63 lines
1.5 KiB
Text
63 lines
1.5 KiB
Text
constant f : Nat → Nat
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constant q : Nat → Prop
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constant r : Nat → Prop
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@[simp] axiom ax1 (p : Prop) : (p ∧ True) ↔ p
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@[simp] axiom ax2 (x : Nat) : q (f x)
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@[simp] axiom ax3 (x : Nat) : ¬ r (f x)
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@[simp] axiom ax4 (p : Prop) : (p ∨ False) ↔ p
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theorem ex1 (x : Nat) (h : q x) : q x ∧ q (f x) := by
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simp [h]
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theorem ex2 (x : Nat) : q (f x) ∨ r (f x) := by
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simp
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@[simp] axiom ax5 (x : Nat) : 0 + x = x
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theorem ex3 (h : 0 + x = y) : f x = f y := by
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simp at h
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simp [h]
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theorem ex4 (x y z : Nat) (h : (x, z).1 = y) : f x = f y := by
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simp at h
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simp [h]
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theorem ex5
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(f : Nat → Nat → Nat)
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(g : Nat → Nat)
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(h₁ : ∀ x, f x x = x)
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(h₂ : ∀ x, g (g x) = x)
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: f (g (g x)) (f x x) = x :=
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by simp [h₁, h₂]
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@[simp] axiom ax6 (x : Nat) : x + 0 = x
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theorem ex6
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(f : Nat → Nat)
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(x y : Nat)
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: (fun (h : y = 0) => y + x) = (fun _ => x + 0) := by
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simp (config := { contextual := true })
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theorem ex7 (x : Nat) : (let y := x + 0; y + y) = x + x := by
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simp
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@[simp] theorem impTrue (p : Sort u) : (p → True) = True :=
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propext <| Iff.intro (fun _ => trivial) (fun _ _ => trivial)
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theorem ex8 (y x : Nat) : y = 0 → x + y = 0 → x = 0 := by
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simp (config := { contextual := true })
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theorem ex9 (y x : Nat) : y = 0 → x + y = 0 → x = 0 := by
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simp
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intro h₁ h₂
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simp [h₁] at h₂
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simp [h₂]
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theorem ex10 (y x : Nat) : y = 0 → x + 0 = 0 → x = 0 := by
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simp
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intro h₁ h₂
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simp [h₂]
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theorem ex11 : ∀ x : Nat, 0 + x + 0 = x := by
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simp
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