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Reconstr.v
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Reconstr.v
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(* Coq v8.6 required *)
(* author: Lukasz Czajka *)
Inductive ReconstrEmptyT : Set := Empty : ReconstrEmptyT.
Create HintDb yhints discriminated.
Ltac getgoal := match goal with [ |- ?G ] => G end.
Ltac notHyp P :=
match goal with
| [ H : ?P1 |- _ ] => constr_eq P P1; fail 1
| _ => idtac
end.
Ltac isProp t :=
lazymatch type of t with
| Prop => idtac
end.
Ltac notProp t :=
lazymatch type of t with
| Prop => fail
| _ => idtac
end.
Ltac checkListLen lst n :=
lazymatch n with
| 0 => constr_eq lst Empty
| S ?k =>
lazymatch lst with
| (?t, ?h) => checkListLen t k
| _ => idtac
end
end.
Ltac noEvars t := tryif has_evar t then fail else idtac.
Ltac natLe m n :=
let t := constr:(Nat.leb m n) in
let b := (eval compute in t) in
match b with
| true => idtac
end.
(* TODO: `isAtom c' fails for a constant c *)
Ltac isAtom t :=
lazymatch t with
| ?A /\ ?B => fail
| ?A \/ ?B => fail
| exists x, _ => fail
| _ _ => idtac
| (_ /\ _) -> False => fail
| (_ \/ _) -> False => fail
| (exists x, _) -> False => fail
| _ _ -> False => idtac
| ?A -> False => is_var A
| _ => is_var t
end.
Ltac isPropAtom t := isAtom t; isProp t.
Ltac inList x lst :=
lazymatch lst with
| (?t, ?y) => tryif constr_eq x y then idtac else inList x t
| x => idtac
| _ => fail
end.
Ltac notInList x lst := tryif inList x lst then fail else idtac.
Ltac all f ls :=
match ls with
| Empty => idtac
| (?LS, ?X) => f X; all f LS
| (_, _) => fail 1
| _ => f ls
end.
Ltac lst_rev lst :=
let rec hlp lst acc :=
match lst with
| Empty => acc
| (?t, ?h) => hlp t (acc, h)
| ?x => constr:((acc, x))
end
in
hlp lst Empty.
Ltac with_hyps p f :=
let rec hlp acc :=
match goal with
| [ H : ?P |- _ ] =>
p P; notInList H acc; hlp (acc, H)
| _ =>
f ltac:(lst_rev acc)
end
in
hlp Empty.
Ltac with_prop_hyps := with_hyps isProp.
Ltac with_atom_hyps := with_hyps isAtom.
Ltac all_hyps f := with_hyps ltac:(fun _ => idtac) ltac:(all f).
Ltac all_prop_hyps f := with_prop_hyps ltac:(all f).
Ltac all_atom_hyps f := with_atom_hyps ltac:(all f).
Ltac countHyps inb :=
let rec hlp n :=
match goal with
| [ H : _ |- _ ] =>
revert H; hlp (S n); intro H
| _ => pose (inb := n)
end
in
hlp 0.
Ltac checkHypsNum n :=
let m := fresh "m" in
countHyps m;
let k := (eval unfold m in m) in
natLe k n; clear m.
Ltac yeasy :=
let rec use_hyp H :=
match type of H with
| _ /\ _ => exact H || destruct_hyp H
| _ => try solve [ inversion H ]
end
with do_intro := let H := fresh in intro H; use_hyp H
with destruct_hyp H := case H; clear H; do_intro; do_intro in
let rec use_hyps :=
match goal with
| H : _ /\ _ |- _ => exact H || (destruct_hyp H; use_hyps)
| H : _ |- _ => solve [ inversion H ]
| _ => idtac
end
in
let do_atom :=
solve [ trivial with eq_true | reflexivity | symmetry; trivial | contradiction ] in
let rec do_ccl n :=
try do_atom;
repeat (do_intro; try do_atom);
lazymatch n with
| O => fail
| S ?k =>
solve [ split; do_ccl k ]
end
in
solve [ do_atom | use_hyps; do_ccl 16 ] ||
fail "Cannot solve this goal".
Ltac tryunfold x :=
let t := eval unfold x in x in
lazymatch t with
| _ _ => unfold x in *
| (fun x => _ _) => unfold x in *
| (fun x y => _ _) => unfold x in *
| (fun x y z => _ _) => unfold x in *
| (fun x y z u => _ _) => unfold x in *
| (fun x y z u w => _ _) => unfold x in *
| (fun x y z u w v => _ _) => unfold x in *
| (forall s, _) => unfold x in *
| (fun x => forall s, _) => unfold x in *
| (fun x y => forall s, _) => unfold x in *
| (fun x y z => forall s, _) => unfold x in *
| (fun x y z u => forall s, _) => unfold x in *
| (fun x y z u w => forall s, _) => unfold x in *
| (fun x y z u w v => forall s, _) => unfold x in *
| _ => idtac
end.
Ltac unfolding defs :=
repeat (autounfold with yhints; unfold iff in *; unfold not in *; all tryunfold defs).
Ltac einst e :=
let tpe := type of e
in
match tpe with
| forall x : ?T, _ =>
notProp T;
let v := fresh "v" in
evar (v : T);
let v2 := eval unfold v in v in
clear v;
einst (e v2)
| _ =>
generalize e
end.
Ltac mcongr tt := try solve [ hnf in *; congruence 8 ].
Ltac trysolve :=
eauto 2 with nocore yhints; try solve [ constructor ]; try subst;
match goal with
| [ |- ?t = ?u ] => mcongr tt
| [ |- ?t <> ?u ] => mcongr tt
| [ |- False ] => mcongr tt
| _ => idtac
end.
Ltac msplit splt simp :=
simp tt;
repeat (progress splt tt; simp tt).
Ltac ydestruct t :=
lazymatch t with
| _ _ => destruct t eqn:?
| _ =>
tryif is_evar t then
destruct t eqn:?
else
(is_var t; destruct t)
end.
Ltac yinversion H := inversion H; try subst; try clear H.
Ltac intro0 f :=
lazymatch goal with
| [ |- forall x : ?T, _ ] =>
tryif isProp T then
let H := fresh "H" in
(tryif notHyp T then
(intro H; try f H)
else
(intro H; try clear H))
else
tryif intro x then
idtac
else
let x1 := fresh x in
intro x1
end.
Ltac simp0 f H :=
let sintro tt := intro0 ltac:(simp0 f) in
let tp := type of H in
lazymatch tp with
| (exists x, _) => elim H; clear H; sintro tt; sintro tt
| ?A = ?A => clear H
| ?A -> ?A => clear H
| ?A -> ?B = ?B => clear H
| ?A /\ ?A => cut A; [ clear H; sintro tt | destruct H; assumption ]
| ?A /\ ?B => elim H; clear H; sintro tt; sintro tt
| ?A /\ ?B -> ?C => cut (A -> B -> C);
[ clear H; sintro tt
| intro; intro; apply H; split; assumption ]
| ?A = ?A -> ?B => cut B; [ clear H; sintro tt | apply H; reflexivity ]
| ?A -> ?A -> ?B => cut (A -> B); [ clear H; sintro tt | intro; apply H; assumption ]
| ?A \/ ?A => cut A; [ clear H; sintro tt | elim H; intro; assumption ]
| ?A \/ ?B -> ?C =>
cut (A -> C); [ cut (B -> C); [ clear H; sintro tt; sintro tt |
intro; apply H; right; assumption ] |
intro; apply H; left; assumption ]
| forall x : ?T1, ?A /\ ?B =>
cut (forall x : T1, A);
[ cut (forall x : T1, B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2), ?A /\ ?B =>
cut (forall (x : T1) (y : T2), A);
[ cut (forall (x : T1) (y : T2), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2) (z : ?T3), ?A /\ ?B =>
cut (forall (x : T1) (y : T2) (z : T3), A);
[ cut (forall (x : T1) (y : T2) (z : T3), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4), ?A /\ ?B =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4), A);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5), ?A /\ ?B =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5), A);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5) (w : ?T6), ?A /\ ?B =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5) (w : T6), A);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5) (w : T6), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5) (w : ?T6) (w1 : ?T7), ?A /\ ?B =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5) (w : T6) (w1 : T7), A);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5) (w : T6) (w1 : T7), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5) (w : ?T6)
(w1 : ?T7) (w2 : ?T8), ?A /\ ?B =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5) (w : T6)
(w1 : T7) (w2 : T8), A);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5) (w : T6)
(w1 : T7) (w2 : T8), B);
[ clear H; sintro tt; sintro tt | apply H ]
| apply H ]
| forall x : ?T1, ?A /\ ?B -> ?C =>
cut (forall x : T1, A -> B -> C);
[ clear H; sintro tt | do 3 intro; apply H; try assumption; split; assumption ]
| forall (x : ?T1) (y : ?T2), ?A /\ ?B -> ?C =>
cut (forall (x : T1) (y : T2), A -> B -> C);
[ clear H; sintro tt | do 4 intro; apply H; try assumption; split; assumption ]
| forall (x : ?T1) (y : ?T2) (z : ?T3), ?A /\ ?B -> ?C =>
cut (forall (x : T1) (y : T2) (z : T3), A -> B -> C);
[ clear H; sintro tt | do 5 intro; apply H; try assumption; split; assumption ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4), ?A /\ ?B -> ?C =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4), A -> B -> C);
[ clear H; sintro tt | do 6 intro; apply H; try assumption; split; assumption ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5), ?A /\ ?B -> ?C =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5), A -> B -> C);
[ clear H; sintro tt | do 7 intro; apply H; try assumption; split; assumption ]
| forall (x : ?T1), ?A \/ ?B -> ?C =>
cut (forall (x : T1), A -> C); [ cut (forall (x : T1), B -> C);
[ clear H; sintro tt; sintro tt |
do 2 intro; apply H with (x := x); right; assumption ] |
do 2 intro; apply H with (x := x); left; assumption ]
| forall (x : ?T1) (y : ?T2), ?A \/ ?B -> ?C =>
cut (forall (x : T1) (y : T2), A -> C);
[ cut (forall (x : T1) (y : T2), B -> C);
[ clear H; sintro tt; sintro tt |
do 3 intro; apply H with (x := x) (y := y); right; assumption ] |
do 3 intro; apply H with (x := x) (y := y); left; assumption ]
| forall (x : ?T1) (y : ?T2) (z : ?T3), ?A \/ ?B -> ?C =>
cut (forall (x : T1) (y : T2) (z : T3), A -> C);
[ cut (forall (x : T1) (y : T2) (z : T3), B -> C);
[ clear H; sintro tt; sintro tt |
do 4 intro; apply H with (x := x) (y := y) (z := z); right; assumption ] |
do 4 intro; apply H with (x := x) (y := y) (z := z); left; assumption ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4), ?A \/ ?B -> ?C =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4), A -> C);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4), B -> C);
[ clear H; sintro tt; sintro tt |
do 5 intro; apply H with (x := x) (y := y) (z := z) (u := u); right; assumption ] |
do 5 intro; apply H with (x := x) (y := y) (z := z) (u := u); left; assumption ]
| forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5), ?A \/ ?B -> ?C =>
cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5), A -> C);
[ cut (forall (x : T1) (y : T2) (z : T3) (u : T4) (v : T5), B -> C);
[ clear H; sintro tt; sintro tt |
do 6 intro; apply H with (x := x) (y := y) (z := z) (u := u) (v := v);
right; assumption ] |
do 6 intro; apply H with (x := x) (y := y) (z := z) (u := u) (v := v);
left; assumption ]
| ?A -> ?B =>
lazymatch goal with
| [ H1 : A |- _ ] => isProp A; cut B; [ clear H; sintro tt | apply H; exact H1 ]
| _ => f H
end
| _ =>
f H
end.
Ltac simp_hyp := simp0 ltac:(fun _ => fail).
Ltac simp_hyps :=
unfold iff in *; unfold not in *;
repeat match goal with
| [ H1 : ?A, H2 : ?A -> ?B |- _ ] =>
assert B by (apply H2; exact H1); clear H2
| [ H : True |- _ ] =>
clear H
| [ H : _ |- _ ] =>
simp_hyp H
end.
Ltac exsimpl :=
match goal with
| [ H : forall (x : ?T1), exists a, _ |- _ ] =>
notProp T1;
einst H; clear H; intro H; elim H; clear H; intro; intro
| [ H : forall (x : ?T1) (y : ?T2), exists a, _ |- _ ] =>
notProp T1; notProp T2;
einst H; clear H; intro H; elim H; clear H; intro; intro
| [ H : forall (x : ?T1) (y : ?T2) (z : ?T3), exists a, _ |- _ ] =>
notProp T1; notProp T2; notProp T3;
einst H; clear H; intro H; elim H; clear H; intro; intro
| [ H : forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4), exists a, _ |- _ ] =>
notProp T1; notProp T2; notProp T3; notProp T4;
einst H; clear H; intro H; elim H; clear H; intro; intro
| [ H : forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5), exists a, _ |- _ ] =>
notProp T1; notProp T2; notProp T3; notProp T4; notProp T5;
einst H; clear H; intro H; elim H; clear H; intro; intro
end.
Ltac isplit :=
match goal with
| [ |- ?A /\ _ ] => assert A; [ idtac | split; [ assumption | idtac ] ]
| [ H : _ \/ _ |- _ ] => elim H; clear H; intro
| [ H : (?a +{ ?b }) |- _ ] => elim H; clear H; intro
| [ H : ({ ?a }+{ ?b }) |- _ ] => elim H; clear H; intro
| [ |- context[match ?X with _ => _ end] ] => ydestruct X
| [ H : context[match ?X with _ => _ end] |- _ ] => ydestruct X
| [ H : forall (x : ?T1), _ \/ _ |- _ ] =>
notProp T1;
einst H; clear H; intro H; elim H; clear H
| [ H : forall (x : ?T1) (y : ?T2), _ \/ _ |- _ ] =>
notProp T1; notProp T2;
einst H; clear H; intro H; elim H; clear H
| [ H : forall (x : ?T1) (y : ?T2) (z : ?T3), _ \/ _ |- _ ] =>
notProp T1; notProp T2; notProp T3;
einst H; clear H; intro H; elim H; clear H
| [ H : forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4), _ \/ _ |- _ ] =>
notProp T1; notProp T2; notProp T3; notProp T4;
einst H; clear H; intro H; elim H; clear H
| [ H : forall (x : ?T1) (y : ?T2) (z : ?T3) (u : ?T4) (v : ?T5), _ \/ _ |- _ ] =>
notProp T1; notProp T2; notProp T3; notProp T4; notProp T5;
einst H; clear H; intro H; elim H; clear H
end.
Ltac eqsolve0 f :=
lazymatch goal with
| [ |- ?A = ?A ] => reflexivity
| [ |- ?A = ?B ] => solve [ unify A B; reflexivity | f tt ]
end.
Ltac rsolve0 tt :=
auto 2 with nocore yhints; try subst; mcongr tt;
match goal with
| [ |- ?A _ = ?A _ ] => apply f_equal; try eqsolve0 rsolve0
| [ |- ?A _ _ = ?A _ _ ] => apply f_equal2; try eqsolve0 rsolve0
| [ |- ?A _ _ _ = ?A _ _ _ ] => apply f_equal3; try eqsolve0 rsolve0
end.
Ltac rsolve :=
msplit ltac:(fun _ => isplit) ltac:(fun _ => intros; simp_hyps; repeat exsimpl);
rsolve0 tt.
Ltac eqsolve2 tt :=
match goal with
| [ |- ?A = ?A ] => reflexivity
| [ |- ?A = ?B ] => unify A B; reflexivity
| [ |- ?A _ = ?A _ ] => apply f_equal; eqsolve2 tt
| [ |- ?A _ _ = ?A _ _ ] => apply f_equal2; eqsolve2 tt
| [ |- ?A _ _ _ = ?A _ _ _ ] => apply f_equal3; eqsolve2 tt
| [ |- ?A _ _ _ _ = ?A _ _ _ _ ] => apply f_equal4; eqsolve2 tt
| [ |- ?A = ?B ] => solve [ rsolve ]
end.
Ltac eqsolve := eqsolve2 tt.
Ltac isolve :=
let rec msolve splt simp :=
msplit splt simp;
lazymatch goal with
| [ H : False |- _ ] => exfalso; exact H
| [ |- _ \/ _ ] =>
trysolve;
try solve [ left; msolve splt simp | right; msolve splt simp ]
| [ |- exists x, _ ] =>
trysolve; try solve [ eexists; msolve splt simp ]
| _ =>
trysolve
end
in
msolve ltac:(fun _ => isplit) ltac:(fun _ => intros; simp_hyps; repeat exsimpl).
Ltac dsolve :=
match goal with
| [ |- ?G ] => notProp G; auto with nocore yhints; try solve [ repeat constructor ]
| _ => auto with nocore yhints; try yeasy
end.
Ltac yisolve := try solve [ unfold iff in *; unfold not in *; unshelve isolve; dsolve ].
Ltac rchange tp :=
lazymatch goal with
| [ |- tp ] => idtac
| [ |- ?G1 = ?G2 ] =>
match tp with
| ?tp1 = ?tp2 =>
let H1 := fresh "H" in
let H2 := fresh "H" in
assert (H1 : G1 = tp1) by eqsolve;
assert (H2 : G2 = tp2) by eqsolve;
try rewrite H1; clear H1;
try rewrite H2; clear H2
| ?tp1 = ?tp2 =>
symmetry;
let H1 := fresh "H" in
let H2 := fresh "H" in
assert (H1 : G1 = tp2) by eqsolve;
assert (H2 : G2 = tp1) by eqsolve;
try rewrite H1; clear H1;
try rewrite H2; clear H2
end
| [ |- ?G ] =>
let H := fresh "H" in
assert (H : G = tp) by eqsolve;
try rewrite H; clear H
end.
Ltac sintuition :=
simp_hyps; intuition (auto with nocore yhints); try yeasy; try subst; mcongr tt;
try solve [ constructor ].
Ltac yintuition := simp_hyps; try subst; cbn in *; sintuition.
Ltac eresolve H1 H2 :=
let H1i := fresh "H" in
einst H1; intro H1i;
let H2i := fresh "H" in
einst H2; intro H2i;
let T1 := type of H1i in
let T2 := type of H2i in
match T2 with
| ?A -> ?B =>
unify T1 A;
let e := fresh "H" in
pose (e := H2i H1i);
let tp := type of e in
generalize e; clear e;
notHyp tp; clear H1i; clear H2i
| ?A1 = ?A2 -> ?B =>
unify T1 (A2 = A1);
let e := fresh "H" in
pose (e := H2i (eq_sym H1i));
let tp := type of e in
generalize e; clear e;
notHyp tp; clear H1i; clear H2i
end.
Ltac resolveGoal :=
repeat match goal with
| [ H : ?A1 |- (?A2 -> ?B) -> _ ] =>
isPropAtom A1; unify A1 A2;
let H0 := fresh "H" in
intro H0; cut B; [ clear H0 | apply H0; exact H ]
end.
Ltac yrewrite H := (erewrite H by isolve) || (erewrite <- H by isolve).
Ltac ysimp0 htrace f :=
simp0 ltac:(fun H =>
try (checkHypsNum 8;
try (isPropAtom ltac:(type of H); yresolvewith0 htrace H);
all_hyps ltac:(fun H0 =>
try (isPropAtom ltac:(type of H0); yresolve0 H0 H0 H));
f H))
with ysimp1 htrace :=
ysimp0 htrace ltac:(fun H =>
try match type of H with
| _ = _ => yrewritingin0 (htrace, H) H
end)
with yintro0 htrace :=
intro0 ltac:(ysimp0 htrace ltac:(fun _ => idtac))
with yintro1 htrace :=
intro0 ltac:(ysimp1 htrace)
with yresolve0 htrace H1 H2 :=
notInList H2 htrace;
eresolve H1 H2;
match goal with
| [ |- (_ -> ?B1) -> ?B2 ] => unify B1 B2
| [ |- (_ -> _ -> ?B1) -> ?B2 ] => unify B1 B2
(* | [ |- (_ -> _ -> _ -> ?B1) -> ?B2 ] => unify B1 B2 *)
| _ => idtac
end;
resolveGoal;
match goal with
| [ |- ?A -> _ ] => noEvars A
end;
yintro0 (htrace, H2)
with yresolvewith0 htrace H1 :=
let A := type of H1 in
repeat match goal with
| [ H2 : A -> ?B |- _ ] => cut B; [ clear H2; yintro0 htrace | apply H2; exact H1 ]
end;
checkListLen htrace 2;
all_prop_hyps ltac:(fun H2 => try yresolve0 (htrace, H1) H1 H2)
with yrewritein0 htrace H H0 :=
notInList H0 htrace;
let H1 := fresh "H" in
einst H0; intro H1; isPropAtom ltac:(type of H1);
(rewrite H in H1 by isolve) || (rewrite <- H in H1 by isolve);
noEvars ltac:(type of H1);
generalize H1; clear H1; yintro0 (htrace, H0)
with yrewritingin0 htrace H :=
let rec hlp hyps n :=
lazymatch n with
| 0 => idtac
| S ?k =>
lazymatch hyps with
| (?t, ?H0) =>
tryif yrewritein0 htrace H H0 then
hlp t k
else
hlp t n
| _ => idtac
end
end
in
with_prop_hyps ltac:(fun hyps => hlp hyps 4).
Ltac ysimp := ysimp1 Empty.
Ltac yintro := yintro1 Empty.
Ltac yresolve := yresolve0 Empty.
Ltac yresolvewith := yresolvewith0 Empty.
Ltac yrewritein := yrewritein0 Empty.
Ltac yintros0 acc :=
lazymatch goal with
| [ |- forall x : ?T, _ ] =>
tryif isProp T then
let H := fresh "H" in
(tryif notHyp T then
(intro H; yintros0 (acc, H))
else
(intro H; try clear H))
else
let x0 := fresh x in
(intro x0; yintros0 (acc, x0))
| _ =>
all ltac:(fun H => try ysimp H) ltac:(lst_rev acc)
end.
Ltac yintros := yintros0 Empty.
Ltac generalizing :=
repeat match goal with
| [ H : _ |- _ ] => generalize H; clear H
end.
Ltac yinduction t :=
repeat match goal with
| [ x : ?T |- _ ] =>
notProp T; tryif constr_eq x t then fail else (generalize x; clear x)
end;
induction t.
Ltac ysplit :=
match goal with
| [ |- ?A /\ _ ] =>
cut A; [ let H := fresh "H" in
intro H; split; [ exact H | ysimp H ] | idtac ]
| [ |- context[match ?X with _ => _ end] ] => ydestruct X
| [ H : context[match ?X with _ => _ end] |- _ ] => ydestruct X
end.
Ltac ysplitting :=
repeat ysplit;
let n := numgoals in
guard n < 12;
yisolve;
let n := numgoals in
guard n < 6.
Ltac yunfold h := unfold h in *; ysplitting.
Ltac yunfolding defs :=
let dounfold h :=
let h2 := eval unfold h in h in
lazymatch h2 with
| (match _ with _ => _ end) => try yunfold h
| (fun x => match _ with _ => _ end) => try yunfold h
| (fun x y => match _ with _ => _ end) => try yunfold h
| (fun x y z => match _ with _ => _ end) => try yunfold h
| (fun x y z u => match _ with _ => _ end) => try yunfold h
| (fun x y z u v => match _ with _ => _ end) => try yunfold h
| (fun x y z u v w => match _ with _ => _ end) => try yunfold h
| _ => idtac
end
in
all dounfold defs; unfolding defs.
Ltac gunfolding defs :=
let dounfold h :=
lazymatch goal with
| [ H : context[h] |- _ ] => idtac
| _ =>
let h2 := eval unfold h in h in
lazymatch h2 with
| (match _ with _ => _ end) => try yunfold h
| (fun x => match _ with _ => _ end) => try yunfold h
| (fun x y => match _ with _ => _ end) => try yunfold h
| (fun x y z => match _ with _ => _ end) => try yunfold h
| (fun x y z u => match _ with _ => _ end) => try yunfold h
| (fun x y z u v => match _ with _ => _ end) => try yunfold h
| (fun x y z u v w => match _ with _ => _ end) => try yunfold h
| _ => idtac
end
end
in
all dounfold defs; unfolding defs.
Ltac rapply e :=
let tpe := type of e
in
lazymatch tpe with
| forall x : ?T, _ =>
tryif isProp T then
let H := fresh "H" in
assert (H : T); [ idtac | rapply (e H) ]
else
let v := fresh "v" in
evar (v : T);
let v2 := eval unfold v in v in
clear v;
rapply (e v2)
| _ =>
rchange tpe; exact e
end.
Ltac orinst H :=
let tpH := type of H
in
lazymatch tpH with
| forall x : ?T, _ =>
tryif isProp T then
let H0 := fresh "H" in
assert (H0 : T); [ clear H |
let H1 := fresh "H" in
generalize (H H0); intro H1; clear H; clear H0;
orinst H1 ]
else
let v := fresh "v" in
evar (v : T);
let v2 := eval unfold v in v in
clear v;
let H1 := fresh "H" in
generalize (H v2); intro H1; clear H;
orinst H1
| _ \/ _ =>
elim H; clear H; yintro
end.
Ltac yapply H :=
lazymatch goal with
| [ H0 : context[_ = _] |- _ ] => rapply H
| _ => simple eapply H
end.
Ltac yelles0 defs n rtrace gtrace :=
lazymatch n with
| O => solve [ isolve ]
| S ?k =>
let G := getgoal in
notInList G gtrace;
match goal with
| [ H : False |- _ ] => exfalso; exact H
| [ H : G |- _ ] => assumption
| [ H : ?A = _ |- ?B = _ -> _ ] =>
unify A B; let H1 := fresh "H" in intro H1; rewrite H in H1; exfalso; congruence 0
| [ H : _ = ?A |- _ = ?B -> _ ] =>
unify A B; let H1 := fresh "H" in intro H1; rewrite H in H1; exfalso; congruence 0
| [ H : _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x, _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y, _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z, _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u, _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v, _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v w, _ -> False |- _ -> False ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : ?P |- ?P -> ?Q ] =>
(let H1 := fresh "H" in intro H1; try clear H1;
move H at bottom; yelles0 defs n rtrace gtrace) || fail 1
| [ |- forall x, _ ] => doyelles defs n || fail 1
| [ |- _ /\ _ ] => doyelles defs n || fail 1
| [ |- context[match ?X with _ => _ end] ] => doyelles defs n || fail 1
| [ H : context[match ?X with _ => _ end] |- _ ] => doyelles defs n || fail 1
| [ |- exists x, _ ] =>
eexists; yelles0 defs n rtrace (gtrace, G)
| [ H : forall x, G |- _ ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y, G |- _ ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z, G |- _ ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u, G |- _ ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v, G |- _ ] =>
simple eapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x y, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ |- _ ] =>
solve [ isolve ] (* TODO: this may be improved? *)
| [ H : forall x y z u v, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v w, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v w p, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v w p p1, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v w p p1 p2, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z u v w p p1 p2 p3, _ _ |- _ _ ] =>
yapply H; yelles0 defs k rtrace (gtrace, G)
| [ H : forall x y z, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x y z u, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x y z u v, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x y z u v w, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x y z u v w p, _ = _ |- _ ] =>
notInList H rtrace;
yrewrite H; yelles0 defs k (rtrace, H) (gtrace, G)
| [ H : forall x, exists e, _ |- _ ] =>
einst H; clear H; yintro; yelles0 defs k Empty Empty
| [ H : forall x y, exists e, _ |- _ ] =>
einst H; clear H; yintro; yelles0 defs k Empty Empty
| [ H : forall x y z, exists e, _ |- _ ] =>
einst H; clear H; yintro; yelles0 defs k Empty Empty
| [ H : forall x y z u, exists e, _ |- _ ] =>
einst H; clear H; yintro; yelles0 defs k Empty Empty
| [ H : forall x y z u v, exists e, _ |- _ ] =>
einst H; clear H; yintro; yelles0 defs k Empty Empty
| [ H : forall x y z u v w, exists e, _ |- _ ] =>
einst H; clear H; yintro; yelles0 defs k Empty Empty
| [ H : _ \/ _ |- _ ] =>
elim H; clear H; yintro; doyelles defs k
| [ |- _ \/ _ ] =>
(left; yelles0 defs k rtrace (gtrace, G)) || (right; yelles0 defs k rtrace (gtrace, G))
| [ H : forall x, _ \/ _ |- _ ] =>
orinst H; yelles0 defs k Empty Empty
| [ H : forall x y, _ \/ _ |- _ ] =>
orinst H; yelles0 defs k Empty Empty
| [ H : forall x y z, _ \/ _ |- _ ] =>
orinst H; yelles0 defs k Empty Empty
| [ H : forall x y z u, _ \/ _ |- _ ] =>
orinst H; yelles0 defs k Empty Empty
| [ H : forall x y z u v, _ \/ _ |- _ ] =>
orinst H; yelles0 defs k Empty Empty
| [ H : forall x y z u v w, _ \/ _ |- _ ] =>
orinst H; yelles0 defs k Empty Empty
end
end
with doyelles defs n :=
yintros; repeat (cbn; try ysplit);
lazymatch n with
| 0 => solve [ isolve ]
| S ?k =>
first [ yelles0 defs n Empty Empty |
match goal with
| [ x : ?T |- _ ] =>
notProp T; ydestruct x; unfolding defs; doyelles defs k
| [ H : ?T |- _ ] =>
isPropAtom T; yinversion H; unfolding defs; doyelles defs k
| [ |- ?A = ?B ] =>
progress (try ydestruct A; try ydestruct B);
unfolding defs;
yelles0 defs k Empty Empty
| [ |- False ] =>
fail 1
| [ H : False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
| [ H : forall x, False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
| [ H : forall x y, False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
| [ H : forall x y z, False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
| [ H : forall x y z u, False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
| [ H : forall x y z u v, False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
| [ H : forall x y z u v w, False |- _ ] =>
(exfalso; yelles0 defs n Empty Empty) || fail 1
end
]
end.
Ltac yelles1 defs n :=
unfolding defs;
repeat (yintros; repeat ysplit);
doyelles defs n.
Ltac yellesd defs n := cbn; unshelve yelles1 defs n; dsolve.
Ltac yelles n := cbn; unshelve yelles1 Empty n; dsolve.
Ltac yauto n := generalizing; yelles n.
Ltac meauto f :=
intros;
multimatch goal with
| [ H : _ |- _ ] => simple eapply H; f tt
| [ H : _ |- _ ] => (erewrite H + erewrite <- H); f tt
end.
Ltac meauto2 f := meauto ltac:(fun _ => meauto f).
Ltac meauto3 f := meauto ltac:(fun _ => meauto2 f).
Ltac ymeauto n := once meauto2 ltac:(fun _ => yelles n).
Ltac yforward H :=
einst H;
progress repeat match goal with
| [ H0 : ?P |- (?Q -> _) -> _ ] =>
unify P Q;
let H1 := fresh "H" in
intro H1; generalize (H1 H0); clear H1
end;
match goal with
| [ |- ?P -> _ ] => noEvars P
end;
yintro.
Ltac yforwarding :=
all_prop_hyps ltac:(fun H => try yforward H).
Ltac forward_reasoning n :=
lazymatch n with
| 0 => idtac
| S ?k => yforwarding; forward_reasoning k
end.
Ltac iauto n :=
let rec doiauto n :=
lazymatch n with
| 0 => solve [ eauto ]
| S ?k =>
match goal with
| [ H : ?T |- _ ] =>
isPropAtom T; yinversion H; cbn in *; doiauto k
| _ => solve [ eauto ]
end
end
in
intros; doiauto n.
Ltac docrush :=
yintuition; cbn in *; simp_hyps; forward_reasoning 2; yisolve; try yelles 1;
try solve [ unshelve (intuition isolve; eauto 10 with yhints); dsolve ];
try congruence;
try iauto 1;
try yelles 2;
try ymeauto 0.
Ltac ycrush := solve [ yisolve | eauto with yhints | docrush ].
Ltac hinit hyps lems defs :=
let rec pose_all ls acc :=
match ls with
| Empty => idtac
| (?LS, ?X) => generalize X; let H := fresh "H" in intro H; pose_all LS (acc, H)
| (_, _) => fail 1
| _ =>
generalize ls;
repeat match goal with
| [ H : ?P |- _ ] => isProp P; notInList H acc; clear H
end;
intro
end
in
pose_all lems hyps;
unfolding defs.
Ltac htrivial hyps lems defs :=
hinit hyps lems defs; sintuition.
Ltac hobvious hyps lems defs :=
htrivial hyps lems defs; simp_hyps; yisolve; try yellesd defs 1.
Ltac heasy hyps lems defs :=
hobvious hyps lems defs;
try solve [ unshelve (intuition isolve; eauto 10 with nocore yhints); dsolve ];
try congruence.
Ltac hsimple hyps lems defs :=
hobvious hyps lems defs;
gunfolding defs;
simp_hyps;
try yellesd defs 2.
Ltac hcrush hyps lems defs :=
hinit hyps lems defs;
try ycrush.
Ltac hyelles n hyps lems defs :=
hobvious hyps lems defs;
try yellesd defs n.
Ltac hexhaustive n hyps lems defs :=
hsimple hyps lems defs;