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Formula and skew circuit properties

Lax461425.CircuitProperties · concepts/Lax461425/CircuitProperties.lean · lax-461425

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    Natural Language Statement

    Definition

    Additional predicates on the existing circuit data.

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    DefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.

    Lean source view on GitHub

    1import Lax461425.Circuits
    2
    3/-!
    4---
    5title: Formula and skew circuit properties
    6type: definition
    7---
    8Additional predicates on the existing circuit data.
    9-/
    10
    11namespace Lax461425.CircuitProperties
    12
    13open Circuits
    14
    15open scoped Classical
    16
    17/-- A formula has no parallel wires and may share inputs, while each noninput gate has
    18at most one outgoing wire. -/
    19def IsFormula {V K : Type*} (C : Circuit V K) : Prop :=
    20 (∀ g h, C.wire g h ≤ 1) ∧
    21 ∀ g, ¬ (C.label g).IsInput → ∑ h, C.wire h g ≤ 1
    22
    23/-- At most one incoming wire of a product gate comes from a noninput gate. -/
    24def IsSkew {V K : Type*} (C : Circuit V K) : Prop :=
    25 ∀ g, C.label g = .mul → ∑ h, (if (C.label h).IsInput then 0 else C.wire g h) ≤ 1
    26
    27end Lax461425.CircuitProperties
    28
    Builds on
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    From Mathlib

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