λbound.filter ? ? (if_p nat_eqType (λx.ltb x bound)) (iota O bound).
lemma iota_ltb : ∀x,p:nat. mem nat_eqType x (iota O p) = ltb x p.
-intros (x p); elim p; simplify; [1:reflexivity]
-rewrite < leb_eqb;
+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); [1: reflexivity]
-rewrite > H; fold simplify (ltb x n); rewrite > H1; reflexivity;
+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;
qed.
lemma mem_filter :
- ∀d1,d2:eqType.(*∀y:d1.*)∀x:d2.∀l:list d1.∀p:d1 → option d2.
+ ∀d1,d2:eqType.∀x:d2.∀l:list d1.∀p:d1 → option d2.
(∀y.mem d1 y l = true →
match (p y) with [None ⇒ false | (Some q) ⇒ cmp d2 x q] = false) →
mem d2 x (filter d1 d2 p l) = false.
intros 5 (d1 d2 x l p);
-elim l; [1: simplify; auto]
-simplify; fold simplify (filter d1 d2 p l1);
+elim l; simplify; [reflexivity]
generalize in match (refl_eq ? (p t));
-generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
-[1: simplify; intros (Hpt); apply H; intros (y Hyl);
- apply H1; simplify; fold simplify (orb (cmp ? y t) (mem ? y l1));
- rewrite > Hyl; rewrite > orbC; reflexivity;
-|2:simplify; intros (Hpt);
- 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 < (H1 t); [2: simplify; rewrite > cmp_refl; reflexivity;]
- rewrite > Hpt; simplify; symmetry; assumption;
- |2: intros (Dxs); simplify; apply H; intros;
- apply H1; simplify; fold simplify (orb (cmp ? y t) (mem ? y l1));
- rewrite > H2; rewrite > orbC; reflexivity;]]
+generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b; intros (Hpt);
+[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]]]
qed.
lemma count_O :
∀d:eqType.∀p:d→bool.∀l:list d.
(∀x:d.mem d x l = true → notb (p x) = true) → count d p l = O.
intros 3 (d p l); elim l; simplify; [1: reflexivity]
-fold simplify (count d p l1); (* XXX simplify troppo *)
generalize in match (refl_eq ? (p t));
generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b);
cases b; simplify;
generalize in match Hn; generalize in match Hn; clear Hn;
unfold segment_enum;
generalize in match bound in ⊢ (% → ? → ? ? (? ? ? (? ? ? ? %)) ?);
- intros 1 (m); elim m (Hm Hn p IH Hm Hn);
- [ destruct Hm;
- | simplify; cases (eqP bool_eqType (ltb p bound) true); simplify;
- (* XXX simplify che spacca troppo *)
- fold simplify (filter nat_eqType (sigma nat_eqType (λx.ltb x bound))
- (if_p nat_eqType (λx.ltb x bound)) (iota O p));
- [1:unfold segment in ⊢ (? ? match ? % ? ? with [true⇒ ?|false⇒ ?] ?);
- unfold nat_eqType in ⊢ (? ? match % with [true⇒ ?|false⇒ ?] ?);
- simplify in ⊢ (? ? match % with [true⇒ ?|false⇒ ?] ?);
- generalize in match (refl_eq ? (eqb n p));
- generalize in match (eqb n p) in ⊢ (? ? ? % → %); intros 1(b);
- cases b; simplify;
- [2:intros (Enp); rewrite > IH; [1,3: auto]
- rewrite < ltb_n_Sm in Hm; rewrite > Enp in Hm;
- generalize in match Hm; cases (ltb n p); intros; [1:reflexivity]
- simplify in H1; destruct H1;
- |1:clear IH; intros (Enp);
- fold simplify (count fsort (cmp fsort {n, Hn})
- (filter ? (sigma nat_eqType (λx.ltb x bound))
- (if_p nat_eqType (λx.ltb x bound)) (iota O p)));
- rewrite > (count_O fsort); [1: 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: auto]
- rewrite < ABS; symmetry; clear ABS;
- generalize in match Hy; clear Hy; rewrite < (b2pT ? ? (eqP nat_eqType ? ?) Eny);
- simplify; intros (Hn); fold simplify (iota O n);
- apply (mem_filter nat_eqType fsort);
- intros (w Hw);
- fold simplify (sort nat_eqType);
- 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;
- 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]
- |2:rewrite > IH; [1:reflexivity|3:assumption]
- rewrite < ltb_n_Sm in Hm;
- cases (b2pT ? ?(orbP ? ?) Hm);[1: assumption]
- rewrite > (b2pT ? ? (eqbP ? ?) H1) in Hn;
- rewrite > Hn in H; cases (H ?); reflexivity;
- ]]]
+ intros 1 (m); elim m (Hm Hn p IH Hm Hn); [ 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;
+ [2:rewrite > IH; [1,3: auto]
+ 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;
+ |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]
+ rewrite < ABS; symmetry; clear ABS;
+ generalize in match Hy; clear Hy;
+ rewrite < (b2pT ? ? (eqP nat_eqType ? ?) 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;
+ 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]
+ |2:rewrite > IH; [1:reflexivity|3:assumption]
+ rewrite < ltb_n_Sm in Hm;
+ cases (b2pT ? ?(orbP ? ?) Hm);[1: assumption]
+ rewrite > (b2pT ? ? (eqbP ? ?) H1) in Hn;
+ rewrite > Hn in H; cases (H ?); reflexivity]]
qed.
+
+let rec uniq (d:eqType) (l:list d) on l : bool ≝
+ match l with
+ [ nil ⇒ true
+ | (cons x tl) ⇒ andb (notb (mem d x tl)) (uniq d tl)].
+
+lemma uniq_mem : ∀d:eqType.∀x:d.∀l:list d.uniq d (x::l) = true → mem d x l = false.
+intros (d x l H); simplify in H; lapply (b2pT ? ? (andbP ? ?) H) as H1; clear H;
+cases H1 (H2 H3); lapply (b2pT ? ?(negbP ?) H2); assumption;
+qed.
+
+lemma andbA : ∀a,b,c.andb a (andb b c) = andb (andb a b) c.
+intros; cases a; cases b; cases c; reflexivity; qed.
+
+lemma andbC : ∀a,b. andb a b = andb b a.
+intros; cases a; cases b; reflexivity; qed.
+
+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]
+rewrite > andbA; rewrite > andbC in ⊢ (? ? (? % ?) ?); rewrite < andbA;
+rewrite < H; rewrite > andbC in ⊢ (? ? ? (? % ?)); rewrite < andbA; reflexivity;
+qed.
+
+lemma count_O_mem : ∀d:eqType.∀x:d.∀l:list d.ltb O (count d (cmp d x) l) = mem d x l.
+intros 3 (d x l); elim l [reflexivity] simplify; rewrite < H; cases (cmp d x t);
+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]
+[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;
+ 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;
+ 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;
+ |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);]]]
+qed.
+
+lemma mem_finType : ∀d:finType.∀x:d. mem d x (enum d) = true.
+intros 1 (d); cases d; simplify; intros; rewrite < count_O_mem;
+rewrite > H; reflexivity;
+qed.
+
+lemma uniq_fintype_enum : ∀d:finType. uniq d (enum d) = true.
+intros; cases d; simplify; apply (p2bT ? ? (uniqP ? ?)); intros; apply H;
+qed.
+
+lemma sub_enumP : ∀d:finType.∀p:d→bool.∀x:sub_eqType d p.
+ count (sub_eqType d p) (cmp ? x) (filter ? ? (if_p ? p) (enum d)) = (S O).
+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;
+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;
+ [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;
+ 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⇒ ?] ?);
+ 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;
+ |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;]
+ |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;]
+qed.
+
+definition sub_finType : ∀d:finType.∀p:d→bool.finType ≝
+ λd:finType.λp:d→bool. mk_finType (sub_eqType d p) (filter ? ? (if_p ? p) (enum d)) (sub_enumP d p).
+
projects / helm.git / blobdiff