# Documentation

Mathlib.Tactic.Tauto

The tauto tactic.

Tries to apply de-Morgan-like rules on a hypothesis.

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• The list of hypothesis left to work on, renamed to be up-to-date with the current goal.

fvars :
• The current goal.

currentGoal : Lean.MVarId

State of the distribNotAt function. We need to carry around the list of remaining hypothesis as fvars so that we can incrementally apply the AssertAfterResult.subst from each step to each of them. Otherwise, they could end up referring to old hypotheses.

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Calls distribNotAt on the head of state.fvars up to nIters times, returning early on failure.

partial def Mathlib.Tactic.Tauto.distribNotAux (fvars : ) (g : Lean.MVarId) :

For each fvar in fvars, calls distribNotAt and carries along the resulting renamings.

Tries to apply de-Morgan-like rules on all hypotheses. Always succeeds, regardless of whether any progress was actually made.

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Config for the tauto tactic. Currently empty. TODO: add closer option.

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Function elaborating Config.

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Matches propositions where we want to apply the constructor tactic in the core loop of tauto.

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Matches propositions where we want to apply the cases tactic in the core loop of tauto.

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The core loop of the tauto tactic. Repeatedly tries to break down propositions until no more progress can be made. Tries assumption and contradiction at every step, to discharge goals as soon as possible. Does not do anything that requires backtracking.

TODO: The Lean 3 version uses more-powerful versions of contradiction and assumption that additionally apply symm and use a fancy union-find data structure to avoid duplicated work.

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Matches propositions where we want to apply the constructor tactic in the finishing stage of tauto.

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Implementation of the tauto tactic.

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tauto breaks down assumptions of the form _ ∧ _∧ _, _ ∨ _∨ _, _ ↔ _↔ _ and ∃ _, _∃ _, _ and splits a goal of the form _ ∧ _∧ _, _ ↔ _↔ _ or ∃ _, _∃ _, _ until it can be discharged using reflexivity or solve_by_elim. This is a finishing tactic: it either closes the goal or raises an error.

The Lean 3 version of this tactic by default attempted to avoid classical reasoning where possible. This Lean 4 version makes no such attempt. The itauto tactic is designed for that purpose.

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