42 lines
1.3 KiB
Text
42 lines
1.3 KiB
Text
inductive formula :=
|
||
| eqf : nat → nat → formula
|
||
| andf : formula → formula → formula
|
||
| impf : formula → formula → formula
|
||
| notf : formula → formula
|
||
| orf : formula → formula → formula
|
||
| allf : (nat → formula) → formula
|
||
|
||
namespace formula
|
||
definition implies (a b : Prop) : Prop := a → b
|
||
|
||
definition denote : formula → Prop
|
||
| denote (eqf n1 n2) := n1 = n2
|
||
| denote (andf f1 f2) := denote f1 ∧ denote f2
|
||
| denote (impf f1 f2) := implies (denote f1) (denote f2)
|
||
| denote (orf f1 f2) := denote f1 ∨ denote f2
|
||
| denote (notf f) := ¬ denote f
|
||
| denote (allf f) := ∀ n : nat, denote (f n)
|
||
|
||
theorem denote_eqf (n1 n2 : nat) : denote (eqf n1 n2) = (n1 = n2) :=
|
||
rfl
|
||
|
||
theorem denote_andf (f1 f2 : formula) : denote (andf f1 f2) = (denote f1 ∧ denote f2) :=
|
||
rfl
|
||
|
||
theorem denote_impf (f1 f2 : formula) : denote (impf f1 f2) = (denote f1 → denote f2) :=
|
||
rfl
|
||
|
||
theorem denote_orf (f1 f2 : formula) : denote (orf f1 f2) = (denote f1 ∨ denote f2) :=
|
||
rfl
|
||
|
||
theorem denote_notf (f : formula) : denote (notf f) = ¬ denote f :=
|
||
rfl
|
||
|
||
theorem denote_allf (f : nat → formula) : denote (allf f) = (∀ n, denote (f n)) :=
|
||
rfl
|
||
|
||
example : denote (allf (λ n₁, allf (λ n₂, impf (eqf n₁ n₂) (eqf n₂ n₁)))) =
|
||
(∀ n₁ n₂ : nat, n₁ = n₂ → n₂ = n₁) :=
|
||
rfl
|
||
|
||
end formula
|