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feat(library/logic/examples/propositional/soundness): cleanup precedence levels

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Leonardo de Moura 2015-03-30 05:42:47 -07:00
parent 5ef88bfbc8
commit 9d34431bb6
1 changed files with 15 additions and 15 deletions
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30
library/logic/examples/propositional/soundness.lean
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@ -24,7 +24,7 @@ namespace PropF
notation `#`:max P:max := Var P
notation A ∨ B := Disj A B
notation A ∧ B := Conj A B
infixr `⇒`:25 := Impl
infixr `⇒`:27 := Impl
notation `⊥` := Bot
definition Neg A := A ⇒ ⊥
@ -32,7 +32,7 @@ namespace PropF
definition Top := ~⊥
notation `⊤` := Top
definition BiImpl A B := A ⇒ B ∧ B ⇒ A
infixr `⇔`:25 := BiImpl
infixr `⇔`:27 := BiImpl
definition valuation := PropVar → bool
@ -53,22 +53,22 @@ namespace PropF
infix `⊨`:80 := Models
definition Valid p := [] ⊨ p
reserve infix `⊢`:80
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
| 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
@ -93,10 +93,10 @@ namespace PropF
(λ Γ A B H w Δ Hs, !OrI₂ (w _ Hs))
(λ Γ A B C H₁ H₂ H₃ w₁ w₂ w₃ Δ Hs, !OrE (w₁ _ Hs) (w₂ _ (cons_sub_cons A Hs)) (w₃ _ (cons_sub_cons B Hs)))
lemma weakening : ∀ Γ Δ A, Γ ⊢ A → (Γ++Δ) ⊢ A :=
lemma weakening : ∀ Γ Δ A, Γ ⊢ A → Γ++Δ ⊢ A :=
λ Γ Δ A H, weakening2 Γ A H (Γ++Δ) (sub_append_left Γ Δ)
lemma deduction : ∀ Γ A B, Γ ⊢ (A ⇒ B) → (A::Γ) ⊢ B :=
lemma deduction : ∀ Γ A B, Γ ⊢ A ⇒ B → A::Γ ⊢ B :=
λ Γ A B H, ImpE _ A _ (!weakening2 H _ (sub_cons A Γ)) (!Nax (mem_cons A Γ))
lemma prov_impl : ∀ A B, Provable (A ⇒ B) → ∀ Γ, Γ ⊢ A → Γ ⊢ B :=