solve_by_elim, apply_rules, and apply_assumption.

solve_by_elim takes a collection of facts from the local context or supplied as arguments by the user, and performs a backtracking depth-first search by attempting to apply these facts to the goal.

It is a highly configurable tactic, with options to control the backtracking, to solve multiple goals simultaneously (with backtracking between goals), or to supply a discharging tactic for unprovable goals.

apply_rules and apply_assumption are much simpler tactics which do not perform backtracking, but are currently implemented in terms of solve_by_elim with backtracking disabled, in order to be able to share the front-end customisation and parsing of user options. It would be reasonable to further separate these in future.

def Lean.Meta.SolveByElim.applyTactics (cfg : optParam Lean.Meta.ApplyConfig { newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true }) (transparency : optParam Lean.Meta.TransparencyMode Lean.Meta.TransparencyMode.default) (lemmas : List Lean.Expr) (g : Lean.MVarId) : Lean.MetaM (Lean.Meta.Iterator (List Lean.MVarId))

applyTactics lemmas goal will return an iterator that applies the lemmas to the goal goal and returns ones that succeed.

Providing this to the backtracking tactic, we can perform backtracking search based on applying a list of lemmas.

applyTactics (trace := `name) will construct trace nodes for ``nameindicating which calls toapply` succeeded or failed.

Equations

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def Lean.Meta.SolveByElim.applyFirst (cfg : optParam Lean.Meta.ApplyConfig { newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true }) (transparency : optParam Lean.Meta.TransparencyMode Lean.Meta.TransparencyMode.default) (lemmas : List Lean.Expr) (g : Lean.MVarId) : Lean.MetaM (List Lean.MVarId)

applyFirst lemmas goal applies the first of the lemmas which can be successfully applied to goal, and fails if none apply.

We use this in apply_rules and apply_assumption where backtracking is not needed.

Equations

• Lean.Meta.SolveByElim.applyFirst cfg transparency lemmas g = do let __do_lift ← Lean.Meta.SolveByElim.applyTactics cfg transparency lemmas g __do_lift.head

structure Lean.Meta.SolveByElim.ApplyRulesConfigextends Lean.Meta.Tactic.Backtrack.BacktrackConfig , Lean.Meta.ApplyConfig : Type

The default maxDepth for apply_rules is higher.

• maxDepth : Nat
• proc : List Lean.MVarId → List Lean.MVarId → Lean.MetaM (Option (List Lean.MVarId))
• suspend : Lean.MVarId → Lean.MetaM Bool
• discharge : Lean.MVarId → Lean.MetaM (Option (List Lean.MVarId))
• commitIndependentGoals : Bool
• newGoals : Lean.Meta.ApplyNewGoals
• synthAssignedInstances : Bool
• allowSynthFailures : Bool
• approx : Bool
• transparency : Lean.Meta.TransparencyMode

  Transparency mode for calls to apply.

• symm : Bool

  Also use symmetric versions (via @[symm]) of local hypotheses.

• exfalso : Bool

  Try proving the goal via exfalso if solve_by_elim otherwise fails. This is only used when operating on a single goal.

structure Lean.Meta.SolveByElim.SolveByElimConfigextends Lean.Meta.SolveByElim.ApplyRulesConfig , Lean.Meta.Tactic.Backtrack.BacktrackConfig , Lean.Meta.ApplyConfig : Type

Configuration structure to control the behaviour of solve_by_elim:

• transparency mode for calls to apply
• whether to use symm on hypotheses and exfalso on the goal as needed,
• see also BacktrackConfig for hooks allowing flow control.

• maxDepth : Nat
• proc : List Lean.MVarId → List Lean.MVarId → Lean.MetaM (Option (List Lean.MVarId))
• suspend : Lean.MVarId → Lean.MetaM Bool
• discharge : Lean.MVarId → Lean.MetaM (Option (List Lean.MVarId))
• commitIndependentGoals : Bool
• newGoals : Lean.Meta.ApplyNewGoals
• synthAssignedInstances : Bool
• allowSynthFailures : Bool
• approx : Bool
• transparency : Lean.Meta.TransparencyMode
• symm : Bool
• exfalso : Bool
• backtracking : Bool

  Enable backtracking search.

• intro : Bool

  Trying calling intro if no lemmas apply.

• constructor : Bool

  Try calling constructor if no lemmas apply.

instance Lean.Meta.SolveByElim.SolveByElimConfig.instCoeBacktrackConfig : Coe Lean.Meta.SolveByElim.SolveByElimConfig Lean.Meta.Tactic.Backtrack.BacktrackConfig

Equations

• Lean.Meta.SolveByElim.SolveByElimConfig.instCoeBacktrackConfig = { coe := fun (x : Lean.Meta.SolveByElim.SolveByElimConfig) => x.toBacktrackConfig }

def Lean.Meta.SolveByElim.SolveByElimConfig.accept (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) (test : Lean.MVarId → Lean.MetaM Bool) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a Config which allows a class of goals to be returned as subgoals.

Equations

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def Lean.Meta.SolveByElim.SolveByElimConfig.mainGoalProc (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) (proc : Lean.MVarId → Lean.MetaM (List Lean.MVarId)) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a Config which runs a tactic on the main goal. If that tactic fails, fall back to the proc behaviour of cfg.

Equations

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def Lean.Meta.SolveByElim.SolveByElimConfig.intros (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a Config which calls intro on each goal before applying lemmas.

Equations

• cfg.intros = cfg.mainGoalProc fun (g : Lean.MVarId) => do let __do_lift ← g.intro1P pure [__do_lift.snd]

def Lean.Meta.SolveByElim.SolveByElimConfig.synthInstance (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) : Lean.Meta.SolveByElim.SolveByElimConfig

Attempt typeclass inference on each goal, before applying lemmas.

Equations

• cfg.synthInstance = cfg.mainGoalProc fun (g : Lean.MVarId) => do let __do_lift ← g.getType let __do_lift ← Lean.Meta.synthInstance __do_lift g.assign __do_lift pure []

def Lean.Meta.SolveByElim.SolveByElimConfig.withDischarge (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) (discharge : Lean.MVarId → Lean.MetaM (Option (List Lean.MVarId))) : Lean.Meta.SolveByElim.SolveByElimConfig

Add a discharging tactic, falling back to the original discharging tactic if it fails. Return none to return the goal as a new subgoal, or some goals to replace it.

Equations

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def Lean.Meta.SolveByElim.SolveByElimConfig.introsAfter (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a SolveByElimConfig which calls intro on any goal for which no lemma applies.

Equations

• cfg.introsAfter = cfg.withDischarge fun (g : Lean.MVarId) => do let __do_lift ← g.intro1P pure (some [__do_lift.snd])

def Lean.Meta.SolveByElim.SolveByElimConfig.constructorAfter (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) : Lean.Meta.SolveByElim.SolveByElimConfig

Call constructor when no lemmas apply.

Equations

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def Lean.Meta.SolveByElim.SolveByElimConfig.synthInstanceAfter (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a Config which calls synthInstance on any goal for which no lemma applies.

Equations

• cfg.synthInstanceAfter = cfg.withDischarge fun (g : Lean.MVarId) => do let __do_lift ← g.getType let __do_lift ← Lean.Meta.synthInstance __do_lift g.assign __do_lift pure (some [])

def Lean.Meta.SolveByElim.SolveByElimConfig.testPartialSolutions (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) (test : List Lean.Expr → Lean.MetaM Bool) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a Config which rejects branches for which test, applied to the instantiations of the original goals, fails or returns false.

Equations

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def Lean.Meta.SolveByElim.SolveByElimConfig.testSolutions (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) (test : List Lean.Expr → Lean.MetaM Bool) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a SolveByElimConfig which rejects complete solutions for which test, applied to the instantiations of the original goals, fails or returns false.

Equations

• cfg.testSolutions test = cfg.testPartialSolutions fun (sols : List Lean.Expr) => if sols.any Lean.Expr.hasMVar = true then pure true else test sols

def Lean.Meta.SolveByElim.SolveByElimConfig.requireUsingAll (cfg : optParam Lean.Meta.SolveByElim.SolveByElimConfig { maxDepth := 6, proc := fun (x x : List Lean.MVarId) => pure none, suspend := fun (x : Lean.MVarId) => pure false, discharge := fun (x : Lean.MVarId) => failure, commitIndependentGoals := false, newGoals := Lean.Meta.ApplyNewGoals.nonDependentFirst, synthAssignedInstances := true, allowSynthFailures := false, approx := true, transparency := Lean.Meta.TransparencyMode.default, symm := true, exfalso := true, backtracking := true, intro := true, constructor := true }) (use : List Lean.Expr) : Lean.Meta.SolveByElim.SolveByElimConfig

Create or modify a Config which only accept solutions for which every expression in use appears as a subexpression.

Equations

• cfg.requireUsingAll use = cfg.testSolutions fun (sols : List Lean.Expr) => pure (use.all fun (e : Lean.Expr) => sols.any fun (s : Lean.Expr) => e.occurs s)

def Lean.Meta.SolveByElim.SolveByElimConfig.processOptions (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) : Lean.Meta.SolveByElim.SolveByElimConfig

Process the intro and constructor options by implementing the discharger tactic.

Equations

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def Lean.Meta.SolveByElim.elabContextLemmas (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) (g : Lean.MVarId) (lemmas : List (Lean.Elab.TermElabM Lean.Expr)) (ctx : Lean.Meta.SolveByElim.SolveByElimConfig → Lean.Elab.TermElabM (List Lean.Expr)) : Lean.MetaM (List Lean.Expr)

Elaborate a list of lemmas and local context. See mkAssumptionSet for an explanation of why this is needed.

Equations

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def Lean.Meta.SolveByElim.applyLemmas (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) (lemmas : List (Lean.Elab.TermElabM Lean.Expr)) (ctx : Lean.Meta.SolveByElim.SolveByElimConfig → Lean.Elab.TermElabM (List Lean.Expr)) (g : Lean.MVarId) : Lean.MetaM (Lean.Meta.Iterator (List Lean.MVarId))

Returns the list of tactics corresponding to applying the available lemmas to the goal.

Equations

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def Lean.Meta.SolveByElim.applyFirstLemma (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) (lemmas : List (Lean.Elab.TermElabM Lean.Expr)) (ctx : Lean.Meta.SolveByElim.SolveByElimConfig → Lean.Elab.TermElabM (List Lean.Expr)) (g : Lean.MVarId) : Lean.MetaM (List Lean.MVarId)

Applies the first possible lemma to the goal.

Equations

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def Lean.Meta.SolveByElim.solveByElim (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) (lemmas : List (Lean.Elab.TermElabM Lean.Expr)) (ctx : Lean.Meta.SolveByElim.SolveByElimConfig → Lean.Elab.TermElabM (List Lean.Expr)) (goals : List Lean.MVarId) : Lean.MetaM (List Lean.MVarId)

Solve a collection of goals by repeatedly applying lemmas, backtracking as necessary.

Arguments:

• cfg : SolveByElimConfig additional configuration options (options for apply, maximum depth, and custom flow control)
• lemmas : List (TermElabM Expr) lemmas to apply. These are thunks in TermElabM to avoid stuck metavariables.
• ctx : TermElabM (List Expr) monadic function returning the local hypotheses to use.
• goals : List MVarId the initial list of goals for solveByElim

Returns a list of suspended goals, if it succeeded on all other subgoals. By default cfg.suspend is false, cfg.discharge fails, and cfg.failAtMaxDepth is true, and so the returned list is always empty. Custom wrappers (e.g. apply_assumption and apply_rules) may modify this behaviour.

Equations

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def Lean.Meta.SolveByElim.solveByElim.run (lemmas : List (Lean.Elab.TermElabM Lean.Expr)) (ctx : Lean.Meta.SolveByElim.SolveByElimConfig → Lean.Elab.TermElabM (List Lean.Expr)) (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) : List Lean.MVarId → Lean.MetaM (List Lean.MVarId)

Run either backtracking search, or repeated application, on the list of goals.

Equations

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def Lean.Meta.SolveByElim.saturateSymm (symm : Bool) (hyps : List Lean.Expr) : Lean.MetaM (List Lean.Expr)

If symm is true, then adds in symmetric versions of each hypothesis.

Equations

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def Lean.MVarId.applyRules (cfg : Lean.Meta.SolveByElim.SolveByElimConfig) (lemmas : List (Lean.Elab.TermElabM Lean.Expr)) (only : optParam Bool false) (g : Lean.MVarId) : Lean.MetaM (List Lean.MVarId)

A MetaM analogue of the apply_rules user tactic.

We pass the lemmas as TermElabM Expr rather than just Expr, so they can be generated fresh for each application, to avoid stuck metavariables.

By default it uses all local hypotheses, but you can disable this with only := true. If you need to remove particular local hypotheses, call solveByElim directly.

Equations

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def Lean.Meta.SolveByElim.mkAssumptionSet (noDefaults : Bool) (star : Bool) (add : List Lean.Term) (remove : List Lean.Term) (use : Array Lean.Ident) : Lean.MetaM (List (Lean.Elab.TermElabM Lean.Expr) × (Lean.Meta.SolveByElim.SolveByElimConfig → Lean.Elab.TermElabM (List Lean.Expr)))

mkAssumptionSet builds a collection of lemmas for use in the backtracking search in solve_by_elim.

• By default, it includes all local hypotheses, along with rfl, trivial, congrFun and congrArg.
• The flag noDefaults removes these.
• The flag star includes all local hypotheses, but not rfl, trivial, congrFun, or congrArg. (It doesn't make sense to use star without noDefaults.)
• The argument add is the list of terms inside the square brackets that did not have - and can be used to add expressions or local hypotheses
• The argument remove is the list of terms inside the square brackets that had a -, and can be used to remove local hypotheses. (It doesn't make sense to remove expressions which are not local hypotheses, to remove local hypotheses unless !noDefaults || star, and it does not make sense to use star unless you remove at least one local hypothesis.)

mkAssumptionSet returns not a List expr, but a List (TermElabM Expr) × TermElabM (List Expr). There are two separate problems that need to be solved.

Stuck metavariables

Lemmas with implicit arguments would be filled in with metavariables if we created the Expr objects immediately, so instead we return thunks that generate the expressions on demand. This is the first component, with type List (TermElabM Expr).

As an example, we have def rfl : ∀ {α : Sort u} {a : α}, a = a, which on elaboration will become @rfl ?m_1 ?m_2.

Because solve_by_elim works by repeated application of lemmas against subgoals, the first time such a lemma is successfully applied, those metavariables will be unified, and thereafter have fixed values. This would make it impossible to apply the lemma a second time with different values of the metavariables.

See https://github.com/leanprover-community/mathlib/issues/2269

Relevant local hypotheses

solve_by_elim* works with multiple goals, and we need to use separate sets of local hypotheses for each goal. The second component of the returned value provides these local hypotheses. (Essentially using getLocalHyps, along with some filtering to remove hypotheses that have been explicitly removed via only or [-h].)

Equations

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def Lean.Meta.SolveByElim.mkAssumptionSet.elab' (t : Lean.Term) : Lean.Elab.TermElabM Lean.Expr

Run elabTerm.

Equations

• Lean.Meta.SolveByElim.mkAssumptionSet.elab' t = Lean.Elab.Term.elabTerm t.raw none