trans tactic

This implements the trans tactic, which can apply transitivity theorems with an optional middle variable argument.

def Trans.simple {α : Sort u_1} {a b c : α} {r : α → α → Sort u_2} [Trans r r r] : r a b → r b c → r a c

Compose using transitivity, homogeneous case.

Equations

• Trans.simple = trans

@[deprecated Trans.trans (since := "2024-10-18")]

def Trans.heq {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {r : α → β → Sort u} {s : β → γ → Sort v} {t : outParam (α → γ → Sort w)} [self : Trans r s t] {a : α} {b : β} {c : γ} : r a b → s b c → t a c

Alias of Trans.trans.

---

Compose two proofs by transitivity, generalized over the relations involved.

Equations

• @Trans.heq = @trans

opaque Batteries.Tactic.transExt : Lean.SimpleScopedEnvExtension (Lean.Name × Array Lean.Meta.DiscrTree.Key) (Lean.Meta.DiscrTree Lean.Name)

Environment extension storing transitivity lemmas

def Batteries.Tactic.getExplicitFuncArg? (e : Lean.Expr) : Lean.MetaM (Option (Lean.Expr × Lean.Expr))

solving e ← mkAppM' f #[x]

Equations

• One or more equations did not get rendered due to their size.
• Batteries.Tactic.getExplicitFuncArg? e = pure none

def Batteries.Tactic.getExplicitRelArg? (tgt f z : Lean.Expr) : Lean.MetaM (Option (Lean.Expr × Lean.Expr))

solving tgt ← mkAppM' rel #[x, z] given tgt = f z

Equations

• One or more equations did not get rendered due to their size.
• Batteries.Tactic.getExplicitRelArg? tgt f z = pure none

def Batteries.Tactic.getExplicitRelArgCore (tgt rel x z : Lean.Expr) : Lean.MetaM (Lean.Expr × Lean.Expr)

refining tgt ← mkAppM' rel #[x, z] dropping more arguments if possible

Equations

• One or more equations did not get rendered due to their size.
• Batteries.Tactic.getExplicitRelArgCore tgt rel x z = pure (rel, x)

inductive Batteries.Tactic.TransRelation : Type

Internal definition for trans tactic. Either a binary relation or a non-dependent arrow.

• app (rel : Lean.Expr) : TransRelation

  Expression for transitive relation.

• implies (name : Lean.Name) (bi : Lean.BinderInfo) : TransRelation

  Constant name for transitive relation.

def Batteries.Tactic.getRel (tgt : Lean.Expr) : Lean.MetaM (Option (TransRelation × Lean.Expr × Lean.Expr))

Finds an explicit binary relation in the argument, if possible.

Equations

• One or more equations did not get rendered due to their size.
• Batteries.Tactic.getRel (Lean.Expr.forallE name binderType body info) = pure (some (Batteries.Tactic.TransRelation.implies name info, binderType, body))
• Batteries.Tactic.getRel tgt = pure none

def Batteries.Tactic.tacticTrans___ : Lean.ParserDescr

trans applies to a goal whose target has the form t ~ u where ~ is a transitive relation, that is, a relation which has a transitivity lemma tagged with the attribute [trans].

• trans s replaces the goal with the two subgoals t ~ s and s ~ u.
• If s is omitted, then a metavariable is used instead.

Additionally, trans also applies to a goal whose target has the form t → u, in which case it replaces the goal with t → s and s → u.

Equations

• One or more equations did not get rendered due to their size.

def Batteries.Tactic.tacticTransitivity___ : Lean.ParserDescr

Synonym for trans tactic.

Equations

• One or more equations did not get rendered due to their size.