80 lines
3.1 KiB
Text
80 lines
3.1 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.
|
|||
|
|
|
|||
|
|
Similar to soundness.lean, but defines Nc in Type.
|
|||
|
|
The idea is to be able to prove soundness using recursive equations.
|
|||
|
|
-/
|
|||
|
|
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
|
|||
|
|
|
|||
|
|
reserve infix `⊢`:26
|
|||
|
|
|
|||
|
|
/- Provability -/
|
|||
|
|
|
|||
|
|
inductive Nc : list PropF → PropF → Type :=
|
|||
|
|
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
|
|||
|
|
open Nc
|
|||
|
|
|
|||
|
|
-- Remark ⌞t⌟ indicates we should not pattern match on t.
|
|||
|
|
-- In the following lemma, we only need to pattern match on Γ ⊢ A,
|
|||
|
|
-- by pattern matching on A, we would be creating 10*6 cases instead of 10.
|
|||
|
|
|
|||
|
|
lemma weakening2 : ∀ {Γ A Δ}, Γ ⊢ A → Γ ⊆ Δ → Δ ⊢ A
|
|||
|
|
| Γ ⌞A⌟ Δ (Nax Γ A Hin) Hs := !Nax (Hs A Hin)
|
|||
|
|
| Γ ⌞A ⇒ B⌟ Δ (ImpI Γ A B H) Hs := !ImpI (weakening2 H (cons_sub_cons A Hs))
|
|||
|
|
| Γ ⌞B⌟ Δ (ImpE Γ A B H₁ H₂) Hs := !ImpE (weakening2 H₁ Hs) (weakening2 H₂ Hs)
|
|||
|
|
| Γ ⌞A⌟ Δ (BotC Γ A H) Hs := !BotC (weakening2 H (cons_sub_cons (~A) Hs))
|
|||
|
|
| Γ ⌞A ∧ B⌟ Δ (AndI Γ A B H₁ H₂) Hs := !AndI (weakening2 H₁ Hs) (weakening2 H₂ Hs)
|
|||
|
|
| Γ ⌞A⌟ Δ (AndE₁ Γ A B H) Hs := !AndE₁ (weakening2 H Hs)
|
|||
|
|
| Γ ⌞B⌟ Δ (AndE₂ Γ A B H) Hs := !AndE₂ (weakening2 H Hs)
|
|||
|
|
| Γ ⌞A ∧ B⌟ Δ (OrI₁ Γ A B H) Hs := !OrI₁ (weakening2 H Hs)
|
|||
|
|
| Γ ⌞A ∨ B⌟ Δ (OrI₂ Γ A B H) Hs := !OrI₂ (weakening2 H Hs)
|
|||
|
|
| Γ ⌞C⌟ Δ (OrE Γ A B C H₁ H₂ H₃) Hs :=
|
|||
|
|
!OrE (weakening2 H₁ Hs) (weakening2 H₂ (cons_sub_cons A Hs)) (weakening2 H₃ (cons_sub_cons B Hs))
|
|||
|
|
|
|||
|
|
end PropF
|