(** Ltac definitions **)
Require Import Coq.Bool.Bool Coq.Reals.Reals Coq.QArith.QArith Coq.QArith.Qreals.
Require Import Daisy.Infra.RealSimps Daisy.Infra.NatSet Daisy.Infra.RationalSimps Daisy.Infra.RealRationalProps.
Ltac iv_assert iv name:=
assert (exists ivlo ivhi, iv = (ivlo, ivhi)) as name by (destruct iv; repeat eexists; auto).
(** Automatic translation and destruction of conjuctinos with andb into Props **)
Ltac andb_to_prop H :=
apply Is_true_eq_left in H;
apply andb_prop_elim in H;
let Left := fresh "L" in
let Right := fresh "R" in
destruct H as [Left Right];
apply Is_true_eq_true in Left;
apply Is_true_eq_true in Right;
try andb_to_prop Left;
try andb_to_prop Right.
Ltac canonize_Q_prop :=
match goal with
| [ H: Qle_bool ?a ?b = true |- _] => rewrite Qle_bool_iff in H
| [ H: Qleb ?a ?b = true |- _ ] => rewrite Qle_bool_iff in H
| [ H: Qeq_bool ?a ?b = true |- _] => rewrite Qeq_bool_iff in H
end.
Ltac canonize_Q_to_R :=
match goal with
| [ H: (?a <= ?b)%Q |- _ ] => apply Qle_Rle in H
| [ H: (?a == ?b)%Q |- _ ] => apply Qeq_eqR in H
| [ H: context [Q2R 0] |- _ ] => rewrite Q2R0_is_0 in H
| [ |- context [Q2R 0] ] => rewrite Q2R0_is_0
end.
Ltac canonize_hyps := repeat canonize_Q_prop; repeat canonize_Q_to_R.
Ltac Q2R_to_head_step :=
match goal with
| [ H: context[Q2R ?a + Q2R ?b] |- _] => rewrite <- Q2R_plus in H
| [ H: context[Q2R ?a - Q2R ?b] |- _] => rewrite <- Q2R_minus in H
| [ H: context[Q2R ?a * Q2R ?b] |- _] => rewrite <- Q2R_minus in H
| [ |- context[Q2R ?a + Q2R ?b]] => rewrite <- Q2R_plus
| [ |- context[Q2R ?a - Q2R ?b]] => rewrite <- Q2R_minus
| [ |- context[Q2R ?a * Q2R ?b]] => rewrite <- Q2R_minus
end.
Ltac Q2R_to_head := repeat Q2R_to_head_step.
Definition optionLift (X Y:Type) (v:option X) (f:X -> Y) (e:Y) :=
match v with
|Some val => f val
| None => e
end.
Lemma optionLift_eq (X Y:Type) v (f:X -> Y) (e:Y):
match v with |Some val => f val | None => e end = optionLift X Y v f e.
Proof.
cbv; auto.
Qed.
Lemma optionLift_cond X (a:bool) (b c :X):
(if a then b else c) = match a with |true => b |false => c end.
Proof.
cbv; auto.
Qed.
Ltac remove_matches := rewrite optionLift_eq in *.
Ltac remove_conds := rewrite <- andb_lazy_alt, optionLift_cond in *.
Ltac match_factorize :=
match goal with
| [ H: ?t = ?u |- context [optionLift _ _ ?t _ _]]
=> rewrite H; cbn
| [ H1: ?t = ?u, H2: context [optionLift _ _ ?t _ _] |- _ ]
=> rewrite H1 in H2; cbn in H2
| [ H: context [optionLift _ _ ?t _ _] |- _ ]
=> destruct t eqn:?; cbn in H; try congruence
| [ |- context [optionLift _ _ ?t _ _] ]
=> destruct t eqn:?; cbn; try congruence
end.
Ltac pair_factorize :=
match goal with
| [H: context[let (_, _) := ?p in _] |- _] => destruct p; cbn in H
end.
Ltac match_simpl :=
try remove_conds;
try remove_matches;
repeat match_factorize;
try pair_factorize.
(* Ltac destruct_if := *)
(* match goal with *)
(* | [H: if ?c then ?a else ?b = _ |- _ ] => *)
(* case_eq ?c; *)
(* let name := fresh "cond" in *)
(* intros name; *)
(* rewrite name in *; *)
(* try congruence *)
(* | [H: _ |- if ?c then ?a else ?b = _] => *)
(* case_eq ?c; *)
(* let name := fresh "cond" in *)
(* intros name; *)
(* rewrite name in *; *)
(* try congruence *)
(* end. *)
(* Ltac match_destr t:= *)
(* match goal with *)
(* | H: context [optionLift (DaisyMap.find ?k ?M) _ _] |- _ => *)
(* destruct (DaisyMap.find (elt:=intv * error) k M); idtac H *)
(* end. *)
Tactic Notation "match_pat" open_constr(pat) tactic(t) :=
match goal with
| [H: ?ty |- _ ] => unify pat ty; t H
end.