(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/decidable_kit/decidable/".
-
(* classical connectives for decidable properties *)
include "decidable_kit/streicher.ma".
unfold Not; intros (H); destruct H;
qed.
+lemma prove_reflect : ∀P:Prop.∀b:bool.
+ (b = true → P) → (b = false → ¬P) → reflect P b.
+intros 2 (P b); cases b; intros; [left|right] [autobatch.|autobatch;]
+qed.
+
(* ### standard connectives/relations with reflection predicate ### *)
definition negb : bool → bool ≝ λb.match b with [ true ⇒ false | false ⇒ true].
λa,b:bool. match a with [ true ⇒ b | false ⇒ false ].
lemma andbP : ∀a,b:bool. reflect (a = true ∧ b = true) (andb a b).
-intros (a b);
-generalize in match (refl_eq ? (andb a b));
-generalize in match (andb a b) in ⊢ (? ? ? % → %); intros 1 (c);
-cases c; intros (H); [ apply reflect_true | apply reflect_false ];
-generalize in match H; clear H;
-cases a; simplify;
-[1: intros (E); rewrite > E; split; reflexivity
-|2: intros (ABS); destruct ABS
-|3: intros (E); rewrite > E; unfold Not; intros (ABS); decompose; destruct H1
-|4: intros (E); unfold Not; intros (ABS); decompose; destruct H]
+intros (a b); apply prove_reflect; cases a; cases b; simplify; intros (H);
+[1,2,3,4: rewrite > H; split; reflexivity;
+|5,6,7,8: unfold Not; intros (H1); cases H1;
+ [destruct H|destruct H3|destruct H2|destruct H2]]
qed.
lemma andbPF : ∀a,b:bool. reflect (a = false ∨ b = false) (negb (andb a b)).
-intros (a b); cases a; cases b; simplify;
-[1: apply reflect_false | *: apply reflect_true ]
-[unfold Not; intros (H); cases H; destruct H1|right|left|left] reflexivity;
+intros (a b); apply prove_reflect; cases a; cases b; simplify; intros (H);
+[1,2,3,4: try rewrite > H; [1,2:right|3,4:left] reflexivity
+|5,6,7,8: unfold Not; intros (H1); [2,3,4: destruct H]; cases H1; destruct H2]
qed.
definition orb : bool → bool → bool ≝
qed.
lemma orbC : ∀a,b. orb a b = orb b a.
-intros (a b); cases a; cases b; auto. qed.
+intros (a b); cases a; cases b; autobatch. qed.
lemma lebP: ∀x,y. reflect (x ≤ y) (leb x y).
intros (x y); generalize in match (leb_to_Prop x y);
lemma lebW : ∀n,m. leb (S n) m = true → leb n m = true.
intros (n m H); lapply (b2pT ? ? (lebP ? ?) H); clear H;
-apply (p2bT ? ? (lebP ? ?)); auto.
+apply (p2bT ? ? (lebP ? ?)); apply lt_to_le; assumption.
qed.
definition ltb ≝ λx,y.leb (S x) y.
qed.
lemma ltb_refl : ∀n.ltb n n = false.
-intros (n); apply (p2bF ? ? (ltbP ? ?)); auto;
+intros (n); apply (p2bF ? ? (ltbP ? ?)); autobatch;
qed.
(* ### = between booleans as <-> in Prop ### *)
lemma leb_eqb : ∀n,m. orb (eqb n m) (leb (S n) m) = leb n m.
intros (n m); apply bool_to_eq; split; intros (H);
-[1:cases (b2pT ? ? (orbP ? ?) H); [2: auto]
- rewrite > (eqb_true_to_eq ? ? H1); auto
+[1:cases (b2pT ? ? (orbP ? ?) H); [2: (*autobatch type;*) apply lebW; assumption; ]
+ rewrite > (eqb_true_to_eq ? ? H1); autobatch
|2:cases (b2pT ? ? (lebP ? ?) H);
[ elim n; [reflexivity|assumption]
| simplify; rewrite > (p2bT ? ? (lebP ? ?) H1); rewrite > orbC ]
(* OUT OF PLACE *)
-lemma ltW : ∀n,m. n < m → n < (S m). intros; auto. qed.
+lemma ltW : ∀n,m. n < m → n < (S m).
+intros; unfold lt; unfold lt in H; autobatch. qed.
lemma ltbW : ∀n,m. ltb n m = true → ltb n (S m) = true.
intros (n m H); letin H1 ≝ (b2pT ? ? (ltbP ? ?) H); clearbody H1;
[1: intros; cases (b2pT ? ? (orbP ? ?) H); [1: apply ltbW; assumption]
rewrite > (eqb_true_to_eq ? ? H1); simplify;
rewrite > leb_refl; reflexivity
-|2: generalize in match m; clear m; elim n 0;
+|2: elim n in m ⊢ % 0;
[1: simplify; intros; cases n1; reflexivity;
|2: intros 1 (m); elim m 0;
[1: intros; apply (p2bT ? ? (orbP ? ?));