57 lines
2.2 KiB
Text
57 lines
2.2 KiB
Text
/-
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Copyright (c) 2022 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura, Joachim Breitner
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-/
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module
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prelude
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import Init.Data.Array.Basic
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import Init.Data.Nat.Linear
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import Init.Data.List.BasicAux
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set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
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set_option linter.indexVariables true -- Enforce naming conventions for index variables.
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namespace Array
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theorem sizeOf_lt_of_mem [SizeOf α] {as : Array α} (h : a ∈ as) : sizeOf a < sizeOf as := by
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cases as with | _ as =>
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exact Nat.lt_trans (List.sizeOf_lt_of_mem h.val) (by simp +arith)
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theorem sizeOf_get [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) : sizeOf as[i] < sizeOf as := by
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cases as with | _ as =>
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simpa using Nat.lt_trans (List.sizeOf_get _ ⟨i, h⟩) (by simp +arith)
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@[simp] theorem sizeOf_getElem [SizeOf α] (as : Array α) (i : Nat) (h : i < as.size) :
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sizeOf (as[i]'h) < sizeOf as := sizeOf_get _ _ h
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/-- This tactic, added to the `decreasing_trivial` toolbox, proves that
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`sizeOf arr[i] < sizeOf arr`, which is useful for well founded recursions
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over a nested inductive like `inductive T | mk : Array T → T`. -/
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macro "array_get_dec" : tactic =>
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`(tactic| first
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-- subsumed by simp
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-- | with_reducible apply sizeOf_get
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-- | with_reducible apply sizeOf_getElem
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| (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_get ..)); simp +arith
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| (with_reducible apply Nat.lt_of_lt_of_le (sizeOf_getElem ..)); simp +arith
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)
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macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_get_dec)
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/-- This tactic, added to the `decreasing_trivial` toolbox, proves that `sizeOf a < sizeOf arr`
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provided that `a ∈ arr` which is useful for well founded recursions over a nested inductive like
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`inductive T | mk : Array T → T`. -/
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-- NB: This is analogue to tactic `sizeOf_list_dec`
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macro "array_mem_dec" : tactic =>
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`(tactic| first
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| with_reducible apply Array.sizeOf_lt_of_mem; assumption; done
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| with_reducible
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apply Nat.lt_of_lt_of_le (Array.sizeOf_lt_of_mem ?h)
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case' h => assumption
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simp +arith)
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macro_rules | `(tactic| decreasing_trivial) => `(tactic| array_mem_dec)
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end Array
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