--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/exadecim.ma".
+include "num/bool_lemmas.ma".
+
+(* *********** *)
+(* ESADECIMALI *)
+(* *********** *)
+
+ndefinition exadecim_destruct1 : Πe2.ΠP:Prop.ΠH:x0 = e2.match e2 with [ x0 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##1: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x0 with [ x0 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct2 : Πe2.ΠP:Prop.ΠH:x1 = e2.match e2 with [ x1 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##2: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x1 with [ x1 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct3 : Πe2.ΠP:Prop.ΠH:x2 = e2.match e2 with [ x2 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##3: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x2 with [ x2 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct4 : Πe2.ΠP:Prop.ΠH:x3 = e2.match e2 with [ x3 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##4: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x3 with [ x3 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct5 : Πe2.ΠP:Prop.ΠH:x4 = e2.match e2 with [ x4 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##5: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x4 with [ x4 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct6 : Πe2.ΠP:Prop.ΠH:x5 = e2.match e2 with [ x5 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##6: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x5 with [ x5 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct7 : Πe2.ΠP:Prop.ΠH:x6 = e2.match e2 with [ x6 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##7: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x6 with [ x6 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct8 : Πe2.ΠP:Prop.ΠH:x7 = e2.match e2 with [ x7 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##8: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x7 with [ x7 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct9 : Πe2.ΠP:Prop.ΠH:x8 = e2.match e2 with [ x8 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##9: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x8 with [ x8 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct10 : Πe2.ΠP:Prop.ΠH:x9 = e2.match e2 with [ x9 ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##10: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match x9 with [ x9 ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct11 : Πe2.ΠP:Prop.ΠH:xA = e2.match e2 with [ xA ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##11: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match xA with [ xA ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct12 : Πe2.ΠP:Prop.ΠH:xB = e2.match e2 with [ xB ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##12: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match xB with [ xB ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct13 : Πe2.ΠP:Prop.ΠH:xC = e2.match e2 with [ xC ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##13: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match xC with [ xC ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct14 : Πe2.ΠP:Prop.ΠH:xD = e2.match e2 with [ xD ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##14: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match xD with [ xD ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct15 : Πe2.ΠP:Prop.ΠH:xE = e2.match e2 with [ xE ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##15: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match xE with [ xE ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct16 : Πe2.ΠP:Prop.ΠH:xF = e2.match e2 with [ xF ⇒ P → P | _ ⇒ P ].
+ #e2; #P; ncases e2; nnormalize; #H;
+ ##[ ##16: napply (λx:P.x)
+ ##| ##*: napply False_ind;
+ nchange with (match xF with [ xF ⇒ False | _ ⇒ True ]);
+ nrewrite > H; nnormalize; napply I
+ ##]
+nqed.
+
+ndefinition exadecim_destruct_aux ≝
+Πe1,e2.ΠP:Prop.ΠH:e1 = e2.
+ match e1 with
+ [ x0 ⇒ match e2 with [ x0 ⇒ P → P | _ ⇒ P ]
+ | x1 ⇒ match e2 with [ x1 ⇒ P → P | _ ⇒ P ]
+ | x2 ⇒ match e2 with [ x2 ⇒ P → P | _ ⇒ P ]
+ | x3 ⇒ match e2 with [ x3 ⇒ P → P | _ ⇒ P ]
+ | x4 ⇒ match e2 with [ x4 ⇒ P → P | _ ⇒ P ]
+ | x5 ⇒ match e2 with [ x5 ⇒ P → P | _ ⇒ P ]
+ | x6 ⇒ match e2 with [ x6 ⇒ P → P | _ ⇒ P ]
+ | x7 ⇒ match e2 with [ x7 ⇒ P → P | _ ⇒ P ]
+ | x8 ⇒ match e2 with [ x8 ⇒ P → P | _ ⇒ P ]
+ | x9 ⇒ match e2 with [ x9 ⇒ P → P | _ ⇒ P ]
+ | xA ⇒ match e2 with [ xA ⇒ P → P | _ ⇒ P ]
+ | xB ⇒ match e2 with [ xB ⇒ P → P | _ ⇒ P ]
+ | xC ⇒ match e2 with [ xC ⇒ P → P | _ ⇒ P ]
+ | xD ⇒ match e2 with [ xD ⇒ P → P | _ ⇒ P ]
+ | xE ⇒ match e2 with [ xE ⇒ P → P | _ ⇒ P ]
+ | xF ⇒ match e2 with [ xF ⇒ P → P | _ ⇒ P ]
+ ].
+
+ndefinition exadecim_destruct : exadecim_destruct_aux.
+ #e1; ncases e1;
+ ##[ ##1: napply exadecim_destruct1
+ ##| ##2: napply exadecim_destruct2
+ ##| ##3: napply exadecim_destruct3
+ ##| ##4: napply exadecim_destruct4
+ ##| ##5: napply exadecim_destruct5
+ ##| ##6: napply exadecim_destruct6
+ ##| ##7: napply exadecim_destruct7
+ ##| ##8: napply exadecim_destruct8
+ ##| ##9: napply exadecim_destruct9
+ ##| ##10: napply exadecim_destruct10
+ ##| ##11: napply exadecim_destruct11
+ ##| ##12: napply exadecim_destruct12
+ ##| ##13: napply exadecim_destruct13
+ ##| ##14: napply exadecim_destruct14
+ ##| ##15: napply exadecim_destruct15
+ ##| ##16: napply exadecim_destruct16
+ ##]
+nqed.
+
+nlemma symmetric_eqex : symmetricT exadecim bool eq_ex.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_andex : symmetricT exadecim exadecim and_ex.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_andex : ∀e1,e2,e3.(and_ex (and_ex e1 e2) e3) = (and_ex e1 (and_ex e2 e3)).
+ #e1; #e2; #e3;
+ nelim e1;
+ ##[ ##1: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##2: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##3: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##4: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##5: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##6: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##7: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##8: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##9: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##10: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##11: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##12: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##13: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##14: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##15: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##16: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma symmetric_orex : symmetricT exadecim exadecim or_ex.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_orex : ∀e1,e2,e3.(or_ex (or_ex e1 e2) e3) = (or_ex e1 (or_ex e2 e3)).
+ #e1; #e2; #e3;
+ nelim e1;
+ ##[ ##1: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##2: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##3: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##4: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##5: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##6: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##7: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##8: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##9: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##10: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##11: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##12: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##13: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##14: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##15: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##16: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma symmetric_xorex : symmetricT exadecim exadecim xor_ex.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_xorex : ∀e1,e2,e3.(xor_ex (xor_ex e1 e2) e3) = (xor_ex e1 (xor_ex e2 e3)).
+ #e1; #e2; #e3;
+ nelim e1;
+ ##[ ##1: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##2: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##3: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##4: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##5: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##6: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##7: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##8: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##9: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##10: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##11: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##12: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##13: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##14: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##15: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##16: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma symmetric_plusex_dc_dc : ∀e1,e2,c.plus_ex_dc_dc e1 e2 c = plus_ex_dc_dc e2 e1 c.
+ #e1; #e2; #c;
+ nelim e1;
+ nelim e2;
+ nelim c;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_dc_dc_to_dc_d : ∀e1,e2,c.fst … (plus_ex_dc_dc e1 e2 c) = plus_ex_dc_d e1 e2 c.
+ #e1; #e2; #c;
+ nelim e1;
+ nelim e2;
+ nelim c;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_dc_dc_to_dc_c : ∀e1,e2,c.snd … (plus_ex_dc_dc e1 e2 c) = plus_ex_dc_c e1 e2 c.
+ #e1; #e2; #c;
+ nelim e1;
+ nelim e2;
+ nelim c;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_dc_dc_to_d_dc : ∀e1,e2.plus_ex_dc_dc e1 e2 false = plus_ex_d_dc e1 e2.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_dc_dc_to_d_d : ∀e1,e2.fst … (plus_ex_dc_dc e1 e2 false) = plus_ex_d_d e1 e2.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_dc_dc_to_d_c : ∀e1,e2.snd … (plus_ex_dc_dc e1 e2 false) = plus_ex_d_c e1 e2.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_plusex_dc_d : ∀e1,e2,c.plus_ex_dc_d e1 e2 c = plus_ex_dc_d e2 e1 c.
+ #e1; #e2; #c;
+ nelim e1;
+ nelim e2;
+ nelim c;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_plusex_dc_c : ∀e1,e2,c.plus_ex_dc_c e1 e2 c = plus_ex_dc_c e2 e1 c.
+ #e1; #e2; #c;
+ nelim e1;
+ nelim e2;
+ nelim c;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_plusex_d_dc : ∀e1,e2.plus_ex_d_dc e1 e2 = plus_ex_d_dc e2 e1.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_d_dc_to_d_d : ∀e1,e2.fst … (plus_ex_d_dc e1 e2) = plus_ex_d_d e1 e2.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma plusex_d_dc_to_d_c : ∀e1,e2.snd … (plus_ex_d_dc e1 e2) = plus_ex_d_c e1 e2.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_plusex_d_d : ∀e1,e2.plus_ex_d_d e1 e2 = plus_ex_d_d e2 e1.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_plusex_d_d : ∀e1,e2,e3.(plus_ex_d_d (plus_ex_d_d e1 e2) e3) = (plus_ex_d_d e1 (plus_ex_d_d e2 e3)).
+ #e1; #e2; #e3;
+ nelim e1;
+ ##[ ##1: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##2: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##3: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##4: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##5: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##6: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##7: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##8: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##9: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##10: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##11: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##12: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##13: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##14: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##15: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##| ##16: nelim e2; nelim e3; nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma symmetric_plusex_d_c : ∀e1,e2.plus_ex_d_c e1 e2 = plus_ex_d_c e2 e1.
+ #e1; #e2;
+ nelim e1;
+ nelim e2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma eqex_to_eq : ∀e1,e2:exadecim.(eq_ex e1 e2 = true) → (e1 = e2).
+ #e1; #e2;
+ ncases e1;
+ ncases e2;
+ nnormalize;
+ ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
+ ##| ##*: #H; napply (bool_destruct … H)
+ ##]
+nqed.
+
+nlemma eq_to_eqex : ∀e1,e2.e1 = e2 → eq_ex e1 e2 = true.
+ #m1; #m2;
+ ncases m1;
+ ncases m2;
+ nnormalize;
+ ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
+ ##| ##*: #H; napply (exadecim_destruct … H)
+ ##]
+nqed.
+
+nlemma neqex_to_neq : ∀e1,e2:exadecim.(eq_ex e1 e2 = false) → (e1 ≠ e2).
+ #e1; #e2;
+ ncases e1;
+ ncases e2;
+ nnormalize;
+ ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply (bool_destruct … H)
+ ##| ##*: #H; #H1; napply (exadecim_destruct … H1)
+ ##]
+nqed.
+
+nlemma neq_to_neqex : ∀e1,e2.e1 ≠ e2 → eq_ex e1 e2 = false.
+ #e1; #e2;
+ ncases e1;
+ ncases e2;
+ nnormalize;
+ ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply False_ind; napply (H (refl_eq …))
+ ##| ##*: #H; napply refl_eq
+ ##]
+nqed.
projects / helm.git / blobdiff