mathlib documentation

Prefix the given attr_name with "simp_attr".

Simp lemmas are used by the "simplifier" family of tactics. simp_lemmas is essentially a pair of tables rb_map (expr_type × name) (priority_list simp_lemma). One of the tables is for congruences and one is for everything else. An individual simp lemma is:

• A kind which can be Refl, Simp or Congr.
• A pair of exprs l ~> r. The rb map is indexed by the name of get_app_fn(l).
• A proof that l = r or l ↔ r.
• A list of the metavariables that must be filled before the proof can be applied.
• A priority number

Make a new table of simp lemmas

Merge the simp_lemma tables.

Remove the given lemmas from the table. Use the names of the lemmas.

Makes the default simp_lemmas table which is composed of all lemmas tagged with simp.

Add a simplification lemma by an expression p. Some conditions on p must hold for it to be added, see list below. If your lemma is not being added, you can see the reasons by setting set_option trace.simp_lemmas true.

• p must have the type Π (h₁ : _) ... (hₙ : _), LHS ~ RHS for some reflexive, transitive relation (usually =).
• Any of the hypotheses hᵢ should either be present in LHS or otherwise a Prop or a typeclass instance.
• LHS should not occur within RHS.
• LHS should not occur within a hypothesis hᵢ.

Add a simplification lemma by it's declaration name. See simp_lemmas.add for more information.

Adds a congruence simp lemma to simp_lemmas. A congruence simp lemma is a lemma that breaks the simplification down into separate problems. For example, to simplify a ∧ b to c ∧ d, we should try to simp a to c and b to d. For examples of congruence simp lemmas look for lemmas with the @[congr] attribute.

lemma if_simp_congr ... (h_c : b ↔ c) (h_t : x = u) (h_e : y = v) : ite b x y = ite c u v := ...
lemma imp_congr_right (h : a → (b ↔ c)) : (a → b) ↔ (a → c) := ...
lemma and_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a ∧ b) ↔ (c ∧ d) := ...

simp_lemmas.rewrite s e prove R apply a simplification lemma from 's'

• 'e' is the expression to be "simplified"
• 'prove' is used to discharge proof obligations.
• 'r' is the equivalence relation being used (e.g., 'eq', 'iff')

• 'md' is the transparency; how aggresively should the simplifier perform reductions.

  Result (new_e, pr) is the new expression 'new_e' and a proof (pr : e R new_e)

simp_lemmas.drewrite s e tries to rewrite 'e' using only refl lemmas in 's'

(Definitional) Simplify the given expression using only reflexivity equality lemmas from the given set of lemmas. The resulting expression is definitionally equal to the input.

The list u contains defintions to be delta-reduced, and projections to be reduced.

Tries to unfold e if it is a constant or a constant application. Remark: this is not a recursive procedure.

If e is a projection application, try to unfold it, otherwise fail.

simplify s e cfg r prove simplify e using s using bottom-up traversal. discharger is a tactic for dischaging new subgoals created by the simplifier. If it fails, the simplifier tries to discharge the subgoal by simplifying it to true.

The parameter to_unfold specifies definitions that should be delta-reduced, and projection applications that should be unfolded.

ext_simplify_core a c s discharger pre post r e:

• a : α - initial user data
• c : simp_config - simp configuration options
• s : simp_lemmas - the set of simp_lemmas to use. Remark: the simplification lemmas are not applied automatically like in the simplify tactic. The caller must use them at pre/post.
• discharger : α → tactic α - tactic for dischaging hypothesis in conditional rewriting rules. The argument 'α' is the current user data.
• pre a s r p e is invoked before visiting the children of subterm 'e'.
  • arguments:
  • a is the current user data
  • s is the updated set of lemmas if 'contextual' is tt,
  • r is the simplification relation being used,
  • p is the "parent" expression (if there is one).
  • e is the current subexpression in question.
  • if it succeeds the result is (new_a, new_e, new_pr, flag) where
  • new_a is the new value for the user data
  • new_e is a new expression s.t. r e new_e
  • new_pr is a proof for e r new_e, If it is none, the proof is assumed to be by reflexivity
  • flag if tt new_e children should be visited, and post invoked.
• (post a s r p e) is invoked after visiting the children of subterm e, The output is similar to (pre a r s p e), but the 'flag' indicates whether the new expression should be revisited or not.
• r is the simplification relation. Usually = or ↔.
• e is the input expression to be simplified.

The method returns (a,e,pr) where

• a is the final user data
• e is the new expression
• pr is the proof that the given expression equals the input expression.