4aa74d9c0b
Defines `mergeSort`, a naive stable merge sort algorithm, replaces it via a `@[csimp]` lemma with something faster at runtime, and proves the following results: * `mergeSort_sorted`: `mergeSort` produces a sorted list. * `mergeSort_perm`: `mergeSort` is a permutation of the input list. * `mergeSort_of_sorted`: `mergeSort` does not change a sorted list. * `mergeSort_cons`: proves `mergeSort le (x :: xs) = l₁ ++ x :: l₂` for some `l₁, l₂` so that `mergeSort le xs = l₁ ++ l₂`, and no `a ∈ l₁` satisfies `le a x`. * `mergeSort_stable`: if `c` is a sorted sublist of `l`, then `c` is still a sublist of `mergeSort le l`.
25 lines
1 KiB
Text
25 lines
1 KiB
Text
open List MergeSort Internal
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unseal mergeSort merge in
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example : mergeSort (· ≤ ·) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5] = [1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 9] :=
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rfl
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unseal mergeSort merge in
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example : mergeSort (fun x y => x/10 ≤ y/10) [3, 100 + 1, 4, 100 + 1, 5, 100 + 9, 2, 10 + 6, 5, 10 + 3, 5] = [3, 4, 5, 2, 5, 5, 16, 13, 101, 101, 109] :=
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rfl
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unseal mergeSortTR.run mergeTR.go in
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example : mergeSortTR (· ≤ ·) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5] = [1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 9] :=
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rfl
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unseal mergeSortTR.run mergeTR.go in
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example : mergeSortTR (fun x y => x/10 ≤ y/10) [3, 100 + 1, 4, 100 + 1, 5, 100 + 9, 2, 10 + 6, 5, 10 + 3, 5] = [3, 4, 5, 2, 5, 5, 16, 13, 101, 101, 109] :=
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rfl
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unseal mergeSortTR₂.run mergeTR.go in
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example : mergeSortTR₂ (· ≤ ·) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5] = [1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 9] :=
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rfl
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unseal mergeSortTR₂.run mergeTR.go in
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example : mergeSortTR₂ (fun x y => x/10 ≤ y/10) [3, 100 + 1, 4, 100 + 1, 5, 100 + 9, 2, 10 + 6, 5, 10 + 3, 5] = [3, 4, 5, 2, 5, 5, 16, 13, 101, 101, 109] :=
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rfl
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