43 lines
1.2 KiB
Text
43 lines
1.2 KiB
Text
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inductive Formula
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| eqf : Nat → Nat → Formula
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| andf : Formula → Formula → Formula
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| impf : Formula → Formula → Formula
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| notf : Formula → Formula
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| orf : Formula → Formula → Formula
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| allf : (Nat → Formula) → Formula
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namespace Formula
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def implies (a b : Prop) : Prop := a → b
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def denote : Formula → Prop
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| eqf n1 n2 => n1 = n2
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| andf f1 f2 => denote f1 ∧ denote f2
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| impf f1 f2 => implies (denote f1) (denote f2)
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| orf f1 f2 => denote f1 ∨ denote f2
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| notf f => ¬ denote f
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| allf f => (n : Nat) → denote (f n)
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theorem denote_eqf (n1 n2 : Nat) : denote (eqf n1 n2) = (n1 = n2) :=
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rfl
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theorem denote_andf (f1 f2 : Formula) : denote (andf f1 f2) = (denote f1 ∧ denote f2) :=
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rfl
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theorem denote_impf (f1 f2 : Formula) : denote (impf f1 f2) = (denote f1 → denote f2) :=
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rfl
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theorem denote_orf (f1 f2 : Formula) : denote (orf f1 f2) = (denote f1 ∨ denote f2) :=
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rfl
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theorem denote_notf (f : Formula) : denote (notf f) = ¬ denote f :=
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rfl
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theorem denote_allf (f : Nat → Formula) : denote (allf f) = (∀ n, denote (f n)) :=
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rfl
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theorem ex : denote (allf (fun n₁ => allf (fun n₂ => impf (eqf n₁ n₂) (eqf n₂ n₁)))) = (∀ (n₁ n₂ : Nat), n₁ = n₂ → n₂ = n₁) :=
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rfl
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end Formula
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