lemma iota_ltb : ∀x,p:nat. mem nat_eqType x (iota O p) = ltb x p.
intros (x p); elim p; simplify;[reflexivity]
-generalize in match (refl_eq ? (cmp ? x n));
-generalize in match (cmp ? x n) in ⊢ (? ? ? % → %); intros 1 (b);
-cases b; simplify; intros (H1); rewrite > H; clear H;
-rewrite < (leb_eqb x n); rewrite > H1; reflexivity;
+apply (cmpP nat_eqType x n); intros (E); rewrite > H; clear H; simplify;
+[1: symmetry; apply (p2bT ? ? (lebP ? ?)); rewrite > E; apply le_n;
+|2: rewrite < (leb_eqb x n); rewrite > E; reflexivity;]
qed.
lemma mem_filter :
[1: apply H; intros (y Hyl);
apply H1; simplify;
rewrite > Hyl; rewrite > orbC; reflexivity;
-|2: generalize in match (refl_eq ? (cmp ? x s));
- generalize in match (cmp ? x s) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- [1: simplify; intros (Exs);
- rewrite > orbC; rewrite > H;
- [2: intros; apply H1; simplify; rewrite > H2; rewrite > orbC; reflexivity
- |1: lapply (H1 t) as H2; [2: simplify; rewrite > cmp_refl; reflexivity]
- rewrite > Hpt in H2; simplify in H2; rewrite > H2 in Exs;
- destruct Exs;]
- |2: intros (Dxs); simplify; rewrite > H;
- [2: intros; apply (H1 y); simplify; rewrite > H2; rewrite > orbC; reflexivity
- |1: rewrite > Dxs; reflexivity]]]
+|2: simplify; apply (cmpP d2 x s); simplify; intros (E);
+ [1: rewrite < (H1 t); simplify; [rewrite > Hpt; rewrite > E]
+ simplify; rewrite > cmp_refl; reflexivity
+ |2: apply H; intros; apply H1; simplify; rewrite > H2;
+ rewrite > orbC; reflexivity]]
qed.
lemma count_O :
generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b);
cases b; simplify;
[2:intros (Hpt); apply H; intros; apply H1; simplify;
- generalize in match (refl_eq ? (cmp d x t));
- generalize in match (cmp d x t) in ⊢ (? ? ? % → %); intros 1 (b1);
- cases b1; simplify; intros; [2:rewrite > H2] autobatch.
+ apply (cmpP d x t); [2: rewrite > H2;]; intros; reflexivity;
|1:intros (H2); lapply (H1 t); [2:simplify; rewrite > cmp_refl; simplify; autobatch]
rewrite > H2 in Hletin; simplify in Hletin; destruct Hletin]
qed.
[ apply (mk_finType fsort enum Hcut)
| intros (x); cases x (n Hn); simplify in Hn; clear x;
generalize in match Hn; generalize in match Hn; clear Hn;
+ unfold enum;
unfold segment_enum;
generalize in match bound in ⊢ (% → ? → ? ? (? ? ? (? ? ? ? %)) ?);
- intros 1 (m); elim m (Hm Hn p IH Hm Hn); [ destruct Hm ]
+ intros 1 (m); elim m (Hm Hn p IH Hm Hn); [ simplify in Hm; destruct Hm ]
simplify; cases (eqP bool_eqType (ltb p bound) true); simplify;
- [1:unfold segment in ⊢ (? ? match ? % ? ? with [true⇒ ?|false⇒ ?] ?);
- unfold nat_eqType in ⊢ (? ? match % with [true⇒ ?|false⇒ ?] ?);
- simplify;
- generalize in match (refl_eq ? (eqb n p));
- generalize in match (eqb n p) in ⊢ (? ? ? % → %); intros 1(b); cases b; clear b;
- intros (Enp); simplify;
+ [1:unfold fsort;
+ unfold segment in ⊢ (? ? match ? % ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
+ unfold nat_eqType in ⊢ (? ? match % with [_ ⇒ ?|_ ⇒ ?] ?);
+ simplify; apply (cmpP nat_eqType n p); intros (Enp); simplify;
[2:rewrite > IH; [1,3: autobatch]
rewrite < ltb_n_Sm in Hm; rewrite > Enp in Hm;
- generalize in match Hm; cases (ltb n p); intros; [reflexivity]
- simplify in H1; destruct H1;
+ rewrite > orbC in Hm; assumption;
|1:clear IH; rewrite > (count_O fsort); [reflexivity]
- intros 1 (x); rewrite < (b2pT ? ? (eqP ? n ?) Enp);
- cases x (y Hy); intros (ABS); clear x;
- unfold segment; unfold notb; simplify;
- generalize in match (refl_eq ? (cmp ? n y));
- generalize in match (cmp ? n y) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- intros (Eny); simplify; [2:reflexivity]
+ intros 1 (x); rewrite < Enp; cases x (y Hy);
+ intros (ABS); clear x; unfold segment; unfold notb; simplify;
+ apply (cmpP ? n y); intros (Eny); simplify; [2:reflexivity]
rewrite < ABS; symmetry; clear ABS;
- generalize in match Hy; clear Hy;
- rewrite < (b2pT ? ? (eqP nat_eqType ? ?) Eny);
+ generalize in match Hy; clear Hy;rewrite < Eny;
simplify; intros (Hn); apply (mem_filter nat_eqType fsort); intros (w Hw);
fold simplify (sort nat_eqType); (* CANONICAL?! *)
cases (in_sub_eq nat_eqType (λx:nat_eqType.ltb x bound) w);
simplify; [2: reflexivity]
- generalize in match H1; clear H1; cases s; clear s; intros (H1);
- unfold segment; simplify; simplify in H1; rewrite > H1;
+ generalize in match H1; clear H1; cases s (r Pr); clear s; intros (H1);
+ unfold fsort; unfold segment; simplify; simplify in H1; rewrite > H1;
rewrite > iota_ltb in Hw; apply (p2bF ? ? (eqP nat_eqType ? ?));
unfold Not; intros (Enw); rewrite > Enw in Hw;
rewrite > ltb_refl in Hw; destruct Hw]
lemma uniq_tail :
∀d:eqType.∀x:d.∀l:list d. uniq d (x::l) = andb (negb (mem d x l)) (uniq d l).
intros (d x l); elim l; simplify; [reflexivity]
-generalize in match (refl_eq ? (cmp d x t));
-generalize in match (cmp d x t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
-intros (E); simplify ; rewrite > E; [reflexivity]
+apply (cmpP d x t); intros (E); simplify ; try rewrite > E; [reflexivity]
rewrite > andbA; rewrite > andbC in ⊢ (? ? (? % ?) ?); rewrite < andbA;
rewrite < H; rewrite > andbC in ⊢ (? ? ? (? % ?)); rewrite < andbA; reflexivity;
qed.
lemma uniqP : ∀d:eqType.∀l:list d.
reflect (∀x:d.mem d x l = true → count d (cmp d x) l = (S O)) (uniq d l).
-intros (d l); apply prove_reflect; elim l; [1: destruct H1 | 3: destruct H]
+intros (d l); apply prove_reflect; elim l; [1: simplify in H1; destruct H1 | 3: simplify in H; destruct H]
[1: generalize in match H2; simplify in H2;
lapply (b2pT ? ? (orbP ? ?) H2) as H3; clear H2;
cases H3; clear H3; intros;
- [2: lapply (uniq_mem ? ? ? H1) as H4;
- generalize in match (refl_eq ? (cmp d x t));
- generalize in match (cmp d x t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- intros (H5);
- [1: simplify; rewrite > H5; simplify; rewrite > count_O; [reflexivity]
- intros (y Hy); rewrite > (b2pT ? ? (eqP d ? ?) H5) in H2 H3 H4 ⊢ %;
- clear H5; clear x; rewrite > H2 in H4; destruct H4;
- |2: simplify; rewrite > H5; simplify; apply H;
+ [2: lapply (uniq_mem ? ? ? H1) as H4; simplify; apply (cmpP d x t);
+ intros (H5); simplify;
+ [1: rewrite > count_O; [reflexivity]
+ intros (y Hy); rewrite > H5 in H2 H3 ⊢ %; clear H5; clear x;
+ rewrite > H2 in H4; destruct H4;
+ |2: simplify; apply H;
rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1);
assumption;]
|1: simplify; rewrite > H2; simplify; rewrite > count_O; [reflexivity]
intros (y Hy); rewrite > (b2pT ? ? (eqP d ? ?) H2) in H3 ⊢ %;
clear H2; clear x; lapply (uniq_mem ? ? ? H1) as H4;
- generalize in match (refl_eq ? (cmp d t y));
- generalize in match (cmp d t y) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b; intros (E);
- [1: rewrite > (b2pT ? ? (eqP d ? ?) E) in H4;
- rewrite > H4 in Hy; destruct Hy;
- |2:intros; reflexivity]]
-|2: rewrite > uniq_tail in H1;
+ apply (cmpP d t y); intros (E); [2: reflexivity].
+ rewrite > E in H4; rewrite > H4 in Hy; destruct Hy;]
+|2: rewrite > uniq_tail in H1;
generalize in match (refl_eq ? (uniq d l1));
generalize in match (uniq d l1) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
[1: intros (E); rewrite > E in H1; rewrite > andbC in H1; simplify in H1;
unfold Not; intros (A); lapply (A t) as A';
[1: simplify in A'; rewrite > cmp_refl in A'; simplify in A';
- destruct A'; clear A'; rewrite < count_O_mem in H1;
- rewrite > Hcut in H1; destruct H1;
+ destruct A'; rewrite < count_O_mem in H1;
+ rewrite > Hcut in H1; simplify in H1; destruct H1;
|2: simplify; rewrite > cmp_refl; reflexivity;]
|2: intros (Ul1); lapply (H Ul1); unfold Not; intros (A); apply Hletin;
intros (r Mrl1); lapply (A r);
- [2: simplify; rewrite > Mrl1; cases (cmp d r t); reflexivity]
- generalize in match (refl_eq ? (cmp d r t));
- generalize in match (cmp d r t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- [1: intros (E); simplify in Hletin1; rewrite > E in Hletin1;
- destruct Hletin1; rewrite < count_O_mem in Mrl1;
- rewrite > Hcut in Mrl1; destruct Mrl1;
- |2: intros; simplify in Hletin1; rewrite > H2 in Hletin1;
- simplify in Hletin1; apply (Hletin1);]]]
+ [2: simplify; rewrite > Mrl1; cases (cmp d r t); reflexivity]
+ generalize in match Hletin1; simplify; apply (cmpP d r t);
+ simplify; intros (E Hc); [2: assumption]
+ destruct Hc; rewrite < count_O_mem in Mrl1;
+ rewrite > Hcut in Mrl1; simplify in Mrl1; destruct Mrl1;]]
qed.
lemma mem_finType : ∀d:finType.∀x:d. mem d x (enum d) = true.
intros (d p x); cases x (t Ht); clear x;
generalize in match (mem_finType d t);
generalize in match (uniq_fintype_enum d);
-elim (enum d); [destruct H1] simplify;
+elim (enum d); [simplify in H1; destruct H1] simplify;
cases (in_sub_eq d p t1); simplify;
[1:generalize in match H3; clear H3; cases s (r Hr); clear s;
simplify; intros (Ert1); generalize in match Hr; clear Hr;
rewrite > Ert1; clear Ert1; clear r; intros (Ht1);
- unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [true⇒ ?|false⇒ ?] ?);
- simplify; generalize in match (refl_eq ? (cmp d t t1));
- generalize in match (cmp d t t1) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- intros (Ett1); simplify;
+ unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
+ simplify; apply (cmpP ? t t1); simplify; intros (Ett1);
[1: cut (count (sub_eqType d p) (cmp (sub_eqType d p) {t,Ht})
(filter d (sigma d p) (if_p d p) l) = O); [1:rewrite > Hcut; reflexivity]
lapply (uniq_mem ? ? ? H1);
generalize in match Ht;
- rewrite > (b2pT ? ? (eqP d ? ?) Ett1); intros (Ht1'); clear Ht1;
+ rewrite > Ett1; intros (Ht1'); clear Ht1;
generalize in match Hletin; elim l; [ reflexivity]
simplify; cases (in_sub_eq d p t2); simplify;
[1: generalize in match H5; cases s; simplify; intros; clear H5;
- unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [true⇒ ?|false⇒ ?] ?);
+ unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
simplify; rewrite > H7; simplify in H4;
- generalize in match H4; clear H4;
- generalize in match (refl_eq ? (cmp d t1 t2));
- generalize in match (cmp d t1 t2) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- simplify; intros; [1: destruct H5] apply H3; assumption;
+ generalize in match H4; clear H4; apply (cmpP ? t1 t2);
+ simplify; intros; [destruct H5] apply H3; assumption;
|2: apply H3;
- generalize in match H4; clear H4; simplify;
- generalize in match (refl_eq ? (cmp d t1 t2));
- generalize in match (cmp d t1 t2) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- simplify; intros; [1: destruct H6] assumption;]
+ generalize in match H4; clear H4; simplify; apply (cmpP ? t1 t2);
+ simplify; intros; [destruct H6] assumption;]
|2: apply H; [ rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
simplify in H2; rewrite > Ett1 in H2; simplify in H2; assumption]
|2:rewrite > H; [1:reflexivity|2: rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
- simplify in H2; generalize in match H2; generalize in match (refl_eq ? (cmp d t t1));
- generalize in match (cmp d t t1) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
- intros (E); [2:assumption]
- lapply (b2pT ? ? (eqP d ? ?) E); clear H; rewrite > Hletin in Ht;
- rewrite > Ht in H3; destruct H3;]
+ simplify in H2; generalize in match H2; apply (cmpP ? t t1);
+ intros (E) [2:assumption] clear H; rewrite > E in Ht; rewrite > H3 in Ht;
+ destruct Ht;]
qed.
definition sub_finType : ∀d:finType.∀p:d→bool.finType ≝