1cf71e54cf
`Nat.succ_eq_add_one` and `Nat.pred_eq_sub_one` are now simp lemmas. For theorems about `Nat.succ` or `Nat.pred` without corresponding theorem for `+ 1` or `- 1`, this adds the corresponding theorem.
62 lines
2.9 KiB
Text
62 lines
2.9 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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Author: Leonardo de Moura
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-/
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prelude
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import Init.SizeOf
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import Init.MetaTypes
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import Init.WF
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/-- Unfold definitions commonly used in well founded relation definitions.
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This is primarily intended for internal use in `decreasing_tactic`. -/
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macro "simp_wf" : tactic =>
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`(tactic| try simp (config := { unfoldPartialApp := true, zetaDelta := true }) [invImage, InvImage, Prod.lex, sizeOfWFRel, measure, Nat.lt_wfRel, WellFoundedRelation.rel])
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/-- Extensible helper tactic for `decreasing_tactic`. This handles the "base case"
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reasoning after applying lexicographic order lemmas.
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It can be extended by adding more macro definitions, e.g.
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```
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macro_rules | `(tactic| decreasing_trivial) => `(tactic| linarith)
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```
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-/
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syntax "decreasing_trivial" : tactic
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macro_rules | `(tactic| decreasing_trivial) => `(tactic| (simp (config := { arith := true, failIfUnchanged := false })) <;> done)
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macro_rules | `(tactic| decreasing_trivial) => `(tactic| omega)
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macro_rules | `(tactic| decreasing_trivial) => `(tactic| assumption)
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/--
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Variant of `decreasing_trivial` that does not use `omega`, intended to be used in core modules
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before `omega` is available.
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-/
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syntax "decreasing_trivial_pre_omega" : tactic
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macro_rules | `(tactic| decreasing_trivial_pre_omega) => `(tactic| apply Nat.sub_succ_lt_self; assumption) -- a - (i+1) < a - i if i < a
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macro_rules | `(tactic| decreasing_trivial_pre_omega) => `(tactic| apply Nat.pred_lt_of_lt; assumption) -- i-1 < i if j < i
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macro_rules | `(tactic| decreasing_trivial_pre_omega) => `(tactic| apply Nat.pred_lt; assumption) -- i-1 < i if i ≠ 0
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/-- Constructs a proof of decreasing along a well founded relation, by applying
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lexicographic order lemmas and using `ts` to solve the base case. If it fails,
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it prints a message to help the user diagnose an ill-founded recursive definition. -/
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macro "decreasing_with " ts:tacticSeq : tactic =>
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`(tactic|
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(simp_wf
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repeat (first | apply Prod.Lex.right | apply Prod.Lex.left)
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repeat (first | apply PSigma.Lex.right | apply PSigma.Lex.left)
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first
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| done
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| $ts
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| fail "failed to prove termination, possible solutions:
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- Use `have`-expressions to prove the remaining goals
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- Use `termination_by` to specify a different well-founded relation
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- Use `decreasing_by` to specify your own tactic for discharging this kind of goal"))
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/-- `decreasing_tactic` is called by default on well-founded recursions in order
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to synthesize a proof that recursive calls decrease along the selected
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well founded relation. It can be locally overridden by using `decreasing_by tac`
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on the recursive definition, and it can also be globally extended by adding
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more definitions for `decreasing_tactic` (or `decreasing_trivial`,
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which this tactic calls). -/
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macro "decreasing_tactic" : tactic =>
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`(tactic| decreasing_with first | decreasing_trivial | subst_vars; decreasing_trivial)
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