52 lines
1.1 KiB
Text
52 lines
1.1 KiB
Text
new_frontend
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def diag : Bool → Bool → Bool → Nat
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| b, true, false => 1
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| false, b, true => 2
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| true, false, b => 3
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| b1, b2, b3 => arbitrary Nat
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theorem diag1 (a : Bool) : diag a true false = 1 :=
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match a with
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| true => rfl
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| false => rfl
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theorem diag2 (a : Bool) : diag false a true = 2 :=
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by cases a; exact rfl; exact rfl
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theorem diag3 (a : Bool) : diag true false a = 3 :=
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by cases a; exact rfl; exact rfl
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theorem diag4_1 : diag false false false = arbitrary Nat :=
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rfl
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theorem diag4_2 : diag true true true = arbitrary Nat :=
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rfl
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def f : Nat → Nat → Nat
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| n, 0 => 0
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| 0, n => 1
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| n, m => arbitrary Nat
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theorem f_zero_right : (a : Nat) → f a 0 = 0
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| 0 => rfl
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| a+1 => rfl
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theorem f_zero_succ (a : Nat) : f 0 (a+1) = 1 :=
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rfl
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theorem f_succ_succ (a b : Nat) : f (a+1) (b+1) = arbitrary Nat :=
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rfl
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def app {α} : List α → List α → List α
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| [], l => l
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| h::t, l => h :: (app t l)
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theorem app_nil {α} (l : List α) : app [] l = l :=
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rfl
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theorem app_cons {α} (h : α) (t l : List α) : app (h :: t) l = h :: (app t l) :=
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rfl
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theorem ex : app [1, 2] [3,4,5] = [1,2,3,4,5] :=
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rfl
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