28 lines
750 B
Text
28 lines
750 B
Text
|
inductive formula :=
|
||
|
|
eqf : nat → nat → formula,
|
||
|
|
andf : formula → formula → formula,
|
||
|
|
impf : formula → formula → formula,
|
||
|
|
allf : (nat → formula) → formula
|
||
|
|
|
||
|
|
check @formula.rec_on
|
||
|
|
|
||
|
|
namespace formula
|
||
|
|
|
||
|
|
definition denote : formula → Prop,
|
||
|
|
denote (eqf n1 n2) := n1 = n2,
|
||
|
|
denote (andf f1 f2) := denote f1 ∧ denote f2,
|
||
|
|
denote (impf f1 f2) := denote f1 → denote f2,
|
||
|
|
denote (allf f) := ∀ n : nat, denote (f n)
|
||
|
|
|
||
|
|
end formula
|
||
|
|
|
||
|
|
definition denote (f : formula) : Prop :=
|
||
|
|
formula.rec_on f
|
||
|
|
(λ n₁ n₂, n₁ = n₂)
|
||
|
|
(λ f₁ f₂ r₁ r₂, r₁ ∧ r₂)
|
||
|
|
(λ f₁ f₂ r₁ r₂, r₁ → r₂)
|
||
|
|
(λ f r, ∀ n : nat, r n)
|
||
|
|
|
||
|
|
open formula
|
||
|
|
eval denote (allf (λ n₁, allf (λ n₂, impf (eqf n₁ n₂) (eqf n₂ n₁))))
|