88 lines
1.4 KiB
Text
88 lines
1.4 KiB
Text
set_option pp.analyze false
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def p (x y : Nat) := x = y
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example (x y : Nat) : p (x + y) (y + x + 0) := by
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conv =>
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whnf
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congr
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. skip
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. whnf; skip
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traceState
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rw [Nat.add_comm]
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rfl
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example (x y : Nat) : p (x + y) (y + x + 0) := by
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conv =>
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whnf
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rhs
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whnf
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traceState
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rw [Nat.add_comm]
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rfl
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example (x y : Nat) : p (x + y) (y + x + 0) := by
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conv =>
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whnf
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lhs
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whnf
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conv =>
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rhs
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whnf
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traceState
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apply Nat.add_comm x y
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example (x y : Nat) : p (x + y) (0 + y + x) := by
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conv =>
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whnf
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rhs
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rw [Nat.zero_add, Nat.add_comm]
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traceState
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skip
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done
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axiom div_self (x : Nat) : x ≠ 0 → x / x = 1
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example (h : x ≠ 0) : x / x + x = x.succ := by
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conv =>
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lhs
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arg 2
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rw [div_self]
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skip
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tactic => assumption
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done
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show 1 + x = x.succ
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rw [Nat.succ_add, Nat.zero_add]
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example (h1 : x ≠ 0) (h2 : y = x / x) : y = 1 := by
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conv at h2 =>
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rhs
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rw [div_self]
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skip
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tactic => assumption
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assumption
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example : id (fun x => 0 + x) = id := by
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conv =>
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lhs
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arg 1
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funext y
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rw [Nat.zero_add]
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def f (x : Nat) :=
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if x > 0 then
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x + 1
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else
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x + 2
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example (g : Nat → Nat) (h₁ : g x = x + 1) (h₂ : x > 0) : g x = f x := by
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conv =>
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rhs
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simp [f, h₂]
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exact h₁
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example (h₁ : f x = x + 1) (h₂ : x > 0) : f x = f x := by
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conv =>
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rhs
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simp [f, h₂]
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exact h₁
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