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