Library Coq.Structures.OrdersTac
The order tactic
Inductive ord : Set := OEQ | OLT | OLE.
Definition trans_ord o o' :=
match o, o' with
| OEQ, _ => o'
| _, OEQ => o
| OLE, OLE => OLE
| _, _ => OLT
end.
The tactic requirements : a total order
- an equivalence eq,
- a strict order lt total and compatible with eq,
- a larger order le synonym for lt\/eq.
Reflexivity rules
Symmetry rules
Lemma eq_sym : forall x y, x==y -> y==x.
Lemma le_antisym : forall x y, x<=y -> y<=x -> x==y.
Lemma neq_sym : forall x y, ~x==y -> ~y==x.
Transitivity rules : first, a generic formulation, then instances
Ltac subst_eqns :=
match goal with
| H : _==_ |- _ => (rewrite H || rewrite <- H); clear H; subst_eqns
| _ => idtac
end.
Definition interp_ord o :=
match o with OEQ => O.eq | OLT => O.lt | OLE => O.le end.
Lemma trans o o' x y z : #o x y -> #o' y z -> #(o+o') x z.
Definition eq_trans x y z : x==y -> y==z -> x==z := @trans OEQ OEQ x y z.
Definition le_trans x y z : x<=y -> y<=z -> x<=z := @trans OLE OLE x y z.
Definition lt_trans x y z : x<y -> y<z -> x<z := @trans OLT OLT x y z.
Definition le_lt_trans x y z : x<=y -> y<z -> x<z := @trans OLE OLT x y z.
Definition lt_le_trans x y z : x<y -> y<=z -> x<z := @trans OLT OLE x y z.
Definition eq_lt x y z : x==y -> y<z -> x<z := @trans OEQ OLT x y z.
Definition lt_eq x y z : x<y -> y==z -> x<z := @trans OLT OEQ x y z.
Definition eq_le x y z : x==y -> y<=z -> x<=z := @trans OEQ OLE x y z.
Definition le_eq x y z : x<=y -> y==z -> x<=z := @trans OLE OEQ x y z.
Lemma eq_neq : forall x y z, x==y -> ~y==z -> ~x==z.
Lemma neq_eq : forall x y z, ~x==y -> y==z -> ~x==z.
(double) negation rules
Lemma not_neq_eq : forall x y, ~~x==y -> x==y.
Lemma not_ge_lt : forall x y, ~y<=x -> x<y.
Lemma not_gt_le : forall x y, ~y<x -> x<=y.
Lemma le_neq_lt : forall x y, x<=y -> ~x==y -> x<y.
End OrderFacts.
Module MakeOrderTac (Import O:EqLtLe)(P:IsTotalOrder O).
Include OrderFacts O P.
Include EqLtLeNotation O.
order_eq : replace x by y in all (in)equations hyps thanks
to equality EQ (where eq has been hidden in order to avoid
self-rewriting), then discard EQ.
Ltac order_rewr x eqn :=
let rewr H t := generalize t; clear H; intro H
in
match goal with
| H : x == _ |- _ => rewr H (eq_trans (eq_sym eqn) H); order_rewr x eqn
| H : _ == x |- _ => rewr H (eq_trans H eqn); order_rewr x eqn
| H : ~x == _ |- _ => rewr H (eq_neq (eq_sym eqn) H); order_rewr x eqn
| H : ~_ == x |- _ => rewr H (neq_eq H eqn); order_rewr x eqn
| H : x < _ |- _ => rewr H (eq_lt (eq_sym eqn) H); order_rewr x eqn
| H : _ < x |- _ => rewr H (lt_eq H eqn); order_rewr x eqn
| H : x <= _ |- _ => rewr H (eq_le (eq_sym eqn) H); order_rewr x eqn
| H : _ <= x |- _ => rewr H (le_eq H eqn); order_rewr x eqn
| _ => clear eqn
end.
Ltac order_eq x y eqn :=
match x with
| y => clear eqn
| _ => change (interp_ord OEQ x y) in eqn; order_rewr x eqn
end.
Goal preparation : We turn all negative hyps into positive ones
and try to prove False from the inverse of the current goal.
These steps require totality of our order. After this preparation,
order only deals with the context, and tries to prove False.
Hypotheses of the form A -> False are also folded in ~A
for convenience (i.e. cope with the mess left by intuition).
Ltac order_prepare :=
match goal with
| H : ?A -> False |- _ => change (~A) in H; order_prepare
| H : ~(?R ?x ?y) |- _ =>
match R with
| eq => fail 1
| _ => (change (~x==y) in H ||
apply not_gt_le in H ||
apply not_ge_lt in H ||
clear H || fail 1); order_prepare
end
| H : ?R ?x ?y |- _ =>
match R with
| eq => fail 1
| lt => fail 1
| le => fail 1
| _ => (change (x==y) in H ||
change (x<y) in H ||
change (x<=y) in H ||
clear H || fail 1); order_prepare
end
| |- ~ _ => intro; order_prepare
| |- _ ?x ?x =>
exact (eq_refl x) || exact (le_refl x) || exfalso
| _ =>
(apply not_neq_eq; intro) ||
(apply not_ge_lt; intro) ||
(apply not_gt_le; intro) || exfalso
end.
We now try to prove False from the various < <= == != hypothesis
Ltac order_loop :=
match goal with
| H : ?x < ?x |- _ => exact (lt_irrefl H)
| H : ~ ?x == ?x |- _ => exact (H (eq_refl x))
| H : ?x <= ?x |- _ => clear H; order_loop
| H : ?x == ?y |- _ => order_eq x y H; order_loop
| H1 : ?x <= ?y, H2 : ?y <= ?x |- _ =>
generalize (le_antisym H1 H2); clear H1 H2; intro; order_loop
| H1: ?x <= ?y, H2: ~ ?x == ?y |- _ =>
generalize (le_neq_lt H1 H2); clear H1 H2; intro; order_loop
| H1: ?x <= ?y, H2: ~ ?y == ?x |- _ =>
generalize (le_neq_lt H1 (neq_sym H2)); clear H1 H2; intro; order_loop
| H1 : ?x < ?y, H2 : ?y < ?z |- _ =>
match goal with
| H : x < z |- _ => fail 1
| _ => generalize (lt_trans H1 H2); intro; order_loop
end
| H1 : ?x <= ?y, H2 : ?y < ?z |- _ =>
match goal with
| H : x < z |- _ => fail 1
| _ => generalize (le_lt_trans H1 H2); intro; order_loop
end
| H1 : ?x < ?y, H2 : ?y <= ?z |- _ =>
match goal with
| H : x < z |- _ => fail 1
| _ => generalize (lt_le_trans H1 H2); intro; order_loop
end
| H1 : ?x <= ?y, H2 : ?y <= ?z |- _ =>
match goal with
| H : x <= z |- _ => fail 1
| _ => generalize (le_trans H1 H2); intro; order_loop
end
| _ => idtac
end.
The complete tactic.
Ltac order :=
intros; order_prepare; order_loop; fail "Order tactic unsuccessful".
End MakeOrderTac.
Module OTF_to_OrderTac (OTF:OrderedTypeFull).
Module TO := OTF_to_TotalOrder OTF.
Include !MakeOrderTac OTF TO.
End OTF_to_OrderTac.
Module OT_to_OrderTac (OT:OrderedType).
Module OTF := OT_to_Full OT.
Include !OTF_to_OrderTac OTF.
End OT_to_OrderTac.