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

@[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

@[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, invert := ref.invert, hgate := ⋯ }

@[simp]

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

@[simp]

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

@[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

@[simp]

theorem Std.Sat.AIG.BinaryInput_invert_lhs {α : Type} [Hashable α] [DecidableEq α] {aig : AIG α} (input : aig.BinaryInput) (linv rinv : Bool) : (input.invert linv rinv).lhs = { gate := input.lhs.gate, invert := linv ^^ input.lhs.invert, hgate := ⋯ }

@[simp]

theorem Std.Sat.AIG.BinaryInput_invert_rhs {α : Type} [Hashable α] [DecidableEq α] {aig : AIG α} (input : aig.BinaryInput) (linv rinv : Bool) : (input.invert linv rinv).rhs = { gate := input.rhs.gate, invert := rinv ^^ input.rhs.invert, hgate := ⋯ }

@[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⟧

@[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, invert := entry.ref.invert, hgate := ⋯ } }⟧ = ⟦assign, entry⟧

@[simp]

theorem Std.Sat.AIG.Ref.denote_flip {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {aig : AIG α} {ref : aig.Ref} {inv : Bool} : ⟦assign, { aig := aig, ref := ref.flip inv }⟧ = (⟦assign, { aig := aig, ref := ref }⟧ ^^ inv)

@[simp]

theorem Std.Sat.AIG.Ref.denote_not {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {aig : AIG α} {ref : aig.Ref} : ⟦assign, { aig := aig, ref := ref.not }⟧ = !⟦assign, { aig := aig, ref := ref }⟧

@[simp]

theorem Std.Sat.AIG.denote_not_invert {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {aig : AIG α} {gate : Nat} {inv : Bool} {hgate : gate < aig.decls.size} : ⟦assign, { aig := aig, ref := { gate := gate, invert := !inv, hgate := hgate } }⟧ = !⟦assign, { aig := aig, ref := { gate := gate, invert := inv, hgate := hgate } }⟧

@[simp]

theorem Std.Sat.AIG.denote_invert_true {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} {aig : AIG α} {gate : Nat} {hgate : gate < aig.decls.size} : ⟦assign, { aig := aig, ref := { gate := gate, invert := true, hgate := hgate } }⟧ = !⟦assign, { aig := aig, ref := { gate := gate, invert := false, hgate := hgate } }⟧

theorem Std.Sat.AIG.denote_idx_false {α : Type} [Hashable α] [DecidableEq α] {start : Nat} {assign : α → Bool} {invert : Bool} {aig : AIG α} {hstart : start < aig.decls.size} (h : aig.decls[start] = Decl.false) : ⟦assign, { aig := aig, ref := { gate := start, invert := invert, hgate := hstart } }⟧ = invert

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

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

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

theorem Std.Sat.AIG.denote_idx_gate {α : Type} [Hashable α] [DecidableEq α] {start : Nat} {assign : α → Bool} {invert : Bool} {lhs rhs : Fanin} {aig : AIG α} {hstart : start < aig.decls.size} (h : aig.decls[start] = Decl.gate lhs rhs) : ⟦assign, { aig := aig, ref := { gate := start, invert := invert, hgate := hstart } }⟧ = (⟦assign, { aig := aig, ref := { gate := lhs.gate, invert := lhs.invert, hgate := ⋯ } }⟧ && ⟦assign, { aig := aig, ref := { gate := rhs.gate, invert := rhs.invert, hgate := ⋯ } }⟧ ^^ invert)

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

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

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

theorem Std.Sat.AIG.mem_def {α : Type} [Hashable α] [DecidableEq α] {aig : AIG α} {a : α} : a ∈ aig ↔ Decl.atom a ∈ aig.decls

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

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

theorem Std.Sat.AIG.denote_getConstant {α : Type} [Hashable α] [DecidableEq α] {aig : AIG α} {assign : α → Bool} {ref : aig.Ref} {b : Bool} : aig.getConstant ref = some b → ⟦assign, { aig := aig, ref := ref }⟧ = b