The Coq Proof Assistant

# Library Coq.Program.Equality

Tactics related to (dependent) equality and proof irrelevance.
Require Export JMeq.
Require Import Coq.Program.Tactics.
Ltac is_ground_goal :=
match goal with
|- ?T => is_ground T
end.
Try to find a contradiction.
#[global]
Hint Extern 10 => is_ground_goal ; progress exfalso : exfalso.
We will use the block definition to separate the goal from the equalities generated by the tactic.
Definition block {A : Type} (a : A) := a.
Ltac block_goal := match goal with [ |- ?T ] => change (block T) end.
Ltac unblock_goal := unfold block in *.
Notation for heterogeneous equality.
Notation " x ~= y " := (@JMeq _ x _ y) (at level 70, no associativity).
Do something on an heterogeneous equality appearing in the context.
Ltac on_JMeq tac :=
match goal with
| [ H : @JMeq ?x ?X ?y ?Y |- _ ] => tac H
end.
Try to apply JMeq_eq to get back a regular equality when the two types are equal.
Ltac simpl_one_JMeq :=
on_JMeq ltac:(fun H => apply JMeq_eq in H).
Repeat it for every possible hypothesis.
Ltac simpl_JMeq := repeat simpl_one_JMeq.
Just simplify an h.eq. without clearing it.
Ltac simpl_one_dep_JMeq :=
on_JMeq
ltac:(fun H => let H' := fresh "H" in
assert (H' := JMeq_eq H)).
Require Import Eqdep.
Simplify dependent equality using sigmas to equality of the second projections if possible. Uses UIP.
Ltac simpl_existT :=
match goal with
[ H : existT _ ?x _ = existT _ ?x _ |- _ ] =>
let Hi := fresh H in assert(Hi:=inj_pairT2 _ _ _ _ _ H) ; clear H
end.
Ltac simpl_existTs := repeat simpl_existT.
Tries to eliminate a call to eq_rect (the substitution principle) by any means available.
Ltac elim_eq_rect :=
match goal with
| [ |- ?t ] =>
match t with
| context [ @eq_rect _ _ _ _ _ ?p ] =>
let P := fresh "P" in
set (P := p); simpl in P ;
((case P ; clear P) || (clearbody P; rewrite (UIP_refl _ _ P); clear P))
| context [ @eq_rect _ _ _ _ _ ?p _ ] =>
let P := fresh "P" in
set (P := p); simpl in P ;
((case P ; clear P) || (clearbody P; rewrite (UIP_refl _ _ P); clear P))
end
end.
Rewrite using uniqueness of identity proofs H = eq_refl.
Ltac simpl_uip :=
match goal with
[ H : ?X = ?X |- _ ] => rewrite (UIP_refl _ _ H) in *; clear H
end.
Simplify equalities appearing in the context and goal.
Ltac simpl_eq := simpl ; unfold eq_rec_r, eq_rec ; repeat (elim_eq_rect ; simpl) ; repeat (simpl_uip ; simpl).
Try to abstract a proof of equality, if no proof of the same equality is present in the context.
Ltac abstract_eq_hyp H' p :=
let ty := type of p in
let tyred := eval simpl in ty in
match tyred with
?X = ?Y =>
match goal with
| [ H : X = Y |- _ ] => fail 1
| _ => set (H':=p) ; try (change p with H') ; clearbody H' ; simpl in H'
end
end.
Apply the tactic tac to proofs of equality appearing as coercion arguments. Just redefine this tactic (using Ltac on_coerce_proof tac ::=) to handle custom coercion operators.
Ltac on_coerce_proof tac T :=
match T with
| context [ eq_rect _ _ _ _ ?p ] => tac p
end.
Ltac on_coerce_proof_gl tac :=
match goal with
[ |- ?T ] => on_coerce_proof tac T
end.
Abstract proofs of equalities of coercions.
Ltac abstract_eq_proof := on_coerce_proof_gl ltac:(fun p => let H := fresh "eqH" in abstract_eq_hyp H p).
Ltac abstract_eq_proofs := repeat abstract_eq_proof.
Factorize proofs, by using proof irrelevance so that two proofs of the same equality in the goal become convertible.
Ltac pi_eq_proof_hyp p :=
let ty := type of p in
let tyred := eval simpl in ty in
match tyred with
?X = ?Y =>
match goal with
| [ H : X = Y |- _ ] =>
match p with
| H => fail 2
| _ => rewrite (UIP _ X Y p H)
end
| _ => fail " No hypothesis with same type "
end
end.
Factorize proofs of equality appearing as coercion arguments.
Ltac pi_eq_proof := on_coerce_proof_gl pi_eq_proof_hyp.
Ltac pi_eq_proofs := repeat pi_eq_proof.
The two preceding tactics in sequence.
Ltac clear_eq_proofs :=
abstract_eq_proofs ; pi_eq_proofs.
Global Hint Rewrite <- eq_rect_eq : refl_id.
The refl_id database should be populated with lemmas of the form coerce_* t eq_refl = t.
Lemma JMeq_eq_refl {A} (x : A) : JMeq_eq (@JMeq_refl _ x) = eq_refl.
Lemma UIP_refl_refl A (x : A) :
Eqdep.EqdepTheory.UIP_refl A x eq_refl = eq_refl.
Lemma inj_pairT2_refl A (x : A) (P : A -> Type) (p : P x) :
Eqdep.EqdepTheory.inj_pairT2 A P x p p eq_refl = eq_refl.
Global Hint Rewrite @JMeq_eq_refl @UIP_refl_refl @inj_pairT2_refl : refl_id.
Ltac rewrite_refl_id := autorewrite with refl_id.
Clear the context and goal of equality proofs.
Ltac clear_eq_ctx :=
rewrite_refl_id ; clear_eq_proofs.
Reapeated elimination of eq_rect applications. Abstracting equalities makes it run much faster than an naive implementation.
Ltac simpl_eqs :=
repeat (elim_eq_rect ; simpl ; clear_eq_ctx).
Clear unused reflexivity proofs.
Ltac clear_refl_eq :=
match goal with [ H : ?X = ?X |- _ ] => clear H end.
Ltac clear_refl_eqs := repeat clear_refl_eq.
Clear unused equality proofs.
Ltac clear_eq :=
match goal with [ H : _ = _ |- _ ] => clear H end.
Ltac clear_eqs := repeat clear_eq.
Combine all the tactics to simplify goals containing coercions.
Ltac simplify_eqs :=
simpl ; simpl_eqs ; clear_eq_ctx ; clear_refl_eqs ;
try subst ; simpl ; repeat simpl_uip ; rewrite_refl_id.
A tactic that tries to remove trivial equality guards in induction hypotheses coming from dependent induction/generalize_eqs invocations.
Ltac simplify_IH_hyps := repeat
match goal with
| [ hyp : context [ block _ ] |- _ ] =>
specialize_eqs hyp
end.
We split substitution tactics in the two directions depending on which names we want to keep corresponding to the generalization performed by the generalize_eqs tactic.
Ltac subst_left_no_fail :=
repeat (match goal with
[ H : ?X = ?Y |- _ ] => subst X
end).
Ltac subst_right_no_fail :=
repeat (match goal with
[ H : ?X = ?Y |- _ ] => subst Y
end).
Ltac inject_left H :=
progress (inversion H ; subst_left_no_fail ; clear_dups) ; clear H.
Ltac inject_right H :=
progress (inversion H ; subst_right_no_fail ; clear_dups) ; clear H.
Ltac autoinjections_left := repeat autoinjection ltac:(inject_left).
Ltac autoinjections_right := repeat autoinjection ltac:(inject_right).
Ltac simpl_depind := subst_no_fail ; autoinjections ; try discriminates ;
simpl_JMeq ; simpl_existTs ; simplify_IH_hyps.
Ltac simpl_depind_l := subst_left_no_fail ; autoinjections_left ; try discriminates ;
simpl_JMeq ; simpl_existTs ; simplify_IH_hyps.
Ltac simpl_depind_r := subst_right_no_fail ; autoinjections_right ; try discriminates ;
simpl_JMeq ; simpl_existTs ; simplify_IH_hyps.
Ltac blocked t := block_goal ; t ; unblock_goal.
The DependentEliminationPackage provides the default dependent elimination principle to be used by the equations resolver. It is especially useful to register the dependent elimination principles for things in Prop which are not automatically generated.
Class DependentEliminationPackage (A : Type) :=
{ elim_type : Type ; elim : elim_type }.
A higher-order tactic to apply a registered eliminator.
Ltac elim_tac tac p :=
let ty := type of p in
let eliminator := eval simpl in (@elim _ (_ : DependentEliminationPackage ty)) in
tac p eliminator.
Specialization to do case analysis or induction. Note: the equations tactic tries case before elim_case: there is no need to register generated induction principles.
Ltac elim_case p := elim_tac ltac:(fun p el => destruct p using el) p.
Ltac elim_ind p := elim_tac ltac:(fun p el => induction p using el) p.
Lemmas used by the simplifier, mainly rephrasings of eq_rect, eq_ind.
Lemma solution_left A (B : A -> Type) (t : A) :
B t -> (forall x, x = t -> B x).
Lemma solution_right A (B : A -> Type) (t : A) :
B t -> (forall x, t = x -> B x).
Lemma deletion A B (t : A) : B -> (t = t -> B).
Lemma simplification_heq A B (x y : A) :
(x = y -> B) -> (JMeq x y -> B).
Definition conditional_eq {A} (x y : A) := eq x y.
Lemma simplification_existT2 A (P : A -> Type) B (p : A) (x y : P p) :
(x = y -> B) -> (conditional_eq (existT P p x) (existT P p y) -> B).
Lemma simplification_existT1 A (P : A -> Type) B (p q : A) (x : P p) (y : P q) :
(p = q -> conditional_eq (existT P p x) (existT P q y) -> B) -> (existT P p x = existT P q y -> B).
Lemma simplification_K A (x : A) (B : x = x -> Type) :
B eq_refl -> (forall p : x = x, B p).
This hint database and the following tactic can be used with autounfold to unfold everything to eq_rects.
#[global]
Hint Unfold solution_left solution_right deletion simplification_heq
simplification_existT1 simplification_existT2 simplification_K
eq_rect_r eq_rec eq_ind : dep_elim.
Using these we can make a simplifier that will perform the unification steps needed to put the goal in normalised form (provided there are only constructor forms). Compare with the lemma 16 of the paper. We don't have a noCycle procedure yet.
Ltac simplify_one_dep_elim_term c :=
match c with
| @JMeq _ _ _ _ -> _ => refine (simplification_heq _ _ _ _ _)
| ?t = ?t -> _ => intros _ || refine (simplification_K _ t _ _)
| eq (existT _ _ _) (existT _ _ _) -> _ =>
refine (simplification_existT1 _ _ _ _ _ _ _ _)
| conditional_eq (existT _ _ _) (existT _ _ _) -> _ =>
refine (simplification_existT2 _ _ _ _ _ _ _) ||
(unfold conditional_eq; intro)
| ?x = ?y -> _ =>
(unfold x) || (unfold y) ||
(let hyp := fresh in intros hyp ;
move hyp before x ; revert_until hyp ; generalize dependent x ;
refine (solution_left _ _ _ _)) ||
(let hyp := fresh in intros hyp ;
move hyp before y ; revert_until hyp ; generalize dependent y ;
refine (solution_right _ _ _ _))
| ?f ?x = ?g ?y -> _ => let H := fresh in progress (intros H ; simple injection H; clear H)
| ?t = ?u -> _ => let hyp := fresh in
intros hyp ; exfalso ; discriminate
| ?x = ?y -> _ => let hyp := fresh in
intros hyp ; (try (clear hyp ; fail 1)) ;
case hyp ; clear hyp
| block ?T => fail 1
| forall x, _ => intro x || (let H := fresh x in rename x into H ; intro x)
| _ => intro
end.
Ltac simplify_one_dep_elim :=
match goal with
| [ |- ?gl ] => simplify_one_dep_elim_term gl
end.
Repeat until no progress is possible. By construction, it should leave the goal with no remaining equalities generated by the generalize_eqs tactic.
Ltac simplify_dep_elim := repeat simplify_one_dep_elim.
Do dependent elimination of the last hypothesis, but not simplifying yet (used internally).
Ltac destruct_last :=
on_last_hyp ltac:(fun id => simpl in id ; generalize_eqs id ; destruct id).
Ltac introduce p := first [
match p with _ =>
generalize dependent p ; intros p
end
| intros until p | intros until 1 | intros ].
Ltac do_case p := introduce p ; (destruct p || elim_case p || (case p ; clear p)).
Ltac do_ind p := introduce p ; (induction p || elim_ind p).
The following tactics allow to do induction on an already instantiated inductive predicate by first generalizing it and adding the proper equalities to the context, in a maner similar to the BasicElim tactic of "Elimination with a motive" by Conor McBride.
The do_depelim higher-order tactic takes an elimination tactic as argument and an hypothesis and starts a dependent elimination using this tactic.
Ltac is_introduced H :=
match goal with
| [ H' : _ |- _ ] => match H' with H => idtac end
end.
Tactic Notation "intro_block" hyp(H) :=
(is_introduced H ; block_goal ; revert_until H ; block_goal) ||
(let H' := fresh H in intros until H' ; block_goal) || (intros ; block_goal).
Tactic Notation "intro_block_id" ident(H) :=
(is_introduced H ; block_goal ; revert_until H; block_goal) ||
(let H' := fresh H in intros until H' ; block_goal) || (intros ; block_goal).
Ltac unblock_dep_elim :=
match goal with
| |- block ?T =>
match T with context [ block _ ] =>
change T ; intros ; unblock_goal
end
| _ => unblock_goal
end.
Ltac simpl_dep_elim := simplify_dep_elim ; simplify_IH_hyps ; unblock_dep_elim.
Ltac do_intros H :=
(try intros until H) ; (intro_block_id H || intro_block H).
Ltac do_depelim_nosimpl tac H := do_intros H ; generalize_eqs H ; tac H.
Ltac do_depelim tac H := do_depelim_nosimpl tac H ; simpl_dep_elim.
Ltac do_depind tac H :=
(try intros until H) ; intro_block H ;
generalize_eqs_vars H ; tac H ; simpl_dep_elim.
To dependent elimination on some hyp.
Ltac depelim id := do_depelim ltac:(fun hyp => do_case hyp) id.
Used internally.
Ltac depelim_nosimpl id := do_depelim_nosimpl ltac:(fun hyp => do_case hyp) id.
To dependent induction on some hyp.
Ltac depind id := do_depind ltac:(fun hyp => do_ind hyp) id.
A variant where generalized variables should be given by the user.
Ltac do_depelim' rev tac H :=
(try intros until H) ; block_goal ;
(try revert_until H ; block_goal) ;
generalize_eqs H ; rev H ; tac H ; simpl_dep_elim.
Calls destruct on the generalized hypothesis, results should be similar to inversion. By default, we don't try to generalize the hyp by its variable indices.
Tactic Notation "dependent" "destruction" ident(H) :=
do_depelim' ltac:(fun hyp => idtac) ltac:(fun hyp => do_case hyp) H.
Tactic Notation "dependent" "destruction" ident(H) "using" constr(c) :=
do_depelim' ltac:(fun hyp => idtac) ltac:(fun hyp => destruct hyp using c) H.
This tactic also generalizes the goal by the given variables before the elimination.
Tactic Notation "dependent" "destruction" ident(H) "generalizing" ne_hyp_list(l) :=
do_depelim' ltac:(fun hyp => revert l) ltac:(fun hyp => do_case hyp) H.
Tactic Notation "dependent" "destruction" ident(H) "generalizing" ne_hyp_list(l) "using" constr(c) :=
do_depelim' ltac:(fun hyp => revert l) ltac:(fun hyp => destruct hyp using c) H.
Then we have wrappers for usual calls to induction. One can customize the induction tactic by writing another wrapper calling do_depelim. We suppose the hyp has to be generalized before calling induction.
Tactic Notation "dependent" "induction" ident(H) :=
do_depind ltac:(fun hyp => do_ind hyp) H.
Tactic Notation "dependent" "induction" ident(H) "using" constr(c) :=
do_depind ltac:(fun hyp => induction hyp using c) H.
This tactic also generalizes the goal by the given variables before the induction.
Tactic Notation "dependent" "induction" ident(H) "generalizing" ne_hyp_list(l) :=
do_depelim' ltac:(fun hyp => revert l) ltac:(fun hyp => do_ind hyp) H.
Tactic Notation "dependent" "induction" ident(H) "generalizing" ne_hyp_list(l) "using" constr(c) :=
do_depelim' ltac:(fun hyp => revert l) ltac:(fun hyp => induction hyp using c) H.
Tactic Notation "dependent" "induction" ident(H) "in" ne_hyp_list(l) :=
do_depelim' ltac:(fun hyp => idtac) ltac:(fun hyp => induction hyp in l) H.
Tactic Notation "dependent" "induction" ident(H) "in" ne_hyp_list(l) "using" constr(c) :=
do_depelim' ltac:(fun hyp => idtac) ltac:(fun hyp => induction hyp in l using c) H.