mod change (-x)

[helm.git] / helm / software / matita / contribs / ng_assembly / num / bool_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/num/bool_lemmas.ma b/helm/software/matita/contribs/ng_assembly/num/bool_lemmas.ma
old mode 100755 (executable)
new mode 100644 (file)
index 25d855d..db5214c
--- a/helm/software/matita/contribs/ng_assembly/num/bool_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/num/bool_lemmas.ma
@@ -15,8 +15,8 @@
 (* ********************************************************************** *)
 (*                          Progetto FreeScale                            *)
 (*                                                                        *)
-(*   Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it                   *)
-(*     Cosimo Oliboni, oliboni@cs.unibo.it                                *)
+(*   Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
+(*   Sviluppo: 2008-2010                                                  *)
 (*                                                                        *)
 (* ********************************************************************** *)
 
@@ -26,32 +26,19 @@ include "num/bool.ma".
 (* BOOLEANI *)
 (* ******** *)
 
+(*
 ndefinition bool_destruct_aux ≝
 Πb1,b2:bool.ΠP:Prop.b1 = b2 →
- match b1 with
-  [ true ⇒ match b2 with [ true ⇒ P → P | false ⇒ P ]
-  | false ⇒ match b2 with [ true ⇒ P | false ⇒ P → P ]
-  ].
+ match eq_bool b1 b2 with [ true ⇒ P → P | false ⇒ P ].
 
 ndefinition bool_destruct : bool_destruct_aux.
- #b1; #b2; #P;
+ #b1; #b2; #P; #H;
+ nrewrite < H;
  nelim b1;
- nelim b2;
  nnormalize;
- #H;
- ##[ ##2: napply False_ind;
-          nchange with (match true with [ true ⇒ False | false ⇒ True]);
-          nrewrite > H;
-          nnormalize;
-          napply I
- ##| ##3: napply False_ind;
-          nchange with (match true with [ true ⇒ False | false ⇒ True]);
-          nrewrite < H;
-          nnormalize;
-          napply I
- ##| ##1,4: napply (λx:P.x)
- ##]
+ napply (λx.x).
 nqed.
+*)
 
 nlemma symmetric_eqbool : symmetricT bool bool eq_bool.
  #b1; #b2;
@@ -118,7 +105,7 @@ nlemma eqbool_to_eq : ∀b1,b2:bool.(eq_bool b1 b2 = true) → (b1 = b2).
  ncases b2;
  nnormalize;
  ##[ ##1,4: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
  ##]
 nqed.
 
@@ -128,7 +115,27 @@ nlemma eq_to_eqbool : ∀b1,b2.b1 = b2 → eq_bool b1 b2 = true.
  ncases b2;
  nnormalize;
  ##[ ##1,4: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma decidable_bool : ∀x,y:bool.decidable (x = y).
+ #x; #y;
+ nnormalize;
+ nelim x;
+ nelim y;
+ ##[ ##1,4: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
+ ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …);
+          nnormalize; #H;
+          napply False_ind;
+          ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma decidable_bexpr : ∀x.(x = true) ∨ (x = false).
+ #x; ncases x;
+ ##[ ##1: napply (or2_intro1 (true = true) (true = false) (refl_eq …))
+ ##| ##2: napply (or2_intro2 (false = true) (false = false) (refl_eq …))
  ##]
 nqed.
 
@@ -137,8 +144,8 @@ nlemma neqbool_to_neq : ∀b1,b2:bool.(eq_bool b1 b2 = false) → (b1 ≠ b2).
  ncases b1;
  ncases b2;
  nnormalize;
- ##[ ##1,4: #H; napply (bool_destruct … H)
- ##| ##*: #H; #H1; napply (bool_destruct … H1)
+ ##[ ##1,4: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##*: #H; #H1; ndestruct (*napply (bool_destruct … H1)*)
  ##]
 nqed.
 
@@ -147,18 +154,46 @@ nlemma neq_to_neqbool : ∀b1,b2.b1 ≠ b2 → eq_bool b1 b2 = false.
  ncases b1;
  ncases b2;
  nnormalize;
- ##[ ##1,4: #H; napply False_ind; napply (H (refl_eq …))
+ ##[ ##1,4: #H; nelim (H (refl_eq …))
  ##| ##*: #H; napply refl_eq
  ##]
 nqed.
 
+nlemma eqfalse_to_neqtrue : ∀x.x = false → x ≠ true.
+ #x; nelim x;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##2: #H; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##]
+nqed.
+
+nlemma eqtrue_to_neqfalse : ∀x.x = true → x ≠ false.
+ #x; nelim x;
+ ##[ ##1: #H; nnormalize; #H1; ndestruct (*napply (bool_destruct … H1)*)
+ ##| ##2: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma neqfalse_to_eqtrue : ∀x.x ≠ false → x = true.
+ #x; nelim x;
+ ##[ ##1: #H; napply refl_eq
+ ##| ##2: nnormalize; #H; nelim (H (refl_eq …))
+ ##]
+nqed.
+
+nlemma neqtrue_to_eqfalse : ∀x.x ≠ true → x = false.
+ #x; nelim x;
+ ##[ ##1: nnormalize; #H; nelim (H (refl_eq …))
+ ##| ##2: #H; napply refl_eq
+ ##]
+nqed.
+
 nlemma andb_true_true_l: ∀b1,b2.(b1 ⊗ b2) = true → b1 = true.
  #b1; #b2;
  ncases b1;
  ncases b2;
  nnormalize;
  ##[ ##1,2: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
  ##]
 nqed.
 
@@ -168,17 +203,113 @@ nlemma andb_true_true_r: ∀b1,b2.(b1 ⊗ b2) = true → b2 = true.
  ncases b2;
  nnormalize;
  ##[ ##1,3: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
  ##]
 nqed.
 
+nlemma andb_false2
+ : ∀b1,b2.(b1 ⊗ b2) = false →
+   (b1 = false) ∨ (b2 = false).
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##2,4: #H; napply (or2_intro2 … H)
+ ##| ##3: #H; napply (or2_intro1 … H)
+ ##]
+nqed.
+
+nlemma andb_false3
+ : ∀b1,b2,b3.(b1 ⊗ b2 ⊗ b3) = false →
+   Or3 (b1 = false) (b2 = false) (b3 = false).
+ #b1; #b2; #b3;
+ ncases b1;
+ ncases b2;
+ ncases b3;
+ nnormalize;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##5,6,7,8: #H; napply (or3_intro1 … H)
+ ##| ##2,4: #H; napply (or3_intro3 … H)
+ ##| ##3: #H; napply (or3_intro2 … H)
+ ##]
+nqed.
+
+nlemma andb_false4
+ : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ b3 ⊗ b4) = false →
+   Or4 (b1 = false) (b2 = false) (b3 = false) (b4 = false).
+ #b1; #b2; #b3; #b4;
+ ncases b1;
+ ncases b2;
+ ncases b3;
+ ncases b4;
+ nnormalize;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##9,10,11,12,13,14,15,16: #H; napply (or4_intro1 … H)
+ ##| ##5,6,7,8: #H; napply (or4_intro2 … H)
+ ##| ##3,4: #H; napply (or4_intro3 … H)
+ ##| ##2: #H; napply (or4_intro4 … H)
+ ##]
+nqed.
+
+nlemma andb_false5
+ : ∀b1,b2,b3,b4,b5.(b1 ⊗ b2 ⊗ b3 ⊗ b4 ⊗ b5) = false →
+   Or5 (b1 = false) (b2 = false) (b3 = false) (b4 = false) (b5 = false).
+ #b1; #b2; #b3; #b4; #b5;
+ ncases b1;
+ ncases b2;
+ ncases b3;
+ ncases b4;
+ ncases b5;
+ nnormalize;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #H; napply (or5_intro1 … H)
+ ##| ##9,10,11,12,13,14,15,16: #H; napply (or5_intro2 … H)
+ ##| ##5,6,7,8: #H; napply (or5_intro3 … H)
+ ##| ##3,4: #H; napply (or5_intro4 … H)
+ ##| ##2: #H; napply (or5_intro5 … H)
+ ##]
+nqed.
+
+nlemma andb_false2_1 : ∀b.(false ⊗ b) = false.
+ #b; nnormalize; napply refl_eq. nqed.
+nlemma andb_false2_2 : ∀b.(b ⊗ false) = false.
+ #b; nelim b; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false3_1 : ∀b1,b2.(false ⊗ b1 ⊗ b2) = false.
+ #b1; #b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false3_2 : ∀b1,b2.(b1 ⊗ false ⊗ b2) = false.
+ #b1; #b2; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false3_3 : ∀b1,b2.(b1 ⊗ b2 ⊗ false) = false.
+ #b1; #b2; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false4_1 : ∀b1,b2,b3.(false ⊗ b1 ⊗ b2 ⊗ b3) = false.
+ #b1; #b2; #b3; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_2 : ∀b1,b2,b3.(b1 ⊗ false ⊗ b2 ⊗ b3) = false.
+ #b1; #b2; #b3; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_3 : ∀b1,b2,b3.(b1 ⊗ b2 ⊗ false ⊗ b3) = false.
+ #b1; #b2; #b3; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_4 : ∀b1,b2,b3.(b1 ⊗ b2 ⊗ b3 ⊗ false) = false.
+ #b1; #b2; #b3; nelim b1; nelim b2; nelim b3; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false5_1 : ∀b1,b2,b3,b4.(false ⊗ b1 ⊗ b2 ⊗ b3 ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_2 : ∀b1,b2,b3,b4.(b1 ⊗ false ⊗ b2 ⊗ b3 ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_3 : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ false ⊗ b3 ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_4 : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ b3 ⊗ false ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nelim b3; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_5 : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ b3 ⊗ b4 ⊗ false) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nelim b3; nelim b4; nnormalize; napply refl_eq. nqed.
+
 nlemma orb_false_false_l : ∀b1,b2:bool.(b1 ⊕ b2) = false → b1 = false.
  #b1; #b2;
  ncases b1;
  ncases b2;
  nnormalize;
  ##[ ##4: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
  ##]
 nqed.
 
@@ -188,6 +319,6 @@ nlemma orb_false_false_r : ∀b1,b2:bool.(b1 ⊕ b2) = false → b2 = false.
  ncases b2;
  nnormalize;
  ##[ ##4: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
  ##]
 nqed.