91 lines
2.7 KiB
Text
91 lines
2.7 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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Define propositional calculus, valuation, provability, validity, prove soundness.
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This file is based on Floris van Doorn Coq files.
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-/
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import data.nat data.list
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open nat bool list decidable
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definition PropVar [reducible] := nat
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inductive PropF :=
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| Var : PropVar → PropF
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| Bot : PropF
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| Conj : PropF → PropF → PropF
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| Disj : PropF → PropF → PropF
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| Impl : PropF → PropF → PropF
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namespace PropF
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notation `#`:max P:max := Var P
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notation A ∨ B := Disj A B
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notation A ∧ B := Conj A B
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infixr `⇒`:27 := Impl
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notation `⊥` := Bot
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definition Neg A := A ⇒ ⊥
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notation ~ A := Neg A
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definition Top := ~⊥
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notation `⊤` := Top
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definition BiImpl A B := A ⇒ B ∧ B ⇒ A
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infixr `⇔`:27 := BiImpl
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definition valuation := PropVar → bool
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definition TrueQ (v : valuation) : PropF → bool
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| TrueQ (# P) := v P
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| TrueQ ⊥ := ff
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| TrueQ (A ∨ B) := TrueQ A || TrueQ B
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| TrueQ (A ∧ B) := TrueQ A && TrueQ B
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| TrueQ (A ⇒ B) := bnot (TrueQ A) || TrueQ B
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definition is_true [reducible] (b : bool) := b = tt
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-- the valuation v satisfies a list of PropF, if forall (A : PropF) in Γ,
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-- (TrueQ v A) is tt (the Boolean true)
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definition Satisfies v Γ := ∀ A, A ∈ Γ → is_true (TrueQ v A)
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definition Models Γ A := ∀ v, Satisfies v Γ → is_true (TrueQ v A)
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infix `⊨`:80 := Models
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definition Valid p := [] ⊨ p
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reserve infix `⊢`:26
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/- Provability -/
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inductive Nc : list PropF → PropF → Prop :=
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infix ⊢ := Nc
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| Nax : ∀ Γ A, A ∈ Γ → Γ ⊢ A
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| ImpI : ∀ Γ A B, A::Γ ⊢ B → Γ ⊢ A ⇒ B
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| ImpE : ∀ Γ A B, Γ ⊢ A ⇒ B → Γ ⊢ A → Γ ⊢ B
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| BotC : ∀ Γ A, (~A)::Γ ⊢ ⊥ → Γ ⊢ A
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| AndI : ∀ Γ A B, Γ ⊢ A → Γ ⊢ B → Γ ⊢ A ∧ B
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| AndE₁ : ∀ Γ A B, Γ ⊢ A ∧ B → Γ ⊢ A
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| AndE₂ : ∀ Γ A B, Γ ⊢ A ∧ B → Γ ⊢ B
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| OrI₁ : ∀ Γ A B, Γ ⊢ A → Γ ⊢ A ∨ B
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| OrI₂ : ∀ Γ A B, Γ ⊢ B → Γ ⊢ A ∨ B
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| OrE : ∀ Γ A B C, Γ ⊢ A ∨ B → A::Γ ⊢ C → B::Γ ⊢ C → Γ ⊢ C
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infix ⊢ := Nc
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definition Provable A := [] ⊢ A
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definition Prop_Soundness := ∀ A, Provable A → Valid A
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definition Prop_Completeness := ∀ A, Valid A → Provable A
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open Nc
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lemma weakening2 (Γ A) (H : Γ ⊢ A) (Δ) (Hs : Γ ⊆ Δ) : Δ ⊢ A :=
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begin
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induction H,
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state,
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exact !Nax (Hs a),
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repeat (apply sorry)
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end
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end PropF
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