internlm/Lean-Github huggingface dataset:
Theorem: AIG.mkAndCached_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.decls.size
Tactic:
omega
Proof state after:
no goals
Theorem: AIG.mkAndCached_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.mkAndCached input).aig.decls.size
⊢ (aig.mkAndCached input).aig.decls[idx] = aig.decls[idx]
Tactic:
simp only [mkAndCached]
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.mkAndCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput false false)).aig.decls[idx] = aig.decls[idx]
Theorem: AIG.mkAndCached_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.mkAndCached 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:
no goals
Theorem: AIG.denote_mkAndCached
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.mkAndCached input, assign⟧ =
    (⟦{ aig := aig, ref := input.lhs }, assign⟧ && ⟦{ aig := aig, ref := input.rhs }, assign⟧)
Tactic:
simp [mkAndCached]
Proof state after:
no goals
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
Tactic:
simp only [mkOrCached]
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 true true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput true true)).ref.cast ⋯, inv := true } }).aig.decls.size
Theorem: AIG.mkOrCached_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 true true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput true 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 true true)).aig.mkConstCached true).aig.decls.size
Theorem: AIG.mkOrCached_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 true 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 true true)).aig.decls.size
Theorem: AIG.mkOrCached_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 true 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.mkOrCached_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.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ (aig.mkOrCached input).aig.decls[idx] = aig.decls[idx]
Tactic:
simp only [mkOrCached]
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.mkOrCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput true true)).ref.cast ⋯, inv := true } }).aig.decls[idx] =
    aig.decls[idx]
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.mkGateCached
            {
              lhs :=
                { ref := ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).ref, inv := false },
              rhs :=
                { ref := (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ ((aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ ((aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.decls.size
Tactic:
rw [AIG.LawfulOperator.decl_eq (f := mkConstCached)]
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.mkOrCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.decls.size
Tactic:
. apply LawfulOperator.lt_size_of_lt_aig_size (f := mkConstCached)
  apply LawfulOperator.lt_size_of_lt_aig_size
  assumption
Proof state after:
no goals
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true true)).aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ idx < aig.decls.size
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ idx < ((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.decls.size
Tactic:
apply LawfulOperator.lt_size_of_lt_aig_size (f := mkConstCached)
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.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true true)).aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true 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.mkOrCached input).aig.decls.size
⊢ idx < aig.decls.size
Theorem: AIG.mkOrCached_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.mkOrCached input).aig.decls.size
⊢ idx < aig.decls.size
Tactic:
assumption
Proof state after:
no goals
Theorem: AIG.denote_mkOrCached
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.mkOrCached input, assign⟧ =
    (⟦{ aig := aig, ref := input.lhs }, assign⟧ || ⟦{ aig := aig, ref := input.rhs }, assign⟧)
Tactic:
rw [← or_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.mkOrCached input, assign⟧ =
    !(!⟦{ aig := aig, ref := input.lhs }, assign⟧ && !⟦{ aig := aig, ref := input.rhs }, assign⟧)
Theorem: AIG.denote_mkOrCached
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.mkOrCached input, assign⟧ =
    !(!⟦{ aig := aig, ref := input.lhs }, assign⟧ && !⟦{ aig := aig, ref := input.rhs }, assign⟧)
Tactic:
simp [mkOrCached, LawfulOperator.denote_input_entry (f := mkConstCached)]
Proof state after:
no goals
Theorem: AIG.mkXorCached_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.mkXorCached input).aig.decls.size
Tactic:
simp only [mkXorCached]
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 false)).aig.mkGateCached
                  ((input.asGateInput true true).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false false)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
                        ((input.asGateInput true true).cast ⋯)).ref,
                  inv := true } }).aig.decls.size
Theorem: AIG.mkXorCached_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 false)).aig.mkGateCached
                  ((input.asGateInput true true).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false false)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
                        ((input.asGateInput true true).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 false)).aig.mkGateCached
            ((input.asGateInput true true).cast ⋯)).aig.decls.size
Theorem: AIG.mkXorCached_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 false)).aig.mkGateCached
            ((input.asGateInput true true).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 false)).aig.decls.size
Theorem: AIG.mkXorCached_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 false)).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.mkXorCached_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.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.mkXorCached input).aig.decls[idx] = aig.decls[idx]
Tactic:
simp only [mkXorCached]
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.mkXorCached input).aig.decls.size
⊢ (((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
                  ((input.asGateInput true true).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false false)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
                        ((input.asGateInput true true).cast ⋯)).ref,
                  inv := true } }).aig.decls[idx] =
    aig.decls[idx]
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.mkGateCached
                  ((input.asGateInput true true).cast ⋯)).aig.mkGateCached
            { lhs := { ref := (aig.mkGateCached (input.asGateInput false false)).ref.cast ⋯, inv := true },
              rhs :=
                {
                  ref :=
                    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
                        ((input.asGateInput true true).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.mkXorCached input).aig.decls.size
⊢ ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).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.mkXorCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).cast ⋯)).aig.decls.size
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.mkGateCached
            ((input.asGateInput true true).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.mkXorCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).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.mkXorCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput false false)).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.mkXorCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false false)).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.mkXorCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).cast ⋯)).aig.decls.size
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]

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.mkXorCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput false false)).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.mkXorCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).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.mkXorCached input).aig.decls.size
⊢ idx <
    ((aig.mkGateCached (input.asGateInput false false)).aig.mkGateCached
            ((input.asGateInput true true).cast ⋯)).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.mkGateCached
            ((input.asGateInput true true).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