
  %3 = llvm.mul %arg38, %0 : i32
  %4 = llvm.xor %2, %1 : i32
  %5 = llvm.xor %3, %2 : i32
  %6 = llvm.xor %5, %1 : i32
  %7 = llvm.or %4, %6 : i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def mul8_low_after := [llvm|
{
^0(%arg26 : i8, %arg27 : i8):
  %0 = llvm.mul %arg26, %arg27 : i8
  "llvm.return"(%0) : (i8) -> ()
}
]

inductive ExpTy where
  | isl : ISL.Ty → ExpTy
  | regFile

def select_replace_udiv_non_speculatable_before := [llvm|
{
^0(%arg1 : i32, %arg2 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "eq" %arg1, %0 : i32
  %2 = llvm.udiv %arg2, %arg1 : i32
  %3 = "llvm.select"(%1, %2, %arg2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def ashrsgt_01_11_exact_after := [llvm|
{
^0(%arg90 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def positive_samevar_shlnuwnsw_before := [llvm|
{
^0(%arg47 : i8, %arg48 : i8):
  %0 = llvm.lshr %arg47, %arg48 : i8
  %1 = llvm.shl %0, %arg48 overflow<nsw,nuw> : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def select_exact_lshr_icmp_const_after := [llvm|
{
^0(%arg146 : i32):
  %0 = llvm.mlir.constant(5 : i32) : i32
  %1 = llvm.lshr %arg146, %0 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def and_zext_demanded_after := [llvm|
{
^0(%arg255 : i16, %arg256 : i32):
  %0 = llvm.mlir.constant(8 : i16) : i16
  %1 = llvm.lshr %arg255, %0 : i16
  %2 = llvm.zext nneg %1 : i16 to i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def ashr_mul_sign_bits_before := [llvm|
{
^0(%arg86 : i8, %arg87 : i8):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.sext %arg86 : i8 to i32
  %2 = llvm.sext %arg87 : i8 to i32
  %3 = llvm.mul %1, %2 : i32
  %4 = llvm.ashr %3, %0 : i32
  %5 = llvm.trunc %4 : i32 to i16
  "llvm.return"(%5) : (i16) -> ()
}
]

def ashr_ult_noexact_before := [llvm|
{
^0(%arg159 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.mlir.constant(10 : i8) : i8
  %2 = llvm.ashr %arg159, %0 : i8
  %3 = llvm.icmp "ult" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def ashrslt_02_08_after := [llvm|
{
^0(%arg245 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

Bool.atLeastTwo in Init.Data.BitVec.Bitblast

def TermBinop.termGadget (t : TermBinop) : CNFA 3 :=
  match t with
  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM
  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM
  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM
  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM
  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM

def src_is_mask_and_fail_mixed_before := [llvm|
{
^0(%arg119 : i8, %arg120 : i8, %arg121 : i8):
  %0 = llvm.mlir.constant(123 : i8) : i8
  %1 = llvm.mlir.constant(-8 : i8) : i8
  %2 = llvm.mlir.constant(-1 : i8) : i8
  %3 = llvm.xor %arg119, %0 : i8
  %4 = llvm.ashr %1, %arg120 : i8
  %5 = llvm.lshr %2, %arg121 : i8
  %6 = llvm.and %4, %5 : i8
  %7 = llvm.and %3, %6 : i8
  %8 = llvm.icmp "eq" %3, %7 : i8
  "llvm.return"(%8) : (i1) -> ()
}
]

def udiv_c_i32_before := [llvm|
{
^0(%arg6 : i8):
  %0 = llvm.mlir.constant(10 : i32) : i32
  %1 = llvm.zext %arg6 : i8 to i32
  %2 = llvm.udiv %0, %1 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

@[simp]
def EnvFin.get_ofEnv (env : Env w) (n : Nat) (i : Fin n) : (EnvFin.ofEnv env n) i = env.getD i 0#w := rfl

def max_of_min_after := [llvm|
{
^0(%arg22 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  "llvm.return"(%0) : (i32) -> ()
}
]

def uge_swap_and_max_commute_before := [llvm|
{
^0(%arg226 : i8, %arg227 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.icmp "ule" %arg227, %arg226 : i8
  %2 = llvm.icmp "eq" %arg226, %0 : i8
  %3 = llvm.and %2, %1 : i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def lshrult_02_09_exact_before := [llvm|
{
^0(%arg124 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(-7 : i4) : i4
  %2 = llvm.lshr exact %arg124, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def and_or2_wrong_operand_after := [llvm|
{
^0(%arg172 : i1, %arg173 : i1, %arg174 : i1, %arg175 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.xor %arg174, %0 : i1
  %2 = llvm.and %arg173, %1 : i1
  %3 = "llvm.select"(%2, %arg172, %arg175) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

structure Item where
 name : String
 tac : Syntax

def icmp_eq_x_invertable_y2_todo_after := [llvm|
{
^0(%arg4 : i8, %arg5 : i1):
  %0 = llvm.mlir.constant(-8 : i8) : i8
  %1 = llvm.mlir.constant(-25 : i8) : i8
  %2 = llvm.mlir.constant(-1 : i8) : i8
  %3 = "llvm.select"(%arg5, %0, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %4 = llvm.or %arg4, %3 : i8
  %5 = llvm.icmp "eq" %4, %2 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

def scalar0_after := [llvm|
{
^0(%arg32 : i4, %arg33 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(-2 : i4) : i4
  %2 = llvm.and %arg32, %0 : i4
  %3 = llvm.and %arg33, %1 : i4
  %4 = llvm.or disjoint %2, %3 : i4
  "llvm.return"(%4) : (i4) -> ()
}
]

def icmp_eq_sext_ne_otherwise_before := [llvm|
{
^0(%arg14 : i32):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.icmp "ne" %arg14, %0 : i32
  %2 = llvm.sext %1 : i1 to i32
  %3 = llvm.icmp "eq" %2, %arg14 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def test5_before := [llvm|
{
^0(%arg2 : i7):
  %0 = llvm.mlir.constant(23 : i7) : i7
  %1 = llvm.or %arg2, %0 : i7
  %2 = llvm.xor %1, %0 : i7
  "llvm.return"(%2) : (i7) -> ()
}
]

def test_imply_not2_after := [llvm|
{
^0(%arg3 : i32, %arg4 : i1):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ne" %arg3, %0 : i32
  %3 = "llvm.select"(%2, %1, %arg4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def shl_add_lshr_after := [llvm|
{
^0(%arg154 : i32, %arg155 : i32, %arg156 : i32):
  %0 = llvm.lshr %arg156, %arg155 : i32
  %1 = llvm.add %0, %arg154 overflow<nuw> : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def tryFactorization_add_mul_nuw_after := [llvm|
{
^0(%arg13 : i32):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.shl %arg13, %0 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

@[simp_llvm]
def sdiv? {w : Nat} (x y : BitVec w) : IntW w :=
  if y == 0 || (w != 1 && x == (BitVec.intMin w) && y == -1) then
    .poison
  else
    .value (x.sdiv y)

def trunc_shl_lshr_infloop_before := [llvm|
{
^0(%arg41 : i64):
  %0 = llvm.mlir.constant(1) : i64
  %1 = llvm.mlir.constant(2) : i64
  %2 = llvm.lshr %arg41, %0 : i64
  %3 = llvm.shl %2, %1 : i64
  %4 = llvm.trunc %3 : i64 to i32
  "llvm.return"(%4) : (i32) -> ()
}
]

Lean.Meta.withLocalDeclsDND in Lean.Meta.Basic

def masked_icmps_mask_notallzeros_bmask_mixed_7b_after := [llvm|
{
^0(%arg76 : i32):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def llvm_const_lower_riscv_li_23 : LLVMPeepholeRewriteRefine 64 [] :=
  {lhs := constLLVM_23, rhs:= liRiscv_23,
   correct := by admit /- proof elided -/

def src_x_or_mask_ne_before := [llvm|
{
^0(%arg3 : i8, %arg4 : i8, %arg5 : i1):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.mlir.constant(0 : i8) : i8
  %2 = llvm.lshr %0, %arg4 : i8
  %3 = "llvm.select"(%arg5, %2, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %4 = llvm.xor %arg3, %0 : i8
  %5 = llvm.or %3, %4 : i8
  %6 = llvm.icmp "ne" %5, %0 : i8
  "llvm.return"(%6) : (i1) -> ()
}
]

def ne_210_before := [llvm|
{
^0(%arg76 : i32, %arg77 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(16 : i32) : i32
  %2 = llvm.trunc %arg76 : i32 to i8
  %3 = llvm.lshr %arg76, %0 : i32
  %4 = llvm.trunc %3 : i32 to i8
  %5 = llvm.lshr %arg76, %1 : i32
  %6 = llvm.trunc %5 : i32 to i8
  %7 = llvm.trunc %arg77 : i32 to i8
  %8 = llvm.lshr %arg77, %0 : i32
  %9 = llvm.trunc %8 : i32 to i8
  %10 = llvm.lshr %arg77, %1 : i32
  %11 = llvm.trunc %10 : i32 to i8
  %12 = llvm.icmp "ne" %2, %7 : i8
  %13 = llvm.icmp "ne" %4, %9 : i8
  %14 = llvm.icmp "ne" %6, %11 : i8
  %15 = llvm.or %12, %13 : i1
  %16 = llvm.or %14, %15 : i1
  "llvm.return"(%16) : (i1) -> ()
}
]

instance : GetElem? (Ctxt Ty) Nat Ty (fun Γ i => i < Γ.length) where
  getElem Γ i h := Γ.toList[i]
  getElem? Γ i  := Γ.toList[i]?

def select_xor_icmp_bad_3_after := [llvm|
{
^0(%arg179 : i32, %arg180 : i32, %arg181 : i32):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.icmp "eq" %arg179, %0 : i32
  %2 = llvm.xor %arg181, %0 : i32
  %3 = "llvm.select"(%1, %2, %arg180) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def trunc_equality_both_after := [llvm|
{
^0(%arg40 : i16, %arg41 : i16):
  %0 = llvm.icmp "eq" %arg40, %arg41 : i16
  "llvm.return"(%0) : (i1) -> ()
}
]

def xor_or_xor_common_op_commute8_before := [llvm|
{
^0(%arg59 : i32, %arg60 : i32, %arg61 : i32):
  %0 = llvm.xor %arg61, %arg59 : i32
  %1 = llvm.or %arg60, %arg59 : i32
  %2 = llvm.xor %1, %0 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def Eqn.denoteFin {w : Nat} (e : Eqn) (envFin : EnvFin w e.numVars) : Int :=
  match e with
  | [] => 0
  | t :: ts =>
    t.denoteFin (envFin.castLe (by admit /- proof elided -/
    )) +
    Eqn.denoteFin ts (envFin.castLe (by admit /- proof elided -/
    ))

def shl_add_add_no_const_before := [llvm|
{
^0(%arg122 : i8, %arg123 : i8, %arg124 : i8, %arg125 : i8):
  %0 = llvm.shl %arg122, %arg124 : i8
  %1 = llvm.shl %arg123, %arg124 : i8
  %2 = llvm.add %1, %arg125 : i8
  %3 = llvm.add %0, %2 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def add32_accept_test : LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 32)] where
  lhs := add32_accept_llvm
  rhs := add32_accept_riscv

def test_shift_and_cmp_changed4_after := [llvm|
{
^0(%arg0 : i8):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def positive_different_trunc_both_logical_before := [llvm|
{
^0(%arg36 : i32):
  %0 = llvm.mlir.constant(-1 : i15) : i15
  %1 = llvm.mlir.constant(128 : i16) : i16
  %2 = llvm.mlir.constant(256 : i16) : i16
  %3 = llvm.mlir.constant(false) : i1
  %4 = llvm.trunc %arg36 : i32 to i15
  %5 = llvm.icmp "sgt" %4, %0 : i15
  %6 = llvm.trunc %arg36 : i32 to i16
  %7 = llvm.add %6, %1 : i16
  %8 = llvm.icmp "ult" %7, %2 : i16
  %9 = "llvm.select"(%5, %8, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%9) : (i1) -> ()
}
]

@[tactic bvAutomataFragmentWidthLegal]
def evalBvAutomataFragmentIllegalWidth : Tactic := fun
| `(tactic| bv_automata_fragment_width_legal) => do
  let g ← getMainGoal
  g.withContext do
    let ws ← findExprBitwidths (← g.getType)
    let ws := ws.toArray
    if h0: ws.size = 0 then throwError m!"found no bitvector in the target: {indentD (← g.getType)}"
    else if hgt: ws.size > 1 then
      let (w1, wExample1) := ws[0]
      let (w2, wExample2) := ws[1]
      let mExample := m!"{w1} → {wExample1}; {w2} → {wExample2}"
      throwError m!"found multiple bitvector widths in the target: {indentD mExample}"
    else
      return ()
| _ => throwUnsupportedSyntax

RV64.ctzw in RISCV.Instructions

def n1_before := [llvm|
{
^0(%arg2 : i8, %arg3 : i8):
  %0 = llvm.mlir.constant(1 : i8) : i8
  %1 = llvm.mlir.constant(-1 : i8) : i8
  %2 = llvm.shl %0, %arg3 : i8
  %3 = llvm.xor %2, %1 : i8
  %4 = llvm.and %3, %arg2 : i8
  %5 = llvm.icmp "ne" %4, %arg2 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

def DCcanAdd := [DCxComb_com| {
  ^entry(%a: !ValueStream_32, %b: !ValueStream_32):
    %unpacka = "DCxComb.unpack" (%a) : (!ValueStream_32) -> (!ValueTokenStream_32)
    %output = "DCxComb.fstVal" (%unpacka) : (!ValueTokenStream_32) -> (!ValueStream_32)
    %token = "DCxComb.sndVal" (%unpacka) : (!ValueTokenStream_32) -> (!TokenStream)
    
    %0 = "DCxComb.fork" (%token) : (!TokenStream) -> (!TokenStream2)
    %fst0 = "DCxComb.fst" (%0) : (!TokenStream2) -> (!TokenStream)
    %snd0 = "DCxComb.snd" (%0) : (!TokenStream2) -> (!TokenStream)
    
    %2 = "DCxComb.shlPar_1" (%output) : (!ValueStream_32) -> (!ValueStream_32)
    %unpackb = "DCxComb.unpack" (%b) : (!ValueStream_32) -> (!ValueTokenStream_32)
    %output1 = "DCxComb.fstVal" (%unpackb) : (!ValueTokenStream_32) -> (!ValueStream_32)
    %token0 = "DCxComb.sndVal" (%unpackb) : (!ValueTokenStream_32) -> (!TokenStream)
    
    %3 = "DCxComb.join" (%fst0, %token0) : (!TokenStream, !TokenStream) -> (!TokenStream)
    %4 = "DCxComb.add" (%2, %output) : (!ValueStream_32, !ValueStream_32) -> (!ValueStream_32)
    
    "return" (%a) : (!ValueStream_32) -> ()
  }]

def test1_before := [llvm|
{
^0(%arg8 : i32, %arg9 : i32):
  %0 = llvm.mlir.constant(5 : i32) : i32
  %1 = llvm.mlir.constant(10 : i32) : i32
  %2 = llvm.xor %arg8, %0 : i32
  %3 = llvm.icmp "eq" %2, %1 : i32
  %4 = llvm.icmp "eq" %2, %arg9 : i32
  %5 = llvm.or %3, %4 : i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def icmp_eq_sext_eq_otherwise_before := [llvm|
{
^0(%arg15 : i32):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.icmp "eq" %arg15, %0 : i32
  %2 = llvm.sext %1 : i1 to i32
  %3 = llvm.icmp "eq" %2, %arg15 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def trunc_shl_1_i32_i64_before := [llvm|
{
^0(%arg35 : i64):
  %0 = llvm.mlir.constant(1) : i64
  %1 = llvm.shl %arg35, %0 : i64
  %2 = llvm.trunc %1 : i64 to i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def llvm_mul_lower_riscv_nsw_flag_32 : LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 32)] where
  lhs := mul_llvm_nsw_32
  rhs := mul_riscv_32

def and_min_7_7_after := [llvm|
{
^0(%arg2 : i8):
  %0 = llvm.mlir.constant(-8 : i8) : i8
  %1 = llvm.and %arg2, %0 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def ashrsgt_01_06_exact_before := [llvm|
{
^0(%arg95 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(6 : i4) : i4
  %2 = llvm.ashr exact %arg95, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def t1_after := [llvm|
{
^0(%arg16 : i8):
  %0 = llvm.mlir.constant(5 : i8) : i8
  %1 = llvm.ashr %arg16, %0 : i8
  %2 = llvm.sext %1 : i8 to i16
  "llvm.return"(%2) : (i16) -> ()
}
]

Lean.Meta.State in Lean.Meta.Basic

def getSimpData (simpsetName : Name) : MetaM (SimpTheorems × Simprocs) := do
  let some ext ← (getSimpExtension? simpsetName)
    | throwError m!"'{simpsetName}' simp attribute not found!"
  let theorems ← ext.getTheorems
  let some ext ← (Simp.getSimprocExtension? simpsetName)
    | throwError m!"'{simpsetName}' simp attribute not found!"
  let simprocs ← ext.getSimprocs
  return (theorems, simprocs)

def ashr_eq_msb_low_second_zero_after := [llvm|
{
^0(%arg175 : i8):
  %0 = llvm.mlir.constant(6 : i8) : i8
  %1 = llvm.icmp "ugt" %arg175, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def canonicalize_logic_first_or0_nswnuw_after := [llvm|
{
^0(%arg38 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(112 : i32) : i32
  %2 = llvm.or %arg38, %0 : i32
  %3 = llvm.add %2, %1 overflow<nsw,nuw> : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def or_ne_notY_eq_1s_before := [llvm|
{
^0(%arg41 : i8, %arg42 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.xor %arg42, %0 : i8
  %2 = llvm.or %arg41, %1 : i8
  %3 = llvm.icmp "ne" %2, %arg41 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def sgt_swap_and_min_commute_logical_after := [llvm|
{
^0(%arg338 : i8, %arg339 : i8):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def ashrslt_03_05_exact_after := [llvm|
{
^0(%arg16 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def or_nosignbit_select_ashr_before := [llvm|
{
^0(%arg8 : i32, %arg9 : i1):
  %0 = llvm.mlir.constant(2147418112 : i32) : i32
  %1 = llvm.mlir.constant(8 : i32) : i32
  %2 = llvm.or %arg8, %0 : i32
  %3 = "llvm.select"(%arg9, %2, %arg8) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %4 = llvm.ashr %3, %1 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def neg_nsw_sgt_0_after := [llvm|
{
^0(%arg17 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "slt" %arg17, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def or_not_and_and_not_and_xor_commute4_before := [llvm|
{
^0(%arg277 : i32, %arg278 : i32, %arg279 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg277 : i32
  %3 = llvm.and %arg278, %arg279 : i32
  %4 = llvm.xor %3, %1 : i32
  %5 = llvm.or %2, %4 : i32
  %6 = llvm.xor %arg278, %arg279 : i32
  %7 = llvm.and %2, %6 : i32
  %8 = llvm.xor %7, %1 : i32
  %9 = llvm.and %5, %8 : i32
  "llvm.return"(%9) : (i32) -> ()
}
]

instance : ToString (Com d Γ eff t)  where toString com  := s!"{com.print}"

def alive_AndOrXor_1294_A__B__A__B___A__B_tgt :=
[sllvm| {
^bb0(%A : i64, %B : i64):
  %v1 = llvm.or %A, %B : i64
  %v2 = llvm.mlir.constant -1 : i64
  %v3 = llvm.xor %A, %v2 : i64
  %v4 = llvm.xor %v3, %B : i64
  %v5 = llvm.and %A, %B : i64
  llvm.return %v5 : i64
}]

def neg_nsw_slt_1_before := [llvm|
{
^0(%arg19 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.mlir.constant(1 : i8) : i8
  %2 = llvm.sub %0, %arg19 overflow<nsw> : i8
  %3 = llvm.icmp "slt" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def slt_zero_eq_ne_0_fail2_before := [llvm|
{
^0(%arg0 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(30 : i32) : i32
  %2 = llvm.icmp "ne" %arg0, %0 : i32
  %3 = llvm.zext %2 : i1 to i32
  %4 = llvm.lshr %arg0, %1 : i32
  %5 = llvm.icmp "eq" %3, %4 : i32
  "llvm.return"(%5) : (i1) -> ()
}
]

def and_or_not_not_commute1_after := [llvm|
{
^0(%arg372 : i32, %arg373 : i32, %arg374 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg373 : i32
  %3 = llvm.or %arg374, %2 : i32
  %4 = llvm.and %3, %arg372 : i32
  %5 = llvm.xor %4, %1 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def one_side_fold_slt_before := [llvm|
{
^0(%arg33 : i32, %arg34 : i32, %arg35 : i32, %arg36 : i1):
  %0 = "llvm.select"(%arg36, %arg33, %arg35) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %1 = "llvm.select"(%arg36, %arg34, %arg35) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %2 = llvm.icmp "slt" %1, %0 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

def main7c_logical_after := [llvm|
{
^0(%arg69 : i32, %arg70 : i32, %arg71 : i32):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.and %arg70, %arg69 : i32
  %2 = llvm.icmp "ne" %arg70, %1 : i32
  %3 = llvm.and %arg71, %arg69 : i32
  %4 = llvm.icmp "ne" %arg71, %3 : i32
  %5 = "llvm.select"(%2, %0, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %6 = llvm.zext %5 : i1 to i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def lshrult_03_04_exact_before := [llvm|
{
^0(%arg113 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(4 : i4) : i4
  %2 = llvm.lshr exact %arg113, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def icmp_sge_riscv_eq_icmp_sge_llvm_32 : LLVMPeepholeRewriteRefine 1 [Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 32)] :=
  {lhs:= icmp_sge_llvm_32, rhs:= icmp_sge_riscv_32}

def irc_constants_APlusC1MinusC2_2_neg1 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (2) : i64
      %c2 = llvm.mlir.constant (-1) : i64
      %0 = llvm.add %a, %c1 : i64
      %1 = llvm.sub %0, %c2 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (2) : i64
      %c2 = llvm.mlir.constant (-1) : i64
      %0 = llvm.sub %c1, %c2 : i64
      %1 = llvm.add %a, %0 : i64
      llvm.return %1 : i64
  }]

Std.Refl in Init.Core

def ugt_umax4_before := [llvm|
{
^0(%arg30 : i32, %arg31 : i32):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.add %arg30, %0 : i32
  %2 = llvm.icmp "ugt" %arg31, %1 : i32
  %3 = "llvm.select"(%2, %arg31, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %4 = llvm.icmp "ult" %1, %3 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def test15_after := [llvm|
{
^0(%arg44 : i8, %arg45 : i8):
  %0 = llvm.mlir.constant(33 : i8) : i8
  %1 = llvm.xor %arg45, %arg44 : i8
  %2 = llvm.xor %arg44, %arg45 : i8
  %3 = llvm.xor %2, %0 : i8
  %4 = llvm.and %1, %3 : i8
  %5 = llvm.mul %4, %3 : i8
  "llvm.return"(%5) : (i8) -> ()
}
]

def slt_zero_slt_i1_fail_before := [llvm|
{
^0(%arg4 : i32, %arg5 : i1):
  %0 = llvm.mlir.constant(31 : i32) : i32
  %1 = llvm.zext %arg5 : i1 to i32
  %2 = llvm.lshr %arg4, %0 : i32
  %3 = llvm.icmp "slt" %1, %2 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def icmp_mul_nsw_sgt_after := [llvm|
{
^0(%arg15 : i8, %arg16 : i8):
  %0 = llvm.icmp "sgt" %arg15, %arg16 : i8
  "llvm.return"(%0) : (i1) -> ()
}
]

def match_signed_after := [llvm|
{
^0(%arg15 : i64):
  %0 = llvm.mlir.constant(172224) : i64
  %1 = llvm.srem %arg15, %0 : i64
  "llvm.return"(%1) : (i64) -> ()
}
]

def shift_trunc_wrong_cmp_after := [llvm|
{
^0(%arg0 : i32):
  %0 = llvm.mlir.constant(24 : i32) : i32
  %1 = llvm.mlir.constant(1 : i8) : i8
  %2 = llvm.lshr %arg0, %0 : i32
  %3 = llvm.trunc %2 overflow<nuw> : i32 to i8
  %4 = llvm.icmp "slt" %3, %1 : i8
  "llvm.return"(%4) : (i1) -> ()
}
]

def lshrugt_01_11_exact_after := [llvm|
{
^0(%arg210 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def positive_samevar_shlnsw_ashrexact_after := [llvm|
{
^0(%arg37 : i8, %arg38 : i8):
  "llvm.return"(%arg37) : (i8) -> ()
}
]

def ashr_lshr_cst2_after := [llvm|
{
^0(%arg103 : i32, %arg104 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.ashr %arg103, %0 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

@[simp_llvm]
def trunc? {w: Nat} (w': Nat) (x: BitVec w) : IntW w' := do
  .value <| (BitVec.truncate w' x)

cmp in Mathlib.Data.Ordering.Basic

def test_and1_before := [llvm|
{
^0(%arg52 : i32, %arg53 : i32):
  %0 = llvm.mlir.constant(2147483647 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.and %arg53, %0 : i32
  %3 = llvm.icmp "sge" %arg52, %1 : i32
  %4 = llvm.icmp "slt" %arg52, %2 : i32
  %5 = llvm.and %3, %4 : i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def t4_before := [llvm|
{
^0(%arg210 : i8, %arg211 : i1):
  %0 = llvm.mlir.constant(-42 : i8) : i8
  %1 = llvm.mlir.constant(44 : i8) : i8
  %2 = "llvm.select"(%arg211, %0, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %3 = llvm.sub %arg210, %2 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def icmp_power2_and_icmp_shifted_mask_2147483648_1610612736_after := [llvm|
{
^0(%arg45 : i32):
  %0 = llvm.mlir.constant(1610612736 : i32) : i32
  %1 = llvm.icmp "ult" %arg45, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def lshrugt_01_02_exact_before := [llvm|
{
^0(%arg219 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(2 : i4) : i4
  %2 = llvm.lshr exact %arg219, %0 : i4
  %3 = llvm.icmp "ugt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def scalar_i32_udiv_and_negC_eq_X_is_constant3_after := [llvm|
{
^0(%arg7 : i32):
  %0 = llvm.mlir.constant(1544 : i32) : i32
  %1 = llvm.icmp "ult" %arg7, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def not_logicalOr_not_op1_after := [llvm|
{
^0(%arg50 : i1, %arg51 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.xor %arg50, %0 : i1
  %3 = "llvm.select"(%2, %arg51, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def masked_icmps_mask_notallzeros_bmask_mixed_swapped_1_after := [llvm|
{
^0(%arg50 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(9 : i32) : i32
  %2 = llvm.and %arg50, %0 : i32
  %3 = llvm.icmp "eq" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def slt_swap_and_not_max_logical_before := [llvm|
{
^0(%arg116 : i8, %arg117 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.icmp "sgt" %arg117, %arg116 : i8
  %3 = llvm.icmp "ne" %arg116, %0 : i8
  %4 = "llvm.select"(%2, %3, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def ashrslt_03_08_after := [llvm|
{
^0(%arg229 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def BoolBinaryRelationKind.toSmtLib : BoolBinaryRelationKind → SexprPBV.BoolBinaryRelationKind
| .eq => .eq

def liftEffect_compose {e1 e2 e3 : EffectKind} {α : Type} [Pure m]
    (h12 : e1 ≤ e2)
    (h23 : e2 ≤ e3)
    (h13 : e1 ≤ e3 := le_trans h12 h23) :
    ((liftEffect (α := α) h23) ∘ (liftEffect h12)) = liftEffect (m := m) h13 :=

def irc_constants_AMinusC1PlusC2_2_0 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (2) : i64
      %c2 = llvm.mlir.constant (0) : i64
      %0 = llvm.sub %a, %c1 : i64
      %1 = llvm.add %0, %c2 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (2) : i64
      %c2 = llvm.mlir.constant (0) : i64
      %0 = llvm.sub %c2, %c1 : i64
      %1 = llvm.add %a, %0 : i64
      llvm.return %1 : i64
  }]

def lshrugt_01_13_exact_after := [llvm|
{
^0(%arg208 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def not_before := [llvm|
{
^0(%arg7 : i8):
  %0 = llvm.mlir.constant(42 : i8) : i8
  %1 = llvm.mlir.constant(-1 : i8) : i8
  %2 = llvm.udiv %arg7, %0 : i8
  %3 = llvm.xor %1, %arg7 : i8
  %4 = llvm.mul %3, %2 : i8
  "llvm.return"(%4) : (i8) -> ()
}
]

def or_xor_tree_0111_after := [llvm|
{
^0(%arg24 : i32, %arg25 : i32, %arg26 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mul %arg24, %0 : i32
  %2 = llvm.mul %arg25, %0 : i32
  %3 = llvm.mul %arg26, %0 : i32
  %4 = llvm.xor %2, %1 : i32
  %5 = llvm.or %4, %3 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def exact_lshr_eq_both_equal_before := [llvm|
{
^0(%arg166 : i8):
  %0 = llvm.mlir.constant(126 : i8) : i8
  %1 = llvm.lshr exact %0, %arg166 : i8
  %2 = llvm.icmp "eq" %1, %0 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

instance : Lean.ToExpr RegIndex := inherit

def test_and3_before := [llvm|
{
^0(%arg44 : i32, %arg45 : i32):
  %0 = llvm.mlir.constant(2147483647 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.and %arg45, %0 : i32
  %3 = llvm.icmp "sgt" %2, %arg44 : i32
  %4 = llvm.icmp "sge" %arg44, %1 : i32
  %5 = llvm.and %3, %4 : i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def udiv_shl_mul_nuw_exact_before := [llvm|
{
^0(%arg142 : i5, %arg143 : i5, %arg144 : i5):
  %0 = llvm.shl %arg142, %arg144 overflow<nuw> : i5
  %1 = llvm.mul %arg142, %arg143 overflow<nuw> : i5
  %2 = llvm.udiv exact %0, %1 : i5
  "llvm.return"(%2) : (i5) -> ()
}
]

def irc_constants_C1Minus2MinusC2_neg1_neg2 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (-1) : i64
      %c2 = llvm.mlir.constant (-2) : i64
      %0 = llvm.sub %c1, %a : i64
      %1 = llvm.sub %0, %c2 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (-1) : i64
      %c2 = llvm.mlir.constant (-2) : i64
      %0 = llvm.sub %c1, %c2 : i64
      %1 = llvm.sub %0, %a : i64
      llvm.return %1 : i64
  }]

def test22_before := [llvm|
{
^0(%arg173 : i32):
  %0 = llvm.mlir.constant(24 : i32) : i32
  %1 = llvm.trunc %arg173 : i32 to i8
  %2 = llvm.sext %1 : i8 to i32
  %3 = llvm.shl %2, %0 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def lshrugt_02_02_after := [llvm|
{
^0(%arg397 : i4):
  %0 = llvm.mlir.constant(-5 : i4) : i4
  %1 = llvm.icmp "ugt" %arg397, %0 : i4
  "llvm.return"(%1) : (i1) -> ()
}
]

def lshrult_01_07_before := [llvm|
{
^0(%arg360 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(7 : i4) : i4
  %2 = llvm.lshr %arg360, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def test19_after := [llvm|
{
^0(%arg7 : i10):
  %0 = llvm.mlir.constant(1 : i3) : i3
  %1 = llvm.mlir.constant(0 : i3) : i3
  %2 = llvm.trunc %arg7 : i10 to i3
  %3 = llvm.and %2, %0 : i3
  %4 = llvm.sub %1, %3 overflow<nsw> : i3
  %5 = llvm.sext %4 : i3 to i10
  "llvm.return"(%5) : (i10) -> ()
}
]

def test3_apint_before := [llvm|
{
^0(%arg43 : i47, %arg44 : i47):
  %0 = llvm.mlir.constant(-1 : i47) : i47
  %1 = llvm.xor %arg43, %0 : i47
  %2 = llvm.xor %arg44, %0 : i47
  %3 = llvm.and %1, %2 : i47
  %4 = llvm.xor %3, %0 : i47
  "llvm.return"(%4) : (i47) -> ()
}
]

def trunc_shl_15_i16_i64_after := [llvm|
{
^0(%arg54 : i64):
  %0 = llvm.mlir.constant(15 : i16) : i16
  %1 = llvm.trunc %arg54 : i64 to i16
  %2 = llvm.shl %1, %0 : i16
  "llvm.return"(%2) : (i16) -> ()
}
]

def dec_mask_neg_i32_after := [llvm|
{
^0(%arg7 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.add %arg7, %0 : i32
  %2 = llvm.xor %arg7, %0 : i32
  %3 = llvm.and %1, %2 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def lshrugt_02_10_exact_before := [llvm|
{
^0(%arg195 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(-6 : i4) : i4
  %2 = llvm.lshr exact %arg195, %0 : i4
  %3 = llvm.icmp "ugt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def test9_before := [llvm|
{
^0(%arg111 : i32, %arg112 : i32):
  %0 = llvm.and %arg111, %arg112 : i32
  %1 = llvm.xor %arg111, %arg112 : i32
  %2 = llvm.xor %0, %1 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def exact_lshr_ne_before := [llvm|
{
^0(%arg157 : i8):
  %0 = llvm.mlir.constant(4 : i8) : i8
  %1 = llvm.mlir.constant(1 : i8) : i8
  %2 = llvm.lshr exact %0, %arg157 : i8
  %3 = llvm.icmp "ne" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def const (a : α) (_b : β) : α := a

def llvm_const_lower_riscv_li_33 : LLVMPeepholeRewriteRefine 64 [] :=
  {lhs := constLLVM_33, rhs:= liRiscv_33,
   correct := by admit /- proof elided -/

def ule_umax2_after := [llvm|
{
^0(%arg50 : i32, %arg51 : i32):
  %0 = llvm.icmp "ule" %arg51, %arg50 : i32
  "llvm.return"(%0) : (i1) -> ()
}
]

def lshrugt_02_07_after := [llvm|
{
^0(%arg392 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def positive_biggerlshr_before := [llvm|
{
^0(%arg60 : i8):
  %0 = llvm.mlir.constant(6 : i8) : i8
  %1 = llvm.mlir.constant(3 : i8) : i8
  %2 = llvm.lshr %arg60, %0 : i8
  %3 = llvm.shl %2, %1 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def constLLVM_6 : Com LLVMPlusRiscV ⟨[]⟩ .pure (.llvm (.bitvec 64)) := [LV| {
  ^entry ():
    %1 = llvm.mlir.constant (-6) : i64
    llvm.return %1 : i64
  }]

def and_not_or_wrong_b_before := [llvm|
{
^0(%arg428 : i32, %arg429 : i32, %arg430 : i32, %arg431 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.and %arg428, %arg429 : i32
  %2 = llvm.xor %1, %0 : i32
  %3 = llvm.or %2, %arg430 : i32
  %4 = llvm.and %arg428, %arg430 : i32
  %5 = llvm.xor %4, %0 : i32
  %6 = llvm.or %5, %arg431 : i32
  %7 = llvm.and %3, %6 : i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def ashr_ugt_11_before := [llvm|
{
^0(%arg103 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(-5 : i4) : i4
  %2 = llvm.ashr %arg103, %0 : i4
  %3 = llvm.icmp "ugt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def alive_Select_1100_src  (w : Nat)   :=
[llvm( w )| {
^bb0(%Y : _, %X : _):
  %v1 = llvm.mlir.constant 1 :  i1
  %v2 = llvm.select %v1, %X, %Y
  llvm.return %v2
}]

def or_basic_before := [llvm|
{
^0(%arg15 : i16):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.mlir.constant(-256 : i16) : i16
  %2 = llvm.mlir.constant(17664 : i16) : i16
  %3 = llvm.trunc %arg15 : i16 to i8
  %4 = llvm.icmp "ne" %3, %0 : i8
  %5 = llvm.and %arg15, %1 : i16
  %6 = llvm.icmp "ne" %5, %2 : i16
  %7 = llvm.or %4, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def different_size_zext_zext_sgt_after := [llvm|
{
^0(%arg16 : i7, %arg17 : i4):
  %0 = llvm.zext %arg17 : i4 to i7
  %1 = llvm.icmp "ugt" %arg16, %0 : i7
  "llvm.return"(%1) : (i1) -> ()
}
]

def positive_with_extra_and_before := [llvm|
{
^0(%arg53 : i32, %arg54 : i1):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.mlir.constant(128 : i32) : i32
  %2 = llvm.mlir.constant(256 : i32) : i32
  %3 = llvm.icmp "sgt" %arg53, %0 : i32
  %4 = llvm.add %arg53, %1 : i32
  %5 = llvm.icmp "ult" %4, %2 : i32
  %6 = llvm.and %3, %arg54 : i1
  %7 = llvm.and %5, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def main7g_after := [llvm|
{
^0(%arg34 : i32, %arg35 : i32, %arg36 : i32, %arg37 : i32, %arg38 : i32):
  %0 = llvm.and %arg35, %arg37 : i32
  %1 = llvm.and %arg36, %arg38 : i32
  %2 = llvm.or %0, %1 : i32
  %3 = llvm.and %arg34, %2 : i32
  %4 = llvm.icmp "ne" %3, %2 : i32
  %5 = llvm.zext %4 : i1 to i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def parseAndGeneralize
  [H : HydrableParseAndGeneralize parsedExpr genPred genExpr]
  (cfg : MedusaSynthGeneralizeConfig)
  (hExpr : Expr) (context: GeneralizeContext): TermElabM MessageData := do
    let targetWidth := 8
    let timeoutMs := 300000

    match_expr hExpr with
    | Eq w lhsExpr rhsExpr =>

          let some width ← H.getWidth w  | throwError m! "Could not determine the rewrite width from {w}"
          let startTime ← Core.liftIOCore IO.monoMsNow

          \n Input expression: {hExpr} has generalization: {pretty}"
                          | GeneralizeContext.Tactic _name =>
                            match cfg.output with
                            | .thmStmt =>
                              let name := Name.mkSimple "foo"
                              pure m! "{H.prettifyAsTheorem name res allVariables widthVals}"
                            | .sexpr =>
                              throwError (H.prettifyAsSexpr res allVariables) widthVals|> format
            | none => throwError m! "Could not generalize {bvLogicalExpr}"
    | _ => throwError m!"The top level constructor is not an equality predicate in {hExpr}"

def ashr_ult_0_before := [llvm|
{
^0(%arg98 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(0 : i4) : i4
  %2 = llvm.ashr %arg98, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def uaddo_commute4_before := [llvm|
{
^0(%arg22 : i32, %arg23 : i32, %arg24 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.xor %arg23, %0 : i32
  %2 = llvm.add %arg23, %arg22 : i32
  %3 = llvm.icmp "ult" %1, %arg22 : i32
  %4 = "llvm.select"(%3, %arg24, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def fold_sext_to_and_wrong_before := [llvm|
{
^0(%arg9 : i8):
  %0 = llvm.mlir.constant(-2147483647 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sext %arg9 : i8 to i32
  %3 = llvm.and %2, %0 : i32
  %4 = llvm.icmp "eq" %3, %1 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def State.setMapping (e x : Expr) : GenM Unit := do
  let s ← get
  let m ← s.mapping.insert e x
  set {s with mapping := m}

def masked_icmps_mask_notallzeros_bmask_mixed_3b_logical_after := [llvm|
{
^0(%arg85 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(3 : i32) : i32
  %3 = llvm.and %arg85, %0 : i32
  %4 = llvm.icmp "ne" %3, %1 : i32
  %5 = llvm.and %arg85, %2 : i32
  %6 = llvm.icmp "eq" %5, %1 : i32
  %7 = llvm.and %4, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def shl_nsw_add_before := [llvm|
{
^0(%arg22 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.shl %0, %arg22 overflow<nsw> : i32
  %3 = llvm.add %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def test_lshr_exact__exact_is_safe_before := [llvm|
{
^0(%arg62 : i32, %arg63 : i64):
  %0 = llvm.mlir.constant(60 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.mlir.constant(0 : i32) : i32
  %3 = llvm.and %arg62, %0 : i32
  %4 = llvm.lshr exact %3, %1 : i32
  %5 = llvm.zext %4 : i32 to i64
  %6 = llvm.icmp "eq" %3, %2 : i32
  %7 = llvm.ashr %arg63, %5 : i64
  %8 = "llvm.select"(%6, %arg63, %7) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i64, i64) -> i64
  "llvm.return"(%8) : (i64) -> ()
}
]

def n11_wrong_pred3_before := [llvm|
{
^0(%arg8 : i8, %arg9 : i8):
  %0 = llvm.add %arg8, %arg9 : i8
  %1 = llvm.icmp "ne" %0, %arg9 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def sub_zext_trunc_lshr_after := [llvm|
{
^0(%arg15 : i64):
  %0 = llvm.mlir.constant(16 : i32) : i32
  %1 = llvm.mlir.constant(31 : i32) : i32
  %2 = llvm.mlir.constant(10 : i32) : i32
  %3 = llvm.trunc %arg15 : i64 to i32
  %4 = llvm.shl %3, %0 : i32
  %5 = llvm.ashr %4, %1 : i32
  %6 = llvm.add %5, %2 overflow<nsw> : i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def lshr_xor_after := [llvm|
{
^0(%arg65 : i8, %arg66 : i8):
  %0 = llvm.mlir.constant(42 : i8) : i8
  %1 = llvm.mlir.constant(3 : i8) : i8
  %2 = llvm.mlir.constant(-8 : i8) : i8
  %3 = llvm.srem %arg65, %0 : i8
  %4 = llvm.shl %3, %1 : i8
  %5 = llvm.and %arg66, %2 : i8
  %6 = llvm.xor %5, %4 : i8
  "llvm.return"(%6) : (i8) -> ()
}
]

HAnd in Init.Prelude

def ashrsgt_02_08_exact_before := [llvm|
{
^0(%arg77 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(-8 : i4) : i4
  %2 = llvm.ashr exact %arg77, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def uge_swap_and_max_logical_before := [llvm|
{
^0(%arg228 : i8, %arg229 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.icmp "ule" %arg229, %arg228 : i8
  %3 = llvm.icmp "eq" %arg228, %0 : i8
  %4 = "llvm.select"(%2, %3, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def sll_i32_test : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64), Ty.llvm (.bitvec 64)] where
  lhs := sll_llvm_i64
  rhs := sll_riscv_i64

def and_xor_commute2_before := [llvm|
{
^0(%arg153 : i32, %arg154 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.udiv %0, %arg153 : i32
  %2 = llvm.udiv %0, %arg154 : i32
  %3 = llvm.and %1, %2 : i32
  %4 = llvm.xor %3, %2 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def p0_after := [llvm|
{
^0(%arg16 : i8, %arg17 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.lshr %arg16, %arg17 : i8
  %2 = llvm.icmp "eq" %1, %0 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

def add_positive_low_bound_reject_test: LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 32)] where
  lhs := add_positive_low_bound_reject_llvm
  rhs := add_positive_low_bound_reject_riscv

def ashrslt_03_04_exact_before := [llvm|
{
^0(%arg17 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(4 : i4) : i4
  %2 = llvm.ashr exact %arg17, %0 : i4
  %3 = llvm.icmp "slt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def one_sdiv { w : Nat} {a : BitVec w} (ha0 : a ≠ 0) (ha1 : a ≠ 1)
    (hao : a ≠ allOnes w) :
    BitVec.sdiv (1#w) a = 0#w :=

def no_masks_with_logical_or2_after := [llvm|
{
^0(%arg7 : i32, %arg8 : i32, %arg9 : i32):
  %0 = llvm.mlir.constant(63 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.mlir.constant(true) : i1
  %3 = llvm.icmp "ne" %arg8, %0 : i32
  %4 = llvm.and %arg7, %arg9 : i32
  %5 = llvm.icmp "ne" %4, %1 : i32
  %6 = "llvm.select"(%5, %2, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%6) : (i1) -> ()
}
]

def cmpeq_xor_cst3_before := [llvm|
{
^0(%arg22 : i32, %arg23 : i32):
  %0 = llvm.mlir.constant(10 : i32) : i32
  %1 = llvm.xor %arg22, %0 : i32
  %2 = llvm.xor %arg23, %0 : i32
  %3 = llvm.icmp "eq" %1, %2 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def test15f_after := [llvm|
{
^0(%arg22 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(256 : i32) : i32
  %2 = llvm.shl %arg22, %0 : i32
  %3 = llvm.and %2, %1 : i32
  %4 = llvm.xor %3, %1 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def ult_to_slt_before := [llvm|
{
^0(%arg32 : i8, %arg33 : i8):
  %0 = llvm.mlir.constant(-128 : i8) : i8
  %1 = llvm.xor %arg32, %0 : i8
  %2 = llvm.xor %arg33, %0 : i8
  %3 = llvm.icmp "ult" %1, %2 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def llvm_zext_lower_riscv_1_to_8 : LLVMPeepholeRewriteRefine 8 [Ty.llvm (.bitvec 1)] :=
  {lhs:= zext_llvm_1_to_8, rhs:= zext_riscv_1_to_8}

def lshr_lshr_pow2_const_negative_nopow2_1_before := [llvm|
{
^0(%arg63 : i16):
  %0 = llvm.mlir.constant(2047 : i16) : i16
  %1 = llvm.mlir.constant(6 : i16) : i16
  %2 = llvm.mlir.constant(4 : i16) : i16
  %3 = llvm.lshr %0, %arg63 : i16
  %4 = llvm.lshr %3, %1 : i16
  %5 = llvm.and %4, %2 : i16
  "llvm.return"(%5) : (i16) -> ()
}
]

def test7_before := [llvm|
{
^0(%arg81 : i64):
  %0 = llvm.mlir.constant(32 : i128) : i128
  %1 = llvm.zext %arg81 : i64 to i128
  %2 = llvm.lshr %1, %0 : i128
  %3 = llvm.trunc %2 : i128 to i92
  "llvm.return"(%3) : (i92) -> ()
}
]

def ult_swap_or_not_max_commute_after := [llvm|
{
^0(%arg34 : i8, %arg35 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.icmp "ne" %arg34, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def zext_sext_sgt_known_nonneg_before := [llvm|
{
^0(%arg24 : i8, %arg25 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.udiv %0, %arg24 : i8
  %2 = llvm.zext %1 : i8 to i32
  %3 = llvm.sext %arg25 : i8 to i32
  %4 = llvm.icmp "sgt" %2, %3 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def lshrugt_03_08_before := [llvm|
{
^0(%arg375 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(-8 : i4) : i4
  %2 = llvm.lshr %arg375, %0 : i4
  %3 = llvm.icmp "ugt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def and_xor_commute2_before := [llvm|
{
^0(%arg153 : i32, %arg154 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.udiv %0, %arg153 : i32
  %2 = llvm.udiv %0, %arg154 : i32
  %3 = llvm.and %1, %2 : i32
  %4 = llvm.xor %3, %2 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def IndexInvariant (f : LoopBody t) : Prop :=
  ∀ (i j : Int) (v : t), f i v = f j v

@[simp_riscv]
def mv_pseudo (rs1_val : BitVec 64) : BitVec 64 :=
  RV64.addi 0 rs1_val

def xor_common_op_commute0_after := [llvm|
{
^0(%arg208 : i8, %arg209 : i8):
  %0 = llvm.or %arg209, %arg208 : i8
  "llvm.return"(%0) : (i8) -> ()
}
]

def test12_after := [llvm|
{
^0(%arg46 : i32, %arg47 : i32):
  %0 = llvm.mlir.constant(1431655765 : i32) : i32
  %1 = llvm.and %arg46, %0 : i32
  %2 = llvm.sub %arg47, %1 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def ashrsgt_03_06_before := [llvm|
{
^0(%arg279 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(6 : i4) : i4
  %2 = llvm.ashr %arg279, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def same_signbit_wrong_type_before := [llvm|
{
^0(%arg34 : i8, %arg35 : i32):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.lshr %arg34, %0 : i8
  %3 = llvm.icmp "sgt" %arg35, %1 : i32
  %4 = llvm.zext %3 : i1 to i8
  %5 = llvm.icmp "ne" %2, %4 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

def udiv_common_divisor_before := [llvm|
{
^0(%arg65 : i1, %arg66 : i5, %arg67 : i5, %arg68 : i5):
  %0 = llvm.udiv %arg67, %arg66 : i5
  %1 = llvm.udiv %arg68, %arg66 : i5
  %2 = "llvm.select"(%arg65, %1, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i5, i5) -> i5
  "llvm.return"(%2) : (i5) -> ()
}
]

def ashr_ult_8_before := [llvm|
{
^0(%arg88 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(-8 : i4) : i4
  %2 = llvm.ashr %arg88, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def select_riscv_select_llvm_32 : LLVMPeepholeRewriteRefine 32
  [Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 1)] where
  lhs := select_llvm_32
  rhs := select_riscv_32

@[simp_denote]
def sdiv {Γ : Ctxt _} (w : ℕ) (l r : Nat)
    (lp : (Γ[l]? = some (InstCombine.MTy.bitvec (ConcreteOrMVar.concrete w)))
      := by admit /- proof elided -/
      )
    (rp : (Γ[r]? = some (InstCombine.MTy.bitvec (ConcreteOrMVar.concrete w)))
      := by admit /- proof elided -/
      ) :
    Expr InstCombine.LLVM Γ .pure (LLVM.Ty.bitvec w) :=
  Expr.mk
    (op := InstCombine.MOp.sdiv w)
    (eff_le := by admit /- proof elided -/
    )
    (ty_eq := rfl)
    (args := .cons ⟨l, lp⟩ <| .cons ⟨r, rp⟩ .nil)
    (regArgs := .nil)

def icmp_select_implied_cond_swapped_select_before := [llvm|
{
^0(%arg56 : i8, %arg57 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "eq" %arg56, %0 : i8
  %2 = "llvm.select"(%1, %arg57, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %3 = llvm.icmp "eq" %2, %arg56 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def lshr_2_add_zext_basic_after := [llvm|
{
^0(%arg37 : i1, %arg38 : i1):
  %0 = llvm.and %arg37, %arg38 : i1
  %1 = llvm.zext %0 : i1 to i2
  "llvm.return"(%1) : (i2) -> ()
}
]

def fold_zext_xor_sandwich_after := [llvm|
{
^0(%arg188 : i1):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.zext %arg188 : i1 to i32
  %2 = llvm.xor %1, %0 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def lshrugt_02_01_exact_before := [llvm|
{
^0(%arg204 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(1 : i4) : i4
  %2 = llvm.lshr exact %arg204, %0 : i4
  %3 = llvm.icmp "ugt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def land_lor_right1_after := [llvm|
{
^0(%arg46 : i1, %arg47 : i1):
  "llvm.return"(%arg46) : (i1) -> ()
}
]

def ashrsgt_01_15_before := [llvm|
{
^0(%arg302 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(-1 : i4) : i4
  %2 = llvm.ashr %arg302, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def negtest_near_pow2_cmpval_ashr_slt_noexact_before := [llvm|
{
^0(%arg3 : i8):
  %0 = llvm.mlir.constant(1 : i8) : i8
  %1 = llvm.mlir.constant(5 : i8) : i8
  %2 = llvm.ashr %arg3, %0 : i8
  %3 = llvm.icmp "slt" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

partial def consumeCloseBracket(c: Bracket)
  (startPos: String.Pos.Raw)
  (i: String.Pos.Raw)
  (brackets: List Bracket)
  (ctx: ParserContext)
  (s: ParserState): ParserState := Id.run do
    match brackets with
    | b::bs =>
      if b == c
      then
        if bs == []
        then
          let parser_fn := Lean.Parser.mkNodeToken `balanced_brackets startPos
          parser_fn ctx (s.setPos (ctx.next i)) 
        else balancedBracketsFnAux startPos (ctx.next i) bs ctx s
      else s.mkError $ "| found Opened `" ++ toString b ++ "` expected to close at `" ++
      toString c ++ "`"
    | _ => s.mkError $ "| found Closed `" ++ toString c ++ "`, but have no opened brackets on stack"

def liRiscv_6 := [LV| {
  ^entry ():
    %0 = "li"() {imm = -6 : !i64} : (!i64) -> (!i64)
    %1 = "builtin.unrealized_conversion_cast"(%0) : (!i64) -> (i64)
    llvm.return %1 : i64
  }]

def ashr_ult_exact_after := [llvm|
{
^0(%arg169 : i8):
  %0 = llvm.mlir.constant(80 : i8) : i8
  %1 = llvm.icmp "ult" %arg169, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def ashrsgt_01_06_before := [llvm|
{
^0(%arg311 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(6 : i4) : i4
  %2 = llvm.ashr %arg311, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def not_signbit_splat_mask1_before := [llvm|
{
^0(%arg109 : i8, %arg110 : i16):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.ashr %arg109, %0 : i8
  %2 = llvm.zext %1 : i8 to i16
  %3 = llvm.and %2, %arg110 : i16
  "llvm.return"(%3) : (i16) -> ()
}
]

def sge_swap_and_max_commute_logical_before := [llvm|
{
^0(%arg240 : i8, %arg241 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.icmp "sle" %arg241, %arg240 : i8
  %3 = llvm.icmp "eq" %arg240, %0 : i8
  %4 = "llvm.select"(%3, %2, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def uadd_sat_flipped_wrong_bounds_after := [llvm|
{
^0(%arg96 : i32):
  %0 = llvm.mlir.constant(-13 : i32) : i32
  %1 = llvm.mlir.constant(9 : i32) : i32
  %2 = llvm.mlir.constant(-1 : i32) : i32
  %3 = llvm.icmp "ugt" %arg96, %0 : i32
  %4 = llvm.add %arg96, %1 : i32
  %5 = "llvm.select"(%3, %2, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def bools_before := [llvm|
{
^0(%arg51 : i1, %arg52 : i1, %arg53 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.xor %arg53, %0 : i1
  %2 = llvm.and %1, %arg51 : i1
  %3 = llvm.and %arg53, %arg52 : i1
  %4 = llvm.or %2, %3 : i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def lshrult_02_02_before := [llvm|
{
^0(%arg349 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.lshr %arg349, %0 : i4
  %2 = llvm.icmp "ult" %1, %0 : i4
  "llvm.return"(%2) : (i1) -> ()
}
]

def mulo_by_2_unsigned : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant (2) : i64
      %0 = llvm.mul %x, %c overflow<nuw> : i64
      llvm.return %0 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i64):
      %0 = llvm.add %x, %x overflow<nuw> : i64
      llvm.return %0 : i64
  }]

def composeQuinaryAux' (p : FSM (Fin 5)) (q₀ q₁ q₂ q₃ q₄: FSM α) : FSM α :=
  p.compose (α)
    (λ _ => α)
    (λ _ i => i)
    (λ v =>
      v.cases q₀ fun v =>
        v.cases q₁ fun v =>
          v.cases q₂ fun v =>
            v.cases q₃ fun v =>
              v.cases q₄ fun v => v.elim0)

def and_zext_eq_odd_commuted_after := [llvm|
{
^0(%arg30 : i32):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.icmp "eq" %arg30, %0 : i32
  %2 = llvm.zext %1 : i1 to i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def test19_after := [llvm|
{
^0(%arg192 : i32, %arg193 : i32):
  "llvm.return"(%arg193) : (i32) -> ()
}
]

@[simp_denote]
def xor_riscv_i32 :=
  [LV| {
    ^entry (%arg0: i32, %arg1: i32):
    %a = "builtin.unrealized_conversion_cast" (%arg0) : (i32) -> (!i64)
    %b = "builtin.unrealized_conversion_cast" (%arg1) : (i32) -> (!i64)
    %0 = xor %a, %b : !i64
    %1 = "builtin.unrealized_conversion_cast" (%0) : (!i64) -> (i32)
    llvm.return %1 : i32
  }]

@[simp_denote]
def add_llvm_nsw_flags_16 := [LV| {
  ^entry (%lhs: i16, %rhs: i16):
    %1 = llvm.add %lhs, %rhs overflow<nsw> : i16
    llvm.return %1 : i16
  }]

def PR42691_8_logical_after := [llvm|
{
^0(%arg328 : i32):
  %0 = llvm.mlir.constant(2147483647 : i32) : i32
  %1 = llvm.mlir.constant(-2147483635 : i32) : i32
  %2 = llvm.add %arg328, %0 : i32
  %3 = llvm.icmp "ult" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def test_invert_demorgan_or2_after := [llvm|
{
^0(%arg15 : i64, %arg16 : i64, %arg17 : i64):
  %0 = llvm.mlir.constant(24) : i64
  %1 = llvm.mlir.constant(60) : i64
  %2 = llvm.icmp "ult" %arg15, %0 : i64
  %3 = llvm.icmp "ult" %arg16, %1 : i64
  %4 = llvm.and %2, %3 : i1
  %5 = llvm.icmp "ult" %arg17, %1 : i64
  %6 = llvm.and %4, %5 : i1
  "llvm.return"(%6) : (i1) -> ()
}
]

def trunc_before := [llvm|
{
^0(%arg15 : i32, %arg16 : i32, %arg17 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.icmp "ult" %arg15, %arg16 : i32
  %2 = "llvm.select"(%1, %0, %arg17) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %3 = llvm.trunc %2 : i32 to i16
  "llvm.return"(%3) : (i16) -> ()
}
]

def ugt_swap_or_not_min_logical_after := [llvm|
{
^0(%arg4 : i8, %arg5 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "ne" %arg4, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def test12_logical_after := [llvm|
{
^0(%arg290 : i32, %arg291 : i32):
  %0 = llvm.icmp "ult" %arg290, %arg291 : i32
  "llvm.return"(%0) : (i1) -> ()
}
]

def or_or_and_pat4_before := [llvm|
{
^0(%arg62 : i8, %arg63 : i8, %arg64 : i8, %arg65 : i8):
  %0 = llvm.mlir.constant(42 : i8) : i8
  %1 = llvm.udiv %0, %arg64 : i8
  %2 = llvm.and %arg62, %arg63 : i8
  %3 = llvm.and %arg65, %arg63 : i8
  %4 = llvm.or %1, %3 : i8
  %5 = llvm.or %4, %2 : i8
  "llvm.return"(%5) : (i8) -> ()
}
]

def foldConstraints (expr: BoolExpr α) (constraints: List (BoolExpr α)) (op: Gate) : BoolExpr α :=
    match constraints with
    | [] => expr
    | x :: xs =>
      foldConstraints (BoolExpr.gate op expr x) xs op

def icmp_slt_0_and_icmp_sge_neg1_i64_fail_after := [llvm|
{
^0(%arg59 : i64):
  %0 = llvm.mlir.constant(-2) : i64
  %1 = llvm.mlir.constant(62) : i64
  %2 = llvm.mlir.constant(1) : i64
  %3 = llvm.mlir.constant(0) : i64
  %4 = llvm.icmp "sgt" %arg59, %0 : i64
  %5 = llvm.lshr %arg59, %1 : i64
  %6 = llvm.and %5, %2 : i64
  %7 = "llvm.select"(%4, %6, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i64, i64) -> i64
  "llvm.return"(%7) : (i64) -> ()
}
]

decidable_of_iff in Init.PropLemmas

def test_invert_demorgan_or3_after := [llvm|
{
^0(%arg13 : i32, %arg14 : i32):
  %0 = llvm.mlir.constant(178206 : i32) : i32
  %1 = llvm.mlir.constant(-196608 : i32) : i32
  %2 = llvm.mlir.constant(-1506 : i32) : i32
  %3 = llvm.mlir.constant(-917760 : i32) : i32
  %4 = llvm.mlir.constant(-716213 : i32) : i32
  %5 = llvm.mlir.constant(-1114112 : i32) : i32
  %6 = llvm.mlir.constant(-196112 : i32) : i32
  %7 = llvm.icmp "ne" %arg13, %0 : i32
  %8 = llvm.add %arg14, %1 : i32
  %9 = llvm.icmp "ult" %8, %2 : i32
  %10 = llvm.add %arg14, %3 : i32
  %11 = llvm.icmp "ult" %10, %4 : i32
  %12 = llvm.add %arg14, %5 : i32
  %13 = llvm.icmp "ult" %12, %6 : i32
  %14 = llvm.and %7, %9 : i1
  %15 = llvm.and %14, %11 : i1
  %16 = llvm.and %15, %13 : i1
  "llvm.return"(%16) : (i1) -> ()
}
]

def and_slt_to_mask_before := [llvm|
{
^0(%arg5 : i8):
  %0 = llvm.mlir.constant(-124 : i8) : i8
  %1 = llvm.mlir.constant(2 : i8) : i8
  %2 = llvm.mlir.constant(0 : i8) : i8
  %3 = llvm.icmp "slt" %arg5, %0 : i8
  %4 = llvm.and %arg5, %1 : i8
  %5 = llvm.icmp "eq" %4, %2 : i8
  %6 = llvm.and %3, %5 : i1
  "llvm.return"(%6) : (i1) -> ()
}
]

def test3a_after := [llvm|
{
^0(%arg100 : i32):
  %0 = llvm.mlir.constant(7 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.and %arg100, %0 : i32
  %3 = llvm.icmp "ne" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def slt_swap_or_not_max_commute_logical_after := [llvm|
{
^0(%arg48 : i8, %arg49 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.icmp "ne" %arg48, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def lshrult_03_05_exact_after := [llvm|
{
^0(%arg112 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def test2_before := [llvm|
{
^0(%arg0 : i999, %arg1 : i999):
  %0 = llvm.mlir.constant(65280 : i999) : i999
  %1 = llvm.and %arg0, %0 : i999
  %2 = llvm.and %arg1, %0 : i999
  %3 = llvm.icmp "ne" %1, %2 : i999
  "llvm.return"(%3) : (i1) -> ()
}
]

def fold_sext_to_and_wrong7_before := [llvm|
{
^0(%arg3 : i8):
  %0 = llvm.mlir.constant(-2147483647 : i32) : i32
  %1 = llvm.mlir.constant(128 : i32) : i32
  %2 = llvm.sext %arg3 : i8 to i32
  %3 = llvm.and %2, %0 : i32
  %4 = llvm.icmp "ne" %3, %1 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def shl_bad_sub2_i32_after := [llvm|
{
^0(%arg10 : i32):
  %0 = llvm.mlir.constant(-31 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.add %arg10, %0 : i32
  %3 = llvm.shl %1, %2 overflow<nuw> : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def OrSextSext : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 32)] where
  lhs := [LV| {
    ^entry (%x: i32, %y: i32):
      %0 = llvm.sext %x : i32 to i64
      %1 = llvm.sext %y : i32 to i64
      %2 = llvm.or %0, %1 : i64
      llvm.return %2 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i32, %y: i32):
      %0 = llvm.or %x, %y : i32
      %1 = llvm.sext %0 : i32 to i64
      llvm.return %1 : i64
  }]

def or_signbit_lshr_after := [llvm|
{
^0(%arg13 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(16776960 : i32) : i32
  %2 = llvm.lshr %arg13, %0 : i32
  %3 = llvm.or %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def pr40493_neg1_before := [llvm|
{
^0(%arg17 : i32):
  %0 = llvm.mlir.constant(11 : i32) : i32
  %1 = llvm.mlir.constant(4 : i32) : i32
  %2 = llvm.mlir.constant(0 : i32) : i32
  %3 = llvm.mul %arg17, %0 : i32
  %4 = llvm.and %3, %1 : i32
  %5 = llvm.icmp "eq" %4, %2 : i32
  "llvm.return"(%5) : (i1) -> ()
}
]

def uge_sext_after := [llvm|
{
^0(%arg134 : i1, %arg135 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "eq" %arg135, %0 : i8
  %2 = llvm.or %1, %arg134 : i1
  "llvm.return"(%2) : (i1) -> ()
}
]

def or_and2_before := [llvm|
{
^0(%arg138 : i1, %arg139 : i1, %arg140 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.xor %arg140, %0 : i1
  %2 = llvm.or %1, %arg138 : i1
  %3 = "llvm.select"(%2, %arg138, %arg139) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def select_constants_and_icmp_eq0_after := [llvm|
{
^0(%arg46 : i1, %arg47 : i1):
  %0 = llvm.xor %arg46, %arg47 : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def ugt_or_not_min_logical_after := [llvm|
{
^0(%arg12 : i8, %arg13 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "ne" %arg12, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def ashrslt_03_02_exact_before := [llvm|
{
^0(%arg19 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(2 : i4) : i4
  %2 = llvm.ashr exact %arg19, %0 : i4
  %3 = llvm.icmp "slt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

ToString in Init.Data.ToString.Basic

def ugt_rem_zero_before := [llvm|
{
^0(%arg167 : i8):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(21 : i8) : i8
  %2 = llvm.mul %arg167, %0 overflow<nuw> : i8
  %3 = llvm.icmp "ugt" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def shrink_or_after := [llvm|
{
^0(%arg11 : i6):
  %0 = llvm.mlir.constant(1 : i3) : i3
  %1 = llvm.trunc %arg11 : i6 to i3
  %2 = llvm.or %1, %0 : i3
  "llvm.return"(%2) : (i3) -> ()
}
]

def nonempty [DecidableEq α] (c : Circuit α) : Bool :=
  (nonemptyAux c c.vars rfl).1

def exactly_one_set_signbit_wrong_pred_before := [llvm|
{
^0(%arg28 : i8, %arg29 : i8):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(-1 : i8) : i8
  %2 = llvm.lshr %arg28, %0 : i8
  %3 = llvm.icmp "sgt" %arg29, %1 : i8
  %4 = llvm.zext %3 : i1 to i8
  %5 = llvm.icmp "sgt" %2, %4 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

@[match_pattern] def sub (w : Width φ)
  (NoWrapFlags: NoWrapFlags := {nsw := false , nuw := false}) : MOp φ
    := .binary w (.sub NoWrapFlags )

def test_ne_0_and_15_add_3_before := [llvm|
{
^0(%arg2 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.mlir.constant(15 : i8) : i8
  %2 = llvm.mlir.constant(0 : i8) : i8
  %3 = llvm.add %arg2, %0 : i8
  %4 = llvm.and %3, %1 : i8
  %5 = llvm.icmp "ne" %4, %2 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

def sub_add_reg_x_sub_x_add_y : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64), Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%x: i64, %y: i64):
      %1 = llvm.add %x, %y : i64
      %2 = llvm.sub %x, %1 : i64
      llvm.return %2 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i64, %y: i64):
      %c = llvm.mlir.constant (0) : i64
      %0 = llvm.sub %c, %y : i64
      llvm.return %0 : i64
  }]

def ashrsgt_03_04_exact_before := [llvm|
{
^0(%arg65 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(4 : i4) : i4
  %2 = llvm.ashr exact %arg65, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def scalar_i32_lshr_and_negC_ne_after := [llvm|
{
^0(%arg31 : i32, %arg32 : i32):
  %0 = llvm.mlir.constant(262143 : i32) : i32
  %1 = llvm.lshr %arg31, %arg32 : i32
  %2 = llvm.icmp "ugt" %1, %0 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

def or_not_and_commute8_after := [llvm|
{
^0(%arg513 : i32, %arg514 : i32, %arg515 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg513 : i32
  %3 = llvm.sdiv %0, %arg514 : i32
  %4 = llvm.xor %3, %arg515 : i32
  %5 = llvm.xor %2, %1 : i32
  %6 = llvm.and %4, %5 : i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def zext_sext_add_icmp_ne_1_after := [llvm|
{
^0(%arg70 : i1, %arg71 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.xor %arg70, %0 : i1
  %2 = llvm.or %arg71, %1 : i1
  "llvm.return"(%2) : (i1) -> ()
}
]

def add_constant_equal_with_the_top_bit_of_demandedbits_insertpt_before := [llvm|
{
^0(%arg4 : i32, %arg5 : i32):
  %0 = llvm.mlir.constant(16 : i32) : i32
  %1 = llvm.mlir.constant(24 : i32) : i32
  %2 = llvm.add %arg4, %0 : i32
  %3 = llvm.or %2, %arg5 : i32
  %4 = llvm.and %3, %1 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def and_or_not_not_commute2_after := [llvm|
{
^0(%arg369 : i32, %arg370 : i32, %arg371 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.or %arg371, %arg370 : i32
  %2 = llvm.and %1, %arg369 : i32
  %3 = llvm.xor %2, %0 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def ashrslt_02_10_before := [llvm|
{
^0(%arg243 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(-6 : i4) : i4
  %2 = llvm.ashr %arg243, %0 : i4
  %3 = llvm.icmp "slt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def add_nuw_const_const_sub_before := [llvm|
{
^0(%arg55 : i8):
  %0 = llvm.mlir.constant(1 : i8) : i8
  %1 = llvm.mlir.constant(-127 : i8) : i8
  %2 = llvm.add %arg55, %0 overflow<nuw> : i8
  %3 = llvm.sub %1, %2 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def shl_xor_before := [llvm|
{
^0(%arg22 : i8):
  %0 = llvm.mlir.constant(16 : i8) : i8
  %1 = llvm.mlir.constant(1 : i8) : i8
  %2 = llvm.mlir.constant(3 : i8) : i8
  %3 = llvm.shl %0, %arg22 : i8
  %4 = llvm.add %arg22, %1 : i8
  %5 = llvm.shl %2, %4 : i8
  %6 = llvm.xor %3, %5 : i8
  "llvm.return"(%6) : (i8) -> ()
}
]

def test11_after := [llvm|
{
^0(%arg184 : i32, %arg185 : i32):
  %0 = llvm.and %arg185, %arg184 : i32
  "llvm.return"(%0) : (i32) -> ()
}
]

def pr4917_3_after := [llvm|
{
^0(%arg25 : i32, %arg26 : i32):
  %0 = llvm.mlir.constant(4294967295) : i64
  %1 = llvm.mlir.constant(111) : i64
  %2 = llvm.zext %arg25 : i32 to i64
  %3 = llvm.zext %arg26 : i32 to i64
  %4 = llvm.mul %2, %3 overflow<nuw> : i64
  %5 = llvm.icmp "ugt" %4, %0 : i64
  %6 = "llvm.select"(%5, %4, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i64, i64) -> i64
  "llvm.return"(%6) : (i64) -> ()
}
]

inductive Module where
  | mk: (functions: List <| Op φ)
        -> (attrs: List <| AttrDefn φ)
        ->  Module

def castCtxt (h_eq : Γ = Δ) : Γ.Var ty → Δ.Var ty
  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩

def masked_icmps_mask_notallzeros_bmask_mixed_swapped_7_logical_after := [llvm|
{
^0(%arg33 : i32):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def scalar_i8_shl_and_negC_eq_after := [llvm|
{
^0(%arg35 : i8, %arg36 : i8):
  %0 = llvm.mlir.constant(4 : i8) : i8
  %1 = llvm.shl %arg35, %arg36 : i8
  %2 = llvm.icmp "ult" %1, %0 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

def ashrsgt_02_01_exact_after := [llvm|
{
^0(%arg84 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def p_constmask_before := [llvm|
{
^0(%arg49 : i32, %arg50 : i32):
  %0 = llvm.mlir.constant(65280 : i32) : i32
  %1 = llvm.mlir.constant(-65281 : i32) : i32
  %2 = llvm.and %arg49, %0 : i32
  %3 = llvm.and %arg50, %1 : i32
  %4 = llvm.or %2, %3 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def bools_logical_commute0_and2_before := [llvm|
{
^0(%arg102 : i1, %arg103 : i1, %arg104 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.xor %arg104, %0 : i1
  %3 = "llvm.select"(%2, %arg102, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %4 = llvm.and %arg104, %arg103 : i1
  %5 = "llvm.select"(%3, %0, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def ashrsgt_03_14_exact_after := [llvm|
{
^0(%arg55 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def test12_before := [llvm|
{
^0(%arg42 : i1):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.icmp "uge" %arg42, %0 : i1
  "llvm.return"(%1) : (i1) -> ()
}
]

def binop_right_to_zero_mul : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant (0) : i64
      %0 = llvm.mul %x, %c: i64
      llvm.return %0 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant (0) : i64
      llvm.return %c : i64
  }]

def exact_lshr_ne_both_equal_after := [llvm|
{
^0(%arg165 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "ne" %arg165, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def a_before := [llvm|
{
^0(%arg48 : i1, %arg49 : i1):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.zext %arg48 : i1 to i32
  %3 = llvm.zext %arg49 : i1 to i32
  %4 = llvm.sub %0, %3 : i32
  %5 = llvm.add %2, %1 : i32
  %6 = llvm.add %5, %4 : i32
  "llvm.return"(%6) : (i32) -> ()
}
]

@[simp_denote]
def sltu_riscv_i32 :=
  [LV| {
    ^entry (%arg0: i32, %arg1: i32):
    %a = "builtin.unrealized_conversion_cast" (%arg0) : (i32) -> (!i64)
    %b = "builtin.unrealized_conversion_cast" (%arg1) : (i32) -> (!i64)
    %0 = "sext.w" (%a) : (!i64) -> (!i64)
    %1 = "sext.w" (%b) : (!i64) -> (!i64)
    %2 = sltu %0, %1 : !i64
    %3 = "builtin.unrealized_conversion_cast" (%2) : (!i64) -> (i32)
    llvm.return %3 : i32
  }]

def select_constants_and_icmp_ne0_common_bit_after := [llvm|
{
^0(%arg18 : i1, %arg19 : i1):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def or_and_not_constant_commute1_before := [llvm|
{
^0(%arg110 : i32, %arg111 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(-2 : i32) : i32
  %2 = llvm.or %arg110, %arg111 : i32
  %3 = llvm.and %0, %2 : i32
  %4 = llvm.and %1, %arg111 : i32
  %5 = llvm.or %3, %4 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

@[simp_denote]
def sltu_riscv :=
  [LV| {
    ^entry (%arg0: i16, %arg1: i16):
    %a0 = "builtin.unrealized_conversion_cast" (%arg0) : (i16) -> (!i64)
    %a1 = "builtin.unrealized_conversion_cast" (%arg1) : (i16) -> (!i64)
    %0 = li (149595403036) : !i64 
    %1 = "lui" (%0) {imm = 16 : !i64} : (!i64) -> (!i64)
    %2 = "addi" (%1) {imm = -1 : !i64} : (!i64) -> (!i64)
    %3 = and %2, %a1 : !i64
    %4 = and %2, %a0 : !i64
    %5 = sltu %4, %3 : !i64
    %6 = "builtin.unrealized_conversion_cast" (%5) : (!i64) -> (i16)
    llvm.return %6 :i16
  }]

def and_logic_and_logic_or_4_before := [llvm|
{
^0(%arg129 : i1, %arg130 : i1, %arg131 : i1):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.and %arg130, %arg129 : i1
  %3 = "llvm.select"(%arg131, %arg129, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %4 = "llvm.select"(%2, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

@[irreducible]
def irreduciblePow (q n : Nat) : Nat := q^n

def shl_or_disjoint_lshr_after := [llvm|
{
^0(%arg109 : i32, %arg110 : i32, %arg111 : i32):
  %0 = llvm.lshr %arg111, %arg110 : i32
  %1 = llvm.or disjoint %0, %arg109 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def a_or_b_after := [llvm|
{
^0(%arg15 : i32, %arg16 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "eq" %arg15, %0 : i32
  %2 = llvm.icmp "eq" %arg16, %0 : i32
  %3 = llvm.xor %1, %2 : i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def sdiv_exact_eq_9_no_of_before := [llvm|
{
^0(%arg8 : i8, %arg9 : i8):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(9 : i8) : i8
  %2 = llvm.and %arg9, %0 : i8
  %3 = llvm.sdiv exact %arg8, %2 : i8
  %4 = llvm.icmp "eq" %3, %1 : i8
  "llvm.return"(%4) : (i1) -> ()
}
]

def t9_highest_bit_after := [llvm|
{
^0(%arg25 : i32, %arg26 : i64, %arg27 : i32):
  %0 = llvm.mlir.constant(63) : i64
  %1 = llvm.mlir.constant(0) : i64
  %2 = llvm.zext %arg25 : i32 to i64
  %3 = llvm.lshr %arg26, %0 : i64
  %4 = llvm.and %3, %2 : i64
  %5 = llvm.icmp "ne" %4, %1 : i64
  "llvm.return"(%5) : (i1) -> ()
}
]

def n5_after := [llvm|
{
^0(%arg9 : i8, %arg10 : i8, %arg11 : i8):
  "llvm.return"(%arg11) : (i8) -> ()
}
]

def positive_biggerashr_shlnsw_ashrexact_after := [llvm|
{
^0(%arg35 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.ashr exact %arg35, %0 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

structure RawCNFA.SimulFun (m : RawCNFA A) (M : NFA A Q) (f : m.states ≃ Q)  where
  accept {q} : ((f.invFun q).val ∈ m.finals ↔ q ∈ M.accept)
  initial {q} : q ∈ M.start ↔ (f.invFun q).val ∈ m.initials
  trans_match {a q q'} : q' ∈ M.step q a ↔ (f.invFun q').val ∈ m.tr (f.invFun q) a

def lowbitmask_casted_shift_after := [llvm|
{
^0(%arg157 : i8):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.sext %arg157 : i8 to i32
  %2 = llvm.lshr %1, %0 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def collectWidthAtom (state : CollectState) (e : Expr) :
    SolverM (MultiWidth.Nondep.WidthExpr × CollectState) := do
    if ← check? then
      if !(← isDefEq (← inferType e) (mkConst ``Nat)) then
        throwError m!"expected width to be a Nat, found: {indentD e}"
    
    if let .some n ← getNatValue? e then
      match (← read).widthAbstraction with
      | .never =>
        return (MultiWidth.Nondep.WidthExpr.const n, state)
      | .generalizeGeq cutoff =>
        if n < cutoff then
          return (MultiWidth.Nondep.WidthExpr.const n, state)
        else
          mkAtom
      | .always => mkAtom
    else
      mkAtom
    where
      mkAtom := do
        let (wix, wToIx) := state.wToIx.findOrInsertVal e
        
        return (.var wix, { state with wToIx := wToIx })

def PR2330_after := [llvm|
{
^0(%arg0 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "ne" %arg0, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def icmp_eq_x_invertable_y2_todo_after := [llvm|
{
^0(%arg4 : i8, %arg5 : i1):
  %0 = llvm.mlir.constant(-8 : i8) : i8
  %1 = llvm.mlir.constant(-25 : i8) : i8
  %2 = llvm.mlir.constant(-1 : i8) : i8
  %3 = "llvm.select"(%arg5, %0, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %4 = llvm.or %arg4, %3 : i8
  %5 = llvm.icmp "eq" %4, %2 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

def sel_false_val_is_a_masked_shl_of_true_val2_before := [llvm|
{
^0(%arg8 : i32, %arg9 : i64):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.mlir.constant(0 : i32) : i32
  %3 = llvm.and %arg8, %0 : i32
  %4 = llvm.shl %3, %1 overflow<nsw,nuw> : i32
  %5 = llvm.zext %4 : i32 to i64
  %6 = llvm.icmp "eq" %4, %2 : i32
  %7 = llvm.ashr %arg9, %5 : i64
  %8 = "llvm.select"(%6, %arg9, %7) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i64, i64) -> i64
  "llvm.return"(%8) : (i64) -> ()
}
]

def neg (x : BitStream) : BitStream :=
  fun n => (negAux x n).1

def twoway_clamp_lt_after := [llvm|
{
^0(%arg15 : i32):
  %0 = llvm.mlir.constant(13767 : i32) : i32
  %1 = llvm.mlir.constant(13768 : i32) : i32
  %2 = llvm.icmp "sgt" %arg15, %0 : i32
  %3 = "llvm.select"(%2, %1, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%3) : (i32) -> ()
}
]

instance : HydrableInstances FpPredicate where

def test2_after := [llvm|
{
^0(%arg308 : i32):
  "llvm.return"(%arg308) : (i32) -> ()
}
]

def icmp_slt_0_and_icmp_ne_neg2_i32_fail_before := [llvm|
{
^0(%arg44 : i32):
  %0 = llvm.mlir.constant(-2 : i32) : i32
  %1 = llvm.mlir.constant(31 : i32) : i32
  %2 = llvm.icmp "ne" %arg44, %0 : i32
  %3 = llvm.zext %2 : i1 to i32
  %4 = llvm.lshr %arg44, %1 : i32
  %5 = llvm.and %4, %3 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def sdiv_icmp4_after := [llvm|
{
^0(%arg12 : i64):
  %0 = llvm.mlir.constant(0) : i64
  %1 = llvm.icmp "eq" %arg12, %0 : i64
  "llvm.return"(%1) : (i1) -> ()
}
]

def constantExprsEnumerationFromCache (previousLevelCache allLhsVars : EnumerativeSearchCache) (lhsSymVars rhsSymVars : Std.HashMap Nat BVExpr.PackedBitVec)
                                          (ops: List (GenBVExpr w → GenBVExpr w → GenBVExpr w))
                                          : GeneralizerStateM ParsedBVExpr GenBVPred (ExpressionSynthesisResult × EnumerativeSearchCache) := do
    let zero := BitVec.ofNat w 0
    let one := BitVec.ofNat w 1
    let minusOne := BitVec.ofInt w (-1)

    let specialConstants : Std.HashMap (GenBVExpr w) BVExpr.PackedBitVec := Std.HashMap.ofList [
      (GenBVExpr.const one, {bv := one}),
      (GenBVExpr.const minusOne, {bv := minusOne})
    ]

    let mut rhsVarByValue : Std.HashMap (BitVec w) Nat := Std.HashMap.emptyWithCapacity
    for (var, value) in rhsSymVars.toArray do
      let h : value.w = w := sorry
      rhsVarByValue := rhsVarByValue.insert (h ▸ value.bv) var

    let mut currentCache := Std.HashMap.emptyWithCapacity

    let mut res : Std.HashMap Nat (List BVExprWrapper) := Std.HashMap.emptyWithCapacity
    for (wrappedBvExpr, packedBV) in previousLevelCache.toArray do
      let packedBVExpr : GenBVExpr packedBV.w := GenBVExpr.const packedBV.bv

      for (lhsVar, lhsVal) in allLhsVars.toArray do
        for op in ops do
          if packedBV.w == lhsVar.width then
            let h : packedBV.w = w ∧ lhsVar.width = w := sorry

            let evaluatedRes := evalBVExpr lhsSymVars (op (h.left ▸ packedBVExpr) (h.right ▸ lhsVar.bvExpr))

            let h' : w = wrappedBvExpr.width := sorry
            let mut newExpr := wrap (op (h' ▸ wrappedBvExpr.bvExpr) (h.right ▸ lhsVar.bvExpr))
            let rhsVarForValue := rhsVarByValue[evaluatedRes]?

            match rhsVarForValue with
            | some rhsVar =>
                let existingCandidates := res.getD rhsVar []
                res := res.insert rhsVar (newExpr::existingCandidates)
            | none =>
              if evaluatedRes == h.left ▸ packedBV.bv then
                newExpr := wrappedBvExpr
              currentCache := currentCache.insert newExpr {bv := evaluatedRes : BVExpr.PackedBitVec}

    pure (res, currentCache)

def test_mul_nuw_nsw__nsw_is_safe_after := [llvm|
{
^0(%arg7 : i32):
  %0 = llvm.mlir.constant(-83886080 : i32) : i32
  %1 = llvm.mlir.constant(9 : i32) : i32
  %2 = llvm.or %arg7, %0 : i32
  %3 = llvm.mul %2, %1 overflow<nsw> : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def t2_symmetry_after := [llvm|
{
^0(%arg37 : i8, %arg38 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.xor %arg37, %0 : i8
  %2 = llvm.icmp "ule" %arg38, %1 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

def select_constants_and_icmp_ne0_before := [llvm|
{
^0(%arg26 : i1, %arg27 : i1):
  %0 = llvm.mlir.constant(2 : i8) : i8
  %1 = llvm.mlir.constant(1 : i8) : i8
  %2 = llvm.mlir.constant(0 : i8) : i8
  %3 = "llvm.select"(%arg26, %0, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %4 = "llvm.select"(%arg27, %0, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  %5 = llvm.and %3, %4 : i8
  %6 = llvm.icmp "ne" %5, %2 : i8
  "llvm.return"(%6) : (i1) -> ()
}
]

def XorAndAnd : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64), Ty.llvm (.bitvec 64), Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%x: i64, %y: i64, %z: i64):
      %0 = llvm.and %x, %z : i64
      %1 = llvm.and %y, %z : i64
      %2 = llvm.xor %0, %1 : i64
      llvm.return %2 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i64, %y: i64, %z: i64):
      %0 = llvm.xor %x, %y : i64
      %1 = llvm.and %0, %z : i64
      llvm.return %1 : i64
  }]

def add_smax_simplify2_after := [llvm|
{
^0(%arg24 : i8):
  %0 = llvm.mlir.constant(127 : i8) : i8
  %1 = llvm.add %arg24, %0 overflow<nsw> : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def sle_smin4_after := [llvm|
{
^0(%arg49 : i32, %arg50 : i32):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def eq_21_comm_and_before := [llvm|
{
^0(%arg122 : i32, %arg123 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(16 : i32) : i32
  %2 = llvm.lshr %arg122, %0 : i32
  %3 = llvm.trunc %2 : i32 to i8
  %4 = llvm.lshr %arg122, %1 : i32
  %5 = llvm.trunc %4 : i32 to i8
  %6 = llvm.lshr %arg123, %0 : i32
  %7 = llvm.trunc %6 : i32 to i8
  %8 = llvm.lshr %arg123, %1 : i32
  %9 = llvm.trunc %8 : i32 to i8
  %10 = llvm.icmp "eq" %3, %7 : i8
  %11 = llvm.icmp "eq" %5, %9 : i8
  %12 = llvm.and %10, %11 : i1
  "llvm.return"(%12) : (i1) -> ()
}
]

def is_rem2_neg_i8_after := [llvm|
{
^0(%arg57 : i8):
  %0 = llvm.mlir.constant(-127 : i8) : i8
  %1 = llvm.and %arg57, %0 : i8
  %2 = llvm.icmp "eq" %1, %0 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

def or_not_and_commute1_before := [llvm|
{
^0(%arg534 : i32, %arg535 : i32, %arg536 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg535 : i32
  %3 = llvm.or %arg534, %2 : i32
  %4 = llvm.xor %3, %1 : i32
  %5 = llvm.and %4, %arg536 : i32
  %6 = llvm.or %arg534, %arg536 : i32
  %7 = llvm.xor %6, %1 : i32
  %8 = llvm.and %2, %7 : i32
  %9 = llvm.or %5, %8 : i32
  "llvm.return"(%9) : (i32) -> ()
}
]

def alive_AndOrXor_2627_tgt  (w : Nat)  :=
[llvm( w )| {
^bb0(%a : _, %c : _, %b : _):
  %v1 = llvm.mlir.constant -1
  %v2 = llvm.xor %a, %v1
  %v3 = llvm.and %v2, %b
  %v4 = llvm.xor %a, %c
  %v5 = llvm.or %a, %b
  %v6 = llvm.xor %v3, %c
  llvm.return %v6
}]

def positive_samevar_after := [llvm|
{
^0(%arg36 : i32, %arg37 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.lshr %0, %arg37 : i32
  %2 = llvm.and %1, %arg36 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def ashr_16_add_zext_basic_before := [llvm|
{
^0(%arg11 : i16, %arg12 : i16):
  %0 = llvm.mlir.constant(16 : i32) : i32
  %1 = llvm.zext %arg11 : i16 to i32
  %2 = llvm.zext %arg12 : i16 to i32
  %3 = llvm.add %1, %2 : i32
  %4 = llvm.lshr %3, %0 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def test_after := [llvm|
{
^0(%arg1 : i32):
  %0 = llvm.mlir.constant(255 : i32) : i32
  %1 = llvm.icmp "ugt" %arg1, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def lshr_exact_ne_0_after := [llvm|
{
^0(%arg121 : i32, %arg122 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "ne" %arg121, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def or_and_or_pat8_after := [llvm|
{
^0(%arg10 : i8, %arg11 : i8, %arg12 : i8, %arg13 : i8):
  %0 = llvm.or %arg13, %arg10 : i8
  %1 = llvm.and %0, %arg11 : i8
  %2 = llvm.or %1, %arg12 : i8
  "llvm.return"(%2) : (i8) -> ()
}
]

def or_zext_zext_before := [llvm|
{
^0(%arg56 : i8, %arg57 : i4):
  %0 = llvm.zext %arg56 : i8 to i16
  %1 = llvm.zext %arg57 : i4 to i16
  %2 = llvm.or %1, %0 : i16
  "llvm.return"(%2) : (i16) -> ()
}
]

def or_xor_common_op_commute5_after := [llvm|
{
^0(%arg120 : i8, %arg121 : i8, %arg122 : i8):
  %0 = llvm.or %arg121, %arg120 : i8
  %1 = llvm.or %0, %arg122 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def EqualUpTo (w : Nat) (x y : BitStream) : Prop :=
  ∀ i < w, x i = y i

def udiv_i32_multiuse_after := [llvm|
{
^0(%arg25 : i8, %arg26 : i8):
  %0 = llvm.zext %arg25 : i8 to i32
  %1 = llvm.zext %arg26 : i8 to i32
  %2 = llvm.udiv %0, %1 : i32
  %3 = llvm.add %0, %1 overflow<nsw,nuw> : i32
  %4 = llvm.mul %2, %3 overflow<nsw,nuw> : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def alive_InstCombineShift__279_src :=
[sllvm| {
^bb0(%X : i64, %C : i64):
  %v1 = llvm.lshr %X, %C : i64
  %v2 = llvm.shl %v1, %C : i64
  llvm.return %v2 : i64
}]

def lshrult_02_11_exact_after := [llvm|
{
^0(%arg122 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def testi64i32_ne_after := [llvm|
{
^0(%arg7 : i64):
  %0 = llvm.mlir.constant(-2147483648) : i64
  %1 = llvm.mlir.constant(-4294967296) : i64
  %2 = llvm.add %arg7, %0 : i64
  %3 = llvm.icmp "ult" %2, %1 : i64
  "llvm.return"(%3) : (i1) -> ()
}
]

def ne_smax4_before := [llvm|
{
^0(%arg41 : i32, %arg42 : i32):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.add %arg41, %0 : i32
  %2 = llvm.icmp "sgt" %arg42, %1 : i32
  %3 = "llvm.select"(%2, %arg42, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %4 = llvm.icmp "ne" %1, %3 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def or_eq_with_one_bit_diff_constants1_logical_before := [llvm|
{
^0(%arg365 : i32):
  %0 = llvm.mlir.constant(50 : i32) : i32
  %1 = llvm.mlir.constant(51 : i32) : i32
  %2 = llvm.mlir.constant(true) : i1
  %3 = llvm.icmp "eq" %arg365, %0 : i32
  %4 = llvm.icmp "eq" %arg365, %1 : i32
  %5 = "llvm.select"(%3, %2, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def uge_umin1_before := [llvm|
{
^0(%arg52 : i32, %arg53 : i32):
  %0 = llvm.icmp "ult" %arg52, %arg53 : i32
  %1 = "llvm.select"(%0, %arg52, %arg53) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %2 = llvm.icmp "uge" %1, %arg52 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

Lean.addAndCompile in Lean.AddDecl

def samesign_inverted_commute3_before := [llvm|
{
^0(%arg97 : i32, %arg98 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.and %arg97, %arg98 : i32
  %3 = llvm.icmp "sgt" %2, %0 : i32
  %4 = llvm.or %arg98, %arg97 : i32
  %5 = llvm.icmp "slt" %4, %1 : i32
  %6 = llvm.and %5, %3 : i1
  "llvm.return"(%6) : (i1) -> ()
}
]

def ult_rem_zero_before := [llvm|
{
^0(%arg174 : i8):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(21 : i8) : i8
  %2 = llvm.mul %arg174, %0 overflow<nuw> : i8
  %3 = llvm.icmp "ult" %2, %1 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def alive_Select_1105_src  (w : Nat)   :=
[llvm( w )| {
^bb0(%Y : _, %X : _):
  %v1 = llvm.mlir.constant 0 :  i1
  %v2 = llvm.select %v1,%X, %Y
  llvm.return %v2
}]

def function : Parser :=
  leading_parser argumentList >> arrow >> termParser

def select_bittest_to_sub_before := [llvm|
{
^0(%arg4 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(4 : i32) : i32
  %3 = llvm.mlir.constant(3 : i32) : i32
  %4 = llvm.and %arg4, %0 : i32
  %5 = llvm.icmp "eq" %4, %1 : i32
  %6 = "llvm.select"(%5, %2, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%6) : (i32) -> ()
}
]

def llvm_const_lower_riscv_li17 : LLVMPeepholeRewriteRefine 64 [] :=
  {lhs := constLLVM17, rhs:= liRiscv17,
   correct := by admit /- proof elided -/

def nonexact_lshr_eq_noexactlog_after := [llvm|
{
^0(%arg136 : i8):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def test4c_before := [llvm|
{
^0(%arg7 : i64):
  %0 = llvm.mlir.constant(63) : i64
  %1 = llvm.mlir.constant(0) : i64
  %2 = llvm.mlir.constant(1 : i32) : i32
  %3 = llvm.ashr %arg7, %0 : i64
  %4 = llvm.sub %1, %arg7 : i64
  %5 = llvm.lshr %4, %0 : i64
  %6 = llvm.or %3, %5 : i64
  %7 = llvm.trunc %6 : i64 to i32
  %8 = llvm.icmp "slt" %7, %2 : i32
  "llvm.return"(%8) : (i1) -> ()
}
]

def test1_before := [llvm|
{
^0(%arg1 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(1431655765 : i32) : i32
  %2 = llvm.mlir.constant(-1 : i32) : i32
  %3 = llvm.ashr %arg1, %0 : i32
  %4 = llvm.and %3, %1 : i32
  %5 = llvm.xor %4, %2 : i32
  %6 = llvm.add %5, %0 overflow<nsw> : i32
  %7 = llvm.add %arg1, %6 overflow<nsw> : i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def add_smin_simplify_after := [llvm|
{
^0(%arg34 : i32):
  %0 = llvm.mlir.constant(-3 : i32) : i32
  %1 = llvm.add %arg34, %0 overflow<nsw> : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def sub_to_add_5 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant 5 : i64
      %1 = llvm.sub %x, %c : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant -5 : i64
      %1 = llvm.add %x, %c : i64
      llvm.return %1 : i64
  }]

def test4_before := [llvm|
{
^0(%arg25 : i32):
  %0 = llvm.mlir.constant(64 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.udiv %0, %arg25 : i32
  %3 = llvm.icmp "ne" %2, %1 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def sle_swap_and_min_logical_after := [llvm|
{
^0(%arg212 : i8, %arg213 : i8):
  %0 = llvm.mlir.constant(-128 : i8) : i8
  %1 = llvm.icmp "eq" %arg212, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def uadd_sat_flipped_wrong_bounds_before := [llvm|
{
^0(%arg96 : i32):
  %0 = llvm.mlir.constant(-12 : i32) : i32
  %1 = llvm.mlir.constant(9 : i32) : i32
  %2 = llvm.mlir.constant(-1 : i32) : i32
  %3 = llvm.icmp "uge" %arg96, %0 : i32
  %4 = llvm.add %arg96, %1 : i32
  %5 = "llvm.select"(%3, %2, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%5) : (i32) -> ()
}
]

inductive AttrDict where
  | mk: List AttrEntry -> AttrDict

def irc_constants_C2MinusAPlusC1_neg2_2 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (-2) : i64
      %c2 = llvm.mlir.constant (2) : i64
      %0 = llvm.add %a, %c1 : i64
      %1 = llvm.sub %c2, %0 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (-2) : i64
      %c2 = llvm.mlir.constant (2) : i64
      %0 = llvm.sub %c2, %c1 : i64
      %1 = llvm.sub %0, %a : i64
      llvm.return %1 : i64
  }]

def and_or3_before := [llvm|
{
^0(%arg168 : i1, %arg169 : i1, %arg170 : i32, %arg171 : i32):
  %0 = llvm.icmp "eq" %arg170, %arg171 : i32
  %1 = llvm.and %arg169, %0 : i1
  %2 = "llvm.select"(%1, %arg168, %arg169) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%2) : (i1) -> ()
}
]

def or_or_and_complex_after := [llvm|
{
^0(%arg82 : i64):
  %0 = llvm.mlir.constant(8) : i64
  %1 = llvm.mlir.constant(71777214294589695) : i64
  %2 = llvm.mlir.constant(-71777214294589696) : i64
  %3 = llvm.lshr %arg82, %0 : i64
  %4 = llvm.shl %arg82, %0 : i64
  %5 = llvm.and %3, %1 : i64
  %6 = llvm.and %4, %2 : i64
  %7 = llvm.or disjoint %5, %6 : i64
  "llvm.return"(%7) : (i64) -> ()
}
]

def ashrslt_02_07_exact_before := [llvm|
{
^0(%arg30 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(7 : i4) : i4
  %2 = llvm.ashr exact %arg30, %0 : i4
  %3 = llvm.icmp "slt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def lshrult_03_04_exact_before := [llvm|
{
^0(%arg113 : i4):
  %0 = llvm.mlir.constant(3 : i4) : i4
  %1 = llvm.mlir.constant(4 : i4) : i4
  %2 = llvm.lshr exact %arg113, %0 : i4
  %3 = llvm.icmp "ult" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def or_logic_or_logic_and_1_after := [llvm|
{
^0(%arg48 : i1, %arg49 : i1, %arg50 : i1):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.mlir.constant(true) : i1
  %2 = "llvm.select"(%arg49, %arg50, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %3 = "llvm.select"(%arg48, %1, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def t9_ashr_exact_flag_preservation_before := [llvm|
{
^0(%arg25 : i32, %arg26 : i32):
  %0 = llvm.mlir.constant(32 : i32) : i32
  %1 = llvm.mlir.constant(-2 : i32) : i32
  %2 = llvm.sub %0, %arg26 : i32
  %3 = llvm.ashr exact %arg25, %2 : i32
  %4 = llvm.add %arg26, %1 : i32
  %5 = llvm.ashr exact %3, %4 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def sle_swap_or_not_min_after := [llvm|
{
^0(%arg280 : i8, %arg281 : i8):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def ashrslt_03_10_exact_after := [llvm|
{
^0(%arg11 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def ult_rem_zero_after := [llvm|
{
^0(%arg174 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.icmp "ult" %arg174, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

instance {test : ConcreteCliTest} : ToString (toType test.ty) where
  toString :=
    match test.ty with
    | bitvec w => inferInstanceAs (ToString (PoisonOr <| BitVec w)) |>.toString

def ugt_swap_and_not_min_logical_after := [llvm|
{
^0(%arg68 : i8, %arg69 : i8):
  %0 = llvm.icmp "ult" %arg69, %arg68 : i8
  "llvm.return"(%0) : (i1) -> ()
}
]

def test_shl_nuw_nsw__nsw_is_safe_before := [llvm|
{
^0(%arg75 : i32):
  %0 = llvm.mlir.constant(-83886080 : i32) : i32
  %1 = llvm.mlir.constant(-83886079 : i32) : i32
  %2 = llvm.mlir.constant(2 : i32) : i32
  %3 = llvm.mlir.constant(-335544316 : i32) : i32
  %4 = llvm.or %arg75, %0 : i32
  %5 = llvm.icmp "eq" %4, %1 : i32
  %6 = llvm.shl %4, %2 overflow<nsw,nuw> : i32
  %7 = "llvm.select"(%5, %3, %6) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %8 = llvm.mul %7, %4 : i32
  %9 = llvm.mul %8, %6 : i32
  "llvm.return"(%9) : (i32) -> ()
}
]

def test9_before := [llvm|
{
^0(%arg52 : i16):
  %0 = llvm.mlir.constant(2 : i16) : i16
  %1 = llvm.mlir.constant(32767 : i16) : i16
  %2 = llvm.mul %arg52, %0 : i16
  %3 = llvm.mul %arg52, %1 : i16
  %4 = llvm.add %2, %3 : i16
  "llvm.return"(%4) : (i16) -> ()
}
]

def fold_sext_to_and_wrong10_before := [llvm|
{
^0(%arg0 : i8):
  %0 = llvm.mlir.constant(-256 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.sext %arg0 : i8 to i32
  %3 = llvm.and %2, %0 : i32
  %4 = llvm.icmp "ne" %3, %1 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def testi16i8_after := [llvm|
{
^0(%arg9 : i16):
  %0 = llvm.mlir.constant(-1 : i16) : i16
  %1 = llvm.mlir.constant(27 : i8) : i8
  %2 = llvm.mlir.constant(-28 : i8) : i8
  %3 = llvm.icmp "sgt" %arg9, %0 : i16
  %4 = "llvm.select"(%3, %1, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  "llvm.return"(%4) : (i8) -> ()
}
]

structure State where
   
  mapping : DiscrTree Expr
  invMapping : Std.HashMap Expr Expr
  deriving Inhabited

Coe in Init.Coe

def both_sides_fold_eq_before := [llvm|
{
^0(%arg37 : i32, %arg38 : i1):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(9 : i32) : i32
  %2 = "llvm.select"(%arg38, %0, %arg37) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %3 = "llvm.select"(%arg38, %1, %arg37) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %4 = llvm.icmp "eq" %3, %2 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def ashr_lshr_exact_both_after := [llvm|
{
^0(%arg129 : i32, %arg130 : i32):
  %0 = llvm.ashr exact %arg129, %arg130 : i32
  "llvm.return"(%0) : (i32) -> ()
}
]

Lean.mkNatLit in Lean.Expr

def ule_bignum_after := [llvm|
{
^0(%arg145 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "eq" %arg145, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def slt_zero_ult_i1_fail2_before := [llvm|
{
^0(%arg6 : i32, %arg7 : i1):
  %0 = llvm.mlir.constant(31 : i32) : i32
  %1 = llvm.zext %arg7 : i1 to i32
  %2 = llvm.ashr %arg6, %0 : i32
  %3 = llvm.icmp "ult" %1, %2 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def uadd_sat_flipped_wrong_bounds7_after := [llvm|
{
^0(%arg90 : i32):
  %0 = llvm.mlir.constant(-11 : i32) : i32
  %1 = llvm.mlir.constant(9 : i32) : i32
  %2 = llvm.mlir.constant(-1 : i32) : i32
  %3 = llvm.icmp "ult" %arg90, %0 : i32
  %4 = llvm.add %arg90, %1 : i32
  %5 = "llvm.select"(%3, %4, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def trunc_equality_nsw_zext_after := [llvm|
{
^0(%arg16 : i32, %arg17 : i8):
  %0 = llvm.zext %arg17 : i8 to i32
  %1 = llvm.icmp "ne" %arg16, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def urem_c_i32_after := [llvm|
{
^0(%arg5 : i8):
  %0 = llvm.mlir.constant(10 : i8) : i8
  %1 = llvm.urem %0, %arg5 : i8
  %2 = llvm.zext nneg %1 : i8 to i32
  "llvm.return"(%2) : (i32) -> ()
}
]

instance {cmp} : Fintype (NFA'.autSignedCmp cmp).σ :=

def alive_AddSub_1043_tgt  (w : Nat)  :=
[llvm( w )| {
^bb0(%C1 : _, %Z : _, %RHS : _):
  %v1 = llvm.not %C1
  %v2 = llvm.or %Z, %v1
  %v3 = llvm.and %Z, %C1
  %v4 = llvm.xor %v3, %C1
  %v5 = llvm.mlir.constant 1
  %v6 = llvm.add %v4, %v5
  %v7 = llvm.sub %RHS, %v2
  llvm.return %v7
}]

def test_sub_4_Y_ne_4_before := [llvm|
{
^0(%arg47 : i8):
  %0 = llvm.mlir.constant(4 : i8) : i8
  %1 = llvm.sub %0, %arg47 : i8
  %2 = llvm.icmp "ne" %1, %0 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

def a_false_implies_b_true2_comm_before := [llvm|
{
^0(%arg9 : i8, %arg10 : i1, %arg11 : i1):
  %0 = llvm.mlir.constant(10 : i8) : i8
  %1 = llvm.mlir.constant(20 : i8) : i8
  %2 = llvm.icmp "ugt" %arg9, %0 : i8
  %3 = llvm.icmp "ult" %arg9, %1 : i8
  %4 = "llvm.select"(%3, %arg10, %arg11) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %5 = llvm.or %4, %2 : i1
  "llvm.return"(%5) : (i1) -> ()
}
]

def no_shift_xor_multiuse_xor_after := [llvm|
{
^0(%arg132 : i32, %arg133 : i32):
  %0 = llvm.mlir.constant(4096 : i32) : i32
  %1 = llvm.and %arg132, %0 : i32
  %2 = llvm.xor %arg133, %0 : i32
  %3 = llvm.xor %1, %arg133 : i32
  %4 = llvm.xor %3, %0 : i32
  %5 = llvm.mul %4, %2 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def irc_constants_C2MinusAPlusC1_1_neg2 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (1) : i64
      %c2 = llvm.mlir.constant (-2) : i64
      %0 = llvm.add %a, %c1 : i64
      %1 = llvm.sub %c2, %0 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%a: i64):
      %c1 = llvm.mlir.constant (1) : i64
      %c2 = llvm.mlir.constant (-2) : i64
      %0 = llvm.sub %c2, %c1 : i64
      %1 = llvm.sub %0, %a : i64
      llvm.return %1 : i64
  }]

def not_and_and_not_after := [llvm|
{
^0(%arg567 : i32, %arg568 : i32, %arg569 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg567 : i32
  %3 = llvm.or %arg568, %arg569 : i32
  %4 = llvm.xor %3, %1 : i32
  %5 = llvm.and %2, %4 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def scalar_i32_lshr_and_negC_eq_X_is_constant2_before := [llvm|
{
^0(%arg8 : i32):
  %0 = llvm.mlir.constant(268435456 : i32) : i32
  %1 = llvm.mlir.constant(-8 : i32) : i32
  %2 = llvm.mlir.constant(0 : i32) : i32
  %3 = llvm.lshr %0, %arg8 : i32
  %4 = llvm.and %3, %1 : i32
  %5 = llvm.icmp "eq" %4, %2 : i32
  "llvm.return"(%5) : (i1) -> ()
}
]

def test23_before := [llvm|
{
^0(%arg172 : i32):
  %0 = llvm.trunc %arg172 : i32 to i16
  %1 = llvm.zext %0 : i16 to i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def liRiscv40 := [LV| {
  ^entry ():
    %0 = "li"() {imm = 40 : !i64} : (!i64) -> (!i64)
    %1 = "builtin.unrealized_conversion_cast"(%0) : (!i64) -> (i64)
    llvm.return %1 : i64
  }]

def ashrsgt_01_10_after := [llvm|
{
^0(%arg307 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def map {α β : Type} (s : Stream α) (f : α → β) : Stream β :=
  fun i => (s i).map f

instance : DialectDenote LLVM := ⟨
  fun o args _ => [Op.denote o args]ₕ
⟩

def select_t_cond : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64), Ty.llvm (.bitvec 1)] where
  lhs := [LV| {
    ^entry (%c: i1, %t: i64):
      %0 = llvm.sext %c: i1 to i64
      %1 = llvm.select %c, %t, %0 : i64
      llvm.return %1 : i64
  }]
  rhs := [LV| {
    ^entry (%c: i1, %t: i64):
      %0 = llvm.sext %c: i1 to i64
      %1 = llvm.freeze %t : i64
      %2 = llvm.and %0, %1 : i64
      llvm.return %2 : i64
  }]

def lshrult_01_12_after := [llvm|
{
^0(%arg355 : i4):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def eq_smax3_after := [llvm|
{
^0(%arg59 : i32, %arg60 : i32):
  %0 = llvm.mlir.constant(3 : i32) : i32
  %1 = llvm.add %arg59, %0 : i32
  %2 = llvm.icmp "sge" %1, %arg60 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

def ashr_mul_times_3_div_2_exact_2_after := [llvm|
{
^0(%arg17 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.ashr exact %arg17, %0 : i32
  %2 = llvm.add %arg17, %1 overflow<nsw> : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def branch (c : ValueStream (BitVec 1)) (x : TokenStream) : TokenStream × TokenStream  :=
  Stream.corec₂ (β := ValueStream (BitVec 1) × TokenStream) (c, x) fun ⟨c, x⟩ =>
    Id.run <| do
      match c 0 with
        | none => (none, none, (c.tail, x))
        | some x₀ =>
          if x₀.msb then
            (some (), none, (c.tail, x.tail))
          else
            (none, some (), (c.tail, x.tail))

def no_shift_xor_multiuse_cmp_xor_before := [llvm|
{
^0(%arg44 : i32, %arg45 : i32, %arg46 : i32, %arg47 : i32):
  %0 = llvm.mlir.constant(4096 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.and %arg44, %0 : i32
  %3 = llvm.icmp "ne" %1, %2 : i32
  %4 = llvm.xor %arg45, %0 : i32
  %5 = "llvm.select"(%3, %arg45, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %6 = "llvm.select"(%3, %arg46, %arg47) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %7 = llvm.mul %5, %6 : i32
  %8 = llvm.mul %7, %4 : i32
  "llvm.return"(%8) : (i32) -> ()
}
]

def select_icmp_eq_and_4096_0_xor_4096_after := [llvm|
{
^0(%arg221 : i32, %arg222 : i32):
  %0 = llvm.mlir.constant(4096 : i32) : i32
  %1 = llvm.and %arg221, %0 : i32
  %2 = llvm.xor %arg222, %1 : i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def fold_sext_to_and_wrong8_after := [llvm|
{
^0(%arg2 : i8):
  %0 = llvm.mlir.constant(true) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def one_side_fold_eq_before := [llvm|
{
^0(%arg25 : i32, %arg26 : i32, %arg27 : i32, %arg28 : i1):
  %0 = "llvm.select"(%arg28, %arg25, %arg27) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %1 = "llvm.select"(%arg28, %arg26, %arg27) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %2 = llvm.icmp "eq" %1, %0 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

def t0_after := [llvm|
{
^0(%arg206 : i8):
  %0 = llvm.mlir.constant(42 : i8) : i8
  %1 = llvm.add %arg206, %0 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def sle_swap_or_min_before := [llvm|
{
^0(%arg150 : i8, %arg151 : i8):
  %0 = llvm.mlir.constant(-128 : i8) : i8
  %1 = llvm.icmp "sge" %arg151, %arg150 : i8
  %2 = llvm.icmp "eq" %arg150, %0 : i8
  %3 = llvm.or %1, %2 : i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def or_not_and_wrong_b_after := [llvm|
{
^0(%arg484 : i32, %arg485 : i32, %arg486 : i32, %arg487 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.or %arg484, %arg485 : i32
  %2 = llvm.xor %1, %0 : i32
  %3 = llvm.and %arg486, %2 : i32
  %4 = llvm.or %arg484, %arg486 : i32
  %5 = llvm.xor %4, %0 : i32
  %6 = llvm.and %arg487, %5 : i32
  %7 = llvm.or %3, %6 : i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def mul_of_sdiv_fail_ub_before := [llvm|
{
^0(%arg1 : i8):
  %0 = llvm.mlir.constant(6 : i8) : i8
  %1 = llvm.mlir.constant(-6 : i8) : i8
  %2 = llvm.sdiv exact %arg1, %0 : i8
  %3 = llvm.mul %2, %1 : i8
  "llvm.return"(%3) : (i8) -> ()
}
]

def State.cseCom {α}
  {lets : Lets d Γstart .pure Γ}
  (s : State d lets)
  (com: Com d Γ .pure α) :
  { com' : Com d Γ .pure α
    // ∀ (V : Ctxt.Valuation Γstart), com.denote (lets.denote V) = com'.denote (lets.denote V) } :=
  match com with
  | .rets vs =>
      let ⟨vs', hvs'⟩ := s.vars2vars vs
      ⟨.rets vs', by admit /- proof elided -/
        ⟩
  | .var e body =>
      let ⟨⟨e', he'⟩, v'?⟩ := s.cseExpr e
      match v'? with
      | .none =>  
        let s' := s.consNewExpr2Cache (e := e')
         
        let ⟨body', hbody'⟩ := s'.cseCom body
        ⟨.var e' body',  by admit /- proof elided -/
            ⟩
      | .some ⟨v', hv'⟩ =>
        let s' := s.consOldExpr2Cache (enew := e') (eold := e) (henew := by admit /- proof elided -/
        )
          (vold := v') (hv := by admit /- proof elided -/
          )
          
        let ⟨body', hbody'⟩ := s'.cseCom body
        
        ⟨.var e body' 
        , by admit /- proof elided -/
        ⟩

noncomputable def R.slice {q n : Nat} (a : R q n) (startIdx endIdx : Nat) : R q n :=
  let coeffIdxs := List.range (endIdx - startIdx)
  let coeffs := coeffIdxs.map (fun i => a.coeff (startIdx + i))
  let accum : R q n → (ZMod q × Nat) → R q n :=
    fun poly (c,i) => poly + R.monomial (n:=n) c i
  coeffs.zip coeffIdxs |>.foldl accum (0 : R q n)

def icmp_ne_zext_ne_non_boolean_after := [llvm|
{
^0(%arg33 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.icmp "ne" %arg33, %0 : i32
  "llvm.return"(%1) : (i1) -> ()
}
]

def ConcreteCliTest.printSignature (test : ConcreteCliTest) : String :=
  s!"{test.context.toList.reverse} → {test.ty}"

def test13_before := [llvm|
{
^0(%arg0 : i1024):
  %0 = llvm.mlir.constant(1016 : i1024) : i1024
  %1 = llvm.mlir.constant(255 : i1024) : i1024
  %2 = llvm.ashr %arg0, %0 : i1024
  %3 = llvm.and %2, %1 : i1024
  "llvm.return"(%3) : (i1024) -> ()
}
]

def test86_after := [llvm|
{
^0(%arg59 : i16):
  %0 = llvm.mlir.constant(4 : i16) : i16
  %1 = llvm.ashr %arg59, %0 : i16
  "llvm.return"(%1) : (i16) -> ()
}
]

@[simps]
def complete (M : NFA α σ) : NFA α (σ ⊕ Unit) where
  start := { x | match x with | .inl x => x ∈ M.start | .inr () => True }
  accept := { x | match x with | .inl x => x ∈ M.accept | .inr () => False }
  step x a := { y |
    match x with
    | .inl x => match y with | .inl y => y ∈ M.step x a | .inr () => False
    | .inr () => y = .inr () }

def test1_before := [llvm|
{
^0(%arg309 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.and %arg309, %0 : i32
  "llvm.return"(%1) : (i32) -> ()
}
]

def or_and3_commuted_after := [llvm|
{
^0(%arg100 : i1, %arg101 : i1, %arg102 : i32, %arg103 : i32):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ne" %arg102, %arg103 : i32
  %3 = "llvm.select"(%2, %arg101, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %4 = "llvm.select"(%arg100, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def uge_swap_and_max_commute_before := [llvm|
{
^0(%arg226 : i8, %arg227 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.icmp "ule" %arg227, %arg226 : i8
  %2 = llvm.icmp "eq" %arg226, %0 : i8
  %3 = llvm.and %2, %1 : i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def shl_nsw_add_negative_before := [llvm|
{
^0(%arg45 : i32):
  %0 = llvm.mlir.constant(-1 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.add %arg45, %0 : i32
  %3 = llvm.shl %1, %2 overflow<nsw> : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def select_icmp_ne_not_pow2_after := [llvm|
{
^0(%arg8 : i8):
  %0 = llvm.mlir.constant(5 : i8) : i8
  %1 = llvm.mlir.constant(0 : i8) : i8
  %2 = llvm.and %arg8, %0 : i8
  %3 = llvm.icmp "eq" %2, %1 : i8
  %4 = llvm.xor %arg8, %0 : i8
  %5 = "llvm.select"(%3, %arg8, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  "llvm.return"(%5) : (i8) -> ()
}
]

def test_invert_demorgan_or2_before := [llvm|
{
^0(%arg15 : i64, %arg16 : i64, %arg17 : i64):
  %0 = llvm.mlir.constant(23) : i64
  %1 = llvm.mlir.constant(59) : i64
  %2 = llvm.mlir.constant(true) : i1
  %3 = llvm.icmp "ugt" %arg15, %0 : i64
  %4 = llvm.icmp "ugt" %arg16, %1 : i64
  %5 = llvm.or %3, %4 : i1
  %6 = llvm.icmp "ugt" %arg17, %1 : i64
  %7 = llvm.or %5, %6 : i1
  %8 = llvm.xor %7, %2 : i1
  "llvm.return"(%8) : (i1) -> ()
}
]

def ule_and_min_logical_after := [llvm|
{
^0(%arg204 : i8, %arg205 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.icmp "eq" %arg204, %0 : i8
  "llvm.return"(%1) : (i1) -> ()
}
]

def eq_smin1_after := [llvm|
{
^0(%arg87 : i32, %arg88 : i32):
  %0 = llvm.icmp "sle" %arg87, %arg88 : i32
  "llvm.return"(%0) : (i1) -> ()
}
]

def scalar_i32_lshr_and_negC_slt_after := [llvm|
{
^0(%arg2 : i32, %arg3 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.lshr %arg2, %arg3 : i32
  %2 = llvm.icmp "slt" %1, %0 : i32
  "llvm.return"(%2) : (i1) -> ()
}
]

RV64.srlw in RISCV.Instructions

def scalar_i8_lshr_and_signbit_eq_before := [llvm|
{
^0(%arg35 : i8, %arg36 : i8):
  %0 = llvm.mlir.constant(-128 : i8) : i8
  %1 = llvm.mlir.constant(0 : i8) : i8
  %2 = llvm.lshr %arg35, %arg36 : i8
  %3 = llvm.and %2, %0 : i8
  %4 = llvm.icmp "eq" %3, %1 : i8
  "llvm.return"(%4) : (i1) -> ()
}
]

def lhs :
    let Γ := ⟨[  Arith.Ty.int,  Arith.Ty.int,
        Arith.Ty.nat,   t]⟩
    Com ScfArith Γ .impure t :=
   
    Com.var (for_ (t := t)
                        ⟨  0, by admit /- proof elided -/
                        ⟩
                        ⟨ 1, by admit /- proof elided -/
                        ⟩
                        ⟨  2, by admit /- proof elided -/
                        ⟩
                        ⟨  3, by admit /- proof elided -/
                        ⟩  rgn) <|
  Com.rets [⟨0, rfl⟩]ₕ

def t1_after := [llvm|
{
^0(%arg28 : i8, %arg29 : i1):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.sub %0, %arg28 : i8
  %2 = "llvm.select"(%arg29, %1, %arg28) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  "llvm.return"(%2) : (i8) -> ()
}
]

def scalar_i32_lshr_and_negC_eq_nonzero_before := [llvm|
{
^0(%arg2 : i32, %arg3 : i32):
  %0 = llvm.mlir.constant(-8 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.lshr %arg2, %arg3 : i32
  %3 = llvm.and %2, %0 : i32
  %4 = llvm.icmp "eq" %3, %1 : i32
  "llvm.return"(%4) : (i1) -> ()
}
]

def bad_shl2_sub_i32_before := [llvm|
{
^0(%arg9 : i32):
  %0 = llvm.mlir.constant(31 : i32) : i32
  %1 = llvm.mlir.constant(1 : i32) : i32
  %2 = llvm.sub %arg9, %0 : i32
  %3 = llvm.shl %1, %2 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def xor_logic_and_logic_or1_after := [llvm|
{
^0(%arg54 : i1, %arg55 : i1, %arg56 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = llvm.xor %arg55, %0 : i1
  %2 = "llvm.select"(%arg54, %1, %arg56) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%2) : (i1) -> ()
}
]

def rem_euclid_pow2_false_arm_folded_before := [llvm|
{
^0(%arg2 : i32):
  %0 = llvm.mlir.constant(2 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(1 : i32) : i32
  %3 = llvm.srem %arg2, %0 : i32
  %4 = llvm.icmp "sge" %3, %1 : i32
  %5 = "llvm.select"(%4, %3, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def constLLVM_2 : Com LLVMPlusRiscV ⟨[]⟩ .pure (.llvm (.bitvec 64)) := [LV| {
  ^entry ():
    %1 = llvm.mlir.constant (-2) : i64
    llvm.return %1 : i64
  }]

def llvm_add_lower_riscv_nuw_flag_16 : LLVMPeepholeRewriteRefine 16 [Ty.llvm (.bitvec 16), Ty.llvm (.bitvec 16)] where
  lhs:= add_llvm_nuw_flags_16
  rhs:= add_riscv_16

def join (x y : TokenStream) : TokenStream  :=
  Stream.corec (β := TokenStream × TokenStream) (x, y) fun ⟨x, y⟩ =>
    match x 0, y 0 with
    | some _, some _ => (some (), (x.tail, y.tail))
    | some _, none => (none, (x, y.tail))
    | none, some _ => (none, (x.tail, y))
    | none, none => (none, (x.tail, y.tail))

def c0_before := [llvm|
{
^0(%arg7 : i8):
  %0 = llvm.mlir.constant(3 : i8) : i8
  %1 = llvm.and %arg7, %0 : i8
  %2 = llvm.icmp "sgt" %1, %arg7 : i8
  "llvm.return"(%2) : (i1) -> ()
}
]

def masked_icmps_mask_notallzeros_bmask_mixed_swapped_7b_logical_after := [llvm|
{
^0(%arg31 : i32):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

@[simp_llvm_option]
def udiv {w : Nat} (x y : IntW w) (flag : ExactFlag := {exact := false}) : IntW w := do
  let x' ← x
  let y' ← y
  if flag.exact ∧ x'.umod y' ≠ 0 then
    .poison
  else
    udiv? x' y'

def masked_or_allzeroes_notoptimised_logical_after := [llvm|
{
^0(%arg116 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(39 : i32) : i32
  %3 = llvm.and %arg116, %0 : i32
  %4 = llvm.icmp "eq" %3, %1 : i32
  %5 = llvm.and %arg116, %2 : i32
  %6 = llvm.icmp "eq" %5, %1 : i32
  %7 = llvm.or %4, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def udiv_pow2_128 : LLVMPeepholeRewriteRefine 64 [Ty.llvm (.bitvec 64)] where
  lhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant (128) : i64
      %0 = llvm.udiv %x, %c : i64
      llvm.return %0 : i64
  }]
  rhs := [LV| {
    ^entry (%x: i64):
      %c = llvm.mlir.constant (7) : i64
      %0 = llvm.lshr %x, %c : i64
      llvm.return %0 : i64
  }]

def llvm_shl_lower_riscv_nsw_nuw_32: LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 32), Ty.llvm (.bitvec 32)] where
  lhs := shl_llvm_nsw_nuw_32
  rhs := shl_riscv_32

instance : Trans (@ExtEq Op) ExtEq ExtEq := ⟨ExtEq.trans⟩

def a_false_implies_b_false2_comm_after := [llvm|
{
^0(%arg0 : i8, %arg1 : i1, %arg2 : i1):
  %0 = llvm.mlir.constant(10 : i8) : i8
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ugt" %arg0, %0 : i8
  %3 = "llvm.select"(%2, %1, %arg2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%3) : (i1) -> ()
}
]

def const_sub_const_sub_before := [llvm|
{
^0(%arg20 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.sub %0, %arg20 : i32
  %3 = llvm.sub %1, %2 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def shl_or_with_or_disjoint_instead_of_add_after := [llvm|
{
^0(%arg1 : i8):
  %0 = llvm.mlir.constant(22 : i8) : i8
  %1 = llvm.shl %0, %arg1 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def icmp_ne_sext_ne_zero_before := [llvm|
{
^0(%arg24 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "ne" %arg24, %0 : i32
  %2 = llvm.sext %1 : i1 to i32
  %3 = llvm.icmp "ne" %2, %arg24 : i32
  "llvm.return"(%3) : (i1) -> ()
}
]

def test35_before := [llvm|
{
^0(%arg50 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.mlir.constant(60 : i32) : i32
  %2 = llvm.mlir.constant(100 : i32) : i32
  %3 = llvm.icmp "sge" %arg50, %0 : i32
  %4 = "llvm.select"(%3, %1, %2) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def lshrugt_02_05_after := [llvm|
{
^0(%arg394 : i4):
  %0 = llvm.mlir.constant(false) : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def select_icmp_eq_pow2_after := [llvm|
{
^0(%arg13 : i8):
  %0 = llvm.mlir.constant(-5 : i8) : i8
  %1 = llvm.and %arg13, %0 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def sgt_or_not_min_logical_before := [llvm|
{
^0(%arg28 : i8, %arg29 : i8):
  %0 = llvm.mlir.constant(-128 : i8) : i8
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "sgt" %arg28, %arg29 : i8
  %3 = llvm.icmp "ne" %arg28, %0 : i8
  %4 = "llvm.select"(%2, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def Expr.mkToFSM (self : Expr) : MetaM Expr :=
  mkAppM ``FSMPredicateSolution.toFSM #[self]

def sra  {Γ : Ctxt _} (e₁ e₂: Ctxt.Var Γ .bv) : Expr RV64 Γ .pure [.bv]  :=
  Expr.mk
    (op := Op.sll)
    (eff_le := by admit /- proof elided -/
    )
    (ty_eq := by admit /- proof elided -/
    )
    (args := .cons e₁ <| .cons e₂ .nil)
    (regArgs := HVector.nil)

def rem_euclid_wrong_operands_select_after := [llvm|
{
^0(%arg8 : i32):
  %0 = llvm.mlir.constant(8 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.srem %arg8, %0 : i32
  %3 = llvm.icmp "slt" %2, %1 : i32
  %4 = llvm.add %2, %0 overflow<nsw> : i32
  %5 = "llvm.select"(%3, %2, %4) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def test14_before := [llvm|
{
^0(%arg36 : i1, %arg37 : i1):
  %0 = llvm.icmp "eq" %arg36, %arg37 : i1
  "llvm.return"(%0) : (i1) -> ()
}
]

def same_signbit_wrong_type_signed_before := [llvm|
{
^0(%arg14 : i8, %arg15 : i32):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.ashr %arg14, %0 : i8
  %3 = llvm.icmp "sgt" %arg15, %1 : i32
  %4 = llvm.sext %3 : i1 to i8
  %5 = llvm.icmp "ne" %2, %4 : i8
  "llvm.return"(%5) : (i1) -> ()
}
]

def attr0Str : AttrEntry 0 := [mlir_attr_entry| sym_name = "add"]

def or_xor_tree_1101_before := [llvm|
{
^0(%arg6 : i32, %arg7 : i32, %arg8 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mul %arg6, %0 : i32
  %2 = llvm.mul %arg7, %0 : i32
  %3 = llvm.mul %arg8, %0 : i32
  %4 = llvm.xor %2, %1 : i32
  %5 = llvm.xor %2, %3 : i32
  %6 = llvm.xor %1, %5 : i32
  %7 = llvm.or %6, %4 : i32
  "llvm.return"(%7) : (i32) -> ()
}
]

def and_logic_and_logic_or_8_after := [llvm|
{
^0(%arg117 : i1, %arg118 : i1, %arg119 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = "llvm.select"(%arg119, %0, %arg118) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %2 = llvm.and %arg117, %1 : i1
  "llvm.return"(%2) : (i1) -> ()
}
]

def a_or_b_before := [llvm|
{
^0(%arg15 : i32, %arg16 : i32):
  %0 = llvm.mlir.constant(0 : i32) : i32
  %1 = llvm.icmp "eq" %arg15, %0 : i32
  %2 = llvm.icmp "ne" %arg16, %0 : i32
  %3 = llvm.and %1, %2 : i1
  %4 = llvm.icmp "ne" %arg15, %0 : i32
  %5 = llvm.icmp "eq" %arg16, %0 : i32
  %6 = llvm.and %4, %5 : i1
  %7 = llvm.or %3, %6 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def add_shl_moreOneUse_sh3add_no_ZBA_test : LLVMPeepholeRewriteRefine 32 [Ty.llvm (.bitvec 32)] where
  lhs := add_shl_moreOneUse_sh3add
  rhs := add_shl_moreOneUse_sh3add_riscv_no_ZBA

def test67_before := [llvm|
{
^0(%arg113 : i1, %arg114 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(24 : i32) : i32
  %2 = llvm.mlir.constant(-16777216 : i32) : i32
  %3 = llvm.mlir.constant(0 : i8) : i8
  %4 = llvm.zext %arg113 : i1 to i32
  %5 = llvm.xor %4, %0 : i32
  %6 = llvm.and %arg114, %5 : i32
  %7 = llvm.shl %6, %1 overflow<nsw,nuw> : i32
  %8 = llvm.xor %7, %2 : i32
  %9 = llvm.ashr exact %8, %1 : i32
  %10 = llvm.trunc %9 : i32 to i8
  %11 = llvm.icmp "eq" %10, %3 : i8
  "llvm.return"(%11) : (i1) -> ()
}
]

def pv_before := [llvm|
{
^0(%arg14 : i8, %arg15 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.lshr %0, %arg15 : i8
  %2 = llvm.and %1, %arg14 : i8
  %3 = llvm.icmp "uge" %2, %arg14 : i8
  "llvm.return"(%3) : (i1) -> ()
}
]

def test1_after := [llvm|
{
^0(%arg8 : i17, %arg9 : i17):
  %0 = llvm.mlir.constant(7 : i17) : i17
  %1 = llvm.and %arg8, %0 : i17
  "llvm.return"(%1) : (i17) -> ()
}
]

def and_or_not_not_commute6_before := [llvm|
{
^0(%arg357 : i32, %arg358 : i32, %arg359 : i32):
  %0 = llvm.mlir.constant(42 : i32) : i32
  %1 = llvm.mlir.constant(-1 : i32) : i32
  %2 = llvm.sdiv %0, %arg358 : i32
  %3 = llvm.and %2, %arg357 : i32
  %4 = llvm.xor %3, %1 : i32
  %5 = llvm.and %arg359, %arg357 : i32
  %6 = llvm.xor %5, %1 : i32
  %7 = llvm.or %2, %6 : i32
  %8 = llvm.and %7, %4 : i32
  "llvm.return"(%8) : (i32) -> ()
}
]

def ult_swap_or_not_max_logical_before := [llvm|
{
^0(%arg36 : i8, %arg37 : i8):
  %0 = llvm.mlir.constant(-1 : i8) : i8
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ugt" %arg37, %arg36 : i8
  %3 = llvm.icmp "ne" %arg36, %0 : i8
  %4 = "llvm.select"(%2, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_5_before := [llvm|
{
^0(%arg16 : i32):
  %0 = llvm.mlir.constant(15 : i32) : i32
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.mlir.constant(8 : i32) : i32
  %3 = llvm.and %arg16, %0 : i32
  %4 = llvm.icmp "eq" %3, %1 : i32
  %5 = llvm.and %arg16, %0 : i32
  %6 = llvm.icmp "ne" %5, %2 : i32
  %7 = llvm.or %6, %4 : i1
  "llvm.return"(%7) : (i1) -> ()
}
]

def testi16i8_after := [llvm|
{
^0(%arg12 : i16):
  %0 = llvm.mlir.constant(128 : i16) : i16
  %1 = llvm.mlir.constant(256 : i16) : i16
  %2 = llvm.add %arg12, %0 : i16
  %3 = llvm.icmp "ult" %2, %1 : i16
  "llvm.return"(%3) : (i1) -> ()
}
]

def or_and3_after := [llvm|
{
^0(%arg104 : i1, %arg105 : i1, %arg106 : i32, %arg107 : i32):
  %0 = llvm.mlir.constant(false) : i1
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ne" %arg106, %arg107 : i32
  %3 = "llvm.select"(%2, %arg105, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  %4 = "llvm.select"(%arg104, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def neg_not_signbit3_after := [llvm|
{
^0(%arg3 : i8):
  %0 = llvm.mlir.constant(7 : i8) : i8
  %1 = llvm.mlir.constant(0 : i32) : i32
  %2 = llvm.ashr %arg3, %0 : i8
  %3 = llvm.zext %2 : i8 to i32
  %4 = llvm.sub %1, %3 overflow<nsw> : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def cond_eq_or_const_before := [llvm|
{
^0(%arg149 : i8, %arg150 : i8):
  %0 = llvm.mlir.constant(10 : i8) : i8
  %1 = llvm.mlir.constant(true) : i1
  %2 = llvm.icmp "ne" %arg149, %0 : i8
  %3 = llvm.icmp "ult" %arg149, %arg150 : i8
  %4 = "llvm.select"(%2, %1, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def positive_samevar_shlnuwnsw_before := [llvm|
{
^0(%arg47 : i8, %arg48 : i8):
  %0 = llvm.ashr %arg47, %arg48 : i8
  %1 = llvm.shl %0, %arg48 overflow<nsw,nuw> : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

def trunc_shl_nuw_31_i32_i64_before := [llvm|
{
^0(%arg56 : i64):
  %0 = llvm.mlir.constant(31) : i64
  %1 = llvm.shl %arg56, %0 overflow<nuw> : i64
  %2 = llvm.trunc %1 : i64 to i32
  "llvm.return"(%2) : (i32) -> ()
}
]

def mul_of_bools_before := [llvm|
{
^0(%arg91 : i32, %arg92 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.and %arg91, %0 : i32
  %3 = llvm.and %arg92, %0 : i32
  %4 = llvm.mul %2, %3 : i32
  %5 = llvm.icmp "ult" %4, %1 : i32
  "llvm.return"(%5) : (i1) -> ()
}
]

noncomputable def R.toTensor {q n} (a : R q n) : List Int :=
  List.range a.repLength |>.map fun i =>
        a.coeff i |>.toInt

def lor_lor_left2_after := [llvm|
{
^0(%arg68 : i1, %arg69 : i1):
  %0 = llvm.mlir.constant(true) : i1
  %1 = "llvm.select"(%arg69, %0, %arg68) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%1) : (i1) -> ()
}
]

def ashrsgt_01_11_exact_before := [llvm|
{
^0(%arg90 : i4):
  %0 = llvm.mlir.constant(1 : i4) : i4
  %1 = llvm.mlir.constant(-5 : i4) : i4
  %2 = llvm.ashr exact %arg90, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def select_icmp_ne_pow2_flipped_after := [llvm|
{
^0(%arg9 : i8):
  %0 = llvm.mlir.constant(4 : i8) : i8
  %1 = llvm.or %arg9, %0 : i8
  "llvm.return"(%1) : (i8) -> ()
}
]

instance : DialectParse Simple 0 where
  mkTy
  | .int .Signless 32 => return .int
  | _ => throw .unsupportedType

  isValidReturn _Γ opStx := return opStx.name == "return"

  mkExpr Γ opStx := do opStx.mkExprOf Γ <|← match opStx.name with
    | "add"   => return .add
    | "const" => do
        let ⟨val, _type⟩ ← opStx.getIntAttr "value"
        return .const val
    | opName => throw <| .unsupportedOp opName

def xor_nosignbit_shl_before := [llvm|
{
^0(%arg18 : i32):
  %0 = llvm.mlir.constant(2147418112 : i32) : i32
  %1 = llvm.mlir.constant(8 : i32) : i32
  %2 = llvm.xor %arg18, %0 : i32
  %3 = llvm.shl %2, %1 : i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def t2_ult_sgt_65536_before := [llvm|
{
^0(%arg60 : i32, %arg61 : i32, %arg62 : i32):
  %0 = llvm.mlir.constant(65535 : i32) : i32
  %1 = llvm.mlir.constant(65536 : i32) : i32
  %2 = llvm.icmp "sgt" %arg60, %0 : i32
  %3 = "llvm.select"(%2, %arg62, %arg61) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  %4 = llvm.icmp "ult" %arg60, %1 : i32
  %5 = "llvm.select"(%4, %arg60, %3) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def multiuse1_after := [llvm|
{
^0(%arg11 : i32):
  %0 = llvm.mlir.constant(1 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.mlir.constant(6 : i32) : i32
  %3 = llvm.mlir.constant(384 : i32) : i32
  %4 = llvm.lshr %arg11, %0 : i32
  %5 = llvm.and %4, %0 : i32
  %6 = llvm.lshr %arg11, %0 : i32
  %7 = llvm.and %6, %1 : i32
  %8 = llvm.shl %arg11, %2 : i32
  %9 = llvm.and %8, %3 : i32
  %10 = llvm.or disjoint %5, %7 : i32
  %11 = llvm.or disjoint %10, %9 : i32
  "llvm.return"(%11) : (i32) -> ()
}
]

noncomputable def nonesUntilSome (x : Stream α) (not_stuck : x ≠ stuck α) : Nat :=
  prop.choose
where
  prop : ∃ i, (x.drop i).head.isSome ∧ ∀ j < i, x.get j = none := by admit /- proof elided -/

def fold_add_udiv_urem_without_noundef_before := [llvm|
{
^0(%arg4 : i32):
  %0 = llvm.mlir.constant(10 : i32) : i32
  %1 = llvm.mlir.constant(4 : i32) : i32
  %2 = llvm.udiv %arg4, %0 : i32
  %3 = llvm.shl %2, %1 : i32
  %4 = llvm.urem %arg4, %0 : i32
  %5 = llvm.add %3, %4 : i32
  "llvm.return"(%5) : (i32) -> ()
}
]

def add_or_and_after := [llvm|
{
^0(%arg19 : i32, %arg20 : i32):
  %0 = llvm.add %arg19, %arg20 : i32
  "llvm.return"(%0) : (i32) -> ()
}
]

def urem_i8_after := [llvm|
{
^0(%arg33 : i8, %arg34 : i8):
  %0 = llvm.urem %arg33, %arg34 : i8
  "llvm.return"(%0) : (i8) -> ()
}
]

def twoway_clamp_lt_after := [llvm|
{
^0(%arg15 : i32):
  %0 = llvm.mlir.constant(13767 : i32) : i32
  %1 = llvm.mlir.constant(13768 : i32) : i32
  %2 = llvm.icmp "sgt" %arg15, %0 : i32
  %3 = "llvm.select"(%2, %1, %0) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i32, i32) -> i32
  "llvm.return"(%3) : (i32) -> ()
}
]

def test_nuw_nsw_and_unsigned_pred_after := [llvm|
{
^0(%arg55 : i64):
  %0 = llvm.mlir.constant(6) : i64
  %1 = llvm.icmp "ugt" %arg55, %0 : i64
  "llvm.return"(%1) : (i1) -> ()
}
]

def test68_xor_after := [llvm|
{
^0(%arg162 : i32, %arg163 : i32):
  %0 = llvm.mlir.constant(6 : i32) : i32
  %1 = llvm.mlir.constant(2 : i32) : i32
  %2 = llvm.lshr %arg162, %0 : i32
  %3 = llvm.and %2, %1 : i32
  %4 = llvm.xor %arg163, %3 : i32
  "llvm.return"(%4) : (i32) -> ()
}
]

def ashrsgt_02_06_before := [llvm|
{
^0(%arg295 : i4):
  %0 = llvm.mlir.constant(2 : i4) : i4
  %1 = llvm.mlir.constant(6 : i4) : i4
  %2 = llvm.ashr %arg295, %0 : i4
  %3 = llvm.icmp "sgt" %2, %1 : i4
  "llvm.return"(%3) : (i1) -> ()
}
]

def ugt_and_min_commute_logical_before := [llvm|
{
^0(%arg330 : i8, %arg331 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.mlir.constant(false) : i1
  %2 = llvm.icmp "ugt" %arg330, %arg331 : i8
  %3 = llvm.icmp "eq" %arg330, %0 : i8
  %4 = "llvm.select"(%3, %2, %1) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i1, i1) -> i1
  "llvm.return"(%4) : (i1) -> ()
}
]

def alive_AddSub_1176_tgt :=
[sllvm| {
^bb0(%a : i64, %b : i64):
  %v1 = llvm.mlir.constant 0 : i64
  %v2 = llvm.sub %v1, %b : i64
  %v3 = llvm.sub %a, %b : i64
  llvm.return %v3 : i64
}]

def select_icmp_slt_zero_smin_flipped_before := [llvm|
{
^0(%arg6 : i8):
  %0 = llvm.mlir.constant(0 : i8) : i8
  %1 = llvm.mlir.constant(-128 : i8) : i8
  %2 = llvm.icmp "slt" %arg6, %0 : i8
  %3 = llvm.xor %arg6, %1 : i8
  %4 = "llvm.select"(%2, %3, %arg6) <{"fastmathFlags" = #llvm.fastmath<none>}> : (i1, i8, i8) -> i8
  "llvm.return"(%4) : (i8) -> ()
}
]

Lean.FVarId in Lean.Expr

Sum in Init.Core
</all_available_defs>

<all_available_lemmas>
Finset.subset_iff in Mathlib.Data.Finset.Defs
List.mem_append in Init.Data.List.Lemmas
List.mem_dedup in Mathlib.Data.List.Dedup
Finset.mem_union in Mathlib.Data.Finset.Lattice.Basic
List.subset_def in Init.Data.List.Sublist
Finset.mem_biUnion in Mathlib.Data.Finset.Union

theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]

theorem ashrslt_01_15_exact_thm (e : IntW 4) :
  icmp IntPred.slt (ashr e (const? 4 1) { «exact» := true }) (const? 4 (-1)) ⊑
    icmp IntPred.slt e (const? 4 (-2))

theorem mul_of_pow2_thm.extracted_1._1 : ∀ (x : BitVec 8) (x_1 : BitVec 32),
  ofBool (510#32 <ᵤ (x_1 &&& 2#32) * zeroExtend 32 x) = 0#1

theorem ashrslt_01_06_proof : ashrslt_01_06_before ⊑ ashrslt_01_06_after

theorem t9_c0_c2_thm.extracted_1._1 : ∀ (x : BitVec 8), 42#8 - x - 24#8 = 18#8 - x

theorem lshrult_01_14_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬1#4 ≥ ↑4 → ofBool (x >>> 1#4 <ᵤ BitVec.ofInt 4 (-2)) = 1#1

theorem shl_fold_or_disjoint_cnt_proof : shl_fold_or_disjoint_cnt_before ⊑ shl_fold_or_disjoint_cnt_after

theorem trunc_unsigned_nsw_zext_proof : trunc_unsigned_nsw_zext_before ⊑ trunc_unsigned_nsw_zext_after

theorem uge_swap_or_not_max_commute_logical_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ¬ofBool (x_1 != -1#8) = 1#1 → ofBool (x ≤ᵤ x_1) = 1#1

theorem test_proof : test_before ⊑ test_after

theorem or_comm (x y : BitStream) : x ||

theorem or_or_logic_and_1_thm.extracted_1._2 : ∀ (x x_1 : BitVec 1), ¬x_1 ||

theorem t13_x_is_one_thm.extracted_1._1 : ∀ (x : BitVec 64) (x_1 : BitVec 32),
  ¬(32#32 - x_1 ≥ ↑32 ∨ zeroExtend 64 (x_1 + BitVec.ofInt 32 (-16)) ≥ ↑64) →
    ofBool (1#32 <<< (32#32 - x_1) &&& truncate 32 (x >>> zeroExtend 64 (x_1 + BitVec.ofInt 32 (-16))) != 0#32) =
      ofBool (x &&& 65536#64 != 0#64)

theorem e_445 :
     - 2 * (x ^^^ y) + 2 *  ~~~x + 11 * (x ||| y) + 2 *  ~~~(x &&&  ~~~x) + 3 *  ~~~(x ||| y) - 13 *  ~~~(x |||  ~~~y) - 11 * (x &&&  ~~~y) - 6 * (x &&& y) = 7 *  ~~~(x ^^^ y)

theorem lshrult_01_11_thm (e : IntW 4) : icmp IntPred.ult (lshr e (const? 4 1)) (const? 4 (-5)) ⊑ const? 1 1

lemma simulFun_sim {m : CNFA n} f :
    m.m.SimulFun M.M f → m.Sim M

theorem zext_or_eq_ult_add_thm (e : IntW 32) :
  zext 32
      (LLVM.or (icmp IntPred.ult (add e (const? 32 (-3))) (const? 32 3)) (icmp IntPred.eq e (const? 32 5))) ⊑
    zext 32 (icmp IntPred.ult (add e (const? 32 (-3))) (const? 32 3))

@[bv_automata_nnf]
theorem ult_iff_ule_eq_false {a b : BitVec w} : a.ule b = false ↔ b.ult a

theorem test91_thm.extracted_1._1 : ∀ (x : BitVec 64),
  ¬48#96 ≥ ↑96 →
    48#96 ≥ ↑96 ∨
        True ∧ signExtend 96 (truncate 64 (signExtend 96 x >>> 48#96)) ≠ signExtend 96 x >>> 48#96 ∨
          True ∧ zeroExtend 96 (truncate 64 (signExtend 96 x >>> 48#96)) ≠ signExtend 96 x >>> 48#96 →
      False

theorem scalar_i16_lshr_and_signbit_eq_thm.extracted_1._1 : ∀ (x x_1 : BitVec 16),
  ¬x ≥ ↑16 → ofBool (x_1 >>> x &&& BitVec.ofInt 16 (-32768) == 0#16) = ofBool (-1#16 <ₛ x_1 >>> x)

theorem no_shift_xor_multiuse_cmp_or_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (0#32 != x_2 &&& 4096#32) = 1#1 → ofBool (x_2 &&& 4096#32 == 0#32) = 1#1 → False

theorem sub_inc_thm (e e_1 : IntW 8) : sub e_1 (add e (const? 8 1)) ⊑ add e_1 (LLVM.xor e (const? 8 (-1)))

theorem positive_samevar_shlnuwnsw_thm (e e_1 : IntW 8) :
  shl (ashr e_1 e) e { «nsw» := true, «nuw» := true } ⊑
    LLVM.and (shl (const? 8 (-1)) e { «nsw» := true, «nuw» := false }) e_1

theorem t4_proof : t4_before ⊑ t4_after

theorem main5e_like_logical_thm.extracted_1._2 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (x_1 &&& x == x_1) = 1#1 → ofBool (x_1 &&& x != x_1) = 1#1 → ¬0#1 = 1#1 → 1#32 = zeroExtend 32 1#1

theorem or_xor_tree_1010_thm (e e_1 e_2 : IntW 32) :
  LLVM.or (LLVM.xor (LLVM.xor (mul e_2 (const? 32 42)) (mul e_1 (const? 32 42))) (mul e (const? 32 42)))
      (LLVM.xor (mul e (const? 32 42)) (mul e_1 (const? 32 42))) ⊑
    LLVM.or (LLVM.xor (mul e (const? 32 42)) (mul e_1 (const? 32 42))) (mul e_2 (const? 32 42))

theorem bv_InstCombineShift__476 :
    ∀ (e e_1 e_2 e_3 : LLVM.IntW w),
      LLVM.shl (LLVM.or (LLVM.and (LLVM.lshr e_1 e_2) e_3) e) e_2 ⊑
        LLVM.or (LLVM.and e_1 (LLVM.shl e_3 e_2)) (LLVM.shl e e_2)

theorem and_nosignbit_select_shl_proof : and_nosignbit_select_shl_before ⊑ and_nosignbit_select_shl_after

theorem e_2219 :
     - 6 * ((d ||| e) &&& (e ^^^ f)) + 2 *  ~~~(d &&& (e ^^^ f)) + 1 * ((e &&& f) ^^^  ~~~( ~~~d &&& (e ^^^ f))) - 7 * ( ~~~(d &&&  ~~~e) &&& (e ^^^ f)) - 3 * ( ~~~d ||| (e &&& f)) - 5 * (f &&&  ~~~(d &&&  ~~~e)) + 1 * (f |||  ~~~(d ^^^ e)) + 4 * ( ~~~d &&& ( ~~~e ||| f)) - 5 * (d &&& e) + 11 * (e ^^^  ~~~( ~~~d &&& ( ~~~e &&& f))) - 11 * (e ^^^  ~~~( ~~~d &&& (e ^^^ f))) + 4 * (d ||| f) - 2 * (e ^^^ (d &&& (e ^^^ f))) - 2 * (f &&&  ~~~(d &&& e)) + 1 * (f ^^^ (d ||| (e &&& f))) - 1 * ((e &&&  ~~~f) |||  ~~~(d ||| ( ~~~e &&& f))) - 1 * (e ^^^  ~~~(d &&&  ~~~f)) + 2 *  ~~~(d ||| ( ~~~e &&& f)) - 1 *  ~~~(d ||| (e &&& f)) - 7 *  ~~~(e &&&  ~~~f) + 5 * ((d &&& e) ^^^ (e ||| f)) - 1 * (d ||| (e ^^^ f)) + 1 * ( ~~~(d &&&  ~~~e) &&&  ~~~(d ^^^ (e ^^^ f))) + 2 * (e ^^^  ~~~(d &&& (e ^^^ f))) + 7 * (e |||  ~~~(d ^^^ f)) + 1 * (d &&& (e ^^^ f)) - 2 * ((d &&& f) ^^^ (d ^^^ ( ~~~e &&& f))) + 3 * ((d &&& e) ||| (e ^^^ f)) - 1 *  ~~~(e ||| f) - 11 * ((d &&& e) ^^^  ~~~(d ^^^ (e &&& f))) - 3 *  ~~~(d ^^^ e) + 7 * ((d &&&  ~~~e) |||  ~~~(e ^^^ f)) - 3 * ( ~~~(d ^^^ e) ||| (d ^^^ f)) + 2 *  ~~~( ~~~d &&& (e &&& f)) + 3 * (f ^^^  ~~~( ~~~d &&& (e ||| f))) - 6 *  ~~~(d &&&  ~~~e) + 5 * (e |||  ~~~f) + 4 *  ~~~(e ^^^ f) - 7 *  ~~~(d ^^^ f) + 1 * (e &&& (d ||| f)) + 2 * (e ||| (d &&& f)) + 1 * (f ||| (d ^^^ e)) + 4 * ( ~~~(d ||| e) ||| (e ^^^ f)) - 5 * ( ~~~(d &&&  ~~~e) &&& (d ^^^ (e ^^^ f))) + 1 * (f ^^^ (d &&& ( ~~~e ||| f))) - 2 *  ~~~f + 41 *  ~~~(d ||| ( ~~~e ||| f)) - 6 *  ~~~( ~~~d ||| (e ||| f)) + 10 *  ~~~( ~~~d ||| ( ~~~e ||| f)) + 24 * ( ~~~d &&& ( ~~~e &&& f)) - 21 * ( ~~~d &&& (e &&& f)) - 10 * (d &&& ( ~~~e &&& f)) + 19 * (d &&& (e &&& f)) =  - 12 *  ~~~(d ||| (e ||| f)) + 11 * (f ^^^  ~~~(d &&& e))

theorem bools2_logical_commute1_and1_and2_thm.extracted_1._1 : ∀ (x x_1 : BitVec 1),
  x_1 &&& x = 1#1 → x = 1#1 → 1#1 = x_1

theorem not_sub_thm (e : IntW 32) :
  LLVM.xor (sub (const? 32 123) e) (const? 32 (-1)) ⊑ add e (const? 32 (-124))

theorem test40_thm.extracted_1._1 : ∀ (x : BitVec 16),
  ¬(9#32 ≥ ↑32 ∨ 8#32 ≥ ↑32) → 9#16 ≥ ↑16 ∨ 8#16 ≥ ↑16 ∨ True ∧ (x >>> 9#16 &&& x <<< 8#16 != 0) = true → False

theorem samesign_inverted_proof : samesign_inverted_before ⊑ samesign_inverted_after

theorem main10_logical_thm.extracted_1._3 : ∀ (x : BitVec 32),
  ¬ofBool (x &&& 64#32 == 0#32) = 1#1 → ofBool (x &&& 192#32 == 0#32) = 1#1 → ¬0#1 = 1#1 → 1#32 = 2#32

theorem lshrugt_01_04_exact_thm (e : IntW 4) :
  icmp IntPred.ugt (lshr e (const? 4 1) { «exact» := true }) (const? 4 4) ⊑
    icmp IntPred.ugt e (const? 4 (-8))

theorem nonexact_lshr_eq_noexactlog_proof : nonexact_lshr_eq_noexactlog_before ⊑ nonexact_lshr_eq_noexactlog_after

theorem t8_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬(True ∧ (0#8).ssubOverflow x = true) → ofBool (0#8 - x == x) = ofBool (x == 0#8)

theorem trunc_ashr_trunc_exact_thm.extracted_1._2 : ∀ (x : BitVec 64),
  ¬(True ∧ truncate 32 x >>> 8#32 <<< 8#32 ≠ truncate 32 x ∨ 8#32 ≥ ↑32) →
    ¬(True ∧ x >>> 8#64 <<< 8#64 ≠ x ∨ 8#64 ≥ ↑64) →
      truncate 8 ((truncate 32 x).sshiftRight' 8#32) = truncate 8 (x >>> 8#64)

theorem logical_and_logical_and_icmps_comm2_thm.extracted_1._3 : ∀ (x x_1 x_2 : BitVec 8),
  ¬ofBool (x_2 &&& 1#8 != 0#8) = 1#1 → ¬x ≥ ↑8 → True ∧ 1#8 <<< x >>> x ≠ 1#8 ∨ x ≥ ↑8 → 0#1 = 1#1 → False

@[simp] theorem HVector.denote_nil
    (T : HVector (fun (t : Ctxt d.Ty × List d.Ty) => Com d t.1 .impure t.2) []) :
    HVector.denote T = HVector.nil

theorem e_226 :
    5 *  ~~~(x &&&  ~~~y) - 3 * (x ||| y) - 7 * y - 12 *  ~~~(x ||| y) - 2 *  ~~~(x |||  ~~~y) - 4 * (x &&&  ~~~y) + 5 * (x &&& y) =  - 7 *  ~~~(x &&& y)

theorem test_shift_and_cmp_changed2_thm (e : IntW 8) :
  icmp IntPred.ult (LLVM.and (shl e (const? 8 5)) (const? 8 (-64))) (const? 8 32) ⊑
    icmp IntPred.eq (LLVM.and e (const? 8 6)) (const? 8 0)

theorem test19_commutative2_thm.extracted_1._2 : ∀ (x x_1 : BitVec 32),
  ¬(x ≥ ↑32 ∨ x_1 ≥ ↑32 ∨ x ≥ ↑32 ∨ 1#32 <<< x + (1#32 <<< x_1 &&& 1#32 <<< x) = 0) →
    ¬(True ∧ 1#32 <<< x >>> x ≠ 1#32 ∨
          x ≥ ↑32 ∨ True ∧ 1#32 <<< x_1 >>> x_1 ≠ 1#32 ∨ x_1 ≥ ↑32 ∨ True ∧ 1#32 <<< x >>> x ≠ 1#32 ∨ x ≥ ↑32) →
      x_1 % (1#32 <<< x + (1#32 <<< x_1 &&& 1#32 <<< x)) = x_1 &&& 1#32 <<< x + (1#32 <<< x_1 &&& 1#32 <<< x) + -1#32

theorem cmpeq_xor_cst1_commuted_proof : cmpeq_xor_cst1_commuted_before ⊑ cmpeq_xor_cst1_commuted_after

theorem icmp_sle3_proof : icmp_sle3_before ⊑ icmp_sle3_after

theorem shl1_trunc_sgt4_thm (e : IntW 32) :
  icmp IntPred.sgt (trunc 16 (shl (const? 32 1) e)) (const? 16 4) ⊑
    icmp IntPred.sgt (trunc 16 (shl (const? 32 1) e { «nsw» := false, «nuw» := true })) (const? 16 4)

theorem a_proof : a_before ⊑ a_after

theorem uge_or_not_max_commute_logical_proof : uge_or_not_max_commute_logical_before ⊑ uge_or_not_max_commute_logical_after

theorem test0_thm.extracted_1._1 : ∀ (x : BitVec 39), x &&& 0#39 = 0#39

@[simp]
theorem toBitVec_sub (a b : BitStream) :
    (a - b).toBitVec w = (a.toBitVec w) - (b.toBitVec w)

theorem main7a_logical_thm.extracted_1._9 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 &&& x_1 == x_2) = 1#1 → ofBool (x_2 &&& x_1 != x_2) = 1#1 → 0#1 = 1#1 → 0#32 = zeroExtend 32 1#1

theorem bitwise_or_logical_or_icmps_comm1_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 8),
  ¬x_1 ≥ ↑8 → ofBool (x == 42#8) = 1#1 → ofBool (x_2 &&& 1#8 <<< x_1 == 0#8) ||

theorem and_xor_or1_proof : and_xor_or1_before ⊑ and_xor_or1_after

theorem relabel_unsat_iff_of_not_NonEmpty {aigα : AIG α} {r : α → β} {hidx1} {hidx2}
    (hαNonempty : ¬ Nonempty α) :
    (aigα.relabel r).UnsatAt idx invert hidx1 ↔ aigα.UnsatAt idx invert hidx2

theorem and_ne_with_diff_one_logical_thm (e : IntW 32) :
  select (icmp IntPred.ne e (const? 32 40)) (icmp IntPred.ne e (const? 32 39)) (const? 1 0) ⊑
    icmp IntPred.ult (add e (const? 32 (-41))) (const? 32 (-2))

theorem e_2305 :
    4 * (e ^^^ (d ||| f)) - 6 * (d ||| (e ||| f)) - 11 * (e ^^^  ~~~(d ||| ( ~~~e &&& f))) + 11 * (d ||| ( ~~~e ||| f)) + 11 * (f ^^^ (d &&& e)) + 1 *  ~~~(d ^^^ (e ^^^ f)) - 1 * (e ^^^ (d &&&  ~~~f)) + 5 *  ~~~(d &&&  ~~~e) + 2 * (d &&&  ~~~f) + 1 * ((d &&& f) ^^^ (d ^^^ (e &&& f))) - 11 * ((d ||| e) &&& (d ^^^ (e ^^^ f))) + 4 * (f ^^^  ~~~(d |||  ~~~e)) - 6 * ( ~~~(d |||  ~~~e) |||  ~~~(d ^^^ (e ^^^ f))) + 4 * ((d ||| e) &&&  ~~~(e ^^^ f)) - 1 * ((e &&&  ~~~f) |||  ~~~( ~~~d ||| ( ~~~e &&& f))) + 2 * (f ^^^ (d &&&  ~~~e)) + 4 *  ~~~(e |||  ~~~f) + 1 * (e ^^^  ~~~( ~~~d &&& ( ~~~e &&& f))) - 7 * (f &&& (d ||| e)) - 1 * (f ^^^ ( ~~~d ||| ( ~~~e &&& f))) - 2 * ((d |||  ~~~e) &&& (e ^^^ f)) - 1 * (e ^^^  ~~~( ~~~d &&& ( ~~~e ||| f))) + 1 * (d ^^^ ( ~~~e ||| f)) - 7 * ((d ^^^ e) &&& (d ^^^ f)) + 2 *  ~~~(e &&& f) + 1 *  ~~~e + 7 * (e ^^^  ~~~( ~~~d ||| (e &&& f))) - 1 *  ~~~( ~~~d ||| ( ~~~e &&& f)) - 1 * ( ~~~f ||| (d ^^^ e)) - 6 * (f ^^^  ~~~(d &&& (e &&& f))) + 3 * (d ^^^ f) + 5 * (e ^^^  ~~~(d &&& f)) + 1 * (e ^^^ (d ||| ( ~~~e &&& f))) + 7 *  ~~~(d &&&  ~~~d) - 2 * ((e &&&  ~~~f) ^^^ ( ~~~d ||| (e ^^^ f))) + 4 * e + 7 * (e ^^^  ~~~( ~~~d &&& (e ^^^ f))) - 1 * (e ^^^ (d ||| (e ^^^ f))) - 7 * ((d &&& e) ^^^ (e ||| f)) - 2 * (f ||| (d ^^^ e)) - 7 * (d ||| (e &&& f)) - 2 * (e &&& f) + 1 * (e ||| (d ^^^ f)) - 2 * (d &&&  ~~~e) - 3 * (f &&&  ~~~(d &&&  ~~~e)) - 1 * (f ^^^ ( ~~~d ||| ( ~~~e ||| f))) + 2 * (e ^^^ ( ~~~d &&& (e ^^^ f))) - 7 * (f ||| (d &&&  ~~~e)) - 2 * f + 1 * ((e &&&  ~~~f) |||  ~~~(d ||| ( ~~~e &&& f))) - 2 * (f ^^^ ( ~~~d &&& (e ||| f))) - 11 * ( ~~~f &&&  ~~~(d ^^^ e)) - 11 * (f ^^^ (d &&& (e ||| f))) + 3 * (f ^^^  ~~~(d ||| ( ~~~e &&& f))) - 1 * (e ^^^  ~~~(d &&& ( ~~~e &&& f))) - 3 * ((d ^^^ e) &&&  ~~~(d ^^^ f)) - 1 * ( ~~~d ||| (e &&& f)) - 4 *  ~~~(d ||| (e ||| f)) + 3 *  ~~~(d ||| ( ~~~e ||| f)) + 11 *  ~~~( ~~~d ||| ( ~~~e ||| f)) - 22 * ( ~~~d &&& ( ~~~e &&& f)) + 12 * ( ~~~d &&& (e &&& f)) - 6 * (d &&& ( ~~~e &&& f)) =  - 15 * (d &&& (e &&& f)) + 3 *  ~~~( ~~~d ||| (e ||| f))

theorem both_thm (e e_1 : IntW 8) :
  icmp IntPred.ult (LLVM.xor (shl (const? 8 (-1)) e_1) (const? 8 (-1)))
      (LLVM.xor (shl (const? 8 (-1)) e) (const? 8 (-1))) ⊑
    icmp IntPred.ugt (shl (const? 8 (-1)) e_1 { «nsw» := true, «nuw» := false })
      (shl (const? 8 (-1)) e { «nsw» := true, «nuw» := false })

theorem sub_sext_mask1_trunc_lshr_thm (e : IntW 64) :
  sub (const? 32 10) (sext 32 (LLVM.and (trunc 8 (lshr e (const? 64 15))) (const? 8 1))) ⊑
    zext 32
      (add (trunc 8 (ashr (shl e (const? 64 48)) (const? 64 63)) { «nsw» := true, «nuw» := false }) (const? 8 10)
        { «nsw» := true, «nuw» := false })

theorem test5_commuted_x_y_thm (e e_1 : IntW 64) :
  LLVM.or (LLVM.xor e_1 e) (LLVM.xor e (const? 64 (-1))) ⊑ LLVM.xor (LLVM.and e_1 e) (const? 64 (-1))

theorem icmp_slt_0_or_icmp_sgt_0_i32_proof : icmp_slt_0_or_icmp_sgt_0_i32_before ⊑ icmp_slt_0_or_icmp_sgt_0_i32_after

theorem shl_and_and_fail_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ¬(4#8 ≥ ↑8 ∨ 5#8 ≥ ↑8) → x_1 <<< 4#8 &&& (x <<< 5#8 &&& 88#8) = x_1 <<< 4#8 &&& (x <<< 5#8 &&& 64#8)

theorem test6_thm (e : IntW 32) :
  icmp IntPred.ugt (LLVM.udiv (const? 32 5) e) (const? 32 0) ⊑ icmp IntPred.ult e (const? 32 6)

theorem test2_thm.extracted_1._1 : ∀ (x x_1 : BitVec 999),
  ofBool (x_1 &&& 65280#999 != x &&& 65280#999) = ofBool ((x_1 ^^^ x) &&& 65280#999 != 0#999)

theorem main5_like_thm (e e_1 : IntW 32) :
  select
      (LLVM.and (icmp IntPred.eq (LLVM.and e_1 (const? 32 7)) (const? 32 7))
        (icmp IntPred.eq (LLVM.and e (const? 32 7)) (const? 32 7)))
      (const? 32 0) (const? 32 1) ⊑
    zext 32 (icmp IntPred.ne (LLVM.and (LLVM.and e_1 e) (const? 32 7)) (const? 32 7))

theorem sext_zext_uge_known_nonneg_op0_wide_thm.extracted_1._2 : ∀ (x : BitVec 8) (x_1 : BitVec 16),
  ¬(True ∧ (x &&& 12#8).msb = true) →
    ofBool (zeroExtend 32 (x &&& 12#8) ≤ᵤ signExtend 32 x_1) = ofBool (zeroExtend 16 (x &&& 12#8) ≤ᵤ x_1)

theorem icmp_power2_and_icmp_shifted_mask_8_6_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x <ᵤ 8#32) &&& ofBool (x &&& 6#32 != 6#32) = ofBool (x <ᵤ 6#32)

theorem land_lor_left2_proof : land_lor_left2_before ⊑ land_lor_left2_after

theorem bitwise_or_logical_or_icmps_comm3_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 8),
  ¬x_1 ≥ ↑8 →
    ofBool (x_2 &&& 1#8 == 0#8) = 1#1 →
      ¬(True ∧ 1#8 <<< x_1 >>> x_1 ≠ 1#8 ∨ x_1 ≥ ↑8 ∨ True ∧ 1#8 <<< x_1 >>> x_1 ≠ 1#8 ∨ x_1 ≥ ↑8) →
        ofBool (x_2 &&& (1#8 <<< x_1 ||| 1#8) != 1#8 <<< x_1 ||| 1#8) = 1#1 →
          ofBool (x_2 &&& 1#8 <<< x_1 == 0#8) ||

theorem e_1517 :
     - 11 * ((b &&&  ~~~c) |||  ~~~(a ||| ( ~~~b &&& c))) - 5 * (c ^^^ (a ||| (b &&& c))) + 18 *  ~~~(a ||| (b ||| c)) + 11 *  ~~~(a ||| ( ~~~b ||| c)) + 12 *  ~~~( ~~~a ||| (b ||| c)) + 16 *  ~~~( ~~~a ||| ( ~~~b ||| c)) + 12 * ( ~~~a &&& ( ~~~b &&& c)) + 11 * ( ~~~a &&& (b &&& c)) + 7 * (a &&& ( ~~~b &&& c)) + 7 * (a &&& (b &&& c)) = 7 * (b ^^^  ~~~(a &&& (b &&& c)))

theorem bitwise_and_logical_and_icmps_comm3_thm.extracted_1._8 : ∀ (x x_1 x_2 : BitVec 8),
  ¬x_1 ≥ ↑8 →
    ¬ofBool (x_2 &&& 1#8 != 0#8) = 1#1 →
      ¬(True ∧ 1#8 <<< x_1 >>> x_1 ≠ 1#8 ∨ x_1 ≥ ↑8 ∨ True ∧ 1#8 <<< x_1 >>> x_1 ≠ 1#8 ∨ x_1 ≥ ↑8) →
        ¬ofBool (x_2 &&& (1#8 <<< x_1 ||| 1#8) == 1#8 <<< x_1 ||| 1#8) = 1#1 →
          ofBool (x_2 &&& 1#8 <<< x_1 != 0#8) &&& 0#1 = 0#1

theorem badimm1_thm (e : IntW 16) :
  select (icmp IntPred.eq (ashr (trunc 8 e) (const? 8 7)) (trunc 8 (lshr e (const? 16 9)))) (trunc 8 e)
      (LLVM.xor (trunc 8 (ashr e (const? 16 15))) (const? 8 127)) ⊑
    select
      (icmp IntPred.eq (ashr (trunc 8 e) (const? 8 7))
        (trunc 8 (lshr e (const? 16 9)) { «nsw» := true, «nuw» := true }))
      (trunc 8 e) (select (icmp IntPred.sgt e (const? 16 (-1))) (const? 8 127) (const? 8 (-128)))

theorem PR42691_3_thm (e : IntW 32) :
  LLVM.or (icmp IntPred.sge e (const? 32 0)) (icmp IntPred.eq e (const? 32 (-2147483648))) ⊑
    icmp IntPred.ult e (const? 32 (-2147483647))

theorem udiv_i32_c_proof : udiv_i32_c_before ⊑ udiv_i32_c_after

theorem e_330 :
    1 *  ~~~(x ^^^ y) - 3 * y - 5 * (x ^^^ y) - 7 *  ~~~x + 4 *  ~~~(x ||| y) + 13 *  ~~~(x |||  ~~~y) + 3 * (x &&&  ~~~y) - 1 * (x &&& y) = 1 *  ~~~(x &&& y) - 3 *  ~~~(x &&&  ~~~x)

theorem t1_ult_slt_0_thm.extracted_1._6 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 <ᵤ 65536#32) = 1#1 → ¬ofBool (65535#32 <ₛ x_1) = 1#1 → ofBool (x_1 <ₛ 0#32) = 1#1 → False

theorem test13_proof : test13_before ⊑ test13_after

theorem icmp_ne_mul_nsw_nonequal_proof : icmp_ne_mul_nsw_nonequal_before ⊑ icmp_ne_mul_nsw_nonequal_after

theorem test_invert_demorgan_logical_and_proof : test_invert_demorgan_logical_and_before ⊑ test_invert_demorgan_logical_and_after

theorem ashrsgt_02_01_exact_proof : ashrsgt_02_01_exact_before ⊑ ashrsgt_02_01_exact_after

theorem or_logic_or_logic_and_8_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 1),
  ¬x_2 = 1#1 → ¬x_1 = 1#1 → 0#1 = x_1 ||

theorem or_logic_or_logic_and_2_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 1),
  x_2 ||

theorem test6_proof : test6_before ⊑ test6_after

theorem and_ne_with_one_bit_diff_constants1_logical_proof : and_ne_with_one_bit_diff_constants1_logical_before ⊑ and_ne_with_one_bit_diff_constants1_logical_after

theorem main7b_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 == x_1 &&& x_2) &&& ofBool (x * 42#32 == x_1 &&& x * 42#32) = 1#1 →
    0#32 = zeroExtend 32 (ofBool (x_1 &&& (x_2 ||| x * 42#32) != x_2 ||| x * 42#32))

theorem alive_Select_700 : forall (w : Nat) (A B : Bitvec 1)
, TSSA.eval
  (Op := Op) (e := e)
  (i := TSSAIndex.STMT (UserType.base (BaseType.bitvec 1)))
  [dsl_bb|
  ^bb
  %v0 := unit: ;
  %v1 := op:const (A) %v0;
  %v2 := op:const (B) %v0;
  %v3 := pair:%v1 %v2;
  %v4 := op:icmp slt  w %v3;
  %v5 := triple:%v4 %v1 %v2;
  %v6 := op:select 1 %v5;
  %v7 := pair:%v6 %v2;
  %v8 := op:icmp slt  w %v7;
  %v9 := triple:%v8 %v6 %v2;
  %v10 := op:select 1 %v9
  dsl_ret %v10
  ]  ⊑
  TSSA.eval
  (Op := Op) (e := e)
  (i := TSSAIndex.STMT (UserType.base (BaseType.bitvec 1)))
  [dsl_bb|
  ^bb
  %v0 := unit: ;
  %v1 := op:const (A) %v0;
  %v2 := op:const (B) %v0;
  %v3 := pair:%v1 %v2;
  %v4 := op:icmp slt  1 %v3;
  %v5 := triple:%v4 %v1 %v2;
  %v6 := op:select 1 %v5;
  %v7 := pair:%v6 %v2;
  %v8 := op:icmp slt  1 %v7;
  %v9 := triple:%v4 %v1 %v2;
  %v10 := op:select 1 %v9
  dsl_ret %v10
  ]

theorem test7_logical_thm.extracted_1._3 : ∀ (x : BitVec 32),
  ¬ofBool (x <ₛ 1#32) = 1#1 → ¬ofBool (x == 0#32) = 1#1 → 0#1 = 1#1 → ofBool (-1#32 <ₛ x) = 0#1

theorem mul_select_sext_proof : mul_select_sext_before ⊑ mul_select_sext_after

theorem foo1_and_proof : foo1_and_before ⊑ foo1_and_after

theorem AvoidCollision_setWidth_one {x : BitVec w} :
    x.setWidth 1 = ofBool (x.getLsbD 0)

theorem test2_proof : test2_before ⊑ test2_after

theorem scalar_i8_lshr_and_negC_eq_not_negatedPowerOf2_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ¬x ≥ ↑8 → ofBool (x_1 >>> x &&& BitVec.ofInt 8 (-3) == 0#8) = ofBool (x_1 &&& BitVec.ofInt 8 (-3) <<< x == 0#8)

theorem band_lor_right1_thm (e e_1 : IntW 1) : select e_1 (const? 1 1) (LLVM.and e_1 e) ⊑ e_1

theorem zext_zext_sgt_proof : zext_zext_sgt_before ⊑ zext_zext_sgt_after

theorem not_true_not_thm.extracted_1._2 : ∀ (x : BitVec 1), x ^^^ 1#1 = 1#1 → ¬x = 1#1 → 1#1 = 0#1 ^^^ 1#1

theorem select_shl_icmp_bad_thm.extracted_1._3 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 == 1#32) = 1#1 → ¬x_2 ≥ ↑32 → 1#32 ≥ ↑32 → False

theorem e_1066 :
    5 * (z ||| t) + 2 *  ~~~z + 1 *  ~~~(z &&&  ~~~z) + 11 * z - 3 *  ~~~(z ||| t) - 9 *  ~~~(z |||  ~~~t) - 17 * (z &&& t) = 18 * (z &&&  ~~~t) - 1 * (z ^^^ t)

theorem test_nuw_nsw_and_signed_pred_proof : test_nuw_nsw_and_signed_pred_before ⊑ test_nuw_nsw_and_signed_pred_after

theorem test19_thm (e : IntW 37) :
  icmp IntPred.eq (ashr e (const? 37 2)) (const? 37 0) ⊑ icmp IntPred.ult e (const? 37 4)

theorem bools_multi_uses2_logical_thm.extracted_1._30 : ∀ (x x_1 x_2 : BitVec 1),
  ¬x_2 ^^^ 1#1 = 1#1 → x_2 = 1#1 → 0#1 = 1#1 → ¬x = 1#1 → True → 0#1 + x = 0#1

theorem select_replace_nested_proof : select_replace_nested_before ⊑ select_replace_nested_after

theorem n2_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ¬x_1 ≥ ↑8 → True ∧ ((-1#8) <<< x_1).sshiftRight' x_1 ≠ -1#8 ∨ x_1 ≥ ↑8 → False

theorem main4c_logical_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x &&& 7#32 != 7#32) = 1#1 → True → 0#32 = zeroExtend 32 (ofBool (x &&& 55#32 == 55#32))

theorem PR42691_5_logical_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x <ₛ 1#32) = 1#1 → 1#1 = ofBool (x + BitVec.ofInt 32 (-2147483647) <ᵤ BitVec.ofInt 32 (-2147483646))

theorem test_eq_0_and_15_add_3_proof : test_eq_0_and_15_add_3_before ⊑ test_eq_0_and_15_add_3_after

theorem ne_basic_equal_minus_7_proof : ne_basic_equal_minus_7_before ⊑ ne_basic_equal_minus_7_after

theorem test2_thm.extracted_1._2 : ∀ (x : BitVec 167),
  ¬(9#577 ≥ ↑577 ∨ 8#577 ≥ ↑577) →
    ¬(9#167 ≥ ↑167 ∨ 8#167 ≥ ↑167) →
      truncate 167 (zeroExtend 577 x >>> 9#577 ||| zeroExtend 577 x <<< 8#577) = x >>> 9#167 ||

theorem test7_thm (e e_1 : IntW 32) :
  LLVM.xor (LLVM.and e_1 (const? 32 7)) (LLVM.and e (const? 32 128)) ⊑
    LLVM.or (LLVM.and e_1 (const? 32 7)) (LLVM.and e (const? 32 128)) { «disjoint» := true }

theorem t2_thm.extracted_1._1 : ∀ (x : BitVec 7),
  ¬3#7 ≥ ↑7 → signExtend 16 (truncate 4 (x >>> 3#7)) = signExtend 16 (x.sshiftRight' 3#7)

theorem test_op_op_thm (e e_1 e_2 : IntW 32) :
  select (icmp IntPred.sgt e_2 (const? 32 0)) (sext 32 (icmp IntPred.sgt e_1 (const? 32 0)))
      (sext 32 (icmp IntPred.sgt e (const? 32 0))) ⊑
    sext 32 (icmp IntPred.sgt (select (icmp IntPred.sgt e_2 (const? 32 0)) e_1 e) (const? 32 0))

theorem select_constants_and_icmp_eq0_zero_tval_proof : select_constants_and_icmp_eq0_zero_tval_before ⊑ select_constants_and_icmp_eq0_zero_tval_after

theorem e_1444 :
     - 1 *  ~~~t - 6 *  ~~~(z &&&  ~~~z) + 5 *  ~~~(z ^^^ t) - 7 *  ~~~z - 5 *  ~~~(z &&& t) + 7 * (z ^^^ t) - 1 * (z |||  ~~~t) + 15 *  ~~~(z ||| t) + 11 *  ~~~(z |||  ~~~t) + 2 * (z &&& t) =  - 6 * (z &&&  ~~~t)

theorem sub_mask1_trunc_lshr_thm (e : IntW 64) :
  sub (const? 8 10) (LLVM.and (trunc 8 (lshr e (const? 64 15))) (const? 8 1)) ⊑
    add (trunc 8 (ashr (shl e (const? 64 48)) (const? 64 63)) { «nsw» := true, «nuw» := false }) (const? 8 10)
      { «nsw» := true, «nuw» := false }

theorem PR30273_three_bools_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 1),
  x_2 = 1#1 →
    ¬x_1 = 1#1 →
      ¬(True ∧ (zeroExtend 32 x).saddOverflow 1#32 = true) →
        True ∧ (zeroExtend 32 x).saddOverflow (zeroExtend 32 x_2) = true ∨
            True ∧ (zeroExtend 32 x).uaddOverflow (zeroExtend 32 x_2) = true →
          False

theorem c_1_1_0_thm.extracted_1._1 : ∀ (x x_1 : BitVec 4),
  True ∧ (x_1 &&& BitVec.ofInt 4 (-2) &&& (x &&& 1#4) != 0) = true → False

theorem trunc_sandwich_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬(28#32 ≥ ↑32 ∨ 2#12 ≥ ↑12) →
    ¬(30#32 ≥ ↑32 ∨
          True ∧ signExtend 32 (truncate 12 (x >>> 30#32)) ≠ x >>> 30#32 ∨
            True ∧ zeroExtend 32 (truncate 12 (x >>> 30#32)) ≠ x >>> 30#32) →
      truncate 12 (x >>> 28#32) >>> 2#12 = truncate 12 (x >>> 30#32)

@[simp]
lemma bitVecToFinFun_rinv (c : carry → Bool) [FinEnum carry]:
    bitVecToFinFun (finFunToBitVec c) = c

theorem differentconsts_thm.extracted_1._5 : ∀ (x : BitVec 32),
  ¬ofBool (x + 16#32 <ᵤ 144#32) = 1#1 →
    ¬ofBool (x <ₛ 128#32) = 1#1 →
      ¬ofBool (127#32 <ₛ x) = 1#1 → ofBool (x <ₛ BitVec.ofInt 32 (-16)) = 1#1 → -1#16 = 256#16

theorem main10_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x &&& 64#32 == 0#32) &&& ofBool (0#8 ≤ₛ truncate 8 x) = 1#1 →
    ¬ofBool (x &&& 192#32 == 0#32) = 1#1 → 2#32 = 1#32

theorem xor_icmp_true_signed_proof : xor_icmp_true_signed_before ⊑ xor_icmp_true_signed_after

@[simp] theorem negAux_zero (x : BitStream) : x.negAux 0 = (x 0, ! (x 0))

theorem and_or2_wrong_operand_thm (e e_1 e_2 e_3 : IntW 1) :
  select (LLVM.and (LLVM.xor e_3 (const? 1 1)) e_2) e_1 e ⊑
    select (LLVM.and e_2 (LLVM.xor e_3 (const? 1 1))) e_1 e

theorem e_433 :
    1 * y + 2 *  ~~~(x &&& y) - 1 * (x |||  ~~~y) - 7 *  ~~~(x &&&  ~~~y) + 1 *  ~~~(x &&&  ~~~x) + 5 *  ~~~(x ||| y) + 3 *  ~~~(x |||  ~~~y) - 2 * (x &&&  ~~~y) =  - 6 * (x &&& y)

theorem test_and3_logical_proof : test_and3_logical_before ⊑ test_and3_logical_after

theorem test_and4_logical_thm (e e_1 : IntW 32) :
  select (icmp IntPred.sge (LLVM.and e_1 (const? 32 2147483647)) e) (icmp IntPred.sge e (const? 32 0))
      (const? 1 0) ⊑
    icmp IntPred.ule e (LLVM.and e_1 (const? 32 2147483647))

theorem trunc_shl_33_i32_i64_thm.extracted_1._1 : ∀ (x : BitVec 64), ¬33#64 ≥ ↑64 → truncate 32 (x <<< 33#64) = 0#32

theorem zext_sext_add_icmp_slt_1_rhs_not_const_thm (e : IntW 8) (e_1 e_2 : IntW 1) :
  icmp IntPred.slt (add (zext 8 e_2) (sext 8 e_1)) e ⊑
    icmp IntPred.slt (add (zext 8 e_2) (sext 8 e_1) { «nsw» := true, «nuw» := false }) e

theorem bools2_logical_commute0_thm.extracted_1._13 : ∀ (x x_1 x_2 : BitVec 1),
  x_2 = 1#1 → x_2 ^^^ 1#1 = 1#1 → x_1 = 1#1 → 1#1 = x_1

theorem and_and_logic_or_1_thm.extracted_1._1 : ∀ (x x_1 : BitVec 1), x_1 &&& x = 1#1 → x = 1#1 → 1#1 = x_1 &&& 1#1

theorem select_icmp_ne_0_and_32_or_4096_thm (e e_1 : IntW 32) :
  select (icmp IntPred.ne (const? 32 0) (LLVM.and e_1 (const? 32 32))) e (LLVM.or e (const? 32 4096)) ⊑
    LLVM.or e (LLVM.xor (LLVM.and (shl e_1 (const? 32 7)) (const? 32 4096)) (const? 32 4096))

theorem shl_add_and_fail_mismatch_shift_proof : shl_add_and_fail_mismatch_shift_before ⊑ shl_add_and_fail_mismatch_shift_after

theorem lshrult_02_14_exact_proof : lshrult_02_14_exact_before ⊑ lshrult_02_14_exact_after

theorem and_xor_common_op_commute2_thm.extracted_1._2 : ∀ (x x_1 : BitVec 32),
  ¬(x_1 = 0 ∨ x = 0 ∨ x = 0) →
    ¬(x = 0 ∨ x_1 = 0) → (43#32 / x_1 ^^^ 42#32 / x) &&& 42#32 / x = 42#32 / x &&& (43#32 / x_1 ^^^ -1#32)

theorem MatchVar.isMonotone_unifyVars  : IsMonotone (unifyVars w v)

theorem select_constants_and_icmp_eq0_thm (e e_1 : IntW 1) :
  icmp IntPred.eq (LLVM.and (select e_1 (const? 8 2) (const? 8 1)) (select e (const? 8 2) (const? 8 1)))
      (const? 8 0) ⊑
    LLVM.xor e_1 e

theorem xor_i8_to_i64_shl_save_and_ne_thm.extracted_1._1 : ∀ (x : BitVec 64) (x_1 : BitVec 8),
  ofBool (x_1 &&& 1#8 != 0#8) = 1#1 → 63#64 ≥ ↑64 → False

theorem test_ashr_exact__exact_is_safe_thm (e : IntW 64) (e_1 : IntW 32) :
  select (icmp IntPred.eq (LLVM.and e_1 (const? 32 (-2147483588))) (const? 32 0)) e
      (ashr e (zext 64 (ashr (LLVM.and e_1 (const? 32 (-2147483588))) (const? 32 2) { «exact» := true }))) ⊑
    ashr e (zext 64 (LLVM.and (ashr e_1 (const? 32 2)) (const? 32 (-536870897))) { «nneg» := true })

theorem evalWith_congrEnv {p : FSM arity}
    {x y : arity → BitStream} {carryState : p.α → Bool} {n : Nat}
    (h : ∀ a i, i ≤ n → x a i = y a i) :
    p.evalWith carryState x n = p.evalWith carryState y n

theorem e_1067 :
     - 1 *  ~~~(z &&&  ~~~t) - 2 * (z ^^^ t) + 4 * z + 2 *  ~~~t + 3 *  ~~~(z |||  ~~~t) - 4 * (z &&&  ~~~t) - 3 * (z &&& t) = 1 *  ~~~(z ||| t)

theorem sext_thm.extracted_1._1 : ∀ (x : BitVec 16),
  (zeroExtend 32 x ^^^ 32768#32) + BitVec.ofInt 32 (-32768) = signExtend 32 x

theorem ashr_ult_3_thm (e : IntW 4) :
  icmp IntPred.ult (ashr e (const? 4 1)) (const? 4 3) ⊑ icmp IntPred.ult e (const? 4 6)

theorem udiv_shl_mul_nuw_swap_thm (e e_1 e_2 : IntW 5) :
  LLVM.udiv (shl e_2 e_1 { «nsw» := false, «nuw» := true }) (mul e e_2 { «nsw» := false, «nuw» := true }) ⊑
    LLVM.udiv (shl (const? 5 1) e_1 { «nsw» := false, «nuw» := true }) e

theorem sgt_swap_or_not_min_commute_thm (e e_1 : IntW 8) :
  LLVM.or (icmp IntPred.ne e_1 (const? 8 (-128))) (icmp IntPred.slt e e_1) ⊑
    icmp IntPred.ne e_1 (const? 8 (-128))

theorem positive_samevar_shlnuw_thm.extracted_1._2 : ∀ (x x_1 : BitVec 8),
  ¬(x ≥ ↑8 ∨ True ∧ x_1.sshiftRight' x <<< x >>> x ≠ x_1.sshiftRight' x ∨ x ≥ ↑8) →
    ¬(True ∧ ((-1#8) <<< x).sshiftRight' x ≠ -1#8 ∨ x ≥ ↑8) → x_1.sshiftRight' x <<< x = (-1#8) <<< x &&& x_1

theorem test_sub_0_Y_eq_0_thm (e : IntW 8) :
  icmp IntPred.eq (sub (const? 8 0) e) (const? 8 0) ⊑ icmp IntPred.eq e (const? 8 0)

theorem e_1108 :
    1 * (z |||  ~~~t) - 1 *  ~~~z + 4 *  ~~~t - 1 *  ~~~(z ^^^ t) - 3 *  ~~~(z ||| t) - 1 *  ~~~(z |||  ~~~t) - 7 * (z &&&  ~~~t) - 2 * (z &&& t) =  - 2 * (z ||| t)

theorem alive_AddSub_1156  (w : Nat)   : alive_AddSub_1156_src w  ⊑ alive_AddSub_1156_tgt w

theorem urem_i32_c_multiuse_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬10#32 = 0 →
    10#32 = 0 ∨
        True ∧ (zeroExtend 32 x % 10#32).saddOverflow (zeroExtend 32 x) = true ∨
          True ∧ (zeroExtend 32 x % 10#32).uaddOverflow (zeroExtend 32 x) = true →
      False

theorem lshrult_03_10_proof : lshrult_03_10_before ⊑ lshrult_03_10_after

theorem e_1571 :
    11 *  ~~~(b |||  ~~~c) - 6 * ((a &&& c) ^^^  ~~~(b &&&  ~~~c)) + 5 * ( ~~~b &&& (a ^^^ c)) + 2 *  ~~~(a ||| ( ~~~b &&& c)) - 2 * (c ^^^ (a ||| (b ||| c))) + 1 * (a ||| (b ^^^ c)) + 1 * ( ~~~(a &&&  ~~~b) &&& (b ^^^ c)) + 7 *  ~~~( ~~~a ||| (b &&& c)) - 6 * ((b &&&  ~~~c) ^^^ ( ~~~a ||| (b ^^^ c))) + 9 *  ~~~(a ||| (b ||| c)) - 2 *  ~~~(a ||| ( ~~~b ||| c)) - 8 *  ~~~( ~~~a ||| ( ~~~b ||| c)) - 7 * ( ~~~a &&& ( ~~~b &&& c)) + 9 * ( ~~~a &&& (b &&& c)) - 14 * (a &&& ( ~~~b &&& c)) - 1 * (a &&& (b &&& c)) =  - 1 * ( ~~~(a ||| b) |||  ~~~(a ^^^ (b ^^^ c))) + 5 *  ~~~( ~~~a ||| (b ||| c))

theorem gt_unsigned_to_small_unsigned_thm (e : IntW 8) :
  icmp IntPred.ugt (zext 32 e) (const? 32 17) ⊑ icmp IntPred.ugt e (const? 8 17)

theorem t4_ugt_slt_65536_thm.extracted_1._8 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (65535#32 <ᵤ x_1) = 1#1 → ofBool (65535#32 <ₛ x_1) = 1#1 → x_1 = x

theorem not_or_or_and_not_and_and_commute2_or_thm (e e_1 e_2 : IntW 32) :
  LLVM.and (LLVM.or (LLVM.or e_2 e_1) (LLVM.xor e (const? 32 (-1))))
      (LLVM.xor (LLVM.and (LLVM.and e_2 e) e_1) (const? 32 (-1))) ⊑
    LLVM.or (LLVM.xor e_2 e_1) (LLVM.xor e (const? 32 (-1)))

@[simp]
lemma bv_to_set_or {m : CNFA n} (x y : BitVec m.m.stateMax) :
    (s ∈ bv_to_set (x ||| y)) ↔ (s ∈ bv_to_set x ∨ s ∈ bv_to_set y)

theorem eq4 (w : Nat) (a b : BitVec w) (h : a &&& b = 0#w) : a + b = a ||

theorem e_888 :
    3 * t + 2 *  ~~~t - 5 *  ~~~(z ||| t) - 4 *  ~~~(z |||  ~~~t) - 5 * (z &&&  ~~~t) - 6 * (z &&& t) =  - 1 *  ~~~(z &&&  ~~~z) - 2 * (z |||  ~~~t)

theorem scalar_i8_shl_ult_const_3_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬7#8 ≥ ↑8 → ofBool (x <<< 7#8 <ᵤ 64#8) = ofBool (x &&& 1#8 == 0#8)

theorem sle_swap_or_not_min_proof : sle_swap_or_not_min_before ⊑ sle_swap_or_not_min_after

theorem src_is_notmask_neg_p2_fail_not_invertable_proof : src_is_notmask_neg_p2_fail_not_invertable_before ⊑ src_is_notmask_neg_p2_fail_not_invertable_after

theorem positive_biggerashr_shlnsw_proof : positive_biggerashr_shlnsw_before ⊑ positive_biggerashr_shlnsw_after

theorem trunc_signed_both_thm.extracted_1._1 : ∀ (x x_1 : BitVec 16),
  ¬(True ∧ signExtend 16 (truncate 8 x_1) ≠ x_1 ∨
        True ∧ zeroExtend 16 (truncate 8 x_1) ≠ x_1 ∨
          True ∧ signExtend 16 (truncate 8 x) ≠ x ∨ True ∧ zeroExtend 16 (truncate 8 x) ≠ x) →
    ofBool (truncate 8 x_1 <ₛ truncate 8 x) = ofBool (x_1 <ₛ x)

theorem icmp_slt_0_or_icmp_eq_100_i32_fail_proof : icmp_slt_0_or_icmp_eq_100_i32_fail_before ⊑ icmp_slt_0_or_icmp_eq_100_i32_fail_after

theorem sext_sext_ult_thm (e e_1 : IntW 8) :
  icmp IntPred.ult (sext 32 e_1) (sext 32 e) ⊑ icmp IntPred.ult e_1 e

theorem icmp_shl_ne_2_proof : icmp_shl_ne_2_before ⊑ icmp_shl_ne_2_after

theorem main4b_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x &&& 7#32 == 7#32) &&& ofBool (x &&& 16#32 != 0#32) = 1#1 →
    0#32 = zeroExtend 32 (ofBool (x &&& 23#32 != 23#32))

theorem test10_thm.extracted_1._1 : ∀ (x : BitVec 16), truncate 16 (signExtend 32 x) = x

theorem test4_proof : test4_before ⊑ test4_after

theorem band_land_left2_thm.extracted_1._2 : ∀ (x x_1 : BitVec 1), ¬x_1 &&& x = 1#1 → 0#1 = x_1 &&& x

theorem e_2411 :
    3 * (e ^^^  ~~~( ~~~d ||| (e &&& f))) - 3 * (e &&& (d |||  ~~~f)) - 6 * (f &&&  ~~~(d &&& e)) + 11 * (f ^^^ (d ||| (e &&& f))) - 1 * (e ^^^ (d &&& (e ||| f))) + 4 *  ~~~e - 3 * (f ^^^  ~~~(d ||| (e &&& f))) - 1 * (e ^^^  ~~~( ~~~d ||| (e ^^^ f))) + 5 * ( ~~~d ||| (e &&& f)) - 5 *  ~~~(d ||| ( ~~~e &&& f)) + 7 * (f ^^^  ~~~(d &&& (e &&& f))) - 1 * (e ^^^  ~~~( ~~~d &&& (e &&& f))) - 2 *  ~~~(d |||  ~~~f) + 11 * (e ||| f) - 1 * (d |||  ~~~f) + 2 * ((e &&& f) ^^^  ~~~(d &&& (e ^^^ f))) + 2 * (d &&& f) - 7 * ((d ||| e) &&&  ~~~(d ^^^ (e ^^^ f))) - 1 * ( ~~~f &&&  ~~~(d ^^^ e)) - 1 * (f ^^^  ~~~( ~~~d &&& (e &&& f))) + 7 *  ~~~(d ||| (e ^^^ f)) - 6 * ( ~~~e |||  ~~~(d ^^^ f)) - 1 * (e ^^^ (d |||  ~~~f)) - 5 * d - 11 * (e ^^^ (d ||| (e &&& f))) - 1 * (f ^^^  ~~~( ~~~d ||| (e &&& f))) - 2 *  ~~~(e |||  ~~~f) + 11 * (f ^^^  ~~~(d ||| ( ~~~e &&& f))) + 2 * f - 5 * (f ^^^  ~~~( ~~~d ||| ( ~~~e &&& f))) + 1 * ((d &&& e) |||  ~~~(d ^^^ (e ^^^ f))) - 2 * ((d ^^^ e) &&&  ~~~(d ^^^ f)) + 4 * (e ^^^  ~~~(d &&& (e &&& f))) + 4 *  ~~~( ~~~d &&& ( ~~~e &&& f)) - 2 * ((d &&& e) ^^^  ~~~(d ^^^ ( ~~~e ||| f))) - 7 *  ~~~(d &&& (e ^^^ f)) + 1 * ( ~~~e &&&  ~~~(d ^^^ f)) + 3 *  ~~~(d ^^^ (e ^^^ f)) + 2 * (e |||  ~~~(d |||  ~~~f)) + 1 * (e ^^^  ~~~(d &&& (e ||| f))) - 11 * (e ^^^ ( ~~~d ||| (e ||| f))) + 3 *  ~~~( ~~~d &&& (e ^^^ f)) - 6 * (e ^^^ f) + 2 * ((e &&& f) ||| ( ~~~d &&& (e ||| f))) - 6 * (e ^^^ (d &&& (e ^^^ f))) - 2 * ((e &&&  ~~~f) |||  ~~~( ~~~d ||| ( ~~~e &&& f))) - 7 * (e ^^^  ~~~(d &&& (e ^^^ f))) + 2 *  ~~~(e ||| f) + 1 * (d ^^^ ( ~~~e ||| f)) - 7 * (e ^^^ ( ~~~d ||| ( ~~~e ||| f))) - 5 * (f ^^^ ( ~~~d &&& ( ~~~e ||| f))) - 11 * (d &&& ( ~~~e ||| f)) + 5 * (f |||  ~~~(d ||| e)) - 5 * (e &&&  ~~~(d &&& f)) + 7 * ((d &&& f) ||| (e &&&  ~~~f)) + 1 * ( ~~~(d |||  ~~~e) ||| (e ^^^ f)) + 2 * (f ^^^ (d &&& (e ||| f))) - 2 * ((d &&& f) ^^^  ~~~(e &&&  ~~~f)) + 7 * (d ^^^ e) + 3 * (e ||| (d ^^^ f)) + 1 *  ~~~(d &&&  ~~~f) - 3 * (e &&&  ~~~(d ^^^ f)) - 5 * (f ^^^  ~~~(d ||| e)) + 1 * (f ^^^ ( ~~~d ||| (e &&& f))) - 11 * ( ~~~d &&& (e ||| f)) + 1 *  ~~~(d ||| (e ||| f)) + 10 *  ~~~(d ||| ( ~~~e ||| f)) + 20 *  ~~~( ~~~d ||| (e ||| f)) - 2 *  ~~~( ~~~d ||| ( ~~~e ||| f)) + 6 * ( ~~~d &&& ( ~~~e &&& f)) + 8 * ( ~~~d &&& (e &&& f)) + 33 * (d &&& ( ~~~e &&& f)) - 16 * (d &&& (e &&& f)) = 1 * ((e &&&  ~~~f) |||  ~~~(d ||| ( ~~~e &&& f)))

theorem sub_to_xor_or_commuted_proof : sub_to_xor_or_commuted_before ⊑ sub_to_xor_or_commuted_after

theorem icmp_sle5_proof : icmp_sle5_before ⊑ icmp_sle5_after

theorem udiv_exact_ne_30_no_of_thm (e e_1 : IntW 8) :
  icmp IntPred.ne (LLVM.udiv e_1 (LLVM.and e (const? 8 7)) { «exact» := true }) (const? 8 30) ⊑
    icmp IntPred.ne (mul (LLVM.and e (const? 8 7)) (const? 8 30) { «nsw» := false, «nuw» := true }) e_1

theorem test2_proof : test2_before ⊑ test2_after

private theorem BitVec.signExtend_sle_signExtend_of_sle {x y : BitVec w}
    {N : Nat} (hN : w ≤ N)
    (h : x.sle y) : (x.signExtend N).sle (y.signExtend N)

theorem sub_shl_same_amount_partial_nsw2_proof : sub_shl_same_amount_partial_nsw2_before ⊑ sub_shl_same_amount_partial_nsw2_after

theorem main5c_like_logical_thm.extracted_1._7 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 &&& 7#32 != 7#32) = 1#1 → ¬ofBool (x_1 &&& 7#32 == 7#32) = 1#1 → True → 0#32 = zeroExtend 32 0#1

theorem narrow_zext_and_proof : narrow_zext_and_before ⊑ narrow_zext_and_after

theorem not_thm.extracted_1._1 : ∀ (x : BitVec 8), ¬42#8 = 0 → (-1#8 ^^^ x) * (x / 42#8) = x / 42#8 * (x ^^^ -1#8)

@[simp]
lemma complete_stepSet {M : NFA α σ} (q : σ) :
    (∀ q, (Sum.inl q ∈ S1) ↔ S2 q) →
    ((.inl q ∈ M.complete.stepSet S1 a) ↔ q ∈ M.stepSet S2 a)

theorem e_2449 :
     - 1 * (e ^^^  ~~~(d &&& (e ||| f))) - 5 * ((d &&& f) ^^^  ~~~(d ^^^ (e &&& f))) - 5 * ((d &&& e) ^^^  ~~~(d ^^^ (e &&& f))) + 1 * ( ~~~d ||| (e ||| f)) + 1 * ( ~~~e |||  ~~~(d ^^^ f)) + 5 * (e &&&  ~~~(d &&&  ~~~f)) + 1 * (e ^^^  ~~~( ~~~d &&& ( ~~~e ||| f))) - 2 * (e &&& (d ^^^ f)) + 1 * ( ~~~d ||| (e &&& f)) - 1 * (d ||| e) - 11 * (d &&& f) - 1 * (e ^^^  ~~~(d &&& ( ~~~e ||| f))) + 1 * ((e &&&  ~~~f) ^^^ ( ~~~d ||| (e ^^^ f))) + 3 * (f ^^^ ( ~~~d &&& ( ~~~e ||| f))) - 2 *  ~~~f + 11 *  ~~~(d &&& f) + 5 * ((d &&& f) ||| (e &&&  ~~~f)) + 4 * (e ^^^  ~~~(d |||  ~~~f)) + 5 * ((d &&& e) |||  ~~~(d ^^^ (e ^^^ f))) - 1 * ((d |||  ~~~e) &&&  ~~~(e ^^^ f)) - 7 * (d ^^^ (e &&& f)) + 11 * (e ^^^  ~~~(d &&& f)) - 7 * (e ^^^ (d ||| (e ||| f))) + 2 * (d &&&  ~~~f) + 4 *  ~~~( ~~~d ||| (e ^^^ f)) + 11 * (d ^^^ f) - 1 * (e &&& (d |||  ~~~f)) + 3 *  ~~~( ~~~d &&& (e ^^^ f)) - 11 * (e ^^^  ~~~(d ||| f)) - 1 * (d ||| (e ||| f)) - 1 * (e ^^^  ~~~(d &&& (e &&& f))) - 1 * (f ^^^  ~~~( ~~~d &&& (e &&& f))) - 1 * e + 1 * ( ~~~(d ||| e) |||  ~~~(d ^^^ (e ^^^ f))) + 2 * (d ^^^ (e ||| f)) + 2 * (e ||| (d &&& f)) + 1 * ((d &&& e) ||| (d ^^^ (e ^^^ f))) + 4 * ( ~~~(d &&&  ~~~e) &&& (d ^^^ (e ^^^ f))) + 7 * ((d &&& f) ^^^ (d ^^^ (e &&& f))) + 2 * (d ^^^ (e ^^^ f)) + 2 * ((d ^^^ e) &&& (d ^^^ f)) - 1 * (f ^^^ (d ||| (e &&& f))) + 7 * ((d &&& e) |||  ~~~(e ^^^ f)) + 5 * (e ^^^ ( ~~~d &&& (e ||| f))) - 3 *  ~~~(e ^^^ f) - 6 * (e ^^^ ( ~~~d ||| (e &&& f))) - 6 * ((d &&&  ~~~e) |||  ~~~(e ^^^ f)) - 7 * (d ||| ( ~~~e ||| f)) - 6 * (e ^^^ ( ~~~d &&& ( ~~~e ||| f))) - 1 * (e ^^^  ~~~( ~~~d ||| ( ~~~e &&& f))) - 3 * (e |||  ~~~(d ||| f)) + 5 * (f ||| (d &&&  ~~~e)) - 6 * (e ||| f) - 6 *  ~~~(d ||| f) + 4 * ((d ^^^ e) &&&  ~~~(d ^^^ f)) - 11 * (d &&& (e ||| f)) - 11 * (d ^^^ ( ~~~e &&& f)) - 7 * (d &&& ( ~~~e ||| f)) - 7 * (f &&&  ~~~(d &&& e)) - 2 * ( ~~~d ||| ( ~~~e ||| f)) + 1 * ( ~~~f &&&  ~~~(d ^^^ e)) - 1 * (f ^^^  ~~~( ~~~d &&& ( ~~~e ||| f))) + 7 * ((d &&& e) ||| (e ^^^ f)) + 7 * (d ^^^ ( ~~~e ||| f)) + 2 * ((e &&&  ~~~f) ^^^ (d ||| (e ^^^ f))) + 2 * (e ^^^  ~~~(d &&& ( ~~~e &&& f))) + 1 * ((d |||  ~~~e) &&&  ~~~(d ^^^ (e ^^^ f))) + 7 * (f ^^^  ~~~(d &&& (e ||| f))) - 5 * ( ~~~(d |||  ~~~e) ||| (d ^^^ (e ^^^ f))) + 7 *  ~~~(d ||| (e ||| f)) - 7 *  ~~~(d ||| ( ~~~e ||| f)) - 4 *  ~~~( ~~~d ||| (e ||| f)) + 18 *  ~~~( ~~~d ||| ( ~~~e ||| f)) - 6 * ( ~~~d &&& ( ~~~e &&& f)) - 7 * ( ~~~d &&& (e &&& f)) + 41 * (d &&& ( ~~~e &&& f)) + 8 * (d &&& (e &&& f)) = 3 * (e ^^^  ~~~(d ||| (e ^^^ f))) + 11 *  ~~~( ~~~d ||| (e &&& f))

theorem test13_proof : test13_before ⊑ test13_after

theorem select_icmp_ne_0_and_4096_or_4096_thm.extracted_1._2 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (0#32 != x_1 &&& 4096#32) = 1#1 → x ||

theorem select_replace_udiv_non_speculatable_proof : select_replace_udiv_non_speculatable_before ⊑ select_replace_udiv_non_speculatable_after

theorem src_x_and_nmask_slt_thm (e e_1 : IntW 8) :
  icmp IntPred.slt (LLVM.and e_1 (shl (const? 8 (-1)) e)) (shl (const? 8 (-1)) e) ⊑
    icmp IntPred.sgt (shl (const? 8 (-1)) e { «nsw» := true, «nuw» := false }) e_1

theorem lshrult_01_15_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 1#4 <<< 1#4 ≠ x ∨ 1#4 ≥ ↑4) → ofBool (x >>> 1#4 <ᵤ -1#4) = 1#1

theorem mul_selectp2_x_propegate_nuw_thm (e : IntW 8) (e_1 : IntW 1) :
  mul (select e_1 (const? 8 2) (const? 8 4)) e { «nsw» := true, «nuw» := true } ⊑
    shl e (select e_1 (const? 8 1) (const? 8 2)) { «nsw» := false, «nuw» := true }

theorem main7a_logical_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 &&& x_1 == x_2) = 1#1 →
    ofBool (x_2 &&& x_1 != x_2) = 1#1 → ofBool (x &&& x_1 == x) = 1#1 → 0#32 = zeroExtend 32 1#1

@[bv_automata_preprocess] theorem BitVec.one_mul (x : BitVec w) : 1#w * x = x

theorem shl_mul_mask_wrong_mul_const_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬8#32 ≥ ↑32 →
    True ∧ (x &&& 255#32).smulOverflow 65535#32 = true ∨
        True ∧ (x &&& 255#32).umulOverflow 65535#32 = true ∨
          True ∧ ((x &&& 255#32) <<< 8#32).sshiftRight' 8#32 ≠ x &&& 255#32 ∨
            True ∧ (x &&& 255#32) <<< 8#32 >>> 8#32 ≠ x &&& 255#32 ∨ 8#32 ≥ ↑32 →
      False

theorem select_1_thm.extracted_1._12 : ∀ (x : BitVec 8) (x_1 : BitVec 1) (x_2 x_3 : BitVec 8) (x_4 : BitVec 1),
  ¬x_4 = 1#1 → ¬x_1 = 1#1 → x ^^^ 123#8 ^^^ -1#8 = x ^^^ BitVec.ofInt 8 (-124)

theorem testi64i32_ne_thm.extracted_1._1 : ∀ (x : BitVec 64),
  ¬(31#32 ≥ ↑32 ∨ 32#64 ≥ ↑64) →
    ofBool ((truncate 32 x).sshiftRight' 31#32 != truncate 32 (x >>> 32#64)) =
      ofBool (x + BitVec.ofInt 64 (-2147483648) <ᵤ BitVec.ofInt 64 (-4294967296))

theorem shl_lshr_pow2_not_const_case2_proof : shl_lshr_pow2_not_const_case2_before ⊑ shl_lshr_pow2_not_const_case2_after

theorem add_shl_same_amount_partial_nuw2_thm (e e_1 e_2 : IntW 6) :
  add (shl e_2 e_1 { «nsw» := false, «nuw» := true }) (shl e e_1) { «nsw» := false, «nuw» := true } ⊑
    shl (add e_2 e) e_1

theorem max_sub_uge_c32_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬ofBool (x ≤ᵤ 2#32) = 1#1 → ofBool (x <ᵤ 3#32) = 1#1 → 0#32 = x + BitVec.ofInt 32 (-2)

theorem reduce_xor_common_op_commute1_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 4),
  x_2 ^^^ x_1 ^^^ x ||

theorem sel_false_val_is_a_masked_lshr_of_true_val2_thm.extracted_1._2 : ∀ (x : BitVec 64) (x_1 : BitVec 32),
  ¬2#32 ≥ ↑32 →
    ofBool ((x_1 &&& 60#32) >>> 2#32 == 0#32) = 1#1 →
      ¬(2#32 ≥ ↑32 ∨ True ∧ (x_1 >>> 2#32 &&& 15#32).msb = true ∨ zeroExtend 64 (x_1 >>> 2#32 &&& 15#32) ≥ ↑64) →
        x = x.sshiftRight' (zeroExtend 64 (x_1 >>> 2#32 &&& 15#32))

theorem main7c_logical_thm.extracted_1._9 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 == x_2 &&& x_1) = 1#1 → ofBool (x_2 != x_2 &&& x_1) = 1#1 → 0#1 = 1#1 → 0#32 = zeroExtend 32 1#1

theorem t12_shift_of_const0_proof : t12_shift_of_const0_before ⊑ t12_shift_of_const0_after

theorem ashrsgt_02_08_exact_proof : ashrsgt_02_08_exact_before ⊑ ashrsgt_02_08_exact_after

theorem pv_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ¬x_1 ≥ ↑8 → ofBool ((-1#8) >>> x_1 &&& x != x) = ofBool ((-1#8) >>> x_1 <ᵤ x)

theorem sle_or_min_commute_thm (e e_1 : IntW 8) :
  LLVM.or (icmp IntPred.eq e_1 (const? 8 (-128))) (icmp IntPred.sle e_1 e) ⊑
    icmp IntPred.sle e_1 e

theorem not_and_and_or_not_or_or_commute1_and_proof : not_and_and_or_not_or_or_commute1_and_before ⊑ not_and_and_or_not_or_or_commute1_and_after

theorem main1_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬ofBool (x &&& 1#32 != 0#32) &&& ofBool (x &&& 2#32 != 0#32) = 1#1 →
    ofBool (x &&& 3#32 == 3#32) = 1#1 → 1#32 = 2#32

theorem bv_276_2 :
    ∀ (e e_1 : LLVM.IntW 5), LLVM.mul (LLVM.udiv e_1 e) (LLVM.sub (LLVM.const? 5 0) e) ⊑ LLVM.sub (LLVM.urem e_1 e) e_1

theorem lsb_mask_sign_zext_commuted_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬31#32 ≥ ↑32 → ((x ^^^ -1#32) &&& x + -1#32) >>> 31#32 = zeroExtend 32 (ofBool (x == 0#32))

theorem test2_proof : test2_before ⊑ test2_after

theorem or_nosignbit_lshr_proof : or_nosignbit_lshr_before ⊑ or_nosignbit_lshr_after

theorem ashrslt_01_15_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬1#4 ≥ ↑4 → ofBool (x.sshiftRight' 1#4 <ₛ -1#4) = ofBool (x <ₛ BitVec.ofInt 4 (-2))

theorem test_invert_demorgan_or3_thm (e e_1 : IntW 32) :
  LLVM.xor
      (LLVM.or
        (LLVM.or
          (LLVM.or (icmp IntPred.eq e_1 (const? 32 178206))
            (icmp IntPred.ult (add e (const? 32 (-195102))) (const? 32 1506)))
          (icmp IntPred.ult (add e (const? 32 (-201547))) (const? 32 716213)))
        (icmp IntPred.ult (add e (const? 32 (-918000))) (const? 32 196112)))
      (const? 1 1) ⊑
    LLVM.and
      (LLVM.and
        (LLVM.and (icmp IntPred.ne e_1 (const? 32 178206))
          (icmp IntPred.ult (add e (const? 32 (-196608))) (const? 32 (-1506))))
        (icmp IntPred.ult (add e (const? 32 (-917760))) (const? 32 (-716213))))
      (icmp IntPred.ult (add e (const? 32 (-1114112))) (const? 32 (-196112)))

theorem bools_multi_uses2_logical_thm.extracted_1._33 : ∀ (x x_1 x_2 : BitVec 1),
  ¬x_2 ^^^ 1#1 = 1#1 → ¬x_2 = 1#1 → 0#1 = 1#1 → x_1 = 1#1 → True → 0#1 + 0#1 = 0#1 ^^^ 0#1

theorem not_signbit_zext_thm.extracted_1._1 : ∀ (x : BitVec 16),
  ¬15#16 ≥ ↑16 → zeroExtend 32 ((x ^^^ -1#16) >>> 15#16) = zeroExtend 32 (ofBool (-1#16 <ₛ x))

theorem uge_umin3_proof : uge_umin3_before ⊑ uge_umin3_after

theorem ne_nuw_rem_zero_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬(True ∧ x.umulOverflow 5#8 = true) → ofBool (x * 5#8 != BitVec.ofInt 8 (-126)) = ofBool (x != 26#8)

theorem not_not_false_proof : not_not_false_before ⊑ not_not_false_after

theorem ne_21_logical_thm (e e_1 : IntW 32) :
  select (icmp IntPred.ne (trunc 8 (lshr e_1 (const? 32 16))) (trunc 8 (lshr e (const? 32 16)))) (const? 1 1)
      (icmp IntPred.ne (trunc 8 (lshr e_1 (const? 32 8))) (trunc 8 (lshr e (const? 32 8)))) ⊑
    icmp IntPred.ne (trunc 16 (lshr e_1 (const? 32 8))) (trunc 16 (lshr e (const? 32 8)))

theorem test27_proof : test27_before ⊑ test27_after

theorem main7e_logical_thm.extracted_1._14 : ∀ (x x_1 x_2 x_3 x_4 : BitVec 32),
  ¬ofBool (x_4 &&& x_3 &&& x_2 == x_4 &&& x_3) = 1#1 →
    ofBool (x_4 &&& x_3 &&& x_2 != x_4 &&& x_3) = 1#1 → ¬0#1 = 1#1 → 1#32 = zeroExtend 32 1#1

theorem ashr_ult_10_proof : ashr_ult_10_before ⊑ ashr_ult_10_after

theorem ashrslt_02_04_proof : ashrslt_02_04_before ⊑ ashrslt_02_04_after

theorem test57_proof : test57_before ⊑ test57_after

@[grind ., simp]
lemma RawCNFA.stateMax_nin_states (m : RawCNFA A) : ¬(m.stateMax ∈ m.states)

theorem test_eq2_thm (e : IntW 32) (e_1 : IntW 16) :
  icmp IntPred.eq (trunc 8 e_1 { «nsw» := true, «nuw» := false }) (trunc 8 e { «nsw» := true, «nuw» := false }) ⊑
    icmp IntPred.eq e_1 (trunc 16 e)

theorem shift_xor_multiuse_cmp_with_and_thm.extracted_1._3 : ∀ (x x_1 x_2 x_3 : BitVec 32),
  ofBool (0#32 != x_3 &&& 4096#32) = 1#1 →
    ofBool (x_3 &&& 4096#32 == 0#32) = 1#1 → x_2 * x_1 = (x_2 &&& BitVec.ofInt 32 (-2049)) * x

theorem uge_or_not_max_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), ofBool (x ≤ᵤ x_1) ||

theorem scalar_i16_lshr_and_signbit_eq_proof : scalar_i16_lshr_and_signbit_eq_before ⊑ scalar_i16_lshr_and_signbit_eq_after

theorem demorgan_nor_use2ab_proof : demorgan_nor_use2ab_before ⊑ demorgan_nor_use2ab_after

theorem trunc_shl_shl_var_proof : trunc_shl_shl_var_before ⊑ trunc_shl_shl_var_after

theorem ult_or_not_max_thm (e e_1 : IntW 8) :
  LLVM.or (icmp IntPred.ult e_1 e) (icmp IntPred.ne e_1 (const? 8 (-1))) ⊑
    icmp IntPred.ne e_1 (const? 8 (-1))

theorem select_constants_and_icmp_eq0_zero_fval_proof : select_constants_and_icmp_eq0_zero_fval_before ⊑ select_constants_and_icmp_eq0_zero_fval_after

theorem absdiff1_proof : absdiff1_before ⊑ absdiff1_after

theorem allSignBits_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 32),
  ¬(31#32 ≥ ↑32 ∨ 31#32 ≥ ↑32) →
    ofBool (x_1 <ₛ 0#32) = 1#1 →
      x_2 &&& x_1.sshiftRight' 31#32 ||

theorem and_orn_xor_commute8_thm (e e_1 : IntW 32) :
  LLVM.and (LLVM.xor (mul e_1 e_1) (mul e e)) (LLVM.or (mul e_1 e_1) (LLVM.xor (mul e e) (const? 32 (-1)))) ⊑
    LLVM.and (mul e_1 e_1) (LLVM.xor (mul e e) (const? 32 (-1)))

theorem add_eq_xor_add_mul_and (x y : BitVec w) :
    x + y = (x ^^^ y) + (x &&& y) + (x &&& y)

theorem exact_ashr_ne_exactdiv_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬(True ∧ BitVec.ofInt 8 (-80) >>> x <<< x ≠ BitVec.ofInt 8 (-80) ∨ x ≥ ↑8) →
    ofBool ((BitVec.ofInt 8 (-80)).sshiftRight' x != BitVec.ofInt 8 (-5)) = ofBool (x != 4#8)

theorem bitwise_and_logical_and_masked_icmp_allones_poison1_proof : bitwise_and_logical_and_masked_icmp_allones_poison1_before ⊑ bitwise_and_logical_and_masked_icmp_allones_poison1_after

theorem substitute_constant_or_ne_swap_sle_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ofBool (x_1 != 42#8) ||

theorem slt_smin2_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (x_1 <ₛ x) = 1#1 → ofBool (x <ₛ x) = ofBool (x_1 <ₛ x)

theorem test_mul__nuw_is_safe_thm.extracted_1._3 : ∀ (x : BitVec 32),
  ¬ofBool (x &&& 268435457#32 == 268435456#32) = 1#1 → True ∧ (x &&& 268435457#32).umulOverflow 9#32 = true → False

theorem n13_proof : n13_before ⊑ n13_after

lemma inter_eq (M N : NFA' n) : M.inter N = NFA'.product And M N

@[simp] theorem dropUntil_toCons : dropUntil (cons ty Γ) (Var.toCons v) = dropUntil Γ v

theorem ule_swap_or_not_min_commute_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ofBool (x_1 != 0#8) ||

theorem test6_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬16#32 ≥ ↑32 → (x ^^^ 1234#32) >>> 16#32 + (x ^^^ 1234#32) = x >>> 16#32 + (x ^^^ 1234#32)

theorem main7c_logical_thm.extracted_1._12 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 == x_2 &&& x_1) = 1#1 →
    ¬ofBool (x_2 != x_2 &&& x_1) = 1#1 → ¬0#1 = 1#1 → 1#32 = zeroExtend 32 (ofBool (x != x &&& x_1))

theorem test2_thm.extracted_1._1 : ∀ (x : BitVec 49),
  ¬(11#49 ≥ ↑49 ∨ 4096#49 <<< 11#49 = 0) → x % 4096#49 <<< 11#49 = x &&& 8388607#49

theorem e_1951 :
    7 * ((a ||| b) &&&  ~~~(b ^^^ c)) + 4 * (b ^^^ ( ~~~a ||| ( ~~~b ||| c))) - 2 * ( ~~~(a &&& b) &&&  ~~~(b ^^^ c)) + 7 *  ~~~(a &&& (b ||| c)) - 5 * ( ~~~a ||| (b ||| c)) + 5 * ( ~~~(a |||  ~~~b) ||| (b ^^^ c)) + 1 * (b &&&  ~~~(a ^^^ c)) + 1 * (a ^^^ (b ||| c)) + 4 * (a ^^^ b) - 11 * ((a &&& b) ^^^  ~~~(a ^^^ (b &&& c))) - 2 * (c &&&  ~~~(a &&& b)) - 11 * ( ~~~(a |||  ~~~b) |||  ~~~(b ^^^ c)) - 2 * b - 2 *  ~~~(a ^^^ ( ~~~b &&& c)) - 6 * ((a &&& c) ||| (b &&&  ~~~c)) - 11 * ((a ^^^ b) ||| (a ^^^ c)) + 1 * ((b &&&  ~~~c) ^^^ ( ~~~a ||| (b ^^^ c))) + 1 * ( ~~~(a &&&  ~~~b) &&&  ~~~(a ^^^ (b ^^^ c))) - 1 * ( ~~~c ||| (a ^^^ b)) + 7 * (a ^^^ (b ^^^ c)) + 5 * (b |||  ~~~(a ||| c)) + 2 * (c ^^^  ~~~( ~~~a ||| ( ~~~b &&& c))) - 2 * ((a &&&  ~~~b) |||  ~~~(b ^^^ c)) + 1 * (c ^^^ (a ||| (b ||| c))) + 3 * ( ~~~a ||| (b ^^^ c)) - 7 * ( ~~~a &&& (b ||| c)) - 6 * (c &&&  ~~~(a ^^^ b)) - 1 *  ~~~(a |||  ~~~b) + 2 * ( ~~~(a &&&  ~~~b) &&&  ~~~(b ^^^ c)) + 4 * (b ^^^ (a &&&  ~~~c)) + 2 * (a ||| (b &&& c)) + 5 *  ~~~(a ||| c) + 3 * (a ||| ( ~~~b &&& c)) + 1 * (a ||| (b ||| c)) + 11 * (a &&& b) + 2 * (a &&& (b ||| c)) + 2 * ((a |||  ~~~b) &&& (b ^^^ c)) - 1 * ((a &&& b) |||  ~~~(b |||  ~~~c)) - 2 * (b ^^^  ~~~(a &&& c)) - 1 * (c ^^^ ( ~~~a ||| (b ||| c))) - 1 * ( ~~~a &&& ( ~~~b ||| c)) - 1 * ((b &&&  ~~~c) |||  ~~~(a ||| ( ~~~b &&& c))) - 5 * ((a ^^^ b) &&& (a ^^^ c)) - 1 * (c ^^^ (a &&& (b ||| c))) - 1 * (a ||| b) - 1 *  ~~~(b |||  ~~~c) + 9 *  ~~~(a ||| (b ||| c)) + 14 *  ~~~(a ||| ( ~~~b ||| c)) - 1 *  ~~~( ~~~a ||| (b ||| c)) - 1 *  ~~~( ~~~a ||| ( ~~~b ||| c)) + 19 * ( ~~~a &&& ( ~~~b &&& c)) + 8 * ( ~~~a &&& (b &&& c)) + 8 * (a &&& ( ~~~b &&& c)) - 4 * (a &&& (b &&& c)) = 7 * ((b &&&  ~~~c) ^^^ (a ||| (b ^^^ c))) - 2 * (c ^^^  ~~~(a ||| (b &&& c)))

theorem main6b_logical_thm.extracted_1._3 : ∀ (x : BitVec 32),
  ¬ofBool (x &&& 7#32 == 3#32) = 1#1 → 0#1 = 1#1 → 0#32 = zeroExtend 32 (ofBool (x &&& 23#32 != 19#32))

theorem lt_unsigned_to_large_negative_thm (e : IntW 8) : icmp IntPred.slt (zext 32 e) (const? 32 (-1024)) ⊑ const? 1 0

theorem bitwise_and_logical_and_icmps_comm3_thm.extracted_1._7 : ∀ (x x_1 x_2 : BitVec 8),
  ¬x_1 ≥ ↑8 →
    ¬ofBool (x_2 &&& 1#8 != 0#8) = 1#1 →
      ¬(True ∧ 1#8 <<< x_1 >>> x_1 ≠ 1#8 ∨ x_1 ≥ ↑8 ∨ True ∧ 1#8 <<< x_1 >>> x_1 ≠ 1#8 ∨ x_1 ≥ ↑8) →
        ofBool (x_2 &&& (1#8 <<< x_1 ||| 1#8) == 1#8 <<< x_1 ||| 1#8) = 1#1 →
          ofBool (x_2 &&& 1#8 <<< x_1 != 0#8) &&& 0#1 = ofBool (x == 42#8)

theorem shl_nsw_sgt_n1_proof : shl_nsw_sgt_n1_before ⊑ shl_nsw_sgt_n1_after

theorem ashrslt_02_09_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 2#4 <<< 2#4 ≠ x ∨ 2#4 ≥ ↑4) → ofBool (x.sshiftRight' 2#4 <ₛ BitVec.ofInt 4 (-7)) = 0#1

theorem ashrslt_01_07_thm (e : IntW 4) : icmp IntPred.slt (ashr e (const? 4 1)) (const? 4 7) ⊑ const? 1 1

theorem e_1920 :
     - 11 *  ~~~(a &&& b) + 2 * (c &&&  ~~~(a &&& b)) - 11 * ((b &&&  ~~~c) |||  ~~~( ~~~a ||| ( ~~~b &&& c))) - 5 * (a ^^^ c) - 2 * ( ~~~(a |||  ~~~b) |||  ~~~(b ^^^ c)) - 5 * (c ^^^  ~~~( ~~~a &&& (b ||| c))) + 1 * ((a ||| b) &&&  ~~~(a ^^^ (b ^^^ c))) - 2 * (a &&& b) + 7 * (b ||| c) + 2 * (b ^^^ ( ~~~a ||| ( ~~~b ||| c))) + 11 * (a ||| (b ||| c)) + 2 * ( ~~~a &&& ( ~~~b ||| c)) - 2 * ((a &&& b) |||  ~~~(b ^^^ c)) - 2 * (c ^^^  ~~~( ~~~a &&& ( ~~~b ||| c))) + 5 * (b &&& (a ||| c)) - 5 * ((a &&& b) |||  ~~~(b |||  ~~~c)) + 7 * ((a &&& c) ^^^  ~~~(a ^^^ (b &&& c))) - 2 * ( ~~~(a ||| b) |||  ~~~(a ^^^ (b ^^^ c))) + 2 * (a &&&  ~~~b) - 1 * ( ~~~(a |||  ~~~b) |||  ~~~(a ^^^ (b ^^^ c))) + 1 * a + 11 *  ~~~(a ||| b) + 2 * (b ^^^ ( ~~~a ||| ( ~~~b &&& c))) + 7 *  ~~~(a ^^^ (b ||| c)) - 1 * (b ^^^  ~~~( ~~~a ||| ( ~~~b &&& c))) + 3 * (a ^^^ ( ~~~b ||| c)) + 4 * ( ~~~c ||| (a ^^^ b)) - 2 * ( ~~~(a &&& b) &&& (a ^^^ (b ^^^ c))) + 1 * (b ^^^ ( ~~~a &&& ( ~~~b ||| c))) - 6 * (a ||| c) + 7 * ( ~~~c |||  ~~~(a ^^^ b)) + 3 * (a &&& ( ~~~b ||| c)) + 11 * (b &&& (a ^^^ c)) - 5 * (a ||| ( ~~~b ||| c)) - 6 * (a &&& (b ^^^ c)) + 3 * ((b &&&  ~~~c) ^^^ (a ||| (b ^^^ c))) + 1 * ( ~~~(a &&& b) &&&  ~~~(b ^^^ c)) - 1 * ((a |||  ~~~b) &&&  ~~~(b ^^^ c)) + 1 * (c |||  ~~~(a ^^^ b)) - 1 * ( ~~~(a ^^^ b) &&& (a ^^^ c)) + 5 *  ~~~( ~~~a &&& (b ^^^ c)) - 2 * ( ~~~(a |||  ~~~b) ||| (b ^^^ c)) + 11 *  ~~~(a ^^^ ( ~~~b ||| c)) - 17 *  ~~~(a ||| (b ||| c)) - 6 *  ~~~(a ||| ( ~~~b ||| c)) + 13 *  ~~~( ~~~a ||| (b ||| c)) - 3 *  ~~~( ~~~a ||| ( ~~~b ||| c)) - 13 * ( ~~~a &&& ( ~~~b &&& c)) - 9 * ( ~~~a &&& (b &&& c)) - 31 * (a &&& ( ~~~b &&& c)) - 23 * (a &&& (b &&& c)) = 3 * (c ^^^ (a ||| (b ||| c))) + 7 * (c ^^^  ~~~(a &&& (b &&& c)))

theorem test_sub_nsw__nsw_is_safe_thm (e : IntW 32) :
  select (icmp IntPred.eq (LLVM.or e (const? 32 (-2147483648))) (const? 32 (-2147483647))) (const? 32 (-1))
      (sub (const? 32 (-2147483648)) (LLVM.or e (const? 32 (-2147483648))) { «nsw» := true, «nuw» := false }) ⊑
    sub (const? 32 (-2147483648)) (LLVM.or e (const? 32 (-2147483648))) { «nsw» := true, «nuw» := false }

theorem bools2_logical_commute1_and1_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 1),
  x_2 &&& x_1 = 1#1 → x_1 = 1#1 → 1#1 = x_2

theorem or_and_or_pat8_proof : or_and_or_pat8_before ⊑ or_and_or_pat8_after

@[grind =, simp]
lemma createSink_finals [LawfulBEq A] {m : RawCNFA A} : m.createSink.2.finals = m.finals

theorem bools_multi_uses1_logical_thm.extracted_1._8 : ∀ (x x_1 : BitVec 1),
  ¬x_1 ^^^ 1#1 = 1#1 → ¬x_1 = 1#1 → 0#1 = 1#1 → 1#1 ^^^ 0#1 = x ^^^ 0#1

theorem add_shl_same_amount_nuw_thm.extracted_1._2 : ∀ (x x_1 x_2 : BitVec 64),
  ¬(True ∧ x_2 <<< x_1 >>> x_1 ≠ x_2 ∨
        x_1 ≥ ↑64 ∨ True ∧ x <<< x_1 >>> x_1 ≠ x ∨ x_1 ≥ ↑64 ∨ True ∧ (x_2 <<< x_1).uaddOverflow (x <<< x_1) = true) →
    ¬(True ∧ x_2.uaddOverflow x = true ∨ True ∧ (x_2 + x) <<< x_1 >>> x_1 ≠ x_2 + x ∨ x_1 ≥ ↑64) →
      x_2 <<< x_1 + x <<< x_1 = (x_2 + x) <<< x_1

theorem t3_ult_sgt_neg1_thm.extracted_1._3 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 + 16#32 <ᵤ 144#32) = 1#1 →
    ¬ofBool (127#32 <ₛ x_1) = 1#1 → ofBool (x_1 <ₛ BitVec.ofInt 32 (-16)) = 1#1 → False

theorem mul_selectp2_x_non_const_thm.extracted_1._2 : ∀ (x : BitVec 8) (x_1 : BitVec 1),
  x_1 = 1#1 → ¬1#8 ≥ ↑8 → 2#8 * x = x <<< 1#8

theorem src_tv_eq_thm.extracted_1._3 : ∀ (x x_1 : BitVec 8) (x_2 : BitVec 1),
  ¬x_2 = 1#1 → ¬(True ∧ x_1.uaddOverflow 1#8 = true) → ofBool (x_1 + 1#8 ||| x == 0#8) = ofBool (x == 0#8) &&& x_2

theorem ashr_x_by_const_cmp_sge_x_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬5#32 ≥ ↑32 → ofBool (x ≤ₛ x.sshiftRight' 5#32) = ofBool (x <ₛ 1#32)

theorem multiuse3_proof : multiuse3_before ⊑ multiuse3_after

theorem t_2_192_thm (e : IntW 8) :
  LLVM.and (select (icmp IntPred.ugt e (const? 8 2)) e (const? 8 2)) (const? 8 (-64)) ⊑
    LLVM.and e (const? 8 (-64))

theorem lt_signed_to_large_negative_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ofBool (signExtend 32 x <ₛ BitVec.ofInt 32 (-1024)) = 0#1

theorem udiv_shl_pair_const_proof : udiv_shl_pair_const_before ⊑ udiv_shl_pair_const_after

theorem masked_icmps_mask_notallzeros_bmask_mixed_swapped_7_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x &&& 15#32 == 8#32) &&& ofBool (x &&& 7#32 != 0#32) = 0#1

theorem reassoc_x2_sub_nuw_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ¬(True ∧ x_1.usubOverflow 4#32 = true ∨
        True ∧ x.usubOverflow 8#32 = true ∨ True ∧ (x_1 - 4#32).usubOverflow (x - 8#32) = true) →
    x_1 - 4#32 - (x - 8#32) = x_1 - x + 4#32

theorem t0_proof : t0_before ⊑ t0_after

theorem test2_proof : test2_before ⊑ test2_after

theorem sub_const_const_sub_proof : sub_const_const_sub_before ⊑ sub_const_const_sub_after

theorem t3_no_extrause_proof : t3_no_extrause_before ⊑ t3_no_extrause_after

theorem e_303 :
    1 *  ~~~x + 11 * (x ^^^ y) - 11 * (x |||  ~~~y) - 2 *  ~~~(x ||| y) - 13 *  ~~~(x |||  ~~~y) - 12 * (x &&&  ~~~y) + 11 * (x &&& y) =  - 1 *  ~~~(x &&& y) - 11 *  ~~~y

theorem e_1079 :
    3 *  ~~~z - 5 * (z ^^^ t) + 5 *  ~~~t - 6 * (z ||| t) - 3 *  ~~~(z &&&  ~~~t) - 8 *  ~~~(z ||| t) + 11 *  ~~~(z |||  ~~~t) + 6 * (z &&& t) =  - 3 * (z &&&  ~~~t) - 3 * (z |||  ~~~t)

theorem logic_and_logic_or_6_proof : logic_and_logic_or_6_before ⊑ logic_and_logic_or_6_after

theorem dec_sub_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), x_1 - x + -1#8 = x_1 + (x ^^^ -1#8)

@[simp]
theorem denote_splitProgramAt [LawfulMonad d.m] {pos : ℕ} {prog : Com d Γ eff t}
    {res : _} (hres : res ∈ splitProgramAt pos prog) :
    res.denote = prog.denote

theorem or_signbit_lshr_proof : or_signbit_lshr_before ⊑ or_signbit_lshr_after

theorem Option.some_bind'' (x : α) (f : α → Option β) : some x >>= f = f x

theorem trunc_shl_shl_infloop_thm (e : IntW 64) :
  trunc 32 (shl (shl e (const? 64 1)) (const? 64 2)) ⊑ shl (trunc 32 e) (const? 32 3)

theorem e_757 :
    3 *  ~~~(z &&&  ~~~t) - 1 *  ~~~(z &&&  ~~~z) - 2 * (z ||| t) + 3 * (z &&&  ~~~t) = 2 *  ~~~(z ||| t)

theorem t0_ult_slt_128_thm.extracted_1._8 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (x_1 + 16#32 <ᵤ 144#32) = 1#1 →
    ¬ofBool (x_1 <ₛ 128#32) = 1#1 →
      ¬ofBool (127#32 <ₛ x_1) = 1#1 → ofBool (x_1 <ₛ BitVec.ofInt 32 (-16)) = 1#1 → False

theorem neg_sgt_1_proof : neg_sgt_1_before ⊑ neg_sgt_1_after

theorem test33b_proof : test33b_before ⊑ test33b_after

theorem icmp_sle2_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬(True ∧ (x <<< 1#8).sshiftRight' 1#8 ≠ x ∨ 1#8 ≥ ↑8) →
    ofBool (x <<< 1#8 ≤ₛ BitVec.ofInt 8 (-127)) = ofBool (x <ₛ BitVec.ofInt 8 (-63))

theorem test9_proof : test9_before ⊑ test9_after

theorem icmp_select_var_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), ofBool (x_1 == 0#8) = 1#1 → ofBool (x == x) = 1#1

theorem a_true_implies_b_false2_thm.extracted_1._4 : ∀ (x : BitVec 1) (x_1 : BitVec 8),
  ¬ofBool (x_1 == 10#8) = 1#1 → ¬ofBool (20#8 <ᵤ x_1) = 1#1 → ofBool (20#8 <ᵤ x_1) &&& x = 0#1

theorem udiv_shl_mul_nuw_exact_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 5),
  ¬(True ∧ x_2 <<< x_1 >>> x_1 ≠ x_2 ∨
        x_1 ≥ ↑5 ∨ True ∧ x_2.umulOverflow x = true ∨ True ∧ (x_2 <<< x_1).umod (x_2 * x) ≠ 0 ∨ x_2 * x = 0) →
    True ∧ 1#5 <<< x_1 >>> x_1 ≠ 1#5 ∨ x_1 ≥ ↑5 ∨ True ∧ (1#5 <<< x_1).umod x ≠ 0 ∨ x = 0 → False

theorem not_cond_proof : not_cond_before ⊑ not_cond_after

theorem select_xor_icmp2_thm (e e_1 e_2 : IntW 32) :
  select (icmp IntPred.ne e_2 (const? 32 0)) e_1 (LLVM.xor e_2 e) ⊑
    select (icmp IntPred.eq e_2 (const? 32 0)) e e_1

theorem add_nextStateCirc_eval :
    (add.nextStateCirc ()).eval =
      fun x => x (inr true) && x (inr false) |

theorem a_false_implies_b_true_proof : a_false_implies_b_true_before ⊑ a_false_implies_b_true_after

theorem shl_lshr_demand1_thm.extracted_1._1 : ∀ (x : BitVec 8), ¬(x ≥ ↑8 ∨ 3#8 ≥ ↑8) → x ≥ ↑8 → False

theorem cond_eq_and_const_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ofBool (x_1 == 10#8) = 1#1 → ofBool (x_1 <ᵤ x) = ofBool (10#8 <ᵤ x)

theorem udiv_i8_thm (e e_1 : IntW 8) : trunc 8 (LLVM.udiv (zext 32 e_1) (zext 32 e)) ⊑ LLVM.udiv e_1 e

theorem udiv_shl_pair3_thm.extracted_1._2 : ∀ (x x_1 x_2 : BitVec 32),
  ¬(True ∧ x_2 <<< x_1 >>> x_1 ≠ x_2 ∨
        x_1 ≥ ↑32 ∨ True ∧ (x_2 <<< x).sshiftRight' x ≠ x_2 ∨ True ∧ x_2 <<< x >>> x ≠ x_2 ∨ x ≥ ↑32 ∨ x_2 <<< x = 0) →
    ¬(True ∧ 1#32 <<< x_1 >>> x_1 ≠ 1#32 ∨ x_1 ≥ ↑32 ∨ x ≥ ↑32) → x_2 <<< x_1 / x_2 <<< x = 1#32 <<< x_1 >>> x

theorem not_xor_eq_and_add_not_or :
    ~~~ (x ^^^ y) = (x &&& y) + ~~~ (x ||| y)

theorem t7_ugt_sgt_neg1_thm.extracted_1._15 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (65535#32 <ᵤ x_2) = 1#1 → ¬ofBool (65535#32 <ₛ x_2) = 1#1 → ofBool (x_2 <ₛ 0#32) = 1#1 → x_2 = x

@[simp, simp_denote, simp_sllvm]
lemma run_pure : StateT.run (pure x : EffectM α) s = .value (x, s)

theorem Factor.denote_eq_toNat_reflect {w : Nat} (xs : Env w) (f : Factor) :
  f.denote xs = (f.reflect xs |>.toNat)

theorem e_313 :
     - 1 * (x |||  ~~~y) + 7 * y + 7 *  ~~~y - 5 *  ~~~(x ||| y) - 6 *  ~~~(x |||  ~~~y) - 5 * (x &&&  ~~~y) - 6 * (x &&& y) = 1 *  ~~~(x &&& y)

theorem udiv_i32_c_multiuse_proof : udiv_i32_c_multiuse_before ⊑ udiv_i32_c_multiuse_after

theorem bv_AddSub_1152 :
    ∀ (e e_1 : LLVM.IntW 1), LLVM.add e_1 e ⊑ LLVM.xor e_1 e

theorem xor_notand_to_or_not3_proof : xor_notand_to_or_not3_before ⊑ xor_notand_to_or_not3_after

theorem icmp_eq_sext_ne_zero_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (signExtend 32 (ofBool (x != 0#32)) == x) = ofBool (x + 1#32 <ᵤ 2#32)

theorem shl_xor_xor_good_mask_thm (e e_1 : IntW 8) :
  LLVM.xor (shl e_1 (const? 8 1)) (LLVM.xor (shl e (const? 8 1)) (const? 8 88)) ⊑
    LLVM.xor (shl (LLVM.xor e e_1) (const? 8 1)) (const? 8 88)

theorem e_244 :
     - 5 * x - 5 *  ~~~x + 11 * (x |||  ~~~y) + 7 * y - 2 *  ~~~(x |||  ~~~y) - 6 * (x &&&  ~~~y) - 13 * (x &&& y) = 6 *  ~~~(x ||| y)

theorem sub_to_xor_proof : sub_to_xor_before ⊑ sub_to_xor_after

theorem test11_thm (e : IntW 737) :
  icmp IntPred.ne (LLVM.and e (const? 737 (-2147483648))) (const? 737 0) ⊑
    icmp IntPred.ugt e (const? 737 2147483647)

theorem test47_proof : test47_before ⊑ test47_after

@[aesop 50% unsafe]
theorem Std.HashMap.getElem?_none_not_mem [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} :
    m[k]? = none → k ∉ m

theorem sgt_or_not_min_commute_logical_thm.extracted_1._2 : ∀ (x x_1 : BitVec 8),
  ofBool (x_1 != BitVec.ofInt 8 (-128)) = 1#1 → 1#1 = ofBool (x_1 != BitVec.ofInt 8 (-128))

theorem p_constmask_thm (e e_1 : IntW 32) :
  add (LLVM.and e_1 (const? 32 65280)) (LLVM.and e (const? 32 (-65281))) ⊑
    LLVM.or (LLVM.and e_1 (const? 32 65280)) (LLVM.and e (const? 32 (-65281))) { «disjoint» := true }

theorem fold_sext_to_and_wrong7_proof : fold_sext_to_and_wrong7_before ⊑ fold_sext_to_and_wrong7_after

theorem ashrsgt_02_05_exact_thm (e : IntW 4) :
  icmp IntPred.sgt (ashr e (const? 4 2) { «exact» := true }) (const? 4 5) ⊑ const? 1 0

theorem e_25 :
     - 2 *  ~~~y + 2 * (x |||  ~~~y) + 1 *  ~~~x - 5 * (x &&&  ~~~y) - 6 * (x &&& y) = 1 *  ~~~(x &&&  ~~~y) - 5 * x

theorem test47_thm (e e_1 : IntW 32) :
  LLVM.and e_1 (LLVM.or e (LLVM.xor e_1 (const? 32 (-1)))) ⊑ LLVM.and e_1 e

theorem a_thm.extracted_1._4 : ∀ (x x_1 : BitVec 1),
  ¬x_1 = 1#1 →
    ¬(True ∧ (1#32).saddOverflow (signExtend 32 x) = true) →
      zeroExtend 32 x_1 + 1#32 + (0#32 - zeroExtend 32 x) = 1#32 + signExtend 32 x

theorem trunc_shl_shl_infloop_thm.extracted_1._1 : ∀ (x : BitVec 64), ¬(1#64 ≥ ↑64 ∨ 2#64 ≥ ↑64) → 3#32 ≥ ↑32 → False

theorem sext_ule_sext_proof : sext_ule_sext_before ⊑ sext_ule_sext_after

theorem and_or2_thm.extracted_1._8 : ∀ (x x_1 x_2 : BitVec 1), ¬(x_2 ^^^ 1#1) &&& x_1 = 1#1 → ¬x_1 = 1#1 → x_1 = 0#1

theorem fold_sext_to_and_proof : fold_sext_to_and_before ⊑ fold_sext_to_and_after

theorem not_and_and_or_not_or_or_commute2_and_thm (e e_1 e_2 : IntW 32) :
  LLVM.or (LLVM.and (LLVM.and e_2 e_1) (LLVM.xor e (const? 32 (-1))))
      (LLVM.xor (LLVM.or (LLVM.or e_2 e) e_1) (const? 32 (-1))) ⊑
    LLVM.xor (LLVM.or (LLVM.xor e_2 e_1) e) (const? 32 (-1))

theorem positive_biggerlshr_shlnsw_lshrexact_thm (e : IntW 8) :
  shl (lshr e (const? 8 6) { «exact» := true }) (const? 8 3) { «nsw» := true, «nuw» := false } ⊑
    lshr e (const? 8 3) { «exact» := true }

theorem sle_or_not_min_thm (e e_1 : IntW 8) :
  LLVM.or (icmp IntPred.sle e_1 e) (icmp IntPred.ne e_1 (const? 8 (-128))) ⊑ const? 1 1

theorem e_254 :
     - 7 * y - 6 *  ~~~(x &&&  ~~~y) - 7 * (x ^^^ y) - 1 *  ~~~(x ^^^ y) + 12 *  ~~~(x ||| y) + 25 *  ~~~(x |||  ~~~y) + 12 * (x &&&  ~~~y) =  - 19 * (x &&& y) + 5 *  ~~~(x &&&  ~~~x)

theorem trunc_sel_smaller_sext_proof : trunc_sel_smaller_sext_before ⊑ trunc_sel_smaller_sext_after

theorem eq_umin3_thm (e e_1 : IntW 32) :
  icmp IntPred.eq (add e_1 (const? 32 3))
      (select (icmp IntPred.ult (add e_1 (const? 32 3)) e) (add e_1 (const? 32 3)) e) ⊑
    icmp IntPred.ule (add e_1 (const? 32 3)) e

theorem add_const_add_const_thm.extracted_1._1 : ∀ (x : BitVec 32), x + 8#32 + 2#32 = x + 10#32

theorem no_masks_with_logical_or_thm.extracted_1._6 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 != 0#32) = 1#1 →
    ¬ofBool (x_2 ||| x != 0#32) = 1#1 → ofBool (x_1 != 63#32) ||

theorem select_icmp_sgt_not_smin_proof : select_icmp_sgt_not_smin_before ⊑ select_icmp_sgt_not_smin_after

theorem positive_sameconst_shlnuwnsw_lshrexact_proof : positive_sameconst_shlnuwnsw_lshrexact_before ⊑ positive_sameconst_shlnuwnsw_lshrexact_after

theorem icmp_power2_and_icmp_shifted_mask_swapped_8_6_thm (e : IntW 32) :
  LLVM.and (icmp IntPred.ne (LLVM.and e (const? 32 6)) (const? 32 6)) (icmp IntPred.ult e (const? 32 8)) ⊑
    icmp IntPred.ult e (const? 32 6)

theorem fsmSext_eval_eq
    (woldFsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep wold))
    (wnewFsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep wnew))
    {wenv : WidthExpr.Env wcard}
    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}
    (hwnew : HNatFSMToBitstream wnewFsm)
    (hwold : HNatFSMToBitstream woldFsm)
    {tctx : Term.Ctx wcard tcard}
    (tenv : Term.Ctx.Env tctx wenv)
    (benv : Term.BoolEnv bcard)
    (nenv : Term.NatEnv ncard)
    (ienv : Term.IntEnv icard)
    (t : Term bcard ncard icard pcard tctx (.bv wold))
    (tFsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t))
    (htfsm : HTermFSMToBitStream tFsm)
    (htenv : HTermEnv fsmEnv tenv benv) :
    (fsmSext tFsm.toFsmZext woldFsm.toFsm  wnewFsm.toFsm).eval fsmEnv = fun i =>
      ((BitStream.ofBitVecZext ((Term.sext t wnew).toBV benv nenv ienv penv tenv))) i

theorem andn_or_cmp_2_logical_proof : andn_or_cmp_2_logical_before ⊑ andn_or_cmp_2_logical_after

theorem main6d_logical_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ofBool (x &&& 7#32 != 3#32) = 1#1 → ¬True → 0#32 = zeroExtend 32 (ofBool (x &&& 23#32 == 19#32))

theorem test5_thm.extracted_1._1 : ∀ (x : BitVec 700),
  (x ||| 288230376151711743#700) ^^^ 288230376151711743#700 = x &&& BitVec.ofInt 700 (-288230376151711744)

theorem select_icmp_x_and_2147483648_ne_0_or_2147483648_proof : select_icmp_x_and_2147483648_ne_0_or_2147483648_before ⊑ select_icmp_x_and_2147483648_ne_0_or_2147483648_after

theorem test_ne_0_and_15_add_1_proof : test_ne_0_and_15_add_1_before ⊑ test_ne_0_and_15_add_1_after

theorem e_1809 :
     - 5 * (b ^^^ (a |||  ~~~c)) - 2 * (b ^^^ ( ~~~a &&& (b ^^^ c))) - 3 *  ~~~( ~~~a &&& (b ||| c)) + 2 * (b |||  ~~~(a |||  ~~~c)) - 2 * (b ^^^ (a ||| ( ~~~b &&& c))) - 1 * (b &&&  ~~~(a &&&  ~~~c)) - 1 * (a ||| (b &&& c)) - 2 * (a ^^^ (b &&& c)) + 5 * (b ^^^  ~~~(a |||  ~~~c)) - 3 *  ~~~(a &&& (b ||| c)) + 3 * (c ^^^ ( ~~~a &&& (b ||| c))) - 5 * (a ^^^ (b ||| c)) + 5 * (b ||| (a ^^^ c)) - 1 * (c &&& (a ||| b)) - 3 * (a &&& b) + 7 * (b ^^^ ( ~~~a ||| (b ^^^ c))) + 2 * ((b &&& c) ||| (a &&& (b ||| c))) + 1 *  ~~~(b &&&  ~~~c) + 1 * (b ^^^ (a &&&  ~~~c)) + 2 *  ~~~(a &&& (b &&& c)) + 2 * (b ^^^  ~~~(a &&& c)) + 7 * (b ^^^  ~~~(a &&& (b ^^^ c))) + 3 * ( ~~~a ||| ( ~~~b ||| c)) - 5 * ((a ||| b) &&& (a ^^^ (b ^^^ c))) + 2 * ((a ^^^ b) &&&  ~~~(a ^^^ c)) + 1 * ( ~~~a &&& (b ^^^ c)) + 1 * (b ^^^  ~~~(a &&& ( ~~~b &&& c))) + 1 * (b ^^^  ~~~( ~~~a &&& (b ^^^ c))) - 3 * (b ^^^  ~~~(a &&& (b ||| c))) + 1 * ((a &&& c) ^^^ (b ||| c)) + 2 *  ~~~(a &&& c) - 1 * (c ^^^  ~~~(a ||| ( ~~~b &&& c))) - 7 * ( ~~~(a |||  ~~~b) ||| (a ^^^ (b ^^^ c))) - 1 * (b &&& c) - 8 *  ~~~(a ||| (b ||| c)) - 4 *  ~~~(a ||| ( ~~~b ||| c)) + 14 *  ~~~( ~~~a ||| (b ||| c)) - 9 *  ~~~( ~~~a ||| ( ~~~b ||| c)) - 16 * ( ~~~a &&& ( ~~~b &&& c)) + 11 * ( ~~~a &&& (b &&& c)) - 6 * (a &&& ( ~~~b &&& c)) = 3 * ((a &&& b) |||  ~~~(b ||| c))

theorem positive_samevar_ashrexact_proof : positive_samevar_ashrexact_before ⊑ positive_samevar_ashrexact_after

theorem PR42691_3_logical_proof : PR42691_3_logical_before ⊑ PR42691_3_logical_after

theorem trunc_unsigned_nsw_sext_thm.extracted_1._1 : ∀ (x : BitVec 8) (x_1 : BitVec 32),
  ¬(True ∧ signExtend 32 (truncate 16 x_1) ≠ x_1) →
    ofBool (truncate 16 x_1 <ᵤ signExtend 16 x) = ofBool (x_1 <ᵤ signExtend 32 x)

theorem test6_proof : test6_before ⊑ test6_after

theorem ashr_eq_msb_low_second_zero_proof : ashr_eq_msb_low_second_zero_before ⊑ ashr_eq_msb_low_second_zero_after

theorem ashrsgt_02_00_thm (e : IntW 4) :
  icmp IntPred.sgt (ashr e (const? 4 2)) (const? 4 0) ⊑ icmp IntPred.sgt e (const? 4 3)

theorem reassoc_mul_nuw_proof : reassoc_mul_nuw_before ⊑ reassoc_mul_nuw_after

theorem not_select_bool_const1_thm.extracted_1._1 : ∀ (x : BitVec 1), ¬x = 1#1 → 1#1 ^^^ 1#1 = 0#1

theorem masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_1_logical_proof : masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_1_logical_before ⊑ masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_1_logical_after

theorem xor_or_thm.extracted_1._1 : ∀ (x : BitVec 8), x ^^^ 32#8 ||

theorem main4c_logical_thm.extracted_1._4 : ∀ (x : BitVec 32),
  ¬ofBool (x &&& 7#32 != 7#32) = 1#1 →
    ¬ofBool (x &&& 48#32 != 48#32) = 1#1 → 1#32 = zeroExtend 32 (ofBool (x &&& 55#32 == 55#32))

theorem foo1_or_signbit_lshr_without_shifting_signbit_logical_proof : foo1_or_signbit_lshr_without_shifting_signbit_logical_before ⊑ foo1_or_signbit_lshr_without_shifting_signbit_logical_after

theorem fold_add_udiv_urem_commuted_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬(10#32 = 0 ∨ 10#32 = 0 ∨ 4#32 ≥ ↑32) → 10#32 = 0 ∨ True ∧ (x / 10#32).umulOverflow 6#32 = true → False

theorem sdiv_eq_smin_proof : sdiv_eq_smin_before ⊑ sdiv_eq_smin_after

theorem lshrugt_02_08_exact_thm (e : IntW 4) :
  icmp IntPred.ugt (lshr e (const? 4 2) { «exact» := true }) (const? 4 (-8)) ⊑ const? 1 0

theorem alive_AndOrXor_2118___A__B__A___A__B : alive_AndOrXor_2118___A__B__A___A__B_src ⊑ alive_AndOrXor_2118___A__B__A___A__B_tgt

theorem eq_nuw_rem_nz_thm (e : IntW 8) :
  icmp IntPred.eq (mul e (const? 8 (-5)) { «nsw» := false, «nuw» := true }) (const? 8 20) ⊑ const? 1 0

theorem gt_unsigned_to_large_negative_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ofBool (BitVec.ofInt 32 (-1024) <ₛ zeroExtend 32 x) = 1#1

theorem p_proof : p_before ⊑ p_after

theorem not_or_or_and_not_and_and_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 32),
  (x_2 ^^^ -1#32 ||| x_1 ||| x) &&& (x_1 &&& x_2 &&& x ^^^ -1#32) = x ^^^ x_1 ||

theorem n15_variable_shamts_proof : n15_variable_shamts_before ⊑ n15_variable_shamts_after

theorem and_nosignbit_select_shl_proof : and_nosignbit_select_shl_before ⊑ and_nosignbit_select_shl_after

theorem n0_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 8),
  ¬x_2 ≥ ↑8 → True ∧ ((-1#8) <<< x_2).sshiftRight' x_2 ≠ -1#8 ∨ x_2 ≥ ↑8 → False

theorem ashrsgt_01_07_thm.extracted_1._1 : ∀ (x : BitVec 4), ¬1#4 ≥ ↑4 → ofBool (7#4 <ₛ x.sshiftRight' 1#4) = 0#1

theorem land_lor_left2_thm.extracted_1._2 : ∀ (x x_1 : BitVec 1), x_1 = 1#1 → x = 1#1 → 1#1 = x

theorem icmp_eq_and_pow2_minus1_shl1_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬x ≥ ↑32 → zeroExtend 32 (ofBool (1#32 <<< x &&& 15#32 == 0#32)) = zeroExtend 32 (ofBool (3#32 <ᵤ x))

theorem positive_biggerashr_thm.extracted_1._2 : ∀ (x : BitVec 8),
  ¬(6#8 ≥ ↑8 ∨ 3#8 ≥ ↑8) → ¬3#8 ≥ ↑8 → x.sshiftRight' 6#8 <<< 3#8 = x.sshiftRight' 3#8 &&& BitVec.ofInt 8 (-8)

theorem p13_commutativity0_thm (e e_1 : IntW 32) :
  select (icmp IntPred.ult e_1 (const? 32 65536)) (const? 32 65535) e ⊑
    select (icmp IntPred.ugt e_1 (const? 32 65535)) e (const? 32 65535)

theorem test_or_eq_a_b_commuted_proof : test_or_eq_a_b_commuted_before ⊑ test_or_eq_a_b_commuted_after

theorem ult_2_proof : ult_2_before ⊑ ult_2_after

theorem PR28476_logical_thm.extracted_1._5 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (x_1 != 0#32) = 1#1 → ofBool (x_1 == 0#32) = 1#1 → zeroExtend 32 0#1 ^^^ 1#32 = zeroExtend 32 1#1

theorem icmp_slt_sext_ne_zero_nofold_thm (e : IntW 32) :
  icmp IntPred.slt (sext 32 (icmp IntPred.ne e (const? 32 0))) e ⊑
    icmp IntPred.sgt e (sext 32 (icmp IntPred.ne e (const? 32 0)))

theorem test_shl_nuw_nsw__none_are_safe_thm.extracted_1._1 : ∀ (x : BitVec 64) (x_1 : BitVec 32),
  ofBool (x_1 &&& BitVec.ofInt 32 (-2) == 0#32) = 1#1 →
    2#32 ≥ ↑32 ∨
        True ∧ (x_1 <<< 2#32 &&& BitVec.ofInt 32 (-8)).msb = true ∨
          zeroExtend 64 (x_1 <<< 2#32 &&& BitVec.ofInt 32 (-8)) ≥ ↑64 →
      False

private theorem falseAfterIncluding_false_iff (n i : Nat) {env : Fin 0 → BitStream} :
  (falseAfterIncluding n).eval env i = false ↔ i ≥ n

theorem positive_with_signbit_logical_thm (e : IntW 32) :
  select (icmp IntPred.sgt e (const? 32 (-1))) (icmp IntPred.ult (add e (const? 32 128)) (const? 32 256))
      (const? 1 0) ⊑
    icmp IntPred.ult e (const? 32 128)

theorem test_shl_nuw_nsw__nuw_is_safe_proof : test_shl_nuw_nsw__nuw_is_safe_before ⊑ test_shl_nuw_nsw__nuw_is_safe_after

theorem sle_and_min_commute_logical_thm.extracted_1._2 : ∀ (x x_1 : BitVec 8),
  ofBool (x_1 == BitVec.ofInt 8 (-128)) = 1#1 → ofBool (x_1 ≤ₛ x) = ofBool (x_1 == BitVec.ofInt 8 (-128))

theorem tryFactorization_xor_lshr_ashr_thm.extracted_1._1 : ∀ (x : BitVec 32), ¬(x ≥ ↑32 ∨ x ≥ ↑32) → x ≥ ↑32 → False

theorem test4_thm (e : IntW 32) :
  sext 64 (LLVM.udiv e (const? 32 3)) ⊑ zext 64 (LLVM.udiv e (const? 32 3)) { «nneg» := true }

theorem demo3 (x y : BitVec 8) :
    (x ^^^ -1#8 ||| 7#8) ^^^ 12#8 = x &&& BitVec.ofInt 8 (-8) ^^^ BitVec.ofInt 8 (-13)

theorem t0_basic_proof : t0_basic_before ⊑ t0_basic_after

theorem and_sel_op0_proof : and_sel_op0_before ⊑ and_sel_op0_after

theorem shl_and_and_proof : shl_and_and_before ⊑ shl_and_and_after

theorem e_192 :
     - 7 *  ~~~(x &&&  ~~~x) + 1 * (x ^^^ y) + 2 *  ~~~(x &&&  ~~~y) + 4 *  ~~~(x ||| y) + 3 *  ~~~(x |||  ~~~y) + 5 * (x &&& y) =  - 1 *  ~~~x - 6 * (x &&&  ~~~y)

theorem test_invert_demorgan_and3_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 + (x ^^^ -1#32) &&& 4095#32 == 0#32) = ofBool (x - x_1 &&& 4095#32 == 4095#32)

theorem t0_commutative_proof : t0_commutative_before ⊑ t0_commutative_after

theorem test11_proof : test11_before ⊑ test11_after

theorem ashrsgt_02_09_exact_thm (e : IntW 4) :
  icmp IntPred.sgt (ashr e (const? 4 2) { «exact» := true }) (const? 4 (-7)) ⊑ const? 1 1

theorem eq_smin4_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (x <ₛ x_1 + 3#32) = 1#1 → ofBool (x_1 + 3#32 == x) = ofBool (x_1 + 3#32 ≤ₛ x)

theorem ashr_1_proof : ashr_1_before ⊑ ashr_1_after

theorem lshr_and_proof : lshr_and_before ⊑ lshr_and_after

theorem and_nosignbit_select_ashr_proof : and_nosignbit_select_ashr_before ⊑ and_nosignbit_select_ashr_after

@[simp]
lemma swapLastTwo_eq2 : swapLastTwo (n := n) n = Fin.last (n+1)

theorem or_or_and_pat4_proof : or_or_and_pat4_before ⊑ or_or_and_pat4_after

theorem test5_thm (e : IntW 32) : trunc 32 (lshr (zext 128 e) (const? 128 16)) ⊑ lshr e (const? 32 16)

theorem udiv_shl_pair1_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 32),
  ¬(True ∧ x_2 <<< x_1 >>> x_1 ≠ x_2 ∨ x_1 ≥ ↑32 ∨ True ∧ x_2 <<< x >>> x ≠ x_2 ∨ x ≥ ↑32 ∨ x_2 <<< x = 0) →
    True ∧ 1#32 <<< x_1 >>> x_1 ≠ 1#32 ∨ x_1 ≥ ↑32 ∨ x ≥ ↑32 → False

theorem e_815 :
     - 5 * (z |||  ~~~t) + 1 *  ~~~t + 4 *  ~~~(z ||| t) + 9 * (z &&&  ~~~t) + 10 * (z &&& t) = 5 * z

theorem select_icmp_ne_0_and_4096_or_4096_proof : select_icmp_ne_0_and_4096_or_4096_before ⊑ select_icmp_ne_0_and_4096_or_4096_after

theorem band_lor_left1_proof : band_lor_left1_before ⊑ band_lor_left1_after

@[simp] theorem Hom.consMap_last {Γ Γ' : Ctxt Ty} (f : Hom Γ Γ') {t : Ty} :
    (f.consMap (Ctxt.Var.last Γ t)) = Ctxt.Var.last Γ' t

@[simp] theorem mapValuation_nil (mapOut : MatchVarResult lets v .nil w mapIn) (V) :
    mapOut.val.mapValuation V w = V v

@[simp] theorem inf_pure_eq      : min e pure = pure

theorem e_513 :
     - 1 *  ~~~(x &&& y) - 1 *  ~~~(x &&&  ~~~y) - 11 * (x |||  ~~~y) + 1 * (x ^^^ y) + 1 *  ~~~y - 1 * y + 17 *  ~~~(x ||| y) + 10 * (x &&&  ~~~y) + 13 * (x &&& y) =  - 7 *  ~~~(x |||  ~~~y) + 5 *  ~~~x

theorem badimm3_thm.extracted_1._2 : ∀ (x : BitVec 16),
  ¬(7#8 ≥ ↑8 ∨ 8#16 ≥ ↑16) →
    ofBool ((truncate 8 x).sshiftRight' 7#8 == truncate 8 (x >>> 8#16)) = 1#1 →
      ¬ofBool (x + 128#16 <ᵤ 256#16) = 1#1 →
        ¬(14#16 ≥ ↑16 ∨ True ∧ signExtend 16 (truncate 8 (x.sshiftRight' 14#16)) ≠ x.sshiftRight' 14#16) →
          truncate 8 x = truncate 8 (x.sshiftRight' 14#16) ^^^ 127#8

theorem e_1440 :
     - 1 * z - 5 *  ~~~z + 5 *  ~~~(z &&&  ~~~z) + 3 * t + 3 *  ~~~t - 1 *  ~~~(z ^^^ t) - 6 * (z ^^^ t) - 2 *  ~~~(z ||| t) + 3 *  ~~~(z |||  ~~~t) - 6 * (z &&& t) = 1 * (z &&&  ~~~t)

theorem test2_proof : test2_before ⊑ test2_after

theorem n2_thm.extracted_1._1 : ∀ (x : BitVec 16),
  ¬15#32 ≥ ↑32 → zeroExtend 32 x <<< 15#32 &&& BitVec.ofInt 32 (-2147483648) = 0#32

theorem test18_proof : test18_before ⊑ test18_after

theorem getElem?_eq_toList_getElem? {i : Nat} : Γ[i]? = Γ.toList[i]?

theorem nonexact_ashr_eq_noexactdiv_proof : nonexact_ashr_eq_noexactdiv_before ⊑ nonexact_ashr_eq_noexactdiv_after

theorem icmp_slt_0_and_icmp_sgt_neg1_i64_thm.extracted_1._1 : ∀ (x : BitVec 64),
  ¬63#64 ≥ ↑64 → x >>> 63#64 &&& zeroExtend 64 (ofBool (-1#64 <ₛ x)) = 0#64

theorem substitute_constant_and_eq_ugt_swap_thm (e e_1 : IntW 8) :
  LLVM.and (icmp IntPred.ugt e_1 e) (icmp IntPred.eq e (const? 8 42)) ⊑
    LLVM.and (icmp IntPred.eq e (const? 8 42)) (icmp IntPred.ugt e_1 (const? 8 42))

theorem lshrugt_03_00_exact_thm (e : IntW 4) :
  icmp IntPred.ugt (lshr e (const? 4 3) { «exact» := true }) (const? 4 0) ⊑
    icmp IntPred.ne e (const? 4 0)

theorem select_icmp_eq_0_and_1_or_1_thm.extracted_1._2 : ∀ (x : BitVec 32) (x_1 : BitVec 64),
  ¬ofBool (x_1 &&& 1#64 == 0#64) = 1#1 → x ||

theorem test12_proof : test12_before ⊑ test12_after

theorem self_EqualUpTo_ofBitVecZext_toBitVec (x : BitStream) :
  EqualUpTo w x (BitStream.ofBitVecZext (x.toBitVec w))

theorem main7f_logical_thm.extracted_1._8 : ∀ (x x_1 x_2 x_3 : BitVec 32),
  ¬ofBool (x_3 &&& x_2 == x_1 &&& (x_3 &&& x_2)) = 1#1 →
    ¬ofBool (x_3 &&& x_2 != x_1 &&& (x_3 &&& x_2)) = 1#1 → ¬0#1 = 1#1 → False

theorem ugt_swap_and_min_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), ofBool (x_1 <ᵤ x) &&& ofBool (x == 0#8) = 0#1

theorem neg_nsw_slt_1_proof : neg_nsw_slt_1_before ⊑ neg_nsw_slt_1_after

theorem xor_or_xor_common_op_commute4_proof : xor_or_xor_common_op_commute4_before ⊑ xor_or_xor_common_op_commute4_after

theorem ne_umax2_thm (e e_1 : IntW 32) :
  icmp IntPred.ne (select (icmp IntPred.ugt e_1 e) e_1 e) e ⊑ icmp IntPred.ult e e_1

theorem shift_xor_multiuse_cmp_with_xor_thm (e e_1 e_2 e_3 : IntW 32) :
  mul (select (icmp IntPred.ne (const? 32 0) (LLVM.and e_3 (const? 32 4096))) e_2 (LLVM.xor e_2 (const? 32 2048)))
      (select (icmp IntPred.ne (const? 32 0) (LLVM.and e_3 (const? 32 4096))) e_1 e) ⊑
    mul
      (select (icmp IntPred.eq (LLVM.and e_3 (const? 32 4096)) (const? 32 0)) (LLVM.xor e_2 (const? 32 2048)) e_2)
      (select (icmp IntPred.eq (LLVM.and e_3 (const? 32 4096)) (const? 32 0)) e e_1)

theorem test24_thm (e : IntW 32) :
  zext 32 (icmp IntPred.eq (lshr (LLVM.and e (const? 32 4)) (const? 32 2)) (const? 32 0)) ⊑
    LLVM.xor (LLVM.and (lshr e (const? 32 2)) (const? 32 1)) (const? 32 1)

theorem ashrsgt_02_04_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 2#4 <<< 2#4 ≠ x ∨ 2#4 ≥ ↑4) → ofBool (4#4 <ₛ x.sshiftRight' 2#4) = 0#1

theorem test17_thm (e : IntW 1) : sub (const? 32 0) (sext 32 e) ⊑ zext 32 e

theorem t_1_10_thm (e : IntW 8) :
  LLVM.and (select (icmp IntPred.ugt e (const? 8 1)) e (const? 8 1)) (const? 8 10) ⊑
    LLVM.and e (const? 8 10)

theorem icmp_select_implied_cond_relational_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ofBool (10#8 <ᵤ x) = 1#1 → ofBool (10#8 <ᵤ x) = 1#1

theorem e_255 :
    3 *  ~~~y + 1 * y + 1 *  ~~~(x ^^^ y) + 2 *  ~~~(x &&& y) - 6 *  ~~~(x ||| y) - 5 * (x &&&  ~~~y) - 2 * (x &&& y) = 3 *  ~~~(x |||  ~~~y)

theorem tryFactorization_add_nuw_mul_nuw_proof : tryFactorization_add_nuw_mul_nuw_before ⊑ tryFactorization_add_nuw_mul_nuw_after

theorem or_and1_commuted_proof : or_and1_commuted_before ⊑ or_and1_commuted_after

theorem slt_or_not_max_commute_thm (e e_1 : IntW 8) :
  LLVM.or (icmp IntPred.ne e_1 (const? 8 127)) (icmp IntPred.slt e_1 e) ⊑
    icmp IntPred.ne e_1 (const? 8 127)

theorem add_of_mul_thm (e e_1 e_2 : IntW 8) :
  add (mul e_2 e_1 { «nsw» := true, «nuw» := false }) (mul e_2 e { «nsw» := true, «nuw» := false })
      { «nsw» := true, «nuw» := false } ⊑
    mul e_2 (add e_1 e)

@[simp]
theorem Eqn.denoteFin_width_zero  (es : Eqn) (env : EnvFin 0 es.numVars) :
    Eqn.denoteFin es env = 0

theorem sgt_smax2_proof : sgt_smax2_before ⊑ sgt_smax2_after

theorem e_1427 :
     - 11 *  ~~~(z &&& t) - 11 *  ~~~(z &&&  ~~~t) - 2 * z - 5 * t - 1 *  ~~~(z ^^^ t) + 1 * (z ||| t) + 23 *  ~~~(z ||| t) + 25 *  ~~~(z |||  ~~~t) + 11 * (z &&&  ~~~t) + 18 * (z &&& t) =  - 1 * (z ^^^ t)

theorem eq_optimized_highbits_cmp_proof : eq_optimized_highbits_cmp_before ⊑ eq_optimized_highbits_cmp_after

@[simp]
theorem union'_accepts (M : NFA α σ) (N : NFA α ς) (hcM: M.Complete) (hcN : N.Complete) :
    (M.union' N).accepts = M.accepts ∪ N.accepts

theorem slt_swap_or_not_max_commute_proof : slt_swap_or_not_max_commute_before ⊑ slt_swap_or_not_max_commute_after

@[simp]
theorem LLVM.lshr?_eq_value {a b : BitVec w} (hb : b < w) :
    LLVM.lshr? a b = .value (BitVec.ushiftRight a b.toNat)

theorem mul_splat_fold_no_nuw_proof : mul_splat_fold_no_nuw_before ⊑ mul_splat_fold_no_nuw_after

theorem e_353 :
     - 2 *  ~~~(x &&&  ~~~x) - 1 * (x ||| y) + 2 *  ~~~x + 7 * y - 1 *  ~~~(x ||| y) - 7 *  ~~~(x |||  ~~~y) + 3 * (x &&&  ~~~y) - 5 * (x &&& y) =  - 1 *  ~~~(x &&&  ~~~y)

theorem ashrslt_02_14_exact_proof : ashrslt_02_14_exact_before ⊑ ashrslt_02_14_exact_after

theorem ashrsgt_03_03_thm (e : IntW 4) : icmp IntPred.sgt (ashr e (const? 4 3)) (const? 4 3) ⊑ const? 1 0

theorem shl_bad_sub_i32_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬32#32 - x ≥ ↑32 → True ∧ 1#32 <<< (32#32 - x) >>> (32#32 - x) ≠ 1#32 ∨ 32#32 - x ≥ ↑32 → False

theorem addhshlhsdivhscalar0_proof : addhshlhsdivhscalar0_before ⊑ addhshlhsdivhscalar0_after

theorem test5_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬30000#32 = 0 →
    ¬(30000#32 = 0 ∨ True ∧ (x % 30000#32).msb = true) → signExtend 64 (x % 30000#32) = zeroExtend 64 (x % 30000#32)

theorem test4_proof : test4_before ⊑ test4_after

theorem not_logicalOr_not_op1_thm.extracted_1._2 : ∀ (x : BitVec 1), x = 1#1 → ¬x ^^^ 1#1 = 1#1 → 1#1 ^^^ 1#1 = 0#1

theorem test8_thm (e : IntW 27) : sub (mul (const? 27 9) e) e ⊑ shl e (const? 27 3)

theorem factorize4_proof : factorize4_before ⊑ factorize4_after

theorem test_ashr__exact_is_safe_thm (e : IntW 64) (e_1 : IntW 32) :
  select (icmp IntPred.eq (LLVM.and e_1 (const? 32 (-2147483588))) (const? 32 0)) e
      (ashr e (zext 64 (ashr (LLVM.and e_1 (const? 32 (-2147483588))) (const? 32 2)))) ⊑
    ashr e (zext 64 (LLVM.and (ashr e_1 (const? 32 2)) (const? 32 (-536870897))) { «nneg» := true })

theorem urem_illegal_type_thm (e e_1 : IntW 9) : urem (zext 32 e_1) (zext 32 e) ⊑ zext 32 (urem e_1 e)

theorem or_not_xor_common_op_commute2_proof : or_not_xor_common_op_commute2_before ⊑ or_not_xor_common_op_commute2_after

theorem test_thm.extracted_1._1 : ∀ (x : BitVec 8), ¬8#32 ≥ ↑32 → 7#8 ≥ ↑8 → False

theorem test1_logical_proof : test1_logical_before ⊑ test1_logical_after

@[bv_automata_nnf↓] theorem not_and (p q : Prop) : (¬(p ∧ q)) = (¬p ∨ ¬q)

theorem icmp_eq_sext_ne_zero_proof : icmp_eq_sext_ne_zero_before ⊑ icmp_eq_sext_ne_zero_after

theorem or_not_and_commute5_thm (e e_1 e_2 : IntW 32) :
  LLVM.or
      (LLVM.and (LLVM.sdiv (const? 32 42) e_2) (LLVM.xor (LLVM.or (LLVM.sdiv (const? 32 42) e_1) e) (const? 32 (-1))))
      (LLVM.and (LLVM.xor (LLVM.or (LLVM.sdiv (const? 32 42) e_1) (LLVM.sdiv (const? 32 42) e_2)) (const? 32 (-1))) e) ⊑
    LLVM.and (LLVM.xor e (LLVM.sdiv (const? 32 42) e_2))
      (LLVM.xor (LLVM.sdiv (const? 32 42) e_1) (const? 32 (-1)))

theorem sext_multi_uses_thm.extracted_1._1 : ∀ (x : BitVec 64) (x_1 : BitVec 1) (x_2 : BitVec 64),
  x_1 = 1#1 → x_2 * signExtend 64 x_1 + ((x ^^^ signExtend 64 x_1) - signExtend 64 x_1) = 0#64 - (x_2 + x)

theorem bools_logical_commute1_thm.extracted_1._14 : ∀ (x x_1 x_2 : BitVec 1),
  ¬x_2 = 1#1 → ¬x_1 = 1#1 → ¬0#1 = 1#1 → 0#1 = x_2

theorem t0_thm (e e_1 e_2 e_3 : IntW 8) (e_4 : IntW 1) :
  select (select (LLVM.xor e_4 (const? 1 1)) (const? 1 1) (icmp IntPred.eq e_3 e_2)) e_1 e ⊑
    select (select e_4 (icmp IntPred.ne e_3 e_2) (const? 1 0)) e e_1

theorem main7e_logical_thm.extracted_1._14 : ∀ (x x_1 x_2 x_3 x_4 : BitVec 32),
  ¬ofBool (x_4 &&& x_3 &&& x_2 == x_4 &&& x_3) = 1#1 →
    ofBool (x_4 &&& x_3 &&& x_2 != x_4 &&& x_3) = 1#1 → ¬0#1 = 1#1 → 1#32 = zeroExtend 32 1#1

theorem test5_proof : test5_before ⊑ test5_after

theorem select_icmp_eq_0_and_1_xor_1_proof : select_icmp_eq_0_and_1_xor_1_before ⊑ select_icmp_eq_0_and_1_xor_1_after

theorem n10_lshr_ashr_proof : n10_lshr_ashr_before ⊑ n10_lshr_ashr_after

theorem testi32i8_proof : testi32i8_before ⊑ testi32i8_after

theorem t2_thm (e : IntW 8) :
  icmp IntPred.slt (sub (const? 8 0) e { «nsw» := true, «nuw» := false }) e ⊑
    icmp IntPred.sgt e (const? 8 0)

theorem test2_thm (e e_1 : IntW 32) : LLVM.xor (LLVM.and e_1 e) (LLVM.or e_1 e) ⊑ LLVM.xor e_1 e

theorem add_nosignbit_shl_thm (e : IntW 32) :
  shl (add e (const? 32 2147418112)) (const? 32 8) ⊑ add (shl e (const? 32 8)) (const? 32 (-16777216))

theorem e_1690 :
     - 7 *  ~~~(a ^^^ c) + 4 * (b ||| c) + 2 * (b ^^^ ( ~~~a &&& ( ~~~b ||| c))) - 1 * ((a &&& b) ^^^ (a ^^^ (b &&& c))) + 11 * ((b &&& c) ||| ( ~~~a &&& (b ||| c))) + 2 * ((a &&& b) ^^^  ~~~(a ^^^ ( ~~~b ||| c))) - 6 * (c &&& (a ||| b)) - 3 * ((a ||| b) &&&  ~~~(a ^^^ (b ^^^ c))) - 7 * (c ^^^ (a ||| (b &&& c))) + 2 * ((b &&&  ~~~c) |||  ~~~(a ||| ( ~~~b &&& c))) + 3 * (a ||| (b &&& c)) + 5 *  ~~~b - 3 * (c ^^^ (a &&& (b ||| c))) - 11 * (b ^^^ ( ~~~a &&& (b ^^^ c))) - 5 *  ~~~(b ||| c) + 5 * ((b &&&  ~~~c) ^^^ ( ~~~a ||| (b ^^^ c))) - 3 * (b ^^^  ~~~( ~~~a ||| ( ~~~b &&& c))) + 2 *  ~~~(a ^^^ (b &&& c)) - 1 * ( ~~~a &&& (b ||| c)) - 5 * ( ~~~a ||| (b ||| c)) - 3 *  ~~~(a ||| (b ||| c)) - 7 *  ~~~(a ||| ( ~~~b ||| c)) + 2 *  ~~~( ~~~a ||| (b ||| c)) + 12 *  ~~~( ~~~a ||| ( ~~~b ||| c)) - 6 * ( ~~~a &&& ( ~~~b &&& c)) + 4 * ( ~~~a &&& (b &&& c)) - 1 * (a &&& ( ~~~b &&& c)) + 8 * (a &&& (b &&& c)) =  - 4 * ( ~~~b ||| (a ^^^ c))

theorem sgt_and_min_logical_proof : sgt_and_min_logical_before ⊑ sgt_and_min_logical_after

theorem test3_thm (e e_1 : IntW 32) :
  LLVM.xor (LLVM.and (LLVM.xor e_1 (const? 32 (-1))) (LLVM.xor e (const? 32 (-1)))) (const? 32 (-1)) ⊑
    LLVM.or e_1 e

theorem pr4917_3_thm (e e_1 : IntW 32) :
  select (icmp IntPred.ugt (mul (zext 64 e_1) (zext 64 e)) (const? 64 4294967295)) (mul (zext 64 e_1) (zext 64 e))
      (const? 64 111) ⊑
    select
      (icmp IntPred.ugt (mul (zext 64 e_1) (zext 64 e) { «nsw» := false, «nuw» := true }) (const? 64 4294967295))
      (mul (zext 64 e_1) (zext 64 e) { «nsw» := false, «nuw» := true }) (const? 64 111)

theorem select_ashr_icmp_bad_thm.extracted_1._3 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 == 1#32) = 1#1 → ¬x_2 ≥ ↑32 → 1#32 ≥ ↑32 → False

theorem e_409 :
     - 7 * (x ||| y) - 3 * y - 3 *  ~~~(x ^^^ y) - 1 *  ~~~x + 10 *  ~~~(x ||| y) + 17 *  ~~~(x |||  ~~~y) + 15 * (x &&&  ~~~y) + 21 * (x &&& y) = 6 *  ~~~(x &&&  ~~~x) + 2 * x

theorem add_congr (e1 : a ≈ʷ b) (e2 : c ≈ʷ d) : (a + c) ≈ʷ (b + d)

theorem e_1324 :
     - 1 * z - 1 * t + 7 * (z ^^^ t) - 11 *  ~~~(z &&&  ~~~z) - 11 *  ~~~(z ^^^ t) + 3 *  ~~~(z &&& t) - 3 * (z ||| t) + 19 *  ~~~(z ||| t) + 5 * (z &&&  ~~~t) =  - 5 *  ~~~(z |||  ~~~t) - 27 * (z &&& t)

theorem not_ashr_not_bitwidth_mask_proof : not_ashr_not_bitwidth_mask_before ⊑ not_ashr_not_bitwidth_mask_after

theorem e_287 :
    2 *  ~~~(x &&&  ~~~y) + 1 * y - 5 *  ~~~(x ^^^ y) - 3 *  ~~~(x ||| y) - 10 *  ~~~(x |||  ~~~y) + 1 * (x &&&  ~~~y) + 3 * (x &&& y) = 1 * (x |||  ~~~y) - 7 *  ~~~x

theorem eq_umax4_proof : eq_umax4_before ⊑ eq_umax4_after

theorem test_add_nsw__none_are_safe_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x == 3#32) = 1#1 → 4#32 = x + 1#32

theorem ashr_ugt_4_proof : ashr_ugt_4_before ⊑ ashr_ugt_4_after

theorem lshrult_02_03_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 2#4 <<< 2#4 ≠ x ∨ 2#4 ≥ ↑4) → ofBool (x >>> 2#4 <ᵤ 3#4) = ofBool (x != BitVec.ofInt 4 (-4))

theorem ashrsgt_01_09_proof : ashrsgt_01_09_before ⊑ ashrsgt_01_09_after

theorem e_924 :
    2 *  ~~~(z &&& t) + 4 * t - 7 *  ~~~(z ||| t) - 11 *  ~~~(z |||  ~~~t) - 2 * (z &&&  ~~~t) - 9 * (z &&& t) =  - 5 *  ~~~(z &&&  ~~~t)

theorem PR42691_6_logical_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x <ᵤ BitVec.ofInt 32 (-2147483647)) = 1#1 → 1#1 = ofBool (x + 1#32 <ᵤ BitVec.ofInt 32 (-2147483646))

theorem PR21222_proof : PR21222_before ⊑ PR21222_after

theorem PR1817_1_logical_proof : PR1817_1_logical_before ⊑ PR1817_1_logical_after

theorem select_icmp_ne_0_and_32_and_not_4096_thm (e e_1 : IntW 32) :
  select (icmp IntPred.ne (const? 32 0) (LLVM.and e_1 (const? 32 32))) e (LLVM.and e (const? 32 (-4097))) ⊑
    select (icmp IntPred.eq (LLVM.and e_1 (const? 32 32)) (const? 32 0)) (LLVM.and e (const? 32 (-4097)))
      e

theorem or_logic_or_logic_and_6_thm.extracted_1._2 : ∀ (x x_1 : BitVec 1),
  ¬x_1 ||

theorem trunc_shl_16_i32_i64_proof : trunc_shl_16_i32_i64_before ⊑ trunc_shl_16_i32_i64_after

theorem e_311 :
     - 2 *  ~~~y + 2 * (x |||  ~~~y) - 6 * x + 1 *  ~~~x - 1 *  ~~~(x ||| y) - 1 *  ~~~(x |||  ~~~y) + 6 * (x &&&  ~~~y) =  - 4 * (x &&& y)

theorem e_558 :
    11 *  ~~~x + 2 * y - 1 *  ~~~(x &&& y) + 2 * x - 7 * (x ||| y) - 9 *  ~~~(x ||| y) - 9 *  ~~~(x |||  ~~~y) + 1 * (x &&&  ~~~y) + 4 * (x &&& y) = 1 *  ~~~(x &&&  ~~~y) - 5 * (x ^^^ y)

theorem shl_lshr_and_exact_proof : shl_lshr_and_exact_before ⊑ shl_lshr_and_exact_after

theorem slt_and_not_max_commute_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ofBool (x_1 != 127#8) &&& ofBool (x_1 <ₛ x) = ofBool (x_1 <ₛ x)

theorem ashr_ult_3_proof : ashr_ult_3_before ⊑ ashr_ult_3_after

theorem bools_logical_commute0_and1_proof : bools_logical_commute0_and1_before ⊑ bools_logical_commute0_and1_after

@[simp, defeq] theorem entries_eqvVar : (eqvVarLeft mapOut).val.entries = mapOut.val.entries

theorem test9_logical_thm (e : IntW 32) :
  select (icmp IntPred.ne (LLVM.and e (const? 32 1073741824)) (const? 32 0))
      (icmp IntPred.sgt e (const? 32 (-1))) (const? 1 0) ⊑
    icmp IntPred.eq (LLVM.and e (const? 32 (-1073741824))) (const? 32 1073741824)

theorem nonexact_ashr_eq_exactdiv_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬x ≥ ↑8 → ofBool ((BitVec.ofInt 8 (-80)).sshiftRight' x == BitVec.ofInt 8 (-5)) = ofBool (x == 4#8)

theorem e_740 :
    7 * (x |||  ~~~y) - 3 *  ~~~(x &&&  ~~~y) - 1 *  ~~~(x &&&  ~~~x) - 11 * x - 1 *  ~~~y + 3 * y + 2 * (x ||| y) - 7 *  ~~~x + 5 *  ~~~(x ||| y) + 6 *  ~~~(x |||  ~~~y) + 4 * (x &&&  ~~~y) =  - 3 * (x &&& y)

theorem test37_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬(True ∧ (x * 7#32 &&& 240#32).msb = true) →
    zeroExtend 64 x * 7#64 &&& 240#64 = zeroExtend 64 (x * 7#32 &&& 240#32)

theorem and_ranges_no_overlap_thm (e : IntW 8) :
  LLVM.and (LLVM.and (icmp IntPred.uge e (const? 8 5)) (icmp IntPred.ule e (const? 8 10)))
      (LLVM.and (icmp IntPred.uge e (const? 8 11)) (icmp IntPred.ule e (const? 8 20))) ⊑
    const? 1 0

theorem e_2065 :
    2 * (f ^^^ ( ~~~d ||| ( ~~~e &&& f))) - 7 * (f ^^^  ~~~(d &&& ( ~~~e &&& f))) - 6 * ((e &&&  ~~~f) |||  ~~~( ~~~d ||| ( ~~~e &&& f))) - 5 * (d ||| (e &&& f)) - 1 * (e ^^^ (d ||| (e &&& f))) - 1 * (f ^^^  ~~~(d ||| ( ~~~e &&& f))) + 1 *  ~~~( ~~~d &&& ( ~~~e ||| f)) - 7 * ( ~~~(d ^^^ e) &&& (d ^^^ f)) - 6 *  ~~~(d ^^^ ( ~~~e ||| f)) - 2 * ((d &&& e) |||  ~~~(d ^^^ (e ^^^ f))) + 4 * ((d ^^^ e) &&& (d ^^^ f)) + 1 * (e &&&  ~~~(d &&&  ~~~f)) - 1 * ( ~~~(d &&&  ~~~e) &&& (e ^^^ f)) - 1 * (f ^^^ (d &&& (e ||| f))) + 1 *  ~~~( ~~~d &&& (e ||| f)) - 1 * ( ~~~(d |||  ~~~e) |||  ~~~(d ^^^ (e ^^^ f))) - 3 * (f ||| (d ^^^ e)) + 7 * ((d &&& e) ^^^ (e |||  ~~~f)) - 3 * (e |||  ~~~f) - 1 * ((d |||  ~~~e) &&& (e ^^^ f)) + 5 * ( ~~~e |||  ~~~(d ^^^ f)) + 2 * ((d &&& e) ^^^  ~~~(d ^^^ ( ~~~e ||| f))) + 5 * ((e &&&  ~~~f) ^^^ (d ||| (e ^^^ f))) + 2 *  ~~~(d ||| f) + 2 * ((d &&& f) ^^^ (d ^^^ (e &&& f))) - 1 * (d ^^^ (e ||| f)) - 7 * (e ^^^  ~~~(d ||| (e ^^^ f))) + 1 * (f ^^^  ~~~( ~~~d &&& (e ||| f))) + 2 * (e ^^^  ~~~( ~~~d ||| (e &&& f))) - 11 * (f ^^^  ~~~( ~~~d ||| (e &&& f))) - 3 * (e ^^^  ~~~(d &&& ( ~~~e &&& f))) + 2 *  ~~~(d ||| (e ||| f)) + 11 *  ~~~(d ||| ( ~~~e ||| f)) + 12 *  ~~~( ~~~d ||| (e ||| f)) + 41 *  ~~~( ~~~d ||| ( ~~~e ||| f)) + 18 * ( ~~~d &&& ( ~~~e &&& f)) + 6 * ( ~~~d &&& (e &&& f)) + 7 * (d &&& ( ~~~e &&& f)) + 21 * (d &&& (e &&& f)) =  - 4 *  ~~~( ~~~d &&& ( ~~~e &&& f))

@[simp] theorem carry'_zero
    (initCarry : Bool) (x y : BitStream) :
  (carry' initCarry x y 0) = initCarry

theorem e_241 :
     - 2 *  ~~~(x ^^^ y) + 4 * (x |||  ~~~y) - 11 *  ~~~(x &&&  ~~~y) - 5 *  ~~~y + 1 * (x ||| y) + 14 *  ~~~(x ||| y) + 10 *  ~~~(x |||  ~~~y) =  - 8 * (x &&& y)

theorem e_1344 :
     - 1 *  ~~~(z &&& t) - 6 * (z ||| t) - 1 *  ~~~(z ^^^ t) - 1 * (z |||  ~~~t) - 1 *  ~~~(z &&&  ~~~t) - 11 * z - 3 * t + 15 *  ~~~(z ||| t) + 11 *  ~~~(z |||  ~~~t) + 23 * (z &&& t) =  - 30 * (z &&&  ~~~t) + 11 *  ~~~t

theorem shift_no_xor_multiuse_xor_thm.extracted_1._4 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (x_1 &&& 1#32 == 0#32) = 1#1 →
    ¬1#32 ≥ ↑32 → (x ^^^ 2#32) * (x ^^^ 2#32) = (x ^^^ x_1 <<< 1#32 &&& 2#32) * (x ^^^ 2#32)

theorem add_shl_same_amount_partial_nsw2_proof : add_shl_same_amount_partial_nsw2_before ⊑ add_shl_same_amount_partial_nsw2_after

theorem t1_sub_of_falseval_thm.extracted_1._1 : ∀ (x : BitVec 8) (x_1 : BitVec 1), ¬x_1 = 1#1 → x - x = 0#8

theorem test4_thm.extracted_1._1 : ∀ (x x_1 : BitVec 108), x_1 - (0#108 - x) = x_1 + x

theorem alive_Select_741 : forall (w : Nat) (A : Bitvec 1)
, TSSA.eval
  (Op := Op) (e := e)
  (i := TSSAIndex.STMT (UserType.base (BaseType.bitvec 1)))
  [dsl_bb|
  ^bb
  %v0 := unit: ;
  %v1 := op:const (A) %v0;
  %v2 := op:const (0) %v0;
  %v3 := pair:%v1 %v2;
  %v4 := op:icmp sgt  w %v3;
  %v5 := op:const (0) %v0;
  %v6 := pair:%v5 %v1;
  %v7 := op:sub 1 %v6;
  %v8 := triple:%v4 %v7 %v1;
  %v9 := op:select w %v8;
  %v10 := op:const (-1) %v0;
  %v11 := pair:%v9 %v10;
  %v12 := op:icmp sgt  w %v11;
  %v13 := op:const (0) %v0;
  %v14 := pair:%v13 %v9;
  %v15 := op:sub w %v14;
  %v16 := triple:%v12 %v15 %v9;
  %v17 := op:select w %v16
  dsl_ret %v17
  ]  ⊑
  TSSA.eval
  (Op := Op) (e := e)
  (i := TSSAIndex.STMT (UserType.base (BaseType.bitvec 1)))
  [dsl_bb|
  ^bb
  %v0 := unit: ;
  %v1 := op:const (A) %v0;
  %v2 := op:const (0) %v0;
  %v3 := pair:%v1 %v2;
  %v4 := op:icmp sgt  1 %v3;
  %v5 := op:const (0) %v0;
  %v6 := pair:%v5 %v1;
  %v7 := op:sub 1 %v6;
  %v8 := triple:%v4 %v7 %v1;
  %v9 := op:select 1 %v8;
  %v10 := op:const (-1) %v0;
  %v11 := pair:%v9 %v10;
  %v12 := op:icmp sgt  1 %v11;
  %v13 := op:const (0) %v0;
  %v14 := pair:%v13 %v9;
  %v15 := op:sub 1 %v14;
  %v16 := triple:%v4 %v7 %v1;
  %v17 := op:select 1 %v16
  dsl_ret %v17
  ]

theorem lshr_pow2_slt_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬x ≥ ↑8 → True ∧ BitVec.ofInt 8 (-128) >>> x <<< x ≠ BitVec.ofInt 8 (-128) ∨ x ≥ ↑8 → False

theorem ashrslt_03_08_exact_proof : ashrslt_03_08_exact_before ⊑ ashrslt_03_08_exact_after

theorem test1_thm (e : IntW 17) : mul e (const? 17 1024) ⊑ shl e (const? 17 10)

@[simp] theorem Valuation.append_inj_left {V V' : Γ.Valuation} {W : Δ.Valuation} :
    (V ++ W) = (V' ++ W) ↔ V = V'

theorem main4e_like_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 &&& x_1 == x_1) &&& ofBool (x_2 &&& x == x) = 1#1 →
    0#32 = zeroExtend 32 (ofBool (x_2 &&& (x_1 ||| x) != x_1 ||| x))

theorem n2_thm.extracted_1._6 : ∀ (x x_1 : BitVec 32) (x_2 : BitVec 1),
  x_2 = 1#1 → ¬x_2 ^^^ 1#1 = 1#1 → 1#1 ^^^ 1#1 = 0#1

theorem e_23 :
     - 2 *  ~~~(x ^^^ y) - 2 *  ~~~(x &&& y) + 1 *  ~~~(x |||  ~~~y) + 1 * (x &&&  ~~~y) + 1 * (x &&& y) =  - 4 *  ~~~(x ||| y) - 1 * (x ||| y)

theorem slt_smin1_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 <ₛ x) = 1#1 → ofBool (x_1 <ₛ x_1) = ofBool (x <ₛ x_1)

theorem lshr_lshr_pow2_const_negative_nopow2_2_proof : lshr_lshr_pow2_const_negative_nopow2_2_before ⊑ lshr_lshr_pow2_const_negative_nopow2_2_after

theorem uge_swap_or_max_logical_proof : uge_swap_or_max_logical_before ⊑ uge_swap_or_max_logical_after

theorem lor_bor_left1_thm (e e_1 : IntW 1) : LLVM.or (select e_1 (const? 1 1) e) e_1 ⊑ select e_1 (const? 1 1) e

theorem trunc_proof : trunc_before ⊑ trunc_after

@[simp]
theorem EnvFin.eq_elim0 (envFin : EnvFin w 0) : envFin = fun i => i.elim0

theorem e_191 :
     - 11 *  ~~~(x ^^^ y) + 11 * y + 5 * (x ||| y) + 4 *  ~~~(x ||| y) - 12 * (x &&&  ~~~y) - 5 * (x &&& y) =  - 7 *  ~~~(x &&& y) + 23 *  ~~~(x |||  ~~~y)

theorem t7_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬(True ∧ (0#8).ssubOverflow x = true) → ofBool (0#8 - x ≤ᵤ x) = ofBool (x <ₛ 1#8)

lemma neg_eq {M : NFA' n} : M.neg = M.determinize.flipAccept

theorem select_mul_icmp_bad_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 == 3#32) = 1#1 → x_1 * x = x * 3#32

theorem t0_commutative_thm.extracted_1._9 : ∀ (x : BitVec 8) (x_1 : BitVec 1) (x_2 x_3 : BitVec 8),
  ¬ofBool (x_3 == x_2) = 1#1 → ¬ofBool (x_3 != x_2) = 1#1 → ¬0#1 = 1#1 → ¬x_1 = 1#1 → False

theorem shl_add_nuw_proof : shl_add_nuw_before ⊑ shl_add_nuw_after

theorem shl_add_nuw_thm (e e_1 : IntW 8) :
  shl (LLVM.and e_1 (const? 8 63)) (LLVM.and e (const? 8 2)) ⊑
    shl (LLVM.and e_1 (const? 8 63)) (LLVM.and e (const? 8 2)) { «nsw» := false, «nuw» := true }

theorem e_2306 :
    5 * ((d |||  ~~~e) &&& (d ^^^ (e ^^^ f))) - 6 * (e &&&  ~~~(d ^^^ f)) - 11 * ((d &&& e) |||  ~~~(e ||| f)) - 2 * ( ~~~(d ^^^ e) &&& (d ^^^ f)) + 1 * ((d ^^^ e) &&& (d ^^^ f)) + 1 * (e ^^^ (d ||| (e ||| f))) - 5 *  ~~~(d &&&  ~~~f) + 2 * (e ^^^ ( ~~~d &&& (e ||| f))) - 11 *  ~~~e + 11 * ( ~~~(d &&&  ~~~e) &&&  ~~~(e ^^^ f)) + 7 * (d ||| (e &&& f)) - 11 * (d ||| f) + 4 * (e ||| (d &&& f)) + 5 *  ~~~f + 1 * (e ||| (d &&&  ~~~f)) - 2 *  ~~~(d &&& f) - 2 * ((d &&&  ~~~e) ||| (d ^^^ (e ^^^ f))) - 7 * ((d &&&  ~~~e) |||  ~~~(e ^^^ f)) + 7 * ((d ^^^ e) ||| (d ^^^ f)) - 1 *  ~~~(d ^^^ (e ^^^ f)) + 1 * (d ^^^ (e ||| f)) - 7 * (d ||| ( ~~~e ||| f)) - 5 * (e ^^^  ~~~(d &&& (e &&& f))) - 3 * ( ~~~(d |||  ~~~e) ||| (d ^^^ (e ^^^ f))) + 2 * (e |||  ~~~(d ||| f)) + 5 *  ~~~(d &&& (e ||| f)) - 11 * (e ||| (d ^^^ f)) + 4 * (f ^^^ ( ~~~d ||| ( ~~~e &&& f))) + 5 *  ~~~(d &&&  ~~~e) - 1 * (e ^^^ ( ~~~d ||| (e ||| f))) + 3 * e + 11 * (e ^^^ ( ~~~d &&& (e ^^^ f))) - 5 * ( ~~~(d ^^^ e) |||  ~~~(d ^^^ f)) + 1 * (e ^^^ (d &&&  ~~~f)) - 2 * ((d &&& f) ||| (e &&&  ~~~f)) - 2 * (e ^^^  ~~~(d &&& ( ~~~e ||| f))) + 2 * (f ^^^ (d &&&  ~~~e)) + 11 * ((d &&&  ~~~e) |||  ~~~(d ^^^ (e ^^^ f))) + 11 * ( ~~~f &&&  ~~~(d ^^^ e)) - 5 * (e ^^^  ~~~(d &&& ( ~~~e &&& f))) - 3 * (f ^^^ (d ||| ( ~~~e ||| f))) - 1 * (e ^^^  ~~~(d &&& f)) + 4 * (f ^^^  ~~~(d &&& ( ~~~e &&& f))) - 2 * ((d &&& e) ^^^  ~~~(d ^^^ (e &&& f))) + 4 * (d &&& ( ~~~e ||| f)) - 3 * ( ~~~(d ||| e) ||| (e ^^^ f)) + 2 * (d &&& (e ||| f)) - 1 * (e &&& f) + 3 *  ~~~(d &&& ( ~~~e ||| f)) - 7 * (e &&&  ~~~(d &&& f)) - 3 * (f |||  ~~~(d |||  ~~~e)) - 5 * ((d &&& e) ^^^ (d ^^^ (e &&& f))) - 1 *  ~~~(e |||  ~~~f) + 1 * (e ^^^ f) - 6 *  ~~~(d ^^^ f) + 6 *  ~~~(d ||| (e ||| f)) + 11 *  ~~~(d ||| ( ~~~e ||| f)) + 30 *  ~~~( ~~~d ||| (e ||| f)) - 11 *  ~~~( ~~~d ||| ( ~~~e ||| f)) + 40 * ( ~~~d &&& ( ~~~e &&& f)) - 12 * ( ~~~d &&& (e &&& f)) + 35 * (d &&& ( ~~~e &&& f)) + 19 * (d &&& (e &&& f)) =  - 11 * ( ~~~(d ^^^ e) &&&  ~~~(d ^^^ f)) + 1 * (d |||  ~~~e)

theorem main5f_like_logical_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 &&& x_1 != x_2) = 1#1 →
    ofBool (x_2 &&& x_1 == x_2) = 1#1 → True → 0#32 = zeroExtend 32 (ofBool (x_2 &&& x == x_2))

theorem mul_of_bool_no_lz_other_op_thm.extracted_1._1 : ∀ (x : BitVec 8) (x_1 : BitVec 32),
  ¬(True ∧ (x_1 &&& 1#32).smulOverflow (signExtend 32 x) = true ∨
        True ∧ (x_1 &&& 1#32).umulOverflow (signExtend 32 x) = true) →
    ofBool (127#32 <ₛ (x_1 &&& 1#32) * signExtend 32 x) = 0#1

theorem or_ranges_single_elem_left_proof : or_ranges_single_elem_left_before ⊑ or_ranges_single_elem_left_after

theorem no_shift_no_xor_multiuse_cmp_with_xor_thm.extracted_1._2 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 &&& 4096#32 == 0#32) = 1#1 → (x_1 ^^^ 4096#32) * x = (x_1 ^^^ x_2 &&& 4096#32) * x

theorem ashr_lshr_exact_both_proof : ashr_lshr_exact_both_before ⊑ ashr_lshr_exact_both_after

theorem cmpeq_xor_cst1_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (x_1 ^^^ 10#32 == x) = ofBool (x_1 ^^^ x == 10#32)

lemma CNFA.determinize_bv_language {m : CNFA n} :
    m.bv_recognizes L → m.determinize.bv_recognizes L

theorem main3e_like_logical_thm.extracted_1._5 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 &&& x_1 == 0#32) = 1#1 →
    ofBool (x_2 &&& x_1 != 0#32) = 1#1 → ofBool (x_2 &&& x == 0#32) = 1#1 → 0#32 = zeroExtend 32 1#1

theorem test_add_nuw_nsw__all_are_safe_proof : test_add_nuw_nsw__all_are_safe_before ⊑ test_add_nuw_nsw__all_are_safe_after

theorem shl_lshr_pow2_const_negative_overflow2_thm (e : IntW 16) :
  LLVM.and (lshr (shl (const? 16 8) e) (const? 16 6)) (const? 16 (-32768)) ⊑ const? 16 0

theorem e_679 :
     - 5 * (x |||  ~~~y) + 4 *  ~~~y + 1 *  ~~~x - 1 * y + 5 * (x ||| y) - 11 *  ~~~(x &&& y) + 6 *  ~~~(x ||| y) + 6 *  ~~~(x |||  ~~~y) + 7 * (x &&&  ~~~y) - 4 * (x &&& y) =  - 5 *  ~~~(x ^^^ y)

theorem bools2_logical_commute1_thm.extracted_1._11 : ∀ (x x_1 : BitVec 1), ¬x_1 = 1#1 → 0#1 = 1#1 → False

theorem bv_AddSub_1043 :
    ∀ (e e_1 e_2 : LLVM.IntW w),
      LLVM.add (LLVM.add (LLVM.xor (LLVM.and e_1 e) e) (LLVM.const? w 1)) e_2 ⊑ LLVM.sub e_2 (LLVM.or e_1 (LLVM.not e))

theorem alive_Select_747 : forall (w : Nat) (A : Bitvec 1)
, TSSA.eval
  (Op := Op) (e := e)
  (i := TSSAIndex.STMT (UserType.base (BaseType.bitvec 1)))
  [dsl_bb|
  ^bb
  %v0 := unit: ;
  %v1 := op:const (A) %v0;
  %v2 := op:const (0) %v0;
  %v3 := pair:%v1 %v2;
  %v4 := op:icmp sgt  w %v3;
  %v5 := op:const (0) %v0;
  %v6 := pair:%v5 %v1;
  %v7 := op:sub 1 %v6;
  %v8 := triple:%v4 %v1 %v7;
  %v9 := op:select w %v8;
  %v10 := op:const (0) %v0;
  %v11 := pair:%v9 %v10;
  %v12 := op:icmp slt  w %v11;
  %v13 := op:const (0) %v0;
  %v14 := pair:%v13 %v9;
  %v15 := op:sub w %v14;
  %v16 := triple:%v12 %v9 %v15;
  %v17 := op:select w %v16
  dsl_ret %v17
  ]  ⊑
  TSSA.eval
  (Op := Op) (e := e)
  (i := TSSAIndex.STMT (UserType.base (BaseType.bitvec 1)))
  [dsl_bb|
  ^bb
  %v0 := unit: ;
  %v1 := op:const (0) %v0;
  %v2 := op:const (A) %v0;
  %v3 := pair:%v1 %v2;
  %v4 := op:sub 1 %v3;
  %v5 := op:const (0) %v0;
  %v6 := pair:%v2 %v5;
  %v7 := op:icmp slt  1 %v6;
  %v8 := op:const (0) %v0;
  %v9 := pair:%v2 %v8;
  %v10 := op:icmp sgt  1 %v9;
  %v11 := triple:%v10 %v2 %v4;
  %v12 := op:select 1 %v11;
  %v13 := op:const (0) %v0;
  %v14 := pair:%v12 %v13;
  %v15 := op:icmp slt  1 %v14;
  %v16 := op:const (0) %v0;
  %v17 := pair:%v16 %v12;
  %v18 := op:sub 1 %v17;
  %v19 := triple:%v7 %v2 %v4;
  %v20 := op:select 1 %v19
  dsl_ret %v20
  ]

theorem masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_2_proof : masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_2_before ⊑ masked_icmps_mask_notallzeros_bmask_mixed_negated_swapped_2_after

theorem sgt_and_not_min_commute_proof : sgt_and_not_min_commute_before ⊑ sgt_and_not_min_commute_after

theorem land_band_left2_thm (e e_1 : IntW 1) : LLVM.and (select e_1 e (const? 1 0)) e ⊑ select e_1 e (const? 1 0)

theorem positive_sameconst_lshrexact_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬(True ∧ x >>> 3#8 <<< 3#8 ≠ x ∨ 3#8 ≥ ↑8 ∨ 3#8 ≥ ↑8) → x >>> 3#8 <<< 3#8 = x

theorem n0_proof : n0_before ⊑ n0_after

omit [Fintype S] in
lemma processOneElem_spec {st : worklist.St A S} (s : State) (sa : S) (k : ℕ) :
    ∀ a sa', (f sa)[k]? = some (a, sa') →
    processOneElem_mot inits final f s sa k st →
    processOneElem_mot inits final f s sa (k+1) (processOneElem A S final s st (a, sa'))

theorem slt_to_ult_thm (e e_1 : IntW 8) :
  icmp IntPred.slt (LLVM.xor e_1 (const? 8 (-128))) (LLVM.xor e (const? 8 (-128))) ⊑
    icmp IntPred.ult e_1 e

theorem ashr_ult_15_proof : ashr_ult_15_before ⊑ ashr_ult_15_after

theorem exact_ashr_eq_both_equal_proof : exact_ashr_eq_both_equal_before ⊑ exact_ashr_eq_both_equal_after

theorem main5f_like_thm.extracted_1._2 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 &&& x_1 != x_2) ||

theorem e_502 :
    1 *  ~~~x + 2 *  ~~~(x &&& y) + 2 * (x ||| y) + 4 * y - 1 *  ~~~y - 1 *  ~~~(x ^^^ y) - 5 *  ~~~(x &&&  ~~~y) + 4 *  ~~~(x ||| y) - 2 *  ~~~(x |||  ~~~y) = 2 * (x ^^^ y) + 1 * (x &&&  ~~~y)

theorem test17_proof : test17_before ⊑ test17_after

theorem PR42691_1_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x <ₛ 0#32) ||

theorem eq_mul_constants_with_tz_thm (e e_1 : IntW 32) :
  icmp IntPred.ne (mul e_1 (const? 32 12)) (mul e (const? 32 12)) ⊑
    icmp IntPred.ne (LLVM.and (LLVM.xor e_1 e) (const? 32 1073741823)) (const? 32 0)

theorem neg_slt_n1_proof : neg_slt_n1_before ⊑ neg_slt_n1_after

theorem ne_smax3_proof : ne_smax3_before ⊑ ne_smax3_after

theorem test24_logical_thm.extracted_1._2 : ∀ (x : BitVec 32), ¬ofBool (1#32 <ₛ x) = 1#1 → 0#1 = ofBool (2#32 <ₛ x)

theorem lshr_or_and_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), ¬(5#8 ≥ ↑8 ∨ 5#8 ≥ ↑8) → 5#8 ≥ ↑8 → False

theorem add_and_or_proof : add_and_or_before ⊑ add_and_or_after

@[simp] theorem get_map (xs : HVector A as) (f : (a : α) → A a → B a) :
    (xs.map f).get i = f _ (xs.get i)

theorem and_not_or_commute4_thm (e e_1 e_2 : IntW 32) :
  LLVM.and (LLVM.or (LLVM.sdiv (const? 32 42) e_2) (LLVM.xor (LLVM.and e_1 e) (const? 32 (-1))))
      (LLVM.or (LLVM.xor (LLVM.and e_1 (LLVM.sdiv (const? 32 42) e_2)) (const? 32 (-1))) e) ⊑
    LLVM.xor (LLVM.and (LLVM.xor e (LLVM.sdiv (const? 32 42) e_2)) e_1) (const? 32 (-1))

theorem sext_true_val_must_be_all_ones_thm.extracted_1._1 : ∀ (x : BitVec 1), x = 1#1 → signExtend 32 x = -1#32

theorem bv_AndOrXor_2607 :
    ∀ (e e_1 : LLVM.IntW w),
      LLVM.xor (LLVM.or e (LLVM.xor e_1 (LLVM.const? w (-1)))) (LLVM.or (LLVM.xor e (LLVM.const? w (-1))) e_1) ⊑
        LLVM.xor e e_1

theorem shl_add_add_thm (e e_1 : IntW 8) :
  add (shl e_1 (const? 8 2)) (add (shl e (const? 8 2)) (const? 8 48)) ⊑
    add (shl (add e e_1) (const? 8 2)) (const? 8 48)

theorem e_891 :
     - 7 * (z |||  ~~~t) - 6 *  ~~~t - 6 * (z ^^^ t) + 7 * (z ||| t) + 13 *  ~~~(z ||| t) - 1 *  ~~~(z |||  ~~~t) =  - 12 * (z &&&  ~~~t)

theorem zext (b : BitVec 8) : (b.zeroExtend 10 |>.zeroExtend 8) = b

theorem lshrugt_02_15_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 2#4 <<< 2#4 ≠ x ∨ 2#4 ≥ ↑4) → ofBool (-1#4 <ᵤ x >>> 2#4) = 0#1

theorem BitVec.odd_mul_eq_shiftLeft_mul_of_eq_mul_two_add_one (w : Nat) (x : BitVec w) (n o : Nat)
    (ho : o = n * 2 + 1) : (BitVec.ofNat w o) * x = x + (BitVec.ofNat w n) * (x <<< (1 : Nat))

@[simp]
theorem Term.toBV_var {wenv : WidthExpr.Env wcard}
    {tctx : Term.Ctx wcard tcard}
    (benv : Term.BoolEnv bcard)
    (tenv : tctx.Env wenv) :
  Term.toBV benv nenv ienv penv tenv (.var v) = tenv v

theorem p2_scalar_shifted_urem_by_const_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ¬(x ≥ ↑32 ∨ 3#32 = 0) → True ∧ (x_1 &&& 1#32) <<< x >>> x ≠ x_1 &&& 1#32 ∨ x ≥ ↑32 ∨ 3#32 = 0 → False

theorem alive_820'   : alive_820'_src ⊑ alive_820'_tgt

theorem shift_xor_multiuse_cmp_proof : shift_xor_multiuse_cmp_before ⊑ shift_xor_multiuse_cmp_after

theorem test5_proof : test5_before ⊑ test5_after

theorem avg_lsb_thm (e e_1 : IntW 8) :
  lshr (add (LLVM.and e_1 (const? 8 1)) (LLVM.and e (const? 8 1)) { «nsw» := true, «nuw» := true }) (const? 8 1) ⊑
    LLVM.and e_1 (LLVM.and e (const? 8 1))

theorem lshr_and_add_fail_thm (e e_1 : IntW 8) :
  add (lshr e_1 (const? 8 1)) (LLVM.and (lshr e (const? 8 1)) (const? 8 123)) ⊑
    add (lshr e_1 (const? 8 1)) (LLVM.and (lshr e (const? 8 1)) (const? 8 123))
      { «nsw» := false, «nuw» := true }

theorem trunc_shl_ashr_infloop_thm.extracted_1._1 : ∀ (x : BitVec 64),
  ¬(3#64 ≥ ↑64 ∨ 2#64 ≥ ↑64) → 1#64 ≥ ↑64 → False

lemma reduce_stepSet {M : NFA α σ} {q : σ} (S1 : Set M.Reachable) (S2 : Set σ) :
    (∀ q, (∃ hq, S1 ⟨q, hq⟩) ↔ S2 q) →
    ((∃ hq,  ⟨q, hq⟩ ∈ M.reduce.stepSet S1 a) ↔ q ∈ M.stepSet S2 a)

theorem test9_proof : test9_before ⊑ test9_after

theorem e_1075 :
     - 11 *  ~~~(z &&& t) + 5 *  ~~~(z &&&  ~~~t) - 11 * z + 4 *  ~~~(z &&&  ~~~z) - 3 *  ~~~(z ||| t) - 3 *  ~~~(z |||  ~~~t) + 18 * (z &&&  ~~~t) =  - 2 * (z &&& t) - 5 *  ~~~z

theorem ashrsgt_03_04_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 3#4 <<< 3#4 ≠ x ∨ 3#4 ≥ ↑4) → ofBool (4#4 <ₛ x.sshiftRight' 3#4) = 0#1

theorem nonexact_lshr_eq_proof : nonexact_lshr_eq_before ⊑ nonexact_lshr_eq_after

theorem add_nsw_const_const_sub_nsw_ov_proof : add_nsw_const_const_sub_nsw_ov_before ⊑ add_nsw_const_const_sub_nsw_ov_after

theorem or_eq_with_diff_one_logical_thm.extracted_1._2 : ∀ (x : BitVec 8),
  ¬ofBool (x == 13#8) = 1#1 → ofBool (x == 14#8) = ofBool (x + BitVec.ofInt 8 (-13) <ᵤ 2#8)

theorem both_sides_fold_slt_thm (e : IntW 32) (e_1 : IntW 1) :
  icmp IntPred.slt (select e_1 (const? 32 9) e) (select e_1 (const? 32 1) e) ⊑ const? 1 0

theorem p_constmask_commutative_proof : p_constmask_commutative_before ⊑ p_constmask_commutative_after

theorem test18_proof : test18_before ⊑ test18_after

theorem xor_icmp_to_icmp_add_proof : xor_icmp_to_icmp_add_before ⊑ xor_icmp_to_icmp_add_after

theorem alive_276 : alive_276_src ⊑ alive_276_tgt

@[simp] theorem size_rets  : (rets v : Com d Γ eff t).size = 0

theorem and_logic_and_logic_or_4_proof : and_logic_and_logic_or_4_before ⊑ and_logic_and_logic_or_4_after

theorem test_sdiv_canonicalize_op0_exact_thm (e e_1 : IntW 32) :
  LLVM.sdiv (sub (const? 32 0) e_1 { «nsw» := true, «nuw» := false }) e { «exact» := true } ⊑
    sub (const? 32 0) (LLVM.sdiv e_1 e { «exact» := true }) { «nsw» := true, «nuw» := false }

theorem shl_or_with_or_disjoint_instead_of_add_thm.extracted_1._2 : ∀ (x : BitVec 8),
  ¬(x ≥ ↑8 ∨ True ∧ (x &&& 1#8 != 0) = true ∨ x ||| 1#8 ≥ ↑8) →
    ¬x ≥ ↑8 → 16#8 <<< x ||

theorem udiv_c_i32_thm.extracted_1._2 : ∀ (x : BitVec 8),
  ¬zeroExtend 32 x = 0 → ¬(x = 0 ∨ True ∧ (10#8 / x).msb = true) → 10#32 / zeroExtend 32 x = zeroExtend 32 (10#8 / x)

theorem e_1269 :
    2 * (z ^^^ t) + 7 * t - 7 *  ~~~(z &&&  ~~~t) + 4 * (z |||  ~~~t) + 11 * (z ||| t) - 1 *  ~~~(z &&&  ~~~z) + 2 *  ~~~(z ||| t) - 18 * (z &&&  ~~~t) - 14 * (z &&& t) = 12 *  ~~~(z |||  ~~~t) - 2 *  ~~~t

theorem mul_mask_notpow2_ne_thm (e : IntW 8) :
  icmp IntPred.ne (LLVM.and (mul e (const? 8 60)) (const? 8 12)) (const? 8 0) ⊑
    icmp IntPred.ne (LLVM.and (mul e (const? 8 12)) (const? 8 12)) (const? 8 0)

theorem max_sub_ult_c12_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ofBool (x <ᵤ 1#32) = 1#1 → ¬ofBool (x == 0#32) = 1#1 → x + BitVec.ofInt 32 (-2) = 0#32

theorem select_constants_and_icmp_eq0_zero_tval_proof : select_constants_and_icmp_eq0_zero_tval_before ⊑ select_constants_and_icmp_eq0_zero_tval_after

theorem match_signed_thm (e : IntW 64) :
  add (add (LLVM.srem e (const? 64 299)) (mul (LLVM.srem (LLVM.sdiv e (const? 64 299)) (const? 64 64)) (const? 64 299)))
      (mul (LLVM.srem (LLVM.sdiv e (const? 64 19136)) (const? 64 9)) (const? 64 19136)) ⊑
    LLVM.srem e (const? 64 172224)

theorem tryFactorization_xor_lshr_ashr_proof : tryFactorization_xor_lshr_ashr_before ⊑ tryFactorization_xor_lshr_ashr_after

theorem AvoidCollision_getLsbD_setWidth (m : Nat) (x : BitVec n) (i : Nat) :
    getLsbD (setWidth m x) i = (decide (i < m) && getLsbD x i)

theorem or_not_xor_common_op_commute2_thm.extracted_1._1 : ∀ (x x_1 x_2 : BitVec 8),
  x_2 ^^^ x_1 ||

theorem select_2_proof : select_2_before ⊑ select_2_after

theorem lshr_16_to_64_add_zext_basic_proof : lshr_16_to_64_add_zext_basic_before ⊑ lshr_16_to_64_add_zext_basic_after

theorem sub_ashr_and_i64_thm.extracted_1._2 : ∀ (x x_1 : BitVec 64),
  ¬(True ∧ x_1.ssubOverflow x = true ∨ 63#64 ≥ ↑64) →
    ¬ofBool (x_1 <ₛ x) = 1#1 → (x_1 - x).sshiftRight' 63#64 &&& x = 0#64

theorem sge_or_max_logical_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8),
  ofBool (x ≤ₛ x_1) = 1#1 → 1#1 = ofBool (x ≤ₛ x_1)

theorem i_proof : i_before ⊑ i_after

theorem lshr_lshr_thm (e : IntW 232) : lshr (lshr e (const? 232 231)) (const? 232 1) ⊑ const? 232 0

theorem bar_proof : bar_before ⊑ bar_after

theorem sext_xor_sub_2_thm.extracted_1._2 : ∀ (x : BitVec 64) (x_1 : BitVec 1),
  ¬x_1 = 1#1 → signExtend 64 x_1 - (x ^^^ signExtend 64 x_1) = 0#64 - x

theorem test8_proof : test8_before ⊑ test8_after

theorem and_ne_with_one_bit_diff_constants3_logical_thm (e : IntW 8) :
  select (icmp IntPred.ne e (const? 8 65)) (icmp IntPred.ne e (const? 8 (-63))) (const? 1 0) ⊑
    icmp IntPred.ne (LLVM.and e (const? 8 127)) (const? 8 65)

theorem sub_from_constant_commute_proof : sub_from_constant_commute_before ⊑ sub_from_constant_commute_after

theorem test_thm.extracted_1._1 : ∀ (x : BitVec 64), ¬1#64 ≥ ↑64 → ofBool (0#64 <ᵤ x >>> 1#64) = ofBool (1#64 <ᵤ x)

theorem fold_sext_to_and_wrong7_thm (e : IntW 8) :
  icmp IntPred.ne (LLVM.and (sext 32 e) (const? 32 (-2147483647))) (const? 32 128) ⊑ const? 1 1

theorem and_or1_wrong_operand_proof : and_or1_wrong_operand_before ⊑ and_or1_wrong_operand_after

theorem t7_ugt_sgt_neg1_thm.extracted_1._14 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (65535#32 <ᵤ x_2) = 1#1 → ofBool (65535#32 <ₛ x_2) = 1#1 → x_2 = x_1

theorem e_1110 :
     - 7 * (z |||  ~~~t) - 11 *  ~~~(z &&& t) + 1 * z - 1 *  ~~~(z ^^^ t) - 6 * t + 19 *  ~~~(z ||| t) + 17 * (z &&&  ~~~t) + 13 * (z &&& t) =  - 17 *  ~~~(z |||  ~~~t)

theorem ashrslt_03_00_thm (e : IntW 4) :
  icmp IntPred.slt (ashr e (const? 4 3)) (const? 4 0) ⊑ icmp IntPred.slt e (const? 4 0)

theorem sle_smax2_proof : sle_smax2_before ⊑ sle_smax2_after

theorem e_1991 :
     - 11 * ((a &&&  ~~~b) ||| (b ^^^ c)) - 2 * (b ^^^ ( ~~~a ||| (b ^^^ c))) - 1 * ((b &&&  ~~~c) |||  ~~~(a ||| ( ~~~b &&& c))) - 2 * (b ^^^ (a &&& ( ~~~b ||| c))) - 6 * (a ^^^ (b ||| c)) + 4 * ( ~~~(a ||| b) |||  ~~~(b ^^^ c)) - 1 * (c &&& (a |||  ~~~b)) - 1 * (a ||| ( ~~~b ||| c)) + 1 * (b ||| c) - 11 *  ~~~(a &&& (b ^^^ c)) - 11 *  ~~~( ~~~a ||| (b ^^^ c)) + 4 * ((a &&& b) |||  ~~~(b ^^^ c)) - 11 *  ~~~(a ^^^ b) - 2 * (a ^^^ ( ~~~b &&& c)) - 1 *  ~~~(a ||| b) + 2 * (c ^^^ (a ||| ( ~~~b ||| c))) - 6 * (b ^^^  ~~~(a ||| (b ^^^ c))) + 7 * ((a &&& c) ||| (b &&&  ~~~c)) + 5 * (c ^^^  ~~~( ~~~a &&& (b ||| c))) + 11 *  ~~~(a ^^^ (b &&& c)) + 1 * ( ~~~(a ^^^ b) &&&  ~~~(a ^^^ c)) + 2 * (c |||  ~~~(a ||| b)) - 5 * ((a ||| b) &&&  ~~~(a ^^^ (b ^^^ c))) - 6 *  ~~~( ~~~a &&& ( ~~~b &&& c)) - 11 * ((a &&& b) ^^^ (a ^^^ ( ~~~b ||| c))) - 3 * (b ||| (a ^^^ c)) + 5 *  ~~~(a &&&  ~~~a) - 7 * (b ||| (a &&&  ~~~c)) - 5 * ( ~~~(a |||  ~~~b) |||  ~~~(b ^^^ c)) + 1 * (c ^^^  ~~~( ~~~a ||| (b &&& c))) + 3 * (c ^^^  ~~~( ~~~a ||| ( ~~~b &&& c))) - 5 * ( ~~~(a ||| b) ||| (a ^^^ (b ^^^ c))) + 1 * (c &&& (a ||| b)) - 1 * (b ^^^  ~~~(a &&& (b ||| c))) - 6 *  ~~~(a ^^^ (b ^^^ c)) + 1 *  ~~~( ~~~a &&& (b ||| c)) + 5 * ((a &&& c) ^^^ (a ^^^ ( ~~~b &&& c))) - 6 * (b ^^^ (a ||| ( ~~~b &&& c))) - 3 * ( ~~~(a ||| b) ||| (b ^^^ c)) - 2 * (c ^^^  ~~~( ~~~a &&& ( ~~~b ||| c))) - 2 * ((a &&& c) ^^^ (b ||| c)) - 11 * (c ^^^ ( ~~~a ||| (b &&& c))) - 5 * ((a &&&  ~~~b) ||| (a ^^^ (b ^^^ c))) - 6 * ( ~~~a ||| (b ||| c)) - 5 * ((a |||  ~~~b) &&& (a ^^^ (b ^^^ c))) + 2 *  ~~~(a &&&  ~~~b) + 3 * (c ^^^  ~~~(a |||  ~~~b)) - 1 * (a &&& (b ^^^ c)) - 2 * ( ~~~(a ||| b) |||  ~~~(a ^^^ (b ^^^ c))) - 7 * ((a ^^^ b) &&& (a ^^^ c)) + 48 *  ~~~(a ||| (b ||| c)) + 69 *  ~~~(a ||| ( ~~~b ||| c)) + 66 *  ~~~( ~~~a ||| (b ||| c)) + 42 *  ~~~( ~~~a ||| ( ~~~b ||| c)) + 55 * ( ~~~a &&& ( ~~~b &&& c)) + 57 * ( ~~~a &&& (b &&& c)) + 40 * (a &&& ( ~~~b &&& c)) + 56 * (a &&& (b &&& c)) =  - 4 *  ~~~(a ||| (b ^^^ c))

theorem test43_thm (e : IntW 8) :
  sext 64 (add (zext 32 e) (const? 32 (-1))) ⊑
    sext 64 (add (zext 32 e) (const? 32 (-1)) { «nsw» := true, «nuw» := false })

theorem Nat.add_lt_is_or {a} (a_lt : a < 2^i) :
    2^i + a = 2^i ||

theorem lshrult_02_12_exact_proof : lshrult_02_12_exact_before ⊑ lshrult_02_12_exact_after

theorem select_and_icmp_zero_proof : select_and_icmp_zero_before ⊑ select_and_icmp_zero_after

@[simp]
lemma dec_spec (bvs' : BitVecs' n) (k : Fin n) (i : Fin bvs'.length) :
    ((dec bvs').bvs.get k).getLsbD i = bvs'[i].getLsbD k

theorem src_is_mask_ashr_proof : src_is_mask_ashr_before ⊑ src_is_mask_ashr_after

theorem test16_thm (e : IntW 16) :
  sext 32 (icmp IntPred.eq (LLVM.and e (const? 16 8)) (const? 16 8)) ⊑
    sext 32 (ashr (shl e (const? 16 12)) (const? 16 15))

theorem test5_thm (e : IntW 7) :
  LLVM.xor (LLVM.or e (const? 7 23)) (const? 7 23) ⊑ LLVM.and e (const? 7 (-24))

theorem icmp_slt_0_and_icmp_sge_neg1_i64_thm.extracted_1._1 : ∀ (x : BitVec 64),
  ¬63#64 ≥ ↑64 → x >>> 63#64 &&& zeroExtend 64 (ofBool (-1#64 ≤ₛ x)) = zeroExtend 64 (ofBool (x == -1#64))

theorem shl_sub_i8_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬7#8 - x ≥ ↑8 → True ∧ BitVec.ofInt 8 (-128) >>> x <<< x ≠ BitVec.ofInt 8 (-128) ∨ x ≥ ↑8 → False

theorem different_size_zext_zext_sgt_thm.extracted_1._1 : ∀ (x : BitVec 4) (x_1 : BitVec 7),
  ofBool (zeroExtend 25 x <ₛ zeroExtend 25 x_1) = ofBool (zeroExtend 7 x <ᵤ x_1)

theorem mul_of_sdiv_fail_ub_proof : mul_of_sdiv_fail_ub_before ⊑ mul_of_sdiv_fail_ub_after

theorem xor_notand_to_or_not4_thm (e e_1 e_2 : IntW 3) :
  LLVM.xor (LLVM.xor (LLVM.and e_2 e_1) (const? 3 (-1))) (LLVM.or e e_2) ⊑
    LLVM.or (LLVM.and e_2 e_1) (LLVM.xor (LLVM.or e e_2) (const? 3 (-1)))

theorem and_to_xor4_thm (e e_1 : IntW 32) :
  LLVM.and (LLVM.xor (LLVM.and e_1 e) (const? 32 (-1))) (LLVM.or e e_1) ⊑ LLVM.xor e e_1

theorem ashrslt_01_12_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 1#4 <<< 1#4 ≠ x ∨ 1#4 ≥ ↑4) → ofBool (x.sshiftRight' 1#4 <ₛ BitVec.ofInt 4 (-4)) = 0#1

theorem e_2476 :
    3 * (e ^^^  ~~~(d &&& (e &&& f))) + 11 * (d |||  ~~~f) + 2 * (f ^^^  ~~~(d &&& (e &&& f))) - 7 * ((d |||  ~~~e) &&& (d ^^^ (e ^^^ f))) - 11 * ((d ^^^ e) ||| (d ^^^ f)) - 1 * (f ^^^ ( ~~~d ||| ( ~~~e &&& f))) + 1 * ((d ||| e) &&& (d ^^^ (e ^^^ f))) + 1 * (e ^^^  ~~~( ~~~d ||| (e ^^^ f))) + 7 * (e ^^^ (d ||| f)) - 11 * (f ^^^  ~~~( ~~~d ||| ( ~~~e &&& f))) - 1 * ( ~~~d ||| (e ^^^ f)) + 2 * (f |||  ~~~(d ||| e)) - 5 *  ~~~( ~~~d &&& (e &&& f)) + 1 * (f ^^^  ~~~( ~~~d &&& (e ||| f))) - 5 * (f ^^^  ~~~( ~~~d ||| (e &&& f))) + 4 * (f ^^^  ~~~(d &&&  ~~~e)) + 5 *  ~~~(d ||| e) - 1 * (e ^^^ ( ~~~d ||| ( ~~~e ||| f))) - 6 * (e ^^^  ~~~( ~~~d &&& ( ~~~e ||| f))) - 2 * (d ||| e) + 3 * ((e &&& f) ||| (d &&& (e ||| f))) + 7 * (e ^^^ ( ~~~d ||| (e ^^^ f))) - 3 * (d &&& ( ~~~e ||| f)) + 1 * ( ~~~e &&&  ~~~(d ^^^ f)) - 1 * ((d &&& e) ^^^  ~~~(d ^^^ ( ~~~e ||| f))) - 7 * ((d &&& f) ^^^ (d ^^^ (e &&& f))) - 2 * (e &&&  ~~~(d &&& f)) - 1 * ( ~~~e &&& (d ^^^ f)) - 11 * (e ^^^ ( ~~~d &&& (e ||| f))) + 3 * (f ^^^ (d ||| (e &&& f))) + 5 *  ~~~(d ||| (e ^^^ f)) + 3 *  ~~~(d ^^^ (e ||| f)) + 1 * ((d &&& e) |||  ~~~(d ^^^ (e ^^^ f))) - 2 * (e ^^^ (d ||| (e ^^^ f))) - 2 * (e |||  ~~~(d |||  ~~~f)) + 1 * ((d ||| e) &&& (e ^^^ f)) - 2 * (d &&& (e ^^^ f)) + 2 * (f ^^^ ( ~~~d ||| (e ||| f))) + 4 *  ~~~( ~~~d &&& ( ~~~e &&& f)) - 2 * (e &&& (d ||| f)) - 1 *  ~~~( ~~~d ||| ( ~~~e &&& f)) + 1 * (e ^^^ (d &&&  ~~~f)) - 5 * (e ^^^ f) - 2 * ( ~~~(d |||  ~~~e) |||  ~~~(d ^^^ (e ^^^ f))) + 2 * (e ^^^  ~~~( ~~~d &&& (e ||| f))) + 11 * ((d &&& e) ^^^  ~~~(d ^^^ (e &&& f))) + 1 *  ~~~(d ^^^ ( ~~~e ||| f)) - 1 * ( ~~~e |||  ~~~(d ^^^ f)) + 1 * (f &&&  ~~~(d ^^^ e)) - 2 * ( ~~~(d ^^^ e) &&& (d ^^^ f)) + 1 * (e ^^^  ~~~(d |||  ~~~f)) - 11 * (d &&& e) - 6 * ((d &&& e) |||  ~~~(e |||  ~~~f)) + 1 * (f ^^^  ~~~(d &&& ( ~~~e ||| f))) - 3 * (f ^^^  ~~~(d &&& ( ~~~e &&& f))) + 3 * (f ^^^  ~~~(d ||| (e &&& f))) + 5 * (e ^^^  ~~~(d ||| (e ^^^ f))) - 5 *  ~~~( ~~~d ||| (e &&& f)) - 1 * ((e &&&  ~~~f) ^^^ (d ||| (e ^^^ f))) - 3 *  ~~~( ~~~d &&& (e ||| f)) + 2 * ((d &&& f) ||| (e &&&  ~~~f)) - 11 * ((d &&& f) ^^^  ~~~(d ^^^ ( ~~~e &&& f))) + 11 * ((e &&& f) ^^^  ~~~(d &&& (e ^^^ f))) + 1 *  ~~~(d &&&  ~~~d) - 3 * (e ^^^ (d ||| ( ~~~e &&& f))) + 7 *  ~~~(d &&& e) + 1 * ((e &&& f) ||| ( ~~~d &&& (e ||| f))) - 1 * (e ^^^ ( ~~~d &&& (e ^^^ f))) - 2 * ((d &&& e) ||| (d ^^^ (e ^^^ f))) - 7 * (e &&&  ~~~f) - 6 * (f ^^^ (d &&& ( ~~~e ||| f))) - 6 * ( ~~~(d &&&  ~~~e) &&& (d ^^^ (e ^^^ f))) - 64 *  ~~~(d ||| (e ||| f)) - 14 *  ~~~(d ||| ( ~~~e ||| f)) + 31 *  ~~~( ~~~d ||| (e ||| f)) + 55 *  ~~~( ~~~d ||| ( ~~~e ||| f)) + 29 * ( ~~~d &&& ( ~~~e &&& f)) + 43 * ( ~~~d &&& (e &&& f)) + 11 * (d &&& ( ~~~e &&& f)) + 35 * (d &&& (e &&& f)) = 1 * (e ||| (d &&& f))

theorem and_logic_and_logic_or_7_thm (e e_1 e_2 : IntW 1) :
  select (select e_2 e_1 (const? 1 0)) (const? 1 1) (LLVM.and e e_2) ⊑
    select e_2 (select e_1 (const? 1 1) e) (const? 1 0)

theorem sub_ashr_and_i32_commute_proof : sub_ashr_and_i32_commute_before ⊑ sub_ashr_and_i32_commute_after

theorem test9_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x &&& 1073741824#32 != 0#32) &&& ofBool (-1#32 <ₛ x) =
    ofBool (x &&& BitVec.ofInt 32 (-1073741824) == 1073741824#32)

theorem sel_false_val_is_a_masked_ashr_of_true_val1_thm.extracted_1._2 : ∀ (x : BitVec 64) (x_1 : BitVec 32),
  ofBool (x_1 &&& BitVec.ofInt 32 (-2147483588) == 0#32) = 1#1 →
    ¬(2#32 ≥ ↑32 ∨
          True ∧ (x_1.sshiftRight' 2#32 &&& BitVec.ofInt 32 (-536870897)).msb = true ∨
            zeroExtend 64 (x_1.sshiftRight' 2#32 &&& BitVec.ofInt 32 (-536870897)) ≥ ↑64) →
      x = x.sshiftRight' (zeroExtend 64 (x_1.sshiftRight' 2#32 &&& BitVec.ofInt 32 (-536870897)))

theorem test22_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬(2147483647#32 == 0 || 32 != 1 && x &&& 2147483647#32 == intMin 32 && 2147483647#32 == -1) = true →
    ¬2147483647#32 = 0 → (x &&& 2147483647#32).srem 2147483647#32 = (x &&& 2147483647#32) % 2147483647#32

theorem not_or_or_and_no_and_commute1_or_thm (e e_1 e_2 : IntW 32) :
  LLVM.and (LLVM.or (LLVM.or e_2 e_1) (LLVM.xor e (const? 32 (-1)))) (LLVM.xor (LLVM.and e_1 e) (const? 32 (-1))) ⊑
    LLVM.or (LLVM.and e_2 (LLVM.xor e_1 (const? 32 (-1)))) (LLVM.xor e (const? 32 (-1)))

theorem or_xor_commute2_thm (e e_1 : IntW 32) :
  LLVM.xor (LLVM.or (LLVM.udiv (const? 32 42) e_1) (LLVM.udiv (const? 32 42) e)) (LLVM.udiv (const? 32 42) e) ⊑
    LLVM.and (LLVM.udiv (const? 32 42) e_1) (LLVM.xor (LLVM.udiv (const? 32 42) e) (const? 32 (-1)))

theorem eq_21_thm (e e_1 : IntW 32) :
  LLVM.and (icmp IntPred.eq (trunc 8 (lshr e_1 (const? 32 16))) (trunc 8 (lshr e (const? 32 16))))
      (icmp IntPred.eq (trunc 8 (lshr e_1 (const? 32 8))) (trunc 8 (lshr e (const? 32 8)))) ⊑
    icmp IntPred.eq (trunc 16 (lshr e_1 (const? 32 8))) (trunc 16 (lshr e (const? 32 8)))

theorem sgt_positive_multip_rem_nz_thm (e : IntW 8) :
  icmp IntPred.sgt (mul e (const? 8 5) { «nsw» := true, «nuw» := false }) (const? 8 21) ⊑
    icmp IntPred.sgt e (const? 8 4)

theorem icmp_ne_zext_ne_one_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (zeroExtend 32 (ofBool (x != 1#32)) != x) = 1#1

theorem trunc_shl_15_i16_i32_proof : trunc_shl_15_i16_i32_before ⊑ trunc_shl_15_i16_i32_after

theorem e_1474 :
     - 11 *  ~~~(z &&&  ~~~t) + 4 *  ~~~z + 11 * (z ^^^ t) + 5 *  ~~~t + 4 * t + 7 * z + 2 *  ~~~(z &&& t) - 1 *  ~~~(z ^^^ t) + 1 *  ~~~(z ||| t) - 9 *  ~~~(z |||  ~~~t) - 24 * (z &&&  ~~~t) = 1 * (z ||| t) - 2 * (z &&& t)

theorem main7c_logical_thm.extracted_1._10 : ∀ (x x_1 x_2 : BitVec 32),
  ¬ofBool (x_2 == x_2 &&& x_1) = 1#1 → ofBool (x_2 != x_2 &&& x_1) = 1#1 → ¬0#1 = 1#1 → 1#32 = zeroExtend 32 1#1

theorem icmp_equality_test_swift_optional_pointers_thm.extracted_1._2 : ∀ (x x_1 : BitVec 64),
  ofBool (x_1 == 0#64) = 1#1 → ¬True → ofBool (x == 0#64) = ofBool (x_1 == x)

@[simp]
theorem R.fromPoly_kernel_eq_zero (x : (ZMod q)[X]) : R.fromPoly (n := n) (f q n * x) = 0

theorem or_simplify_uge_thm (e e_1 : IntW 8) :
  icmp IntPred.uge (LLVM.and e_1 (const? 8 127))
      (LLVM.or (LLVM.or e (const? 8 (-127))) (LLVM.and e_1 (const? 8 127))) ⊑
    const? 1 0

theorem lshrult_01_11_proof : lshrult_01_11_before ⊑ lshrult_01_11_after

theorem test31_thm (e : IntW 1) :
  LLVM.and (shl (zext 32 e) (const? 32 4)) (const? 32 16) ⊑ select e (const? 32 16) (const? 32 0)

theorem demorgan_nor_proof : demorgan_nor_before ⊑ demorgan_nor_after

theorem lor_band_left2_proof : lor_band_left2_before ⊑ lor_band_left2_after

theorem src_fv_ne_thm.extracted_1._2 : ∀ (x x_1 : BitVec 8) (x_2 : BitVec 1),
  x_2 = 1#1 → ¬(True ∧ x_1.uaddOverflow 1#8 = true) → ofBool (x_1 + 1#8 ||| x != 0#8) = ofBool (x != 0#8) ||

theorem icmp_select_var_both_fold_thm.extracted_1._2 : ∀ (x x_1 : BitVec 8),
  ¬ofBool (x_1 == 0#8) = 1#1 → ofBool (2#8 == x ||| 1#8) = ofBool (x_1 == 0#8)

theorem test2_proof : test2_before ⊑ test2_after

theorem add_nuw_const_const_sub_thm (e : IntW 8) :
  sub (const? 8 (-127)) (add e (const? 8 1) { «nsw» := false, «nuw» := true }) ⊑ sub (const? 8 (-128)) e

theorem test28_thm.extracted_1._1 : ∀ (x : BitVec 32),
  x + BitVec.ofInt 32 (-2147483647) ^^^ BitVec.ofInt 32 (-2147483648) = x + 1#32

theorem e_64 :
     - 1 * (x |||  ~~~y) - 1 *  ~~~(x ^^^ y) + 5 *  ~~~(x ||| y) + 3 *  ~~~(x |||  ~~~y) + 2 * (x &&& y) =  - 1 * (x &&&  ~~~y) + 3 *  ~~~x

theorem udiv_eq_umax_proof : udiv_eq_umax_before ⊑ udiv_eq_umax_after

theorem not_or_or_and_not_and_and_commute2_thm (e e_1 e_2 : IntW 32) :
  LLVM.and (LLVM.or (LLVM.or (LLVM.xor e_2 (const? 32 (-1))) e_1) (LLVM.sdiv (const? 32 42) e))
      (LLVM.xor (LLVM.and (LLVM.sdiv (const? 32 42) e) (LLVM.and e_1 e_2)) (const? 32 (-1))) ⊑
    LLVM.or (LLVM.xor (LLVM.sdiv (const? 32 42) e) e_1) (LLVM.xor e_2 (const? 32 (-1)))

theorem select_1_thm.extracted_1._4 : ∀ (x : BitVec 8) (x_1 x_2 : BitVec 1),
  ¬x_2 = 1#1 → x_1 = 1#1 → x ^^^ -1#8 ^^^ -1#8 = x

theorem srem_common_dividend_defined_cond_proof : srem_common_dividend_defined_cond_before ⊑ srem_common_dividend_defined_cond_after

theorem neg_or_lshr_i32_commute_proof : neg_or_lshr_i32_commute_before ⊑ neg_or_lshr_i32_commute_after

theorem xor_icmp_true_signed_thm.extracted_1._1 : ∀ (x : BitVec 32), ofBool (5#32 <ₛ x) ^^^ ofBool (x <ₛ 6#32) = 1#1

theorem icmp_ne_zext_eq_one_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (zeroExtend 32 (ofBool (x == 1#32)) != x) = ofBool (1#32 <ᵤ x)

theorem cmp_x_and_negp2_with_eq_proof : cmp_x_and_negp2_with_eq_before ⊑ cmp_x_and_negp2_with_eq_after

theorem n10_proof : n10_before ⊑ n10_after

theorem e_1125 :
    4 * t + 1 * (z ^^^ t) + 7 *  ~~~(z &&&  ~~~z) - 11 *  ~~~t - 1 *  ~~~z - 11 *  ~~~(z |||  ~~~t) + 3 * (z &&&  ~~~t) - 11 * (z &&& t) =  - 5 *  ~~~(z ||| t)

theorem ne_umax1_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (x <ᵤ x_1) = 1#1 → ofBool (x_1 != x_1) = ofBool (x_1 <ᵤ x)

theorem positive_biggerLshr_lshrexact_thm (e : IntW 32) :
  lshr (shl e (const? 32 5)) (const? 32 10) { «exact» := true } ⊑
    LLVM.and (lshr e (const? 32 5) { «exact» := true }) (const? 32 4194303)

theorem masked_icmps_mask_notallzeros_bmask_mixed_swapped_0_logical_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬ofBool (x &&& 3#32 == 1#32) = 1#1 → 0#1 = ofBool (x &&& 3#32 == 1#32) &&& ofBool (x &&& 12#32 != 0#32)

theorem ashrsgt_02_12_exact_proof : ashrsgt_02_12_exact_before ⊑ ashrsgt_02_12_exact_after

theorem icmp_mul_nsw_sle_proof : icmp_mul_nsw_sle_before ⊑ icmp_mul_nsw_sle_after

theorem xor_or_thm.extracted_1._4 : ∀ (x : BitVec 32) (x_1 : BitVec 1),
  x_1 = 1#1 → ¬x_1 ^^^ 1#1 = 1#1 → 1#1 ^^^ 1#1 = 0#1

theorem and_and_thm (e : IntW 32) :
  select (icmp IntPred.eq (LLVM.and e (const? 32 2)) (const? 32 0)) (LLVM.and e (const? 32 1)) (const? 32 1) ⊑
    zext 32 (icmp IntPred.ne (LLVM.and e (const? 32 3)) (const? 32 0))

theorem test_nsw_and_signed_pred_proof : test_nsw_and_signed_pred_before ⊑ test_nsw_and_signed_pred_after

theorem slt_zero_slt_i1_fail_thm.extracted_1._1 : ∀ (x : BitVec 32) (x_1 : BitVec 1),
  ¬31#32 ≥ ↑32 → ofBool (zeroExtend 32 x_1 <ₛ x >>> 31#32) = ofBool (x <ₛ 0#32) &&& (x_1 ^^^ 1#1)

theorem mul_sel_op0_thm (e : IntW 32) (e_1 : IntW 1) :
  mul (select e_1 (const? 32 0) (LLVM.udiv (const? 32 42) e { «exact» := true })) e ⊑
    select e_1 (const? 32 0) (const? 32 42)

theorem test_lshr__exact_is_unsafe_thm.extracted_1._3 : ∀ (x : BitVec 64) (x_1 : BitVec 32),
  ¬ofBool (x_1 &&& 63#32 == 0#32) = 1#1 →
    ¬(2#32 ≥ ↑32 ∨ zeroExtend 64 ((x_1 &&& 63#32) >>> 2#32) ≥ ↑64) →
      2#32 ≥ ↑32 ∨ True ∧ (x_1 >>> 2#32 &&& 15#32).msb = true ∨ zeroExtend 64 (x_1 >>> 2#32 &&& 15#32) ≥ ↑64 → False

theorem zext_masked_bit_nonzero_to_smaller_bitwidth_proof : zext_masked_bit_nonzero_to_smaller_bitwidth_before ⊑ zext_masked_bit_nonzero_to_smaller_bitwidth_after

theorem test16_thm (e e_1 : IntW 8) :
  mul (LLVM.and (LLVM.xor (LLVM.xor e_1 (const? 8 33)) e) (LLVM.xor e e_1)) (LLVM.xor (LLVM.xor e_1 (const? 8 33)) e) ⊑
    mul (LLVM.and (LLVM.xor (LLVM.xor e_1 e) (const? 8 33)) (LLVM.xor e e_1))
      (LLVM.xor (LLVM.xor e_1 e) (const? 8 33))

theorem test3_thm.extracted_1._1 : ∀ (x : BitVec 23), x &&& 127#23 &&& 128#23 = 0#23

theorem t12_shift_of_const0_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ¬x_1 ≥ ↑32 → True ∧ 1#32 <<< x_1 >>> x_1 ≠ 1#32 ∨ x_1 ≥ ↑32 → False

theorem test3_thm.extracted_1._4 : ∀ (x : BitVec 1) (x_1 : BitVec 599),
  ¬x = 1#1 → ¬4096#599 = 0 → ¬12#599 ≥ ↑599 → x_1 / 4096#599 = x_1 >>> 12#599

theorem lshr_or_and_thm.extracted_1._2 : ∀ (x x_1 : BitVec 8),
  ¬(5#8 ≥ ↑8 ∨ 5#8 ≥ ↑8) →
    ¬5#8 ≥ ↑8 → (x_1 >>> 5#8 ||| BitVec.ofInt 8 (-58)) &&& x >>> 5#8 = (x &&& (x_1 ||| BitVec.ofInt 8 (-64))) >>> 5#8

theorem test7_proof : test7_before ⊑ test7_after

theorem test1_thm (e : IntW 32) :
  icmp IntPred.uge e (const? 32 1) ⊑ icmp IntPred.ne e (const? 32 0)

theorem ashrsgt_01_10_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 1#4 <<< 1#4 ≠ x ∨ 1#4 ≥ ↑4) → ofBool (BitVec.ofInt 4 (-6) <ₛ x.sshiftRight' 1#4) = 1#1

theorem masked_or_allones_logical_proof : masked_or_allones_logical_before ⊑ masked_or_allones_logical_after

theorem positive_different_trunc_both_logical_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (-1#15 <ₛ truncate 15 x) = 1#1 →
    ¬ofBool (x &&& 16384#32 == 0#32) = 1#1 → ofBool (truncate 16 x + 128#16 <ᵤ 256#16) = 0#1

@[simp] theorem Valuation.comap_cons_consMap {Γ Γ_out : Ctxt Ty}
    (V : Γ_out.Valuation) {t} (x : ⟦t⟧) (map : Γ.Hom Γ_out) :
    Valuation.comap (x ::ᵥ V) (Ctxt.Hom.consMap map)
    = (x ::ᵥ (Valuation.comap V map))

theorem and_or1_thm.extracted_1._4 : ∀ (x x_1 x_2 : BitVec 1),
  x_2 ^^^ 1#1 ||

theorem test10_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  x_1 ^^^ x ||

theorem lshr_mul_times_3_div_2_exact_2_proof : lshr_mul_times_3_div_2_exact_2_before ⊑ lshr_mul_times_3_div_2_exact_2_after

theorem ashrslt_02_15_proof : ashrslt_02_15_before ⊑ ashrslt_02_15_after

theorem main6c_logical_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (x &&& 7#32 != 3#32) = 1#1 → True → 0#32 = zeroExtend 32 (ofBool (x &&& 55#32 == 19#32))

theorem uadd_sat_flipped_wrong_bounds_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬ofBool (BitVec.ofInt 32 (-12) ≤ᵤ x) = 1#1 → ofBool (BitVec.ofInt 32 (-13) <ᵤ x) = 1#1 → x + 9#32 = -1#32

theorem e_1702 :
    2 *  ~~~(a &&& (b &&& c)) - 1 * (a &&&  ~~~c) - 7 *  ~~~(a ^^^ (b ^^^ c)) + 11 * (b ^^^ (a ||| c)) - 6 * (a &&& (b ^^^ c)) + 1 *  ~~~(a ^^^ (b &&& c)) + 11 * ((a &&& b) ^^^  ~~~(a ^^^ (b &&& c))) + 7 * (b ^^^  ~~~(a ||| c)) - 2 * ((a ||| b) &&&  ~~~(a ^^^ (b ^^^ c))) - 2 * ((b &&& c) ^^^  ~~~( ~~~a &&& (b ^^^ c))) + 2 * (b ^^^  ~~~(a &&&  ~~~c)) - 5 * a + 11 * (b ^^^  ~~~( ~~~a ||| (b &&& c))) + 7 * (a ||| (b &&& c)) - 1 * (c ^^^  ~~~( ~~~a ||| (b &&& c))) - 1 * (b ||| (a &&&  ~~~c)) + 3 * ( ~~~(a |||  ~~~b) ||| (a ^^^ (b ^^^ c))) - 1 *  ~~~(a &&&  ~~~c) + 5 * ((a &&& c) ^^^ (a ^^^ ( ~~~b &&& c))) + 4 * (c ^^^ (a ||| b)) + 7 * (c |||  ~~~(a ||| b)) - 23 *  ~~~(a ||| (b ||| c)) - 41 *  ~~~(a ||| ( ~~~b ||| c)) - 31 *  ~~~( ~~~a ||| (b ||| c)) - 16 *  ~~~( ~~~a ||| ( ~~~b ||| c)) - 42 * ( ~~~a &&& ( ~~~b &&& c)) - 25 * ( ~~~a &&& (b &&& c)) - 27 * (a &&& ( ~~~b &&& c)) - 28 * (a &&& (b &&& c)) = 2 * (b ^^^  ~~~(a &&& (b ^^^ c))) - 5 * ((a |||  ~~~b) &&&  ~~~(a ^^^ (b ^^^ c)))

theorem test35_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬(True ∧ (0#32 - x &&& 240#32).msb = true) →
    0#64 - zeroExtend 64 x &&& 240#64 = zeroExtend 64 (0#32 - x &&& 240#32)

theorem band_lor_right1_thm.extracted_1._2 : ∀ (x x_1 : BitVec 1), x_1 = 1#1 → 1#1 = x_1

theorem ashrslt_02_04_exact_proof : ashrslt_02_04_exact_before ⊑ ashrslt_02_04_exact_after

theorem not_or_xor_thm.extracted_1._1 : ∀ (x : BitVec 8),
  (x ^^^ -1#8 ||| 7#8) ^^^ 12#8 = x &&& BitVec.ofInt 8 (-8) ^^^ BitVec.ofInt 8 (-13)

theorem n3_constmask_samemask_thm (e e_1 : IntW 32) :
  add (LLVM.and e_1 (const? 32 65280)) (LLVM.and e (const? 32 65280)) ⊑
    add (LLVM.and e_1 (const? 32 65280)) (LLVM.and e (const? 32 65280)) { «nsw» := true, «nuw» := true }

theorem test0_thm (e : IntW 32) :
  icmp IntPred.sgt (LLVM.xor e (const? 32 (-2147483648))) (const? 32 (-1)) ⊑
    icmp IntPred.slt e (const? 32 0)

theorem or_ranges_single_elem_left_proof : or_ranges_single_elem_left_before ⊑ or_ranges_single_elem_left_after

theorem shl_add_thm.extracted_1._2 : ∀ (x : BitVec 32),
  ¬x ≥ ↑32 →
    ¬(True ∧ ((-1#32) <<< x).sshiftRight' x ≠ -1#32 ∨ x ≥ ↑32) → 1#32 <<< x + -1#32 = (-1#32) <<< x ^^^ -1#32

theorem ashrsgt_01_09_thm (e : IntW 4) : icmp IntPred.sgt (ashr e (const? 4 1)) (const? 4 (-7)) ⊑ const? 1 1

theorem lshrugt_03_07_exact_proof : lshrugt_03_07_exact_before ⊑ lshrugt_03_07_exact_after

theorem e_849 :
    1 * (z |||  ~~~t) + 3 *  ~~~(z &&& t) + 1 * t - 4 *  ~~~(z ||| t) - 4 * (z &&&  ~~~t) - 2 * (z &&& t) = 4 *  ~~~(z |||  ~~~t)

theorem e_394 :
     - 1 *  ~~~(x ^^^ y) - 5 *  ~~~(x &&& y) - 2 *  ~~~y + 3 * x + 11 *  ~~~(x ||| y) + 3 *  ~~~(x |||  ~~~y) + 7 * (x &&&  ~~~y) + 1 * (x &&& y) = 5 * (x |||  ~~~y) - 2 *  ~~~(x &&&  ~~~x)

theorem commutativity_constant_14_vary_proof : commutativity_constant_14_vary_before ⊑ commutativity_constant_14_vary_after

theorem zext_nneg_thm.extracted_1._1 : ∀ (x : BitVec 16),
  ¬(True ∧ (x &&& 32767#16).msb = true) →
    zeroExtend 24 (x &&& 32767#16) &&& 8388607#24 = zeroExtend 24 (x &&& 32767#16)

theorem ashrslt_01_11_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬1#4 ≥ ↑4 → ofBool (x.sshiftRight' 1#4 <ₛ BitVec.ofInt 4 (-5)) = 0#1

theorem and_not_or_commute8_thm.extracted_1._2 : ∀ (x x_1 x_2 : BitVec 32),
  ¬((x_2 == 0 || 32 != 1 && 42#32 == intMin 32 && x_2 == -1) = true ∨
        (x_1 == 0 || 32 != 1 && 42#32 == intMin 32 && x_1 == -1) = true ∨
          (x_1 == 0 || 32 != 1 && 42#32 == intMin 32 && x_1 == -1) = true ∨
            (x_2 == 0 || 32 != 1 && 42#32 == intMin 32 && x_2 == -1) = true) →
    ¬((x_1 == 0 || 32 != 1 && 42#32 == intMin 32 && x_1 == -1) = true ∨
          (x_2 == 0 || 32 != 1 && 42#32 == intMin 32 && x_2 == -1) = true) →
      ((42#32).sdiv x_2 &&& (42#32).sdiv x_1 ^^^ -1#32 ||| x) &&&
          ((42#32).sdiv x_1 ||| x &&& (42#32).sdiv x_2 ^^^ -1#32) =
        ((42#32).sdiv x_1 ^^^ x) &&& (42#32).sdiv x_2 ^^^ -1#32

theorem ashr_slt_exact_near_pow2_cmpval_thm (e : IntW 8) :
  icmp IntPred.slt (ashr e (const? 8 1) { «exact» := true }) (const? 8 5) ⊑
    icmp IntPred.slt e (const? 8 9)

theorem add_smax_simplify_proof : add_smax_simplify_before ⊑ add_smax_simplify_after

theorem or_xor_tree_1010_proof : or_xor_tree_1010_before ⊑ or_xor_tree_1010_after

theorem t0_thm.extracted_1._8 : ∀ (x x_1 x_2 x_3 : BitVec 32),
  ¬ofBool (x_3 == x_2) = 1#1 → ¬ofBool (x_3 != x_2) = 1#1 → 0#1 ^^^ 1#1 = ofBool (x_1 != x)

theorem e_168 :
     - 1 *  ~~~x + 2 *  ~~~y - 2 * (x ^^^ y) - 12 *  ~~~(x ||| y) - 11 * (x &&&  ~~~y) - 11 * (x &&& y) =  - 11 * (x |||  ~~~y) - 3 *  ~~~(x |||  ~~~y)

theorem select_non_const_sides_thm.extracted_1._4 : ∀ (x x_1 : BitVec 64) (x_2 : BitVec 1),
  ¬x_2 = 1#1 → x - zeroExtend 64 x_2 = x

theorem test15d_thm.extracted_1._1 : ∀ (x : BitVec 32), ofBool (x &&& 16#32 != 0#32) = 1#1 → 16#32 = x &&& 16#32

theorem e_438 :
    7 * (x |||  ~~~y) - 5 *  ~~~x + 3 * x - 1 *  ~~~(x ^^^ y) - 11 * (x ||| y) - 2 *  ~~~(x ||| y) + 16 *  ~~~(x |||  ~~~y) + 2 * (x &&& y) =  - 1 *  ~~~y

theorem test_thm.extracted_1._1 : ∀ (x : BitVec 31),
  ¬15#32 ≥ ↑32 → True ∧ (zeroExtend 32 x).uaddOverflow 16384#32 = true ∨ 15#32 ≥ ↑32 → False

theorem e_481 :
     - 11 * y + 5 *  ~~~(x ^^^ y) - 6 *  ~~~(x &&&  ~~~y) - 2 *  ~~~(x &&&  ~~~x) + 1 * (x |||  ~~~y) - 2 * (x ^^^ y) + 19 *  ~~~(x |||  ~~~y) + 1 * (x &&&  ~~~y) + 13 * (x &&& y) =  - 2 *  ~~~(x &&& y)

theorem t0_proof : t0_before ⊑ t0_after

theorem select_icmp_ne_0_and_4096_and_not_32_thm.extracted_1._2 : ∀ (x x_1 : BitVec 32),
  ¬ofBool (0#32 != x_1 &&& 4096#32) = 1#1 → ¬ofBool (x_1 &&& 4096#32 == 0#32) = 1#1 → x &&& BitVec.ofInt 32 (-33) = x

theorem e_686 :
    1 *  ~~~(x ^^^ y) + 5 * y - 7 *  ~~~(x &&&  ~~~y) + 3 * (x ||| y) + 4 * x - 1 *  ~~~x - 11 *  ~~~(x &&& y) + 11 *  ~~~(x |||  ~~~y) + 4 * (x &&&  ~~~y) - 6 * (x &&& y) =  - 18 *  ~~~(x ||| y)

theorem t4_ugt_slt_128_thm (e e_1 e_2 : IntW 32) :
  select (icmp IntPred.ugt (add e_2 (const? 32 16)) (const? 32 143))
      (select (icmp IntPred.slt e_2 (const? 32 128)) e_1 e) e_2 ⊑
    select (icmp IntPred.sgt e_2 (const? 32 127)) e
      (select (icmp IntPred.slt e_2 (const? 32 (-16))) e_1 e_2)

theorem ashrsgt_02_05_exact_proof : ashrsgt_02_05_exact_before ⊑ ashrsgt_02_05_exact_after

theorem and_zext_eq_even_commuted_thm.extracted_1._1 : ∀ (x : BitVec 32),
  zeroExtend 32 (ofBool (x == 2#32)) &&& x = 0#32

theorem icmp_shl_ne_2_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬((x == 0 || 8 != 1 && 42#8 == intMin 8 && x == -1) = true ∨
        (x == 0 || 8 != 1 && 42#8 == intMin 8 && x == -1) = true ∨ 1#8 ≥ ↑8) →
    (x == 0 || 8 != 1 && 42#8 == intMin 8 && x == -1) = true → False

theorem test3_proof : test3_before ⊑ test3_after

theorem xor_to_xor4_thm (e e_1 : IntW 32) : LLVM.xor (LLVM.or e_1 e) (LLVM.and e e_1) ⊑ LLVM.xor e e_1

theorem urem_illegal_type_c_thm (e : IntW 9) :
  urem (zext 32 e) (const? 32 10) ⊑ zext 32 (urem e (const? 9 10)) { «nneg» := true }

theorem syncMap3_flip23 {f : α → β → γ → δ} :
  syncMap₃ f xs ys zs = syncMap₃ (fun x z y => f x y z) xs zs ys

theorem test4_thm.extracted_1._4 : ∀ (x : BitVec 1023),
  ¬ofBool (x <ₛ 0#1023) = 1#1 → ¬1022#1023 ≥ ↑1023 → 0#1023 = x.sshiftRight' 1022#1023

theorem samesign_commute3_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (-1#32 <ₛ x_1 ||| x) ||

theorem e_2085 :
     - 7 *  ~~~(d ||| ( ~~~e &&& f)) - 3 * (f ^^^  ~~~(d ||| (e &&& f))) - 1 * (e ^^^  ~~~(d ||| (e &&& f))) + 1 * (f ^^^  ~~~( ~~~d ||| ( ~~~e &&& f))) - 2 * (d &&&  ~~~e) + 1 * (e ^^^ (d &&&  ~~~f)) + 5 * ((d ||| e) &&& (d ^^^ (e ^^^ f))) - 5 *  ~~~(d ^^^ ( ~~~e ||| f)) + 4 * (e ^^^ (d ||| (e ^^^ f))) - 1 * (d &&& ( ~~~e ||| f)) - 11 *  ~~~(e |||  ~~~f) - 2 * (e ^^^  ~~~( ~~~d &&& ( ~~~e ||| f))) - 2 * ((d &&& e) |||  ~~~(e ||| f)) + 1 * (f ^^^  ~~~(d |||  ~~~e)) - 3 * ((d ||| e) &&& (e ^^^ f)) + 2 *  ~~~e + 7 *  ~~~( ~~~d &&& ( ~~~e &&& f)) - 6 * (d ||| e) + 2 *  ~~~(d &&& (e ||| f)) - 6 * ((d &&& e) ||| (e ^^^ f)) + 2 * (e ^^^ (d ||| ( ~~~e ||| f))) - 2 * (f ^^^  ~~~(d &&&  ~~~e)) + 3 * (e ^^^  ~~~(d &&& (e &&& f))) - 3 * (f &&&  ~~~(d ^^^ e)) + 2 * ( ~~~(d &&&  ~~~e) &&&  ~~~(d ^^^ (e ^^^ f))) - 6 * ((d &&& f) ^^^ (e ||| f)) - 1 * (e ^^^ ( ~~~d ||| ( ~~~e &&& f))) - 1 * d - 5 * ((d &&& f) ^^^  ~~~(e &&&  ~~~f)) - 7 * (d ||| (e &&& f)) - 1 * (f ^^^ (d ||| (e ||| f))) - 1 * (e ^^^ ( ~~~d ||| (e ||| f))) + 3 * (e ||| (d &&&  ~~~f)) - 1 *  ~~~(d ||| (e ||| f)) + 13 *  ~~~(d ||| ( ~~~e ||| f)) + 2 *  ~~~( ~~~d ||| (e ||| f)) + 18 *  ~~~( ~~~d ||| ( ~~~e ||| f)) + 14 * ( ~~~d &&& ( ~~~e &&& f)) + 17 * ( ~~~d &&& (e &&& f)) + 31 * (d &&& ( ~~~e &&& f)) + 16 * (d &&& (e &&& f)) =  - 5 * ((d &&& e) ^^^  ~~~(d ^^^ (e &&& f)))

theorem main5e_like_proof : main5e_like_before ⊑ main5e_like_after

theorem t_2_63_or_thm (e : IntW 8) :
  LLVM.or (select (icmp IntPred.ugt e (const? 8 2)) e (const? 8 2)) (const? 8 63) ⊑
    LLVM.or e (const? 8 63)

theorem test31_proof : test31_before ⊑ test31_after

theorem add_or_sub_comb_i8_negative_y_add_thm (e e_1 : IntW 8) :
  add (LLVM.or (sub (const? 8 0) e_1) e_1) e ⊑ add (LLVM.or e_1 (sub (const? 8 0) e_1)) e

theorem udiv_common_divisor_thm (e e_1 e_2 : IntW 5) (e_3 : IntW 1) :
  select e_3 (LLVM.udiv e_2 e_1) (LLVM.udiv e e_1) ⊑ LLVM.udiv (select e_3 e_2 e) e_1

theorem n11_wrong_pred3_thm (e e_1 : IntW 8) :
  icmp IntPred.ne (add e_1 e) e ⊑ icmp IntPred.ne e_1 (const? 8 0)

theorem shift_xor_multiuse_xor_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32),
  ofBool (0#32 != x_1 &&& 4096#32) = 1#1 →
    ofBool (x_1 &&& 4096#32 == 0#32) = 1#1 → x * (x ^^^ 2048#32) = (x ^^^ 2048#32) * (x ^^^ 2048#32)

@[simp] theorem getValue_value [Inhabited α] : (value a).getValue = a

theorem bad_add2_thm.extracted_1._1 : ∀ (x : BitVec 32), ¬x ≥ ↑32 → True ∧ 1#32 <<< x >>> x ≠ 1#32 ∨ x ≥ ↑32 → False

theorem test14_proof : test14_before ⊑ test14_after

theorem testi8i8_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ¬7#8 ≥ ↑8 → ofBool (-1#8 <ₛ x) = 1#1 → x.sshiftRight' 7#8 ^^^ 127#8 = 127#8

theorem sdiv_shl_pair_const_thm (e : IntW 32) :
  LLVM.sdiv (shl e (const? 32 2) { «nsw» := true, «nuw» := false })
      (shl e (const? 32 1) { «nsw» := true, «nuw» := false }) ⊑
    const? 32 2

theorem e_931 :
     - 11 *  ~~~(z &&&  ~~~z) - 5 *  ~~~t + 7 *  ~~~z + 5 *  ~~~(z |||  ~~~t) + 16 * (z &&&  ~~~t) + 12 * (z &&& t) =  - 9 *  ~~~(z ||| t) + 1 * t

theorem ule_swap_or_not_min_logical_proof : ule_swap_or_not_min_logical_before ⊑ ule_swap_or_not_min_logical_after

theorem e_1191 :
    2 *  ~~~z + 11 *  ~~~t - 7 *  ~~~(z ^^^ t) + 3 *  ~~~(z &&&  ~~~z) - 1 * z - 9 *  ~~~(z ||| t) - 5 *  ~~~(z |||  ~~~t) - 13 * (z &&&  ~~~t) =  - 5 * (z &&& t)

theorem icmp_ne_sext_eq_allones_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ofBool (signExtend 32 (ofBool (x == -1#32)) != x) = ofBool (x + -1#32 <ᵤ BitVec.ofInt 32 (-2))

theorem shl_fold_or_disjoint_cnt_proof : shl_fold_or_disjoint_cnt_before ⊑ shl_fold_or_disjoint_cnt_after

theorem syncMap₂_eq_syncMap₂ {f : α → β → γ}
    (hxs : xs ~ xs') (hys : ys ~ ys') :
    syncMap₂ f xs ys ~ syncMap₂ f xs' ys'

theorem positive_biggerShl_proof : positive_biggerShl_before ⊑ positive_biggerShl_after

theorem a_false_implies_b_false2_comm_proof : a_false_implies_b_false2_comm_before ⊑ a_false_implies_b_false2_comm_after

theorem n7_wrong_pred2_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), ofBool (x_1 - x == x_1) = ofBool (x == 0#8)

theorem different_size_zext_zext_slt_thm.extracted_1._1 : ∀ (x : BitVec 4) (x_1 : BitVec 7),
  ofBool (zeroExtend 25 x_1 <ₛ zeroExtend 25 x) = ofBool (x_1 <ᵤ zeroExtend 7 x)

theorem udiv_shl_mul_nuw_proof : udiv_shl_mul_nuw_before ⊑ udiv_shl_mul_nuw_after

theorem masked_icmps_mask_notallzeros_bmask_mixed_2_proof : masked_icmps_mask_notallzeros_bmask_mixed_2_before ⊑ masked_icmps_mask_notallzeros_bmask_mixed_2_after

theorem bv_AndOrXor_135 :
    ∀ (e e_1 e_2 : LLVM.IntW w), LLVM.and (LLVM.xor e e_1) e_2 ⊑ LLVM.xor (LLVM.and e e_2) (LLVM.and e_1 e_2)

theorem lshrult_02_13_proof : lshrult_02_13_before ⊑ lshrult_02_13_after

@[simp]
lemma dec_spec' (bvs' : BitVecs' n) (k : Fin n) (i : Fin bvs'.length) :
    ((dec bvs').bvs.get k).getLsbD i = bvs'[i].getLsbD k

theorem lt_signed_to_large_negative_thm.extracted_1._1 : ∀ (x : BitVec 8),
  ofBool (signExtend 32 x <ₛ BitVec.ofInt 32 (-1024)) = 0#1

theorem uge_zext_thm (e : IntW 8) (e_1 : IntW 1) :
  icmp IntPred.uge (zext 8 e_1) e ⊑ icmp IntPred.ule e (zext 8 e_1)

theorem sge_swap_or_max_proof : sge_swap_or_max_before ⊑ sge_swap_or_max_after

theorem test35_with_trunc_thm.extracted_1._1 : ∀ (x : BitVec 64),
  ofBool (0#32 ≤ₛ truncate 32 x) = 1#1 → ¬ofBool (x &&& 2147483648#64 == 0#64) = 1#1 → 60#32 = 100#32

theorem lshrult_01_04_proof : lshrult_01_04_before ⊑ lshrult_01_04_after

theorem lshrugt_01_10_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 1#4 <<< 1#4 ≠ x ∨ 1#4 ≥ ↑4) → ofBool (BitVec.ofInt 4 (-6) <ᵤ x >>> 1#4) = 0#1

theorem test1_thm.extracted_1._1 : ∀ (x x_1 : BitVec 32), ¬x = 0 → ofBool (x_1 / x == 0#32) = ofBool (x_1 <ᵤ x)

theorem lshrugt_01_01_proof : lshrugt_01_01_before ⊑ lshrugt_01_01_after

theorem ripple_nsw4_proof : ripple_nsw4_before ⊑ ripple_nsw4_after

theorem lshr_and_add_fail_proof : lshr_and_add_fail_before ⊑ lshr_and_add_fail_after

theorem main7f_proof : main7f_before ⊑ main7f_after

theorem p2_slt_65536_thm.extracted_1._1 : ∀ (x : BitVec 32),
  ¬ofBool (x <ₛ 65536#32) = 1#1 → ¬ofBool (65535#32 <ₛ x) = 1#1 → False

theorem sge_swap_and_max_commute_logical_proof : sge_swap_and_max_commute_logical_before ⊑ sge_swap_and_max_commute_logical_after

theorem select_ashr_thm.extracted_1._3 : ∀ (x x_1 : BitVec 128) (x_2 : BitVec 1), ¬x_2 = 1#1 → 0#128 ≥ ↑128 → False

theorem a_false_implies_b_true2_comm_thm.extracted_1._7 : ∀ (x x_1 : BitVec 1) (x_2 : BitVec 8),
  ¬ofBool (x_2 <ᵤ 20#8) = 1#1 → ofBool (10#8 <ᵤ x_2) = 1#1 → x ||

theorem trunc_shl_15_i16_i64_thm.extracted_1._1 : ∀ (x : BitVec 64), ¬15#64 ≥ ↑64 → 15#16 ≥ ↑16 → False

theorem add_mask_ashr28_i32_thm (e : IntW 32) :
  add (LLVM.and (ashr e (const? 32 28)) (const? 32 8)) (ashr e (const? 32 28)) ⊑
    LLVM.and (lshr e (const? 32 28)) (const? 32 7)

theorem shl_add_and_thm.extracted_1._1 : ∀ (x x_1 : BitVec 8), ¬(1#8 ≥ ↑8 ∨ 1#8 ≥ ↑8) → 1#8 ≥ ↑8 → False

theorem e_1561 :
    4 * (a &&& (b ||| c)) + 11 * (a ^^^ ( ~~~b &&& c)) - 2 * (b ||| (a &&&  ~~~c)) - 7 * (c &&&  ~~~(a ^^^ b)) - 3 *  ~~~(a ^^^ b) - 5 * (b ||| (a ^^^ c)) - 3 *  ~~~a + 6 *  ~~~(a ||| (b ||| c)) + 10 *  ~~~(a ||| ( ~~~b ||| c)) - 4 *  ~~~( ~~~a ||| (b ||| c)) - 5 *  ~~~( ~~~a ||| ( ~~~b ||| c)) + 10 * ( ~~~a &&& (b &&& c)) - 4 * (a &&& ( ~~~b &&& c)) + 2 * (a &&& (b &&& c)) =  - 7 * ( ~~~a &&& ( ~~~b &&& c))

theorem main5f_like_logical_thm.extracted_1._8 : ∀ (x x_1 x_2 : BitVec 32),
  ofBool (x_2 &&& x_1 != x_2) = 1#1 → ¬ofBool (x_2 &&& x_1 == x_2) = 1#1 → ¬True → 0#32 = zeroExtend 32 0#1

theorem test6a_thm.extracted_1._1 : ∀ (x : BitVec 55), ¬1#55 ≥ ↑55 → (x * 3#55) <<< 1#55 = x * 6#55

theorem main6_proof : main6_before ⊑ main6_after

theorem lshrugt_02_07_exact_thm.extracted_1._1 : ∀ (x : BitVec 4),
  ¬(True ∧ x >>> 2#4 <<< 2#4 ≠ x ∨ 2#4 ≥ ↑4) → ofBool (7#4 <ᵤ x >>> 2#4) = 0#1

theorem testtrunclowhigh_thm (e e_1 : IntW 16) (e_2 : IntW 32) :
  select (icmp IntPred.ult (add e_2 (const? 32 128)) (const? 32 256)) (trunc 16 e_2)
      (select (icmp IntPred.sgt e_2 (const? 32 (-1))) e_1 e) ⊑
    select (icmp IntPred.ult (add e_2 (const? 32 128)) (const? 32 256)) (trunc 16 e_2)
      (select (icmp IntPred.slt e_2 (const? 32 0)) e e_1)

theorem ashrslt_01_15_proof : ashrslt_01_15_before ⊑ ashrslt_01_15_after

theorem shl_const_op1_sub_const_op0_proof : shl_const_op1_sub_const_op0_before ⊑ shl_const_op1_sub_const_op0_after

theorem test36_proof : test36_before ⊑ test36_after

theorem test8_proof : test8_before ⊑ test8_after
</all_available_lemmas>

<local_ctx>
import Mathlib.Data.Finset.Card

import Mathlib.Data.List.Pi

import Mathlib.Data.Finset.Union

import Mathlib.Data.Fin.Basic

import Mathlib.Data.Finset.Basic

import Mathlib.Data.Finset.Defs

import Mathlib.Data.Fintype.Basic

open Std Sat AIG

inductive Circuit (α : Type u) : Type u
  | tru : Circuit α
  | fals : Circuit α
   
  | var : (positive: Bool) → α → Circuit α
  | and : Circuit α → Circuit α → Circuit α
  | or : Circuit α → Circuit α → Circuit α
  | xor : Circuit α → Circuit α → Circuit α
deriving Repr, DecidableEq

open Lean in

namespace Circuit

variable {α : Type u} {β : Type v}

def vars [DecidableEq α] : Circuit α → List α
  | tru => []
  | fals => []
  | var _ x => [x]
  | and c₁ c₂ => (vars c₁ ++ vars c₂).dedup
  | or c₁ c₂ => (vars c₁ ++ vars c₂).dedup
  | xor c₁ c₂ => (vars c₁ ++ vars c₂).dedup

def varsFinset [DecidableEq α] (c : Circuit α) : Finset α :=
  ⟨c.vars, nodup_vars c⟩

def simplifyAnd : Circuit α → Circuit α → Circuit α
  | tru, c => c
  | c, tru => c
  | fals, _ => fals
  | _, fals => fals
  | c₁, c₂ => and c₁ c₂

def simplifyOr : Circuit α → Circuit α → Circuit α
  | tru, _ => tru
  | _, tru => tru
  | fals, c => c
  | c, fals => c
  | c₁, c₂ => or c₁ c₂

def simplifyNot : Circuit α → Circuit α
  | tru => fals
  | fals => tru
  | xor a b => xor (simplifyNot a) b
  | and a b => or (simplifyNot a) (simplifyNot b)
  | or a b => and (simplifyNot a) (simplifyNot b)
  | var b a => var (!b) a

@[simp]
def simplifyXor : Circuit α → Circuit α → Circuit α
  | fals, c => c
  | c, fals => c
  | tru, c => ~~~ c
  | c, tru => ~~~ c
  | c₁, c₂ => xor c₁ c₂

def assignVars [DecidableEq α] :
    ∀ (c : Circuit α) (_f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool), Circuit β
  | tru, _ => tru
  | fals, _ => fals
  | var b x, f =>
    Sum.elim
      (var b)
      (λ c : Bool => if Bool.xor b c then fals else tru)
      (f x (by admit /- proof elided -/
      ))
  | and c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/
  ))) &&&
                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/
                    )))
  | or c₁ c₂, f =>  (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/
  ))) |||
                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/
                    )))
  | xor c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/
  ))) ^^^
                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/
                    )))
</local_ctx>

<target_theorem>
lemma varsFinset_assignVars [DecidableEq α] [DecidableEq β] :
    ∀ (c : Circuit α) (f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool),
      (c.assignVars f).varsFinset ⊆ c.varsFinset.biUnion
        (fun a => if ha : a ∈ c.vars
                  then
                    match f a ha with
                    | Sum.inl b => {b}
                    | Sum.inr _ => ∅
                  else ∅) :=
</target_theorem>

