Std.Sat.AIG.Lemmas

This module provides a basic theory around the naive AIG node creation functions. It is mostly fundamental work for the cached versions later on.

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@[simp]

theorem Std.Sat.AIG.Ref.gate_cast {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (ref : aig1.Ref) (h : aig1.decls.size ≤ aig2.decls.size) :

(ref.cast h).gate = ref.gate

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@[simp]

theorem Std.Sat.AIG.Ref.cast_eq {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (ref : aig1.Ref) (h : aig1.decls.size ≤ aig2.decls.size) :

ref.cast h = { gate := ref.gate, hgate := ⋯ }

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@[simp]

theorem Std.Sat.AIG.Fanin.ref_cast {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (fanin : aig1.Fanin) (h : aig1.decls.size ≤ aig2.decls.size) :

(fanin.cast h).ref = fanin.ref.cast h

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@[simp]

theorem Std.Sat.AIG.Fanin.inv_cast {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (fanin : aig1.Fanin) (h : aig1.decls.size ≤ aig2.decls.size) :

(fanin.cast h).inv = fanin.inv

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@[simp]

theorem Std.Sat.AIG.GateInput.lhs_cast {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (input : aig1.GateInput) (h : aig1.decls.size ≤ aig2.decls.size) :

(input.cast h).lhs = input.lhs.cast h

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@[simp]

theorem Std.Sat.AIG.GateInput.rhs_cast {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (input : aig1.GateInput) (h : aig1.decls.size ≤ aig2.decls.size) :

(input.cast h).rhs = input.rhs.cast h

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@[simp]

theorem Std.Sat.AIG.BinaryInput.each_cast {α : Type} [Hashable α] [DecidableEq α] {aig1 aig2 : AIG α} (lhs rhs : aig1.Ref) (h1 h2 : aig1.decls.size ≤ aig2.decls.size) :

{ lhs := lhs.cast h1, rhs := rhs.cast h2 } = { lhs := lhs, rhs := rhs }.cast h2

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@[simp]

theorem Std.Sat.AIG.denote_projected_entry {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {entry : Entrypoint α} :

⟦assign, { aig := entry.aig, ref := entry.ref }⟧ = ⟦assign, entry⟧

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@[simp]

theorem Std.Sat.AIG.denote_projected_entry' {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {entry : Entrypoint α} :

⟦assign, { aig := entry.aig, ref := { gate := entry.ref.gate, hgate := ⋯ } }⟧ = ⟦assign, entry⟧

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theorem Std.Sat.AIG.mkGate_le_size {α : Type} [Hashable α] [DecidableEq α] (aig : AIG α) (input : aig.GateInput) :

aig.decls.size ≤ (aig.mkGate input).aig.decls.size

AIG.mkGate never shrinks the underlying AIG.

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theorem Std.Sat.AIG.mkGate_decl_eq {α : Type} [Hashable α] [DecidableEq α] (idx : Nat) (aig : AIG α) (input : aig.GateInput) {h : idx < aig.decls.size} :

let_fun this := ⋯; (aig.mkGate input).aig.decls [idx] = aig.decls [idx]

The AIG produced by AIG.mkGate agrees with the input AIG on all indices that are valid for both.

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instance Std.Sat.AIG.instLawfulOperatorGateInputMkGate {α : Type} [Hashable α] [DecidableEq α] :

LawfulOperator α GateInput mkGate

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@[simp]

theorem Std.Sat.AIG.denote_mkGate {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {aig : AIG α} {input : aig.GateInput} :

⟦assign, aig.mkGate input⟧ = ((⟦assign, { aig := aig, ref := input.lhs.ref }⟧ ^^ input.lhs.inv) && (⟦assign, { aig := aig, ref := input.rhs.ref }⟧ ^^ input.rhs.inv))

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theorem Std.Sat.AIG.mkAtom_le_size {α : Type} [Hashable α] [DecidableEq α] (aig : AIG α) (var : α) :

aig.decls.size ≤ (aig.mkAtom var).aig.decls.size

AIG.mkAtom never shrinks the underlying AIG.

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theorem Std.Sat.AIG.mkAtom_decl_eq {α : Type} [Hashable α] [DecidableEq α] (aig : AIG α) (var : α) (idx : Nat) {h : idx < aig.decls.size} {hbound : idx < (aig.mkAtom var).aig.decls.size} :

(aig.mkAtom var).aig.decls [idx] = aig.decls [idx]

The AIG produced by AIG.mkAtom agrees with the input AIG on all indices that are valid for both.

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instance Std.Sat.AIG.instLawfulOperatorMkAtom {α : Type} [Hashable α] [DecidableEq α] :

LawfulOperator α (fun (x : AIG α) => α) mkAtom

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@[simp]

theorem Std.Sat.AIG.denote_mkAtom {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {var : α} {aig : AIG α} :

⟦assign, aig.mkAtom var⟧ = assign var

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theorem Std.Sat.AIG.mkConst_le_size {α : Type} [Hashable α] [DecidableEq α] (aig : AIG α) (val : Bool) :

aig.decls.size ≤ (aig.mkConst val).aig.decls.size

AIG.mkConst never shrinks the underlying AIG.

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theorem Std.Sat.AIG.mkConst_decl_eq {α : Type} [Hashable α] [DecidableEq α] (aig : AIG α) (val : Bool) (idx : Nat) {h : idx < aig.decls.size} :

let_fun this := ⋯; (aig.mkConst val).aig.decls [idx] = aig.decls [idx]

The AIG produced by AIG.mkConst agrees with the input AIG on all indices that are valid for both.

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instance Std.Sat.AIG.instLawfulOperatorBoolMkConst {α : Type} [Hashable α] [DecidableEq α] :

LawfulOperator α (fun (x : AIG α) => Bool) mkConst

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@[simp]

theorem Std.Sat.AIG.denote_mkConst {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {val : Bool} {aig : AIG α} :

⟦assign, aig.mkConst val⟧ = val

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theorem Std.Sat.AIG.denote_idx_const {α : Type} [Hashable α] [DecidableEq α] {start : Nat} {assign : α → Bool} {b : Bool} {aig : AIG α} {hstart : start < aig.decls.size} (h : aig.decls[start] = Decl.const b) :

⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = b

If an index contains a Decl.const we know how to denote it.

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theorem Std.Sat.AIG.denote_idx_atom {α : Type} [Hashable α] [DecidableEq α] {start : Nat} {assign : α → Bool} {a : α} {aig : AIG α} {hstart : start < aig.decls.size} (h : aig.decls[start] = Decl.atom a) :

⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = assign a

If an index contains a Decl.atom we know how to denote it.

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theorem Std.Sat.AIG.denote_idx_gate {α : Type} [Hashable α] [DecidableEq α] {start : Nat} {assign : α → Bool} {lhs : Nat} {linv : Bool} {rhs : Nat} {rinv : Bool} {aig : AIG α} {hstart : start < aig.decls.size} (h : aig.decls[start] = Decl.gate lhs rhs linv rinv) :

⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = ((⟦assign, { aig := aig, ref := { gate := lhs, hgate := ⋯ } }⟧ ^^ linv) && (⟦assign, { aig := aig, ref := { gate := rhs, hgate := ⋯ } }⟧ ^^ rinv))

If an index contains a Decl.gate we know how to denote it.

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theorem Std.Sat.AIG.idx_trichotomy {α : Type} [Hashable α] [DecidableEq α] {start : Nat} (aig : AIG α) (hstart : start < aig.decls.size) {prop : Prop} (hconst : ∀ (b : Bool), aig.decls[start] = Decl.const b → prop) (hatom : ∀ (a : α), aig.decls[start] = Decl.atom a → prop) (hgate : ∀ (lhs rhs : Nat) (linv rinv : Bool), aig.decls[start] = Decl.gate lhs rhs linv rinv → prop) :

prop

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theorem Std.Sat.AIG.denote_idx_trichotomy {α : Type} [Hashable α] [DecidableEq α] {start : Nat} {assign : α → Bool} {res : Bool} {aig : AIG α} {hstart : start < aig.decls.size} (hconst : ∀ (b : Bool), aig.decls[start] = Decl.const b → ⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = res) (hatom : ∀ (a : α), aig.decls[start] = Decl.atom a → ⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = res) (hgate : ∀ (lhs rhs : Nat) (linv rinv : Bool), aig.decls[start] = Decl.gate lhs rhs linv rinv → ⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = res) :

⟦assign, { aig := aig, ref := { gate := start, hgate := hstart } }⟧ = res

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theorem Std.Sat.AIG.mem_def {α : Type} [Hashable α] [DecidableEq α] {aig : AIG α} {a : α} :

a ∈ aig ↔ Decl.atom a ∈ aig.decls

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@[irreducible]

theorem Std.Sat.AIG.denote_congr {α : Type} [Hashable α] [DecidableEq α] (assign1 assign2 : α → Bool) (aig : AIG α) (idx : Nat) (hidx : idx < aig.decls.size) (h : ∀ (a : α), a ∈ aig → assign1 a = assign2 a) :

⟦assign1, { aig := aig, ref := { gate := idx, hgate := hidx } }⟧ = ⟦assign2, { aig := aig, ref := { gate := idx, hgate := hidx } }⟧

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theorem Std.Sat.AIG.of_isConstant {α : Type} [Hashable α] [DecidableEq α] {aig : AIG α} {assign : α → Bool} {ref : aig.Ref} {b : Bool} :

aig.isConstant ref b = true → ⟦assign, { aig := aig, ref := ref }⟧ = b