internlm/Lean-Github huggingface dataset:
Theorem: AIG.mkXorCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput false false)).aig.decls[idx] = aig.decls[idx]
Tactic:
rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)]
Proof state after:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false false)).aig.decls.size
Theorem: AIG.mkXorCached_decl_eq
Proof state before:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false false)).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size
Proof state after:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < aig.decls.size
Theorem: AIG.mkXorCached_decl_eq
Proof state before:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < aig.decls.size
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.mkXorCached_decl_eq
Proof state before:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).cast ⋯)).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size
Proof state after:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false false)).aig.decls.size
Theorem: AIG.mkXorCached_decl_eq
Proof state before:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false false)).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size
Proof state after:
case h2.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < aig.decls.size
Theorem: AIG.mkXorCached_decl_eq
Proof state before:
case h2.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkXorCached input).aig.decls.size
⊢ idx < aig.decls.size
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.denote_mkXorCached
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.BinaryInput
⊢ ⟦aig.mkXorCached input, assign⟧ =
    xor ⟦{ aig := aig, ref := input.lhs }, assign⟧ ⟦{ aig := aig, ref := input.rhs }, assign⟧
Tactic:
rw [← xor_as_aig]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.BinaryInput
⊢ ⟦aig.mkXorCached input, assign⟧ =
    (!(⟦{ aig := aig, ref := input.lhs }, assign⟧ && ⟦{ aig := aig, ref := input.rhs }, assign⟧) &&
      !(!⟦{ aig := aig, ref := input.lhs }, assign⟧ && !⟦{ aig := aig, ref := input.rhs }, assign⟧))
Theorem: AIG.denote_mkXorCached
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.BinaryInput
⊢ ⟦aig.mkXorCached input, assign⟧ =
    (!(⟦{ aig := aig, ref := input.lhs }, assign⟧ && ⟦{ aig := aig, ref := input.rhs }, assign⟧) &&
      !(!⟦{ aig := aig, ref := input.lhs }, assign⟧ && !⟦{ aig := aig, ref := input.rhs }, assign⟧))
Tactic:
simp [
  mkXorCached,
  LawfulOperator.denote_mem_prefix (f := mkGateCached) input.lhs.hgate,
  LawfulOperator.denote_mem_prefix (f := mkGateCached) input.rhs.hgate
]
Proof state after:
no goals
Theorem: AIG.mkBEqCached_le_size
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkBEqCached input).aig.decls.size
Tactic:
simp only [mkBEqCached]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
    (((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                  ((input.asGateInput true false).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                        ((input.asGateInput true false).cast ⋯)).ref,
                  inv := true } }).aig.decls.size
Theorem: AIG.mkBEqCached_le_size
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
    (((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                  ((input.asGateInput true false).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                        ((input.asGateInput true false).cast ⋯)).ref,
                  inv := true } }).aig.decls.size
Tactic:
apply LawfulOperator.le_size_of_le_aig_size
Proof state after:
case h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Theorem: AIG.mkBEqCached_le_size
Proof state before:
case h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Tactic:
apply LawfulOperator.le_size_of_le_aig_size
Proof state after:
case h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Theorem: AIG.mkBEqCached_le_size
Proof state before:
case h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Tactic:
apply LawfulOperator.le_size_of_le_aig_size
Proof state after:
case h.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ aig.decls.size
Theorem: AIG.mkBEqCached_le_size
Proof state before:
case h.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ aig.decls.size
Tactic:
omega
Proof state after:
no goals
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ (aig.mkBEqCached input).aig.decls[idx] = aig.decls[idx]
Tactic:
simp only [mkBEqCached]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                  ((input.asGateInput true false).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                        ((input.asGateInput true false).cast ⋯)).ref,
                  inv := true } }).aig.decls[idx] =
    aig.decls[idx]
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                  ((input.asGateInput true false).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
                        ((input.asGateInput true false).cast ⋯)).ref,
                  inv := true } }).aig.decls[idx] =
    aig.decls[idx]
Tactic:
rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls[idx] =
    aig.decls[idx]

case h1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls[idx] =
    aig.decls[idx]

case h1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Tactic:
rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput false true)).aig.decls[idx] = aig.decls[idx]

case h1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false true)).aig.decls.size

case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput false true)).aig.decls[idx] = aig.decls[idx]

case h1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false true)).aig.decls.size

case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Tactic:
. rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)]
  apply LawfulOperator.lt_size_of_lt_aig_size
  assumption
Proof state after:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Tactic:
. apply LawfulOperator.lt_size_of_lt_aig_size
  apply LawfulOperator.lt_size_of_lt_aig_size
  assumption
Proof state after:
no goals
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput false true)).aig.decls[idx] = aig.decls[idx]
Tactic:
rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)]
Proof state after:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size
Proof state after:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < aig.decls.size
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
case h2
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
            ((input.asGateInput true false).cast ⋯)).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size
Proof state after:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
case h2.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size
Proof state after:
case h2.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < aig.decls.size
Theorem: AIG.mkBEqCached_decl_eq
Proof state before:
case h2.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkBEqCached input).aig.decls.size
⊢ idx < aig.decls.size
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.denote_mkBEqCached
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.BinaryInput
⊢ ⟦aig.mkBEqCached input, assign⟧ =
    (⟦{ aig := aig, ref := input.lhs }, assign⟧ == ⟦{ aig := aig, ref := input.rhs }, assign⟧)
Tactic:
rw [← beq_as_aig]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.BinaryInput
⊢ ⟦aig.mkBEqCached input, assign⟧ =
    (!(⟦{ aig := aig, ref := input.lhs }, assign⟧ && !⟦{ aig := aig, ref := input.rhs }, assign⟧) &&
      !(!⟦{ aig := aig, ref := input.lhs }, assign⟧ && ⟦{ aig := aig, ref := input.rhs }, assign⟧))
Theorem: AIG.denote_mkBEqCached
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : aig.BinaryInput
⊢ ⟦aig.mkBEqCached input, assign⟧ =
    (!(⟦{ aig := aig, ref := input.lhs }, assign⟧ && !⟦{ aig := aig, ref := input.rhs }, assign⟧) &&
      !(!⟦{ aig := aig, ref := input.lhs }, assign⟧ && ⟦{ aig := aig, ref := input.rhs }, assign⟧))
Tactic:
simp [
  mkBEqCached,
  LawfulOperator.denote_mem_prefix (f := mkGateCached) input.lhs.hgate,
  LawfulOperator.denote_mem_prefix (f := mkGateCached) input.rhs.hgate
]
Proof state after:
no goals
Theorem: AIG.mkImpCached_le_size
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkImpCached input).aig.decls.size
Tactic:
simp only [mkImpCached]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
    (((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true } }).aig.decls.size
Theorem: AIG.mkImpCached_le_size
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
    (((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true } }).aig.decls.size
Tactic:
apply LawfulOperator.le_size_of_le_aig_size
Proof state after:
case h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.decls.size
Theorem: AIG.mkImpCached_le_size
Proof state before:
case h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.decls.size
Tactic:
apply LawfulOperator.le_size_of_le_aig_size (f := mkConstCached)
Proof state after:
case h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Theorem: AIG.mkImpCached_le_size
Proof state before:
case h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkGateCached (input.asGateInput false true)).aig.decls.size
Tactic:
apply LawfulOperator.le_size_of_le_aig_size
Proof state after:
case h.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ aig.decls.size
Theorem: AIG.mkImpCached_le_size
Proof state before:
case h.h.h
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ aig.decls.size
Tactic:
omega
Proof state after:
no goals
Theorem: AIG.mkImpCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkImpCached input).aig.decls.size
⊢ (aig.mkImpCached input).aig.decls[idx] = aig.decls[idx]
Tactic:
simp only [mkImpCached]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkImpCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true } }).aig.decls[idx] =
    aig.decls[idx]
Theorem: AIG.mkImpCached_decl_eq
Proof state before:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkImpCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput false true)).ref.cast ⋯, inv := true } }).aig.decls[idx] =
    aig.decls[idx]
Tactic:
rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)]
Proof state after:
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkImpCached input).aig.decls.size
⊢ ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.decls[idx] = aig.decls[idx]

case h1
α : Type
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkImpCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput false true)).aig.mkConstCached true).aig.decls.size