50 lines
1.5 KiB
Text
50 lines
1.5 KiB
Text
axiom appendNil {α} (as : List α) : as ++ [] = as
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axiom appendAssoc {α} (as bs cs : List α) : (as ++ bs) ++ cs = as ++ (bs ++ cs)
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axiom reverseEq {α} (as : List α) : as.reverse.reverse = as
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theorem ex1 {α} (as bs : List α) : as.reverse.reverse ++ [] ++ [] ++ bs ++ bs = as ++ (bs ++ bs) := by
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rw [appendNil, appendNil, reverseEq];
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trace_state;
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rw [←appendAssoc];
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theorem ex2 {α} (as bs : List α) : as.reverse.reverse ++ [] ++ [] ++ bs ++ bs = as ++ (bs ++ bs) := by
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rewrite [reverseEq, reverseEq]; -- Error on second reverseEq
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done
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axiom zeroAdd (x : Nat) : 0 + x = x
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theorem ex2a (x y z) (h₁ : 0 + x = y) (h₂ : 0 + y = z) : x = z := by
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rewrite [zeroAdd] at h₁ h₂;
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trace_state;
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subst x;
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subst y;
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exact rfl
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theorem ex3 (x y z) (h₁ : 0 + x = y) (h₂ : 0 + y = z) : x = z := by
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rewrite [zeroAdd] at *;
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subst x;
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subst y;
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exact rfl
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theorem ex4 (x y z) (h₁ : 0 + x = y) (h₂ : 0 + y = z) : x = z := by
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rewrite [appendAssoc] at *; -- Error
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done
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theorem ex5 (m n k : Nat) (h : 0 + n = m) (h : k = m) : k = n := by
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rw [zeroAdd] at *;
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trace_state; -- `h` is still a name for `h : k = m`
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refine Eq.trans h ?hole;
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apply Eq.symm;
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assumption
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theorem ex6 (p q r : Prop) (h₁ : q → r) (h₂ : p ↔ q) (h₃ : p) : r := by
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rw [←h₂] at h₁;
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exact h₁ h₃
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theorem ex7 (p q r : Prop) (h₁ : q → r) (h₂ : p ↔ q) (h₃ : p) : r := by
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rw [h₂] at h₃;
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exact h₁ h₃
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example (α : Type) (p : Prop) (a b c : α) (h : p → a = b) : a = c := by
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rw [h _] -- should manifest goal `⊢ p`, like `rw [h]` would
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