7fa1a8b114
This PR eliminates uses of `intros x y z` (with arguments) and updates the `intros` docstring to suggest that `intro x y z` should be used instead. The `intros` tactic is historical, and can be traced all the way back to Lean 2, when `intro` could only introduce a single hypothesis. Since 2020, the `intro` tactic has superceded it. The `intros` tactic (without arguments) is currently still useful.
130 lines
2.6 KiB
Text
130 lines
2.6 KiB
Text
inductive Le (m : Nat) : Nat → Prop
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| base : Le m m
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| succ : (n : Nat) → Le m n → Le m n.succ
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theorem ex1 (m : Nat) : Le m 0 → m = 0 := by
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intro h
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cases h
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rfl
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theorem ex2 (m n : Nat) : Le m n → Le m.succ n.succ := by
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intro h
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induction h with
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| base => apply Le.base
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| succ n m ih =>
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apply Le.succ
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apply ih
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theorem ex3 (m : Nat) : Le 0 m := by
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induction m with
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| zero => apply Le.base
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| succ m ih =>
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apply Le.succ
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apply ih
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theorem ex4 (m : Nat) : ¬ Le m.succ 0 := by
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intro h
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cases h
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theorem ex5 {m n : Nat} : Le m n.succ → m = n.succ ∨ Le m n := by
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intro h
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cases h with
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| base => apply Or.inl; rfl
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| succ => apply Or.inr; assumption
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theorem ex6 {m n : Nat} : Le m.succ n.succ → Le m n := by
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revert m
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induction n with
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| zero =>
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intro m h;
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cases h with
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| base => apply Le.base
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| succ n h => exact absurd h (ex4 _)
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| succ n ih =>
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intro m h
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have aux := ih (m := m)
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cases ex5 h with
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| inl h =>
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injection h with h
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subst h
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apply Le.base
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| inr h =>
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apply Le.succ
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exact ih h
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theorem ex7 {m n o : Nat} : Le m n → Le n o → Le m o := by
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intro h
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induction h with
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| base => intros; assumption
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| succ n s ih =>
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intro h₂
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apply ih
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apply ex6
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apply Le.succ
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assumption
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theorem ex8 {m n : Nat} : Le m.succ n → Le m n := by
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intro h
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apply ex6
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apply Le.succ
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assumption
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theorem ex9 {m n : Nat} : Le m n → m = n ∨ Le m.succ n := by
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intro h
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cases h with
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| base => apply Or.inl; rfl
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| succ n s =>
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apply Or.inr
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apply ex2
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assumption
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/-
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theorem ex10 (n : Nat) : ¬ Le n.succ n := by
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intro h
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cases h -- TODO: improve cases tactic
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done
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-/
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theorem ex10 (n : Nat) : n.succ ≠ n := by
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induction n with
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| zero => intro h; injection h; done
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| succ n ih => intro h; injection h with h; apply ih h
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theorem ex11 (n : Nat) : ¬ Le n.succ n := by
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induction n with
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| zero => intro h; cases h; done
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| succ n ih =>
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intro h
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have aux := ex6 h
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exact absurd aux ih
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done
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theorem ex12 (m n : Nat) : Le m n → Le n m → m = n := by
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revert m
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induction n with
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| zero => intro m h1 h2; apply ex1; assumption; done
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| succ n ih =>
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intro m h1 h2
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have ih := ih m
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cases ex5 h1 with
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| inl h => assumption
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| inr h =>
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have ih := ih h
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have h3 := ex8 h2
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have ih := ih h3
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subst ih
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apply absurd h2 (ex11 _)
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done
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inductive Foo : Nat → Prop where
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| foo : Foo 0
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| bar : Foo 0
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| baz : Foo 1
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example (f : Foo 0) : True := by
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cases f with
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| _ => trivial
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example (f : Foo n) (h : n = 0) : True := by
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induction f with simp at h
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| _ => trivial
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