Std.Sat.AIG.LawfulOperator

The lawful operator framework provides free theorems around the typeclass LawfulOperator. Its definition is based on section 3.3 of the AIGNET paper.

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structure Std.Sat.AIG.IsPrefix {α : Type} (decls1 : Array (Std.Sat.AIG.Decl α)) (decls2 : Array (Std.Sat.AIG.Decl α)) :

Prop

decls1 is a prefix of decls2

of :: (

size_le : decls1.size ≤ decls2.size
The prefix may never be longer than the other array.
idx_eq : ∀ (idx : Nat) (h : idx < decls1.size), decls2[idx] = decls1[idx]
The prefix and the other array must agree on all elements up until the bound of the prefix

)

Instances For

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theorem Std.Sat.AIG.IsPrefix.size_le {α : Type} {decls1 : Array (Std.Sat.AIG.Decl α)} {decls2 : Array (Std.Sat.AIG.Decl α)} (self : Std.Sat.AIG.IsPrefix decls1 decls2) :

decls1.size ≤ decls2.size

The prefix may never be longer than the other array.

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theorem Std.Sat.AIG.IsPrefix.idx_eq {α : Type} {decls1 : Array (Std.Sat.AIG.Decl α)} {decls2 : Array (Std.Sat.AIG.Decl α)} (self : Std.Sat.AIG.IsPrefix decls1 decls2) (idx : Nat) (h : idx < decls1.size) :

decls2[idx] = decls1[idx]

The prefix and the other array must agree on all elements up until the bound of the prefix

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@[irreducible]

theorem Std.Sat.AIG.denote.go_eq_of_isPrefix {α : Type} {assign : α → Bool} (decls1 : Array (Std.Sat.AIG.Decl α)) (decls2 : Array (Std.Sat.AIG.Decl α)) (start : Nat) {hdag1 : Std.Sat.AIG.IsDAG α decls1} {hdag2 : Std.Sat.AIG.IsDAG α decls2} {hbounds1 : start < decls1.size} {hbounds2 : start < decls2.size} (hprefix : Std.Sat.AIG.IsPrefix decls1 decls2) :

Std.Sat.AIG.denote.go start decls2 assign hbounds2 hdag2 = Std.Sat.AIG.denote.go start decls1 assign hbounds1 hdag1

If decls1 is a prefix of decls2 and we start evaluating decls2 at an index in bounds of decls1 we can evaluate at decls1.

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theorem Std.Sat.AIG.denote.eq_of_isPrefix {α : Type} [Hashable α] [DecidableEq α] {assign : α → Bool} (entry : Std.Sat.AIG.Entrypoint α) (newAIG : Std.Sat.AIG α) (hprefix : Std.Sat.AIG.IsPrefix entry.aig.decls newAIG.decls) :

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

If decls1 is a prefix of decls2 and we start evaluating decls2 at an index in bounds of decls1 we can evaluate at decls1.

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@[reducible, inline]

abbrev Std.Sat.AIG.ExtendingEntrypoint {α : Type} [Hashable α] [DecidableEq α] (aig : Std.Sat.AIG α) :

Type

Equations

aig.ExtendingEntrypoint = { entry : Std.Sat.AIG.Entrypoint α // aig.decls.size ≤ entry.aig.decls.size }

Instances For

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@[reducible, inline]

abbrev Std.Sat.AIG.ExtendingRefVecEntry {α : Type} [Hashable α] [DecidableEq α] (aig : Std.Sat.AIG α) (len : Nat) :

Type

Equations

aig.ExtendingRefVecEntry len = { ref : Std.Sat.AIG.RefVecEntry α len // aig.decls.size ≤ ref.aig.decls.size }

Instances For

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class Std.Sat.AIG.LawfulOperator (α : Type) [Hashable α] [DecidableEq α] (β : Std.Sat.AIG α → Type) (f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α) :

Type

A function f that takes some aig : AIG α and an argument of type β aig is called a lawful AIG operator if it only extends the AIG but never modifies already existing nodes.

This guarantees that applying such a function will not change the semantics of any existing parts of the circuit, allowing us to perform local reasoning on the AIG.

le_size : ∀ (aig : Std.Sat.AIG α) (input : β aig), aig.decls.size ≤ (f aig input).aig.decls.size
decl_eq : ∀ (aig : Std.Sat.AIG α) (input : β aig) (idx : Nat) (h1 : idx < aig.decls.size) (h2 : idx < (f aig input).aig.decls.size), (f aig input).aig.decls[idx] = aig.decls[idx]

Instances

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theorem Std.Sat.AIG.LawfulOperator.le_size {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [self : Std.Sat.AIG.LawfulOperator α β f] (aig : Std.Sat.AIG α) (input : β aig) :

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

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theorem Std.Sat.AIG.LawfulOperator.decl_eq {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [self : Std.Sat.AIG.LawfulOperator α β f] (aig : Std.Sat.AIG α) (input : β aig) (idx : Nat) (h1 : idx < aig.decls.size) (h2 : idx < (f aig input).aig.decls.size) :

(f aig input).aig.decls[idx] = aig.decls[idx]

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theorem Std.Sat.AIG.LawfulOperator.isPrefix_aig {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] (aig : Std.Sat.AIG α) (input : β aig) :

Std.Sat.AIG.IsPrefix aig.decls (f aig input).aig.decls

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theorem Std.Sat.AIG.LawfulOperator.lt_size {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] (entry : Std.Sat.AIG.Entrypoint α) (input : β entry.aig) :

entry.ref.gate < (f entry.aig input).aig.decls.size

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theorem Std.Sat.AIG.LawfulOperator.lt_size_of_lt_aig_size {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] {x : Nat} (aig : Std.Sat.AIG α) (input : β aig) (h : x < aig.decls.size) :

x < (f aig input).aig.decls.size

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theorem Std.Sat.AIG.LawfulOperator.le_size_of_le_aig_size {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] {x : Nat} (aig : Std.Sat.AIG α) (input : β aig) (h : x ≤ aig.decls.size) :

x ≤ (f aig input).aig.decls.size

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@[simp]

theorem Std.Sat.AIG.LawfulOperator.denote_input_entry {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] {assign : α → Bool} (entry : Std.Sat.AIG.Entrypoint α) {input : β entry.aig} {h : entry.ref.gate < (f entry.aig input).aig.decls.size} :

⟦assign, { aig := (f entry.aig input).aig, ref := { gate := entry.ref.gate, hgate := h } }⟧ = ⟦assign, entry⟧

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@[simp]

theorem Std.Sat.AIG.LawfulOperator.denote_cast_entry {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] {assign : α → Bool} (entry : Std.Sat.AIG.Entrypoint α) {input : β entry.aig} {h : entry.aig.decls.size ≤ (f entry.aig input).aig.decls.size} :

⟦assign, { aig := (f entry.aig input).aig, ref := entry.ref.cast h }⟧ = ⟦assign, entry⟧

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theorem Std.Sat.AIG.LawfulOperator.denote_mem_prefix {α : Type} [Hashable α] [DecidableEq α] {β : Std.Sat.AIG α → Type} {f : (aig : Std.Sat.AIG α) → β aig → Std.Sat.AIG.Entrypoint α} [Std.Sat.AIG.LawfulOperator α β f] {assign : α → Bool} {start : Nat} {aig : Std.Sat.AIG α} {input : β aig} (h : start < aig.decls.size) :

⟦assign, { aig := (f aig input).aig, ref := { gate := start, hgate := ⋯ } }⟧ = ⟦assign, { aig := aig, ref := { gate := start, hgate := h } }⟧