17b133369d
This PR removes the auto-generated `binductionOn` and `ibelow` implementations for inductive types in favor of the improved `brecOn` implementation from #7639.
86 lines
2.6 KiB
Text
86 lines
2.6 KiB
Text
import Lean
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inductive Tree
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| leaf : Tree
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| node : Tree → Tree → Tree
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abbrev notSubtree (x : Tree) (t : Tree) : Prop :=
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t.rec True fun l r l_ih r_ih => (x ≠ l ∧ l_ih) ∧ (x ≠ r ∧ r_ih)
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infix:50 "≮" => notSubtree
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theorem Tree.acyclic (x t : Tree) : x = t → x ≮ t := by
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let rec right (x s : Tree) (b : Tree) (h : x ≮ b) : node s x ≠ b ∧ node s x ≮ b := by
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match b, h with
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| leaf, h =>
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apply And.intro _ trivial
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intro h; injection h
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| node l r, h =>
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have ihl : x ≮ l → node s x ≠ l ∧ node s x ≮ l := right x s l
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have ihr : x ≮ r → node s x ≠ r ∧ node s x ≮ r := right x s r
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have hl : x ≠ l ∧ x ≮ l := h.1
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have hr : x ≠ r ∧ x ≮ r := h.2
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have ihl : node s x ≠ l ∧ node s x ≮ l := ihl hl.2
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have ihr : node s x ≠ r ∧ node s x ≮ r := ihr hr.2
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apply And.intro
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focus
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intro h
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injection h with _ h
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exact absurd h hr.1
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done
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focus
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apply And.intro
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apply ihl
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apply ihr
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let rec left (x t : Tree) (b : Tree) (h : x ≮ b) : node x t ≠ b ∧ node x t ≮ b := by
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match b, h with
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| leaf, h =>
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apply And.intro _ trivial
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intro h; injection h
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| node l r, h =>
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have ihl : x ≮ l → node x t ≠ l ∧ node x t ≮ l := left x t l
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have ihr : x ≮ r → node x t ≠ r ∧ node x t ≮ r := left x t r
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have hl : x ≠ l ∧ x ≮ l := h.1
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have hr : x ≠ r ∧ x ≮ r := h.2
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have ihl : node x t ≠ l ∧ node x t ≮ l := ihl hl.2
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have ihr : node x t ≠ r ∧ node x t ≮ r := ihr hr.2
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apply And.intro
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focus
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intro h
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injection h with h _
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exact absurd h hl.1
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done
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focus
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apply And.intro
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apply ihl
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apply ihr
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let rec aux : (x : Tree) → x ≮ x
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| leaf => trivial
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| node l r => by
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have ih₁ : l ≮ l := aux l
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have ih₂ : r ≮ r := aux r
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show (node l r ≠ l ∧ node l r ≮ l) ∧ (node l r ≠ r ∧ node l r ≮ r)
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apply And.intro
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focus
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apply left
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assumption
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focus
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apply right
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assumption
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intro h
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subst h
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apply aux
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open Tree
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theorem ex1 (x : Tree) : x ≠ node leaf (node x leaf) := by
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intro h
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exact absurd rfl $ Tree.acyclic _ _ h |>.2.2.1.1
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theorem ex2 (x : Tree) : x ≠ node x leaf := by
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intro h
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exact absurd rfl $ Tree.acyclic _ _ h |>.1.1
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theorem ex3 (x y : Tree) : x ≠ node y x := by
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intro h
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exact absurd rfl $ Tree.acyclic _ _ h |>.2.1
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