30 lines
1.2 KiB
Text
30 lines
1.2 KiB
Text
import Lean.LibrarySuggestions.Default
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theorem sq_inj (x y : Nat) (h : x ^ 2 = y ^ 2) : x = y := by
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-- Puzzle for anyone bored: a quick Mathlib-free proof?
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sorry
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example (a b c d e : Nat) (_ : a = b) (_ : b = c) (_ : c ^ 2 = d ^ 2) (_ : d = e) : a = e := by
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grind =>
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-- We can verify that `grind` can see certain facts:
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have : a = c
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-- We can ask for all the known equivalence classes,
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-- or classes involving certain terms:
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show_eqcs a
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-- We can add additional facts, giving proofs inline.
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have : c = d := by grind? +suggestions
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-- These facts are automatically used to extend equivalence classes,
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-- so in this case the `have` statement itself closes the goal.
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example (a b c d e : Nat) (_ : a = b) (_ : b = c) (_ : c ^ 2 = d ^ 2) (_ : d = e) : a = e := by
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grind [sq_inj]
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example (x y : Rat) (_ : x^2 = 1) (_ : x + y = 1) : y ≤ 2 := by
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fail_if_success grind
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grind =>
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-- We can also use `have` statements as clues to guide `grind`.
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have : x = 1 ∨ x = -1
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-- Here `grind` can both prove the `have` statement (via a Grobner argument)
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-- and finish off afterwards (using linear arithmetic),
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-- even though it can't close the original goal by itself.
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finish
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