From: Cosimo Oliboni <??>
Date: Thu, 4 Feb 2010 03:42:47 +0000 (+0000)
Subject: freescale porting
X-Git-Tag: make_still_working~3059
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=bd112857523fc543c78cd29b74417585033ec464;p=helm.git
freescale porting
---
diff --git a/helm/software/matita/contribs/ng_assembly2/common/ascii.ma b/helm/software/matita/contribs/ng_assembly2/common/ascii.ma
new file mode 100755
index 000000000..c1f80c50c
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/ascii.ma
@@ -0,0 +1,101 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/ascii_base.ma".
+include "common/comp.ma".
+include "num/bool_lemmas.ma".
+
+(* ************************** *)
+(* DEFINIZIONE ASCII MINIMALE *)
+(* ************************** *)
+
+ndefinition ascii_destruct_aux â
+Î c1,c2.Î P:Prop.c1 = c2 â
+ match eq_ascii c1 c2 with [ true â P â P | false â P ].
+
+nlemma ascii_destruct : ascii_destruct_aux.
+ #c1; #c2; #P; #H;
+ nrewrite < H;
+ nelim c1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+nlemma eq_to_eqascii : ân1,n2.n1 = n2 â eq_ascii n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqascii_to_neq : ân1,n2.eq_ascii n1 n2 = false â n1 â n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_ascii n1 n2 = true) â¦);
+ ##[ ##1: napply (eq_to_eqascii n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue ⦠H)
+ ##]
+nqed.
+
+(* !!! per brevita... *)
+naxiom eqascii_to_eq : âc1,c2.eq_ascii c1 c2 = true â c1 = c2.
+
+nlemma neq_to_neqascii : ân1,n2.n1 â n2 â eq_ascii n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_ascii n1 n2));
+ napply (not_to_not (eq_ascii n1 n2 = true) (n1 = n2) ? H);
+ napply (eqascii_to_eq n1 n2).
+nqed.
+
+nlemma decidable_ascii : âx,y:ascii.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_ascii x y = true) (eq_ascii x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x â y) (eqascii_to_eq ⦠H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x â y) (neqascii_to_neq ⦠H))
+ ##]
+nqed.
+
+nlemma symmetric_eqascii : symmetricT ascii bool eq_ascii.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_ascii n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqascii n1 n2 H);
+ napply (symmetric_eq ? (eq_ascii n2 n1) false);
+ napply (neq_to_neqascii n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma ascii_is_comparable : comparable.
+ @ ascii
+ ##[ napply ch_0
+ ##| napply forall_ascii
+ ##| napply eq_ascii
+ ##| napply eqascii_to_eq
+ ##| napply eq_to_eqascii
+ ##| napply neqascii_to_neq
+ ##| napply neq_to_neqascii
+ ##| napply decidable_ascii
+ ##| napply symmetric_eqascii
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr ascii_is_comparable â¡ ascii.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/ascii_base.ma b/helm/software/matita/contribs/ng_assembly2/common/ascii_base.ma
new file mode 100755
index 000000000..bf42120d4
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/ascii_base.ma
@@ -0,0 +1,147 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/bool.ma".
+
+(* ************************** *)
+(* DEFINIZIONE ASCII MINIMALE *)
+(* ************************** *)
+
+ninductive ascii : Type â
+(* numeri *)
+ ch_0: ascii
+| ch_1: ascii
+| ch_2: ascii
+| ch_3: ascii
+| ch_4: ascii
+| ch_5: ascii
+| ch_6: ascii
+| ch_7: ascii
+| ch_8: ascii
+| ch_9: ascii
+
+(* simboli *)
+| ch__: ascii
+
+(* maiuscole *)
+| ch_A: ascii
+| ch_B: ascii
+| ch_C: ascii
+| ch_D: ascii
+| ch_E: ascii
+| ch_F: ascii
+| ch_G: ascii
+| ch_H: ascii
+| ch_I: ascii
+| ch_J: ascii
+| ch_K: ascii
+| ch_L: ascii
+| ch_M: ascii
+| ch_N: ascii
+| ch_O: ascii
+| ch_P: ascii
+| ch_Q: ascii
+| ch_R: ascii
+| ch_S: ascii
+| ch_T: ascii
+| ch_U: ascii
+| ch_V: ascii
+| ch_W: ascii
+| ch_X: ascii
+| ch_Y: ascii
+| ch_Z: ascii
+
+(* minuscole *)
+| ch_a: ascii
+| ch_b: ascii
+| ch_c: ascii
+| ch_d: ascii
+| ch_e: ascii
+| ch_f: ascii
+| ch_g: ascii
+| ch_h: ascii
+| ch_i: ascii
+| ch_j: ascii
+| ch_k: ascii
+| ch_l: ascii
+| ch_m: ascii
+| ch_n: ascii
+| ch_o: ascii
+| ch_p: ascii
+| ch_q: ascii
+| ch_r: ascii
+| ch_s: ascii
+| ch_t: ascii
+| ch_u: ascii
+| ch_v: ascii
+| ch_w: ascii
+| ch_x: ascii
+| ch_y: ascii
+| ch_z: ascii.
+
+(* iteratore sugli ascii *)
+ndefinition forall_ascii â λP.
+ P ch_0 â P ch_1 â P ch_2 â P ch_3 â P ch_4 â P ch_5 â P ch_6 â P ch_7 â
+ P ch_8 â P ch_9 â P ch__ â P ch_A â P ch_B â P ch_C â P ch_D â P ch_E â
+ P ch_F â P ch_G â P ch_H â P ch_I â P ch_J â P ch_K â P ch_L â P ch_M â
+ P ch_N â P ch_O â P ch_P â P ch_Q â P ch_R â P ch_S â P ch_T â P ch_U â
+ P ch_V â P ch_W â P ch_X â P ch_Y â P ch_Z â P ch_a â P ch_b â P ch_c â
+ P ch_d â P ch_e â P ch_f â P ch_g â P ch_h â P ch_i â P ch_j â P ch_k â
+ P ch_l â P ch_m â P ch_n â P ch_o â P ch_p â P ch_q â P ch_r â P ch_s â
+ P ch_t â P ch_u â P ch_v â P ch_w â P ch_x â P ch_y â P ch_z.
+
+(* confronto fra ascii *)
+ndefinition eq_ascii â
+λc,c':ascii.match c with
+ [ ch_0 â match c' with [ ch_0 â true | _ â false ] | ch_1 â match c' with [ ch_1 â true | _ â false ]
+ | ch_2 â match c' with [ ch_2 â true | _ â false ] | ch_3 â match c' with [ ch_3 â true | _ â false ]
+ | ch_4 â match c' with [ ch_4 â true | _ â false ] | ch_5 â match c' with [ ch_5 â true | _ â false ]
+ | ch_6 â match c' with [ ch_6 â true | _ â false ] | ch_7 â match c' with [ ch_7 â true | _ â false ]
+ | ch_8 â match c' with [ ch_8 â true | _ â false ] | ch_9 â match c' with [ ch_9 â true | _ â false ]
+ | ch__ â match c' with [ ch__ â true | _ â false ] | ch_A â match c' with [ ch_A â true | _ â false ]
+ | ch_B â match c' with [ ch_B â true | _ â false ] | ch_C â match c' with [ ch_C â true | _ â false ]
+ | ch_D â match c' with [ ch_D â true | _ â false ] | ch_E â match c' with [ ch_E â true | _ â false ]
+ | ch_F â match c' with [ ch_F â true | _ â false ] | ch_G â match c' with [ ch_G â true | _ â false ]
+ | ch_H â match c' with [ ch_H â true | _ â false ] | ch_I â match c' with [ ch_I â true | _ â false ]
+ | ch_J â match c' with [ ch_J â true | _ â false ] | ch_K â match c' with [ ch_K â true | _ â false ]
+ | ch_L â match c' with [ ch_L â true | _ â false ] | ch_M â match c' with [ ch_M â true | _ â false ]
+ | ch_N â match c' with [ ch_N â true | _ â false ] | ch_O â match c' with [ ch_O â true | _ â false ]
+ | ch_P â match c' with [ ch_P â true | _ â false ] | ch_Q â match c' with [ ch_Q â true | _ â false ]
+ | ch_R â match c' with [ ch_R â true | _ â false ] | ch_S â match c' with [ ch_S â true | _ â false ]
+ | ch_T â match c' with [ ch_T â true | _ â false ] | ch_U â match c' with [ ch_U â true | _ â false ]
+ | ch_V â match c' with [ ch_V â true | _ â false ] | ch_W â match c' with [ ch_W â true | _ â false ]
+ | ch_X â match c' with [ ch_X â true | _ â false ] | ch_Y â match c' with [ ch_Y â true | _ â false ]
+ | ch_Z â match c' with [ ch_Z â true | _ â false ] | ch_a â match c' with [ ch_a â true | _ â false ]
+ | ch_b â match c' with [ ch_b â true | _ â false ] | ch_c â match c' with [ ch_c â true | _ â false ]
+ | ch_d â match c' with [ ch_d â true | _ â false ] | ch_e â match c' with [ ch_e â true | _ â false ]
+ | ch_f â match c' with [ ch_f â true | _ â false ] | ch_g â match c' with [ ch_g â true | _ â false ]
+ | ch_h â match c' with [ ch_h â true | _ â false ] | ch_i â match c' with [ ch_i â true | _ â false ]
+ | ch_j â match c' with [ ch_j â true | _ â false ] | ch_k â match c' with [ ch_k â true | _ â false ]
+ | ch_l â match c' with [ ch_l â true | _ â false ] | ch_m â match c' with [ ch_m â true | _ â false ]
+ | ch_n â match c' with [ ch_n â true | _ â false ] | ch_o â match c' with [ ch_o â true | _ â false ]
+ | ch_p â match c' with [ ch_p â true | _ â false ] | ch_q â match c' with [ ch_q â true | _ â false ]
+ | ch_r â match c' with [ ch_r â true | _ â false ] | ch_s â match c' with [ ch_s â true | _ â false ]
+ | ch_t â match c' with [ ch_t â true | _ â false ] | ch_u â match c' with [ ch_u â true | _ â false ]
+ | ch_v â match c' with [ ch_v â true | _ â false ] | ch_w â match c' with [ ch_w â true | _ â false ]
+ | ch_x â match c' with [ ch_x â true | _ â false ] | ch_y â match c' with [ ch_y â true | _ â false ]
+ | ch_z â match c' with [ ch_z â true | _ â false ]
+ ].
diff --git a/helm/software/matita/contribs/ng_assembly2/common/comp.ma b/helm/software/matita/contribs/ng_assembly2/common/comp.ma
new file mode 100755
index 000000000..75300217c
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/comp.ma
@@ -0,0 +1,40 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/hints_declaration.ma".
+include "num/bool.ma".
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+
+nrecord comparable : Type[1] â
+ {
+ carr :> Type[0];
+ zeroc : carr;
+ forallc : (carr â bool) â bool;
+ eqc : carr â carr â bool;
+ eqc_to_eq : âx,y.(eqc x y = true) â (x = y);
+ eq_to_eqc : âx,y.(x = y) â (eqc x y = true);
+ neqc_to_neq : âx,y.(eqc x y = false) â (x â y);
+ neq_to_neqc : âx,y.(x â y) â (eqc x y = false);
+ decidable_c : âx,y:carr.decidable (x = y);
+ symmetric_eqc : symmetricT carr bool eqc
+ }.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/hints_declaration.ma b/helm/software/matita/contribs/ng_assembly2/common/hints_declaration.ma
new file mode 100644
index 000000000..354358c0e
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/hints_declaration.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/pts.ma".
+
+(*
+
+Notation for hint declaration
+==============================
+
+The idea is to write a context, with abstraction first, then
+recursive calls (let-in) and finally the two equivalent terms.
+The context can be empty. Note the ; to begin the second part of
+the context (necessary even if the first part is empty).
+
+ unification hint PREC \coloneq
+ ID : TY, ..., ID : TY
+ ; ID \equest T, ..., ID \equest T
+ \vdash T1 \equiv T2
+
+With unidoce and some ASCII art it looks like the following:
+
+ unification hint PREC â ID : TY, ..., ID : TY;
+ ID â T, ..., ID â T
+ (*---------------------*) â¢
+ T1 â¡ T2
+
+*)
+
+(* it seems unbelivable, but it works! *)
+notation > "â (list0 ( (list1 (ident x) sep , ) opt (: T) ) sep ,) opt (; (list1 (ident U â term 90 V ) sep ,)) ⢠term 19 Px â¡ term 19 Py"
+ with precedence 90
+ for @{ ${ fold right
+ @{ ${ default
+ @{ ${ fold right
+ @{ 'hint_decl $Px $Py }
+ rec acc1 @{ let ( ${ident U} : ?) â $V in $acc1} } }
+ @{ 'hint_decl $Px $Py }
+ }
+ }
+ rec acc @{
+ ${ fold right @{ $acc } rec acc2
+ @{ â${ident x}:${ default @{ $T } @{ ? } }.$acc2 } }
+ }
+ }}.
+
+ndefinition hint_declaration_Type0 â λA:Type[0].λa,b:A.Prop.
+ndefinition hint_declaration_Type1 â λA:Type[1].λa,b:A.Prop.
+ndefinition hint_declaration_Type2 â λa,b:Type[2].Prop.
+ndefinition hint_declaration_CProp0 â λA:CProp[0].λa,b:A.Prop.
+ndefinition hint_declaration_CProp1 â λA:CProp[1].λa,b:A.Prop.
+ndefinition hint_declaration_CProp2 â λa,b:CProp[2].Prop.
+
+interpretation "hint_decl_Type2" 'hint_decl a b = (hint_declaration_Type2 a b).
+interpretation "hint_decl_CProp2" 'hint_decl a b = (hint_declaration_CProp2 a b).
+interpretation "hint_decl_Type1" 'hint_decl a b = (hint_declaration_Type1 ? a b).
+interpretation "hint_decl_CProp1" 'hint_decl a b = (hint_declaration_CProp1 ? a b).
+interpretation "hint_decl_CProp0" 'hint_decl a b = (hint_declaration_CProp0 ? a b).
+interpretation "hint_decl_Type0" 'hint_decl a b = (hint_declaration_Type0 ? a b).
diff --git a/helm/software/matita/contribs/ng_assembly2/common/list.ma b/helm/software/matita/contribs/ng_assembly2/common/list.ma
new file mode 100644
index 000000000..a1077f697
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/list.ma
@@ -0,0 +1,590 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "common/option.ma".
+include "common/nat.ma".
+
+(* *************** *)
+(* LISTE GENERICHE *)
+(* *************** *)
+
+ninductive list (A:Type) : Type â
+ nil: list A
+| cons: A â list A â list A.
+
+nlet rec append A (l1: list A) l2 on l1 â
+ match l1 with
+ [ nil â l2
+ | (cons hd tl) â cons A hd (append A tl l2) ].
+
+notation "hvbox(hd break :: tl)"
+ right associative with precedence 47
+ for @{'cons $hd $tl}.
+
+notation "[ list0 x sep ; ]"
+ non associative with precedence 90
+ for ${fold right @'nil rec acc @{'cons $x $acc}}.
+
+notation "hvbox(l1 break @ l2)"
+ right associative with precedence 47
+ for @{'append $l1 $l2 }.
+
+interpretation "nil" 'nil = (nil ?).
+interpretation "cons" 'cons hd tl = (cons ? hd tl).
+interpretation "append" 'append l1 l2 = (append ? l1 l2).
+
+nlemma list_destruct_1 : âT.âx1,x2:T.ây1,y2:list T.cons T x1 y1 = cons T x2 y2 â x1 = x2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match cons T x2 y2 with [ nil â False | cons a _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma list_destruct_2 : âT.âx1,x2:T.ây1,y2:list T.cons T x1 y1 = cons T x2 y2 â y1 = y2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match cons T x2 y2 with [ nil â False | cons _ b â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma list_destruct_cons_nil : âT.âx:T.ây:list T.cons T x y = nil T â False.
+ #T; #x; #y; #H;
+ nchange with (match cons T x y with [ nil â True | cons a b â False ]);
+ nrewrite > H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma list_destruct_nil_cons : âT.âx:T.ây:list T.nil T = cons T x y â False.
+ #T; #x; #y; #H;
+ nchange with (match cons T x y with [ nil â True | cons a b â False ]);
+ nrewrite < H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma append_nil : âT:Type.âl:list T.(l@[]) = l.
+ #T; #l;
+ nelim l;
+ nnormalize;
+ ##[ ##1: napply refl_eq
+ ##| ##2: #x; #y; #H;
+ nrewrite > H;
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma associative_list : âT.associative (list T) (append T).
+ #T; #x; #y; #z;
+ nelim x;
+ nnormalize;
+ ##[ ##1: napply refl_eq
+ ##| ##2: #a; #b; #H;
+ nrewrite > H;
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma cons_append_commute : âT:Type.âl1,l2:list T.âa:T.a :: (l1 @ l2) = (a :: l1) @ l2.
+ #T; #l1; #l2; #a;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma append_cons_commute : âT:Type.âa:T.âl,l1:list T.l @ (a::l1) = (l@[a]) @ l1.
+ #T; #a; #l; #l1;
+ nrewrite > (associative_list T l [a] l1);
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+(* listlen *)
+nlet rec len_list (T:Type) (l:list T) on l â
+ match l with [ nil â O | cons _ t â S (len_list T t) ].
+
+(* vuota? *)
+ndefinition is_empty_list â
+λT:Type.λl:list T.match l with [ nil â True | cons _ _ â False ].
+
+ndefinition isb_empty_list â
+λT:Type.λl:list T.match l with [ nil â true | cons _ _ â false ].
+
+ndefinition isnot_empty_list â
+λT:Type.λl:list T.match l with [ nil â False | cons _ _ â True ].
+
+ndefinition isnotb_empty_list â
+λT:Type.λl:list T.match l with [ nil â false | cons _ _ â true ].
+
+(* reverse *)
+nlet rec reverse_list (T:Type) (l:list T) on l â
+ match l with
+ [ nil â nil T
+ | cons h t â (reverse_list T t)@[h]
+ ].
+
+(* getFirst *)
+ndefinition get_first_list â
+λT:Type.λl:list T.match l with
+ [ nil â None ?
+ | cons h _ â Some ? h ].
+
+(* getLast *)
+ndefinition get_last_list â
+λT:Type.λl:list T.match reverse_list T l with
+ [ nil â None ?
+ | cons h _ â Some ? h ].
+
+(* cutFirst *)
+ndefinition cut_first_list â
+λT:Type.λl:list T.match l with
+ [ nil â nil T
+ | cons _ t â t ].
+
+(* cutLast *)
+ndefinition cut_last_list â
+λT:Type.λl:list T.match reverse_list T l with
+ [ nil â nil T
+ | cons _ t â reverse_list T t ].
+
+(* apply f *)
+nlet rec apply_f_list (T1,T2:Type) (l:list T1) (f:T1 â T2) on l â
+match l with
+ [ nil â nil T2
+ | cons h t â cons T2 (f h) (apply_f_list T1 T2 t f) ].
+
+(* fold right *)
+nlet rec fold_right_list (T1,T2:Type) (f:T1 â T2 â T2) (acc:T2) (l:list T1) on l â
+ match l with
+ [ nil â acc
+ | cons h t â f h (fold_right_list T1 T2 f acc t)
+ ].
+
+(* double fold right *)
+nlemma fold_right_list2_aux1 :
+âT.âh,t.len_list T [] = len_list T (h::t) â False.
+ #T; #h; #t;
+ nnormalize;
+ #H;
+ ndestruct (*napply (nat_destruct_0_S ? H)*).
+nqed.
+
+nlemma fold_right_list2_aux2 :
+âT.âh,t.len_list T (h::t) = len_list T [] â False.
+ #T; #h; #t;
+ nnormalize;
+ #H;
+ ndestruct (*napply (nat_destruct_S_0 ? H)*).
+nqed.
+
+nlemma fold_right_list2_aux3 :
+âT.âh,h',t,t'.len_list T (h::t) = len_list T (h'::t') â len_list T t = len_list T t'.
+ #T; #h; #h'; #t; #t';
+ nelim t;
+ nelim t';
+ ##[ ##1: nnormalize; #H; napply refl_eq
+ ##| ##2: #a; #l'; #H; #H1;
+ nchange in H1:(%) with ((S O) = (S (S (len_list T l'))));
+ ndestruct (*nelim (nat_destruct_0_S ? (nat_destruct_S_S ⦠H1))*)
+ ##| ##3: #a; #l'; #H; #H1;
+ nchange in H1:(%) with ((S (S (len_list T l'))) = (S O));
+ ndestruct (*nelim (nat_destruct_S_0 ? (nat_destruct_S_S ⦠H1))*)
+ ##| ##4: #a; #l; #H; #a1; #l1; #H1; #H2;
+ nchange in H2:(%) with ((S (S (len_list T l1))) = (S (S (len_list T l))));
+ nchange with ((S (len_list T l1)) = (S (len_list T l)));
+ nrewrite > (nat_destruct_S_S ⦠H2);
+ napply refl_eq
+ ##]
+nqed.
+
+nlet rec fold_right_list2 (T1,T2:Type) (f:T1 â T1 â T2 â T2) (acc:T2) (l1:list T1) on l1 â
+ match l1
+ return λl1.Î l2.len_list T1 l1 = len_list T1 l2 â T2
+ with
+ [ nil â λl2.match l2 return λl2.len_list T1 [] = len_list T1 l2 â T2 with
+ [ nil â λp:len_list T1 [] = len_list T1 [].acc
+ | cons h t â λp:len_list T1 [] = len_list T1 (h::t).
+ False_rect_Type0 ? (fold_right_list2_aux1 T1 h t p)
+ ]
+ | cons h t â λl2.match l2 return λl2.len_list T1 (h::t) = len_list T1 l2 â T2 with
+ [ nil â λp:len_list T1 (h::t) = len_list T1 [].
+ False_rect_Type0 ? (fold_right_list2_aux2 T1 h t p)
+ | cons h' t' â λp:len_list T1 (h::t) = len_list T1 (h'::t').
+ f h h' (fold_right_list2 T1 T2 f acc t t' (fold_right_list2_aux3 T1 h h' t t' p))
+ ]
+ ].
+
+nlet rec bfold_right_list2 (T1:Type) (f:T1 â T1 â bool) (l1,l2:list T1) on l1 â
+ match l1 with
+ [ nil â match l2 with
+ [ nil â true | cons h t â false ]
+ | cons h t â match l2 with
+ [ nil â false | cons h' t' â (f h h') â (bfold_right_list2 T1 f t t')
+ ]
+ ].
+
+(* nth elem *)
+nlet rec nth_list (T:Type) (l:list T) (n:nat) on l â
+ match l with
+ [ nil â None ?
+ | cons h t â match n with
+ [ O â Some ? h | S n' â nth_list T t n' ]
+ ].
+
+(* abs elem *)
+ndefinition abs_list_aux1 : âT:Type.ân.((len_list T []) > n) = true â False.
+ #T; nnormalize; #n; #H; ndestruct (*napply (bool_destruct ⦠H)*). nqed.
+
+ndefinition abs_list_aux2 : âT:Type.âh:T.ât:list T.ân.((len_list T (h::t)) > (S n) = true) â ((len_list T t) > n) = true.
+ #T; #h; #t; #n; nnormalize; #H; napply H. nqed.
+
+nlet rec abs_list (T:Type) (l:list T) on l â
+ match l
+ return λl.Î n.(((len_list T l) > n) = true) â T
+ with
+ [ nil â λn.λp:(((len_list T []) > n) = true).False_rect_Type0 ? (abs_list_aux1 T n p)
+ | cons h t â λn.
+ match n with
+ [ O â λp:(((len_list T (h::t)) > O) = true).h
+ | S n' â λp:(((len_list T (h::t)) > (S n')) = true).
+ abs_list T t n' (abs_list_aux2 T h t n' p)
+ ]
+ ].
+
+(* esempio: abs_list ? [ 1; 2; 3 ; 4 ] 0 (refl_eq â¦) = 1. *)
+
+nlemma symmetric_lenlist : âT.âl1,l2:list T.len_list T l1 = len_list T l2 â len_list T l2 = len_list T l1.
+ #T; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2; nnormalize;
+ ##[ ##1: #H; napply refl_eq
+ ##| ##2: #h; #t; #H; ndestruct (*nelim (nat_destruct_0_S ? H)*)
+ ##]
+ ##| ##2: #h; #l2; ncases l2; nnormalize;
+ ##[ ##1: #H; #l; #H1; nrewrite < H1; napply refl_eq
+ ##| ##2: #h; #l; #H; #l3; #H1; nrewrite < H1; napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma symmetric_foldrightlist2_aux :
+âT1,T2:Type.âf:T1 â T1 â T2 â T2.
+ (âx,y,z.f x y z = f y x z) â
+ (âacc:T2.âl1,l2:list T1.
+ âH1:(len_list T1 l1 = len_list T1 l2).
+ âH2:(len_list T1 l2 = len_list T1 l1).
+ (fold_right_list2 T1 T2 f acc l1 l2 H1 = fold_right_list2 T1 T2 f acc l2 l1 H2)).
+ #T1; #T2; #f; #H; #acc; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: nnormalize; #H1; #H2; napply refl_eq
+ ##| ##2: #h; #l; #H1; #H2;
+ nchange in H1:(%) with (O = (S (len_list ? l)));
+ ndestruct (*nelim (nat_destruct_0_S ? H1)*)
+ ##]
+ ##| ##2: #h3; #l3; #H1; #l2; ncases l2;
+ ##[ ##1: #H2; #H3; nchange in H2:(%) with ((S (len_list ? l3)) = O);
+ ndestruct (*nelim (nat_destruct_S_0 ? H1)*)
+ ##| ##2: #h4; #l4; #H2; #H3;
+ nchange in H2:(%) with ((S (len_list ? l3)) = (S (len_list ? l4)));
+ nchange in H3:(%) with ((S (len_list ? l4)) = (S (len_list ? l3)));
+ nchange with ((f h3 h4 (fold_right_list2 T1 T2 f acc l3 l4 (fold_right_list2_aux3 T1 h3 h4 l3 l4 ?))) =
+ (f h4 h3 (fold_right_list2 T1 T2 f acc l4 l3 (fold_right_list2_aux3 T1 h4 h3 l4 l3 ?))));
+ nrewrite < (H1 l4 (fold_right_list2_aux3 T1 h3 h4 l3 l4 H2) (fold_right_list2_aux3 T1 h4 h3 l4 l3 H3));
+ nrewrite > (H h3 h4 (fold_right_list2 T1 T2 f acc l3 l4 ?));
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma symmetric_foldrightlist2 :
+âT1,T2:Type.âf:T1 â T1 â T2 â T2.
+ (âx,y,z.f x y z = f y x z) â
+ (âacc:T2.âl1,l2:list T1.âH:len_list T1 l1 = len_list T1 l2.
+ fold_right_list2 T1 T2 f acc l1 l2 H = fold_right_list2 T1 T2 f acc l2 l1 (symmetric_lenlist T1 l1 l2 H)).
+ #T1; #T2; #f; #H; #acc; #l1; #l2; #H1;
+ nrewrite > (symmetric_foldrightlist2_aux T1 T2 f H acc l1 l2 H1 (symmetric_lenlist T1 l1 l2 H1));
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_bfoldrightlist2 :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.f x y = f y x) â
+ (âl1,l2:list T1.
+ bfold_right_list2 T1 f l1 l2 = bfold_right_list2 T1 f l2 l1).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: nnormalize; napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize; napply refl_eq
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: nnormalize; napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize;
+ nrewrite > (H1 ll2);
+ nrewrite > (H hh1 hh2);
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma bfoldrightlist2_to_eq :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.(f x y = true â x = y)) â
+ (âl1,l2:list T1.
+ (bfold_right_list2 T1 f l1 l2 = true â l1 = l2)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: #H1; napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize; #H1;
+ ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: nnormalize; #H2;
+ ndestruct (*napply (bool_destruct ⦠H2)*)
+ ##| ##2: #hh2; #ll2; #H2;
+ nchange in H2:(%) with (((f hh1 hh2)â(bfold_right_list2 T f ll1 ll2)) = true);
+ nrewrite > (H hh1 hh2 (andb_true_true_l ⦠H2));
+ nrewrite > (H1 ll2 (andb_true_true_r ⦠H2));
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma eq_to_bfoldrightlist2 :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.(x = y â f x y = true)) â
+ (âl1,l2:list T1.
+ (l1 = l2 â bfold_right_list2 T1 f l1 l2 = true)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: #H1; nnormalize; napply refl_eq
+ ##| ##2: #hh2; #ll2; #H1;
+ (* !!! ndestruct: assert false *)
+ nelim (list_destruct_nil_cons ??? H1)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #H2;
+ (* !!! ndestruct: assert false *)
+ nelim (list_destruct_cons_nil ??? H2)
+ ##| ##2: #hh2; #ll2; #H2; nnormalize;
+ nrewrite > (list_destruct_1 ⦠H2);
+ nrewrite > (H hh2 hh2 (refl_eq â¦));
+ nnormalize;
+ nrewrite > (H1 ll2 (list_destruct_2 ⦠H2));
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma bfoldrightlist2_to_lenlist :
+âT.âf:T â T â bool.
+ (âl1,l2:list T.bfold_right_list2 T f l1 l2 = true â len_list T l1 = len_list T l2).
+ #T; #f; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: nnormalize; #H; napply refl_eq
+ ##| ##2: nnormalize; #hh; #tt; #H;
+ ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+ ##| ##2: #hh; #tt; #H; #l2; ncases l2;
+ ##[ ##1: nnormalize; #H1;
+ ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##| ##2: #hh1; #tt1; #H1; nnormalize;
+ nrewrite > (H tt1 ?);
+ ##[ ##1: napply refl_eq
+ ##| ##2: nchange in H1:(%) with ((? â (bfold_right_list2 T f tt tt1)) = true);
+ napply (andb_true_true_r ⦠H1)
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma decidable_list :
+âT.(âx,y:T.decidable (x = y)) â
+ (âx,y:list T.decidable (x = y)).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro1 (? = ?) (? â ?) (refl_eq â¦))
+ ##| ##2: #hh2; #tt2; nnormalize; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H1;
+ (* !!! ndestruct: assert false *)
+ napply (list_destruct_nil_cons T ⦠H1)
+ ##]
+ ##| ##2: #hh1; #tt1; #H1; #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (list_destruct_cons_nil T ⦠H2)
+ ##| ##2: #hh2; #tt2; nnormalize; napply (or2_elim (hh1 = hh2) (hh1 â hh2) ? (H â¦));
+ ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H3; napply (H2 (list_destruct_1 T ⦠H3))
+ ##| ##1: #H2; napply (or2_elim (tt1 = tt2) (tt1 â tt2) ? (H1 tt2));
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H4; napply (H3 (list_destruct_2 T ⦠H4))
+ ##| ##1: #H3; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H2; nrewrite > H3; napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma nbfoldrightlist2_to_neq :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.(f x y = false â x â y)) â
+ (âl1,l2:list T1.
+ (bfold_right_list2 T1 f l1 l2 = false â l1 â l2)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: nnormalize; #H1;
+ ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##| ##2: #hh2; #ll2; #H1; nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (list_destruct_nil_cons T ⦠H2)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #H2; nnormalize; #H3;
+ (* !!! ndestruct: assert false *)
+ napply (list_destruct_cons_nil T ⦠H3)
+ ##| ##2: #hh2; #ll2; #H2; nnormalize; #H3;
+ nchange in H2:(%) with (((f hh1 hh2)â(bfold_right_list2 T f ll1 ll2)) = false);
+ napply (H1 ll2 ? (list_destruct_2 T ⦠H3));
+ napply (or2_elim ??? (andb_false2 ⦠H2) );
+ ##[ ##1: #H4; napply (absurd (hh1 = hh2) â¦);
+ ##[ ##1: nrewrite > (list_destruct_1 T ⦠H3); napply refl_eq
+ ##| ##2: napply (H ⦠H4)
+ ##]
+ ##| ##2: #H4; napply H4
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma list_destruct :
+âT.(âx,y:T.decidable (x = y)) â
+ (âh1,h2:T.âl1,l2:list T.
+ (h1::l1) â (h2::l2) â h1 â h2 ⨠l1 â l2).
+ #T; #H; #h1; #h2; #l1; nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: #H1; napply (or2_intro1 (h1 â h2) ([] â []) â¦);
+ nnormalize; #H2; nrewrite > H2 in H1:(%);
+ nnormalize; #H1; napply (H1 (refl_eq â¦))
+ ##| ##2: #hh2; #ll2; #H1; napply (or2_intro2 (h1 â h2) ([] â (hh2::ll2)) â¦);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (list_destruct_nil_cons T ⦠H2)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #H2; napply (or2_intro2 (h1 â h2) ((hh1::ll1) â []) â¦);
+ nnormalize; #H3;
+ (* !!! ndestruct: assert false *)
+ napply (list_destruct_cons_nil T ⦠H3)
+ ##| ##2: #hh2; #ll2; #H2;
+ napply (or2_elim (h1 = h2) (h1 â h2) ? (H h1 h2) â¦);
+ ##[ ##2: #H3; napply (or2_intro1 (h1 â h2) ((hh1::ll1) â (hh2::ll2)) H3)
+ ##| ##1: #H3; napply (or2_intro2 (h1 â h2) ((hh1::ll1) â (hh2::ll2) â¦));
+ nrewrite > H3 in H2:(%); #H2;
+ nnormalize; #H4; nrewrite > (list_destruct_1 T ⦠H4) in H2:(%); #H2;
+ nrewrite > (list_destruct_2 T ⦠H4) in H2:(%); #H2;
+ napply (H2 (refl_eq â¦))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_nbfoldrightlist2 :
+âT:Type.âf:T â T â bool.
+ (âx,y:T.decidable (x = y)) â
+ (âx,y.(x â y â f x y = false)) â
+ (âl1,l2:list T.
+ (l1 â l2 â bfold_right_list2 T f l1 l2 = false)).
+ #T; #f; #H; #H1; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+ ##[ ##1: nnormalize; #H2; nelim (H2 (refl_eq â¦))
+ ##| ##2: #hh2; #ll2; nnormalize; #H2; napply refl_eq
+ ##]
+ ##| ##2: #hh1; #ll1; #H2; #l2; ncases l2;
+ ##[ ##1: nnormalize; #H3; napply refl_eq
+ ##| ##2: #hh2; #ll2; #H3;
+ nchange with (((f hh1 hh2)â(bfold_right_list2 T f ll1 ll2)) = false);
+ napply (or2_elim (hh1 â hh2) (ll1 â ll2) ? (list_destruct T H ⦠H3) â¦);
+ ##[ ##1: #H4; nrewrite > (H1 hh1 hh2 H4); nnormalize; napply refl_eq
+ ##| ##2: #H4; nrewrite > (H2 ll2 H4);
+ nrewrite > (symmetric_andbool (f hh1 hh2) false);
+ nnormalize; napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma isbemptylist_to_isemptylist : âT,l.isb_empty_list T l = true â is_empty_list T l.
+ #T; #l;
+ ncases l;
+ nnormalize;
+ ##[ ##1: #H; napply I
+ ##| ##2: #x; #l; #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma isnotbemptylist_to_isnotemptylist : âT,l.isnotb_empty_list T l = true â isnot_empty_list T l.
+ #T; #l;
+ ncases l;
+ nnormalize;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##| ##2: #x; #l; #H; napply I
+ ##]
+nqed.
+
+nlemma list_is_comparable : comparable â comparable.
+ #T; napply (mk_comparable (list T));
+ ##[ napply (nil ?)
+ ##| napply (λx.false)
+ ##| napply (bfold_right_list2 T (eqc T))
+ ##| napply (bfoldrightlist2_to_eq ⦠(eqc T));
+ napply (eqc_to_eq T)
+ ##| napply (eq_to_bfoldrightlist2 ⦠(eqc T));
+ napply (eq_to_eqc T)
+ ##| napply (nbfoldrightlist2_to_neq ⦠(eqc T));
+ napply (neqc_to_neq T)
+ ##| napply (neq_to_nbfoldrightlist2 ⦠(eqc T));
+ ##[ napply (decidable_c T)
+ ##| napply (neq_to_neqc T)
+ ##]
+ ##| napply decidable_list;
+ napply (decidable_c T)
+ ##| napply symmetric_bfoldrightlist2;
+ napply (symmetric_eqc T)
+ ##]
+nqed.
+
+unification hint 0 â S: comparable;
+ T â (carr S),
+ X â (list_is_comparable S)
+ (*********************************************) â¢
+ carr X â¡ list T.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/nat.ma b/helm/software/matita/contribs/ng_assembly2/common/nat.ma
new file mode 100755
index 000000000..df3a8990a
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/nat.ma
@@ -0,0 +1,388 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "num/bool_lemmas.ma".
+
+(* ******** *)
+(* NATURALI *)
+(* ******** *)
+
+ninductive nat : Type â
+ O : nat
+| S : nat â nat.
+
+(*
+interpretation "Natural numbers" 'N = nat.
+
+default "natural numbers" cic:/matita/common/nat/nat.ind.
+
+alias num (instance 0) = "natural number".
+*)
+
+nlet rec nat_it (T:Type) (f:T â T) (arg:T) (n:nat) on n â
+ match n with
+ [ O â arg
+ | S n' â nat_it T f (f arg) n'
+ ].
+
+ndefinition nat1 â S O.
+ndefinition nat2 â S nat1.
+ndefinition nat3 â S nat2.
+ndefinition nat4 â S nat3.
+ndefinition nat5 â S nat4.
+ndefinition nat6 â S nat5.
+ndefinition nat7 â S nat6.
+ndefinition nat8 â S nat7.
+ndefinition nat9 â S nat8.
+ndefinition nat10 â S nat9.
+ndefinition nat11 â S nat10.
+ndefinition nat12 â S nat11.
+ndefinition nat13 â S nat12.
+ndefinition nat14 â S nat13.
+ndefinition nat15 â S nat14.
+ndefinition nat16 â S nat15.
+ndefinition nat17 â S nat16.
+ndefinition nat18 â S nat17.
+ndefinition nat19 â S nat18.
+ndefinition nat20 â S nat19.
+ndefinition nat21 â S nat20.
+ndefinition nat22 â S nat21.
+ndefinition nat23 â S nat22.
+ndefinition nat24 â S nat23.
+ndefinition nat25 â S nat24.
+ndefinition nat26 â S nat25.
+ndefinition nat27 â S nat26.
+ndefinition nat28 â S nat27.
+ndefinition nat29 â S nat28.
+ndefinition nat30 â S nat29.
+ndefinition nat31 â S nat30.
+ndefinition nat32 â S nat31.
+ndefinition nat33 â S nat32.
+ndefinition nat34 â S nat33.
+ndefinition nat35 â S nat34.
+ndefinition nat36 â S nat35.
+ndefinition nat37 â S nat36.
+ndefinition nat38 â S nat37.
+ndefinition nat39 â S nat38.
+ndefinition nat40 â S nat39.
+ndefinition nat41 â S nat40.
+ndefinition nat42 â S nat41.
+ndefinition nat43 â S nat42.
+ndefinition nat44 â S nat43.
+ndefinition nat45 â S nat44.
+ndefinition nat46 â S nat45.
+ndefinition nat47 â S nat46.
+ndefinition nat48 â S nat47.
+ndefinition nat49 â S nat48.
+ndefinition nat50 â S nat49.
+ndefinition nat51 â S nat50.
+ndefinition nat52 â S nat51.
+ndefinition nat53 â S nat52.
+ndefinition nat54 â S nat53.
+ndefinition nat55 â S nat54.
+ndefinition nat56 â S nat55.
+ndefinition nat57 â S nat56.
+ndefinition nat58 â S nat57.
+ndefinition nat59 â S nat58.
+ndefinition nat60 â S nat59.
+ndefinition nat61 â S nat60.
+ndefinition nat62 â S nat61.
+ndefinition nat63 â S nat62.
+ndefinition nat64 â S nat63.
+ndefinition nat65 â S nat64.
+ndefinition nat66 â S nat65.
+ndefinition nat67 â S nat66.
+ndefinition nat68 â S nat67.
+ndefinition nat69 â S nat68.
+ndefinition nat70 â S nat69.
+ndefinition nat71 â S nat70.
+ndefinition nat72 â S nat71.
+ndefinition nat73 â S nat72.
+ndefinition nat74 â S nat73.
+ndefinition nat75 â S nat74.
+ndefinition nat76 â S nat75.
+ndefinition nat77 â S nat76.
+ndefinition nat78 â S nat77.
+ndefinition nat79 â S nat78.
+ndefinition nat80 â S nat79.
+ndefinition nat81 â S nat80.
+ndefinition nat82 â S nat81.
+ndefinition nat83 â S nat82.
+ndefinition nat84 â S nat83.
+ndefinition nat85 â S nat84.
+ndefinition nat86 â S nat85.
+ndefinition nat87 â S nat86.
+ndefinition nat88 â S nat87.
+ndefinition nat89 â S nat88.
+ndefinition nat90 â S nat89.
+ndefinition nat91 â S nat90.
+ndefinition nat92 â S nat91.
+ndefinition nat93 â S nat92.
+ndefinition nat94 â S nat93.
+ndefinition nat95 â S nat94.
+ndefinition nat96 â S nat95.
+ndefinition nat97 â S nat96.
+ndefinition nat98 â S nat97.
+ndefinition nat99 â S nat98.
+ndefinition nat100 â S nat99.
+
+nlet rec eq_nat (n1,n2:nat) on n1 â
+ match n1 with
+ [ O â match n2 with [ O â true | S _ â false ]
+ | S n1' â match n2 with [ O â false | S n2' â eq_nat n1' n2' ]
+ ].
+
+nlet rec le_nat n m â
+ match n with
+ [ O â true
+ | (S p) â match m with
+ [ O â false | (S q) â le_nat p q ]
+ ].
+
+interpretation "natural 'less or equal to'" 'leq x y = (le_nat x y).
+
+ndefinition lt_nat â λn1,n2:nat.(le_nat n1 n2) â (â (eq_nat n1 n2)).
+
+interpretation "natural 'less than'" 'lt x y = (lt_nat x y).
+
+ndefinition ge_nat â λn1,n2:nat.(â (le_nat n1 n2)) â (eq_nat n1 n2).
+
+interpretation "natural 'greater or equal to'" 'geq x y = (ge_nat x y).
+
+ndefinition gt_nat â λn1,n2:nat.â (le_nat n1 n2).
+
+interpretation "natural 'greater than'" 'gt x y = (gt_nat x y).
+
+nlet rec plus (n1,n2:nat) on n1 â
+ match n1 with
+ [ O â n2
+ | (S n1') â S (plus n1' n2) ].
+
+interpretation "natural plus" 'plus x y = (plus x y).
+
+nlet rec times (n1,n2:nat) on n1 â
+ match n1 with
+ [ O â O
+ | (S n1') â n2 + (times n1' n2) ].
+
+interpretation "natural times" 'times x y = (times x y).
+
+nlet rec minus n m â
+ match n with
+ [ O â O
+ | (S p) â
+ match m with
+ [O â (S p)
+ | (S q) â minus p q ]].
+
+interpretation "natural minus" 'minus x y = (minus x y).
+
+nlet rec div_aux p m n : nat â
+match (le_nat m n) with
+[ true â O
+| false â
+ match p with
+ [ O â O
+ | (S q) â S (div_aux q (m-(S n)) n)]].
+
+ndefinition div : nat â nat â nat â
+λn,m.match m with
+ [ O â S n
+ | (S p) â div_aux n n p].
+
+interpretation "natural divide" 'divide x y = (div x y).
+
+ndefinition pred â λn.match n with [ O â O | S n â n ].
+
+ndefinition nat128 â nat64 + nat64.
+ndefinition nat256 â nat128 + nat128.
+ndefinition nat512 â nat256 + nat256.
+ndefinition nat1024 â nat512 + nat512.
+ndefinition nat2048 â nat1024 + nat1024.
+ndefinition nat4096 â nat2048 + nat2048.
+ndefinition nat8192 â nat4096 + nat4096.
+ndefinition nat16384 â nat8192 + nat8192.
+ndefinition nat32768 â nat16384 + nat16384.
+ndefinition nat65536 â nat32768 + nat32768.
+
+nlemma nat_destruct_S_S : ân1,n2:nat.S n1 = S n2 â n1 = n2.
+ #n1; #n2; #H;
+ nchange with (match S n2 with [ O â False | S a â n1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma nat_destruct_0_S : ân:nat.O = S n â False.
+ #n; #H;
+ nchange with (match S n with [ O â True | S a â False ]);
+ nrewrite < H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma nat_destruct_S_0 : ân:nat.S n = O â False.
+ #n; #H;
+ nchange with (match S n with [ O â True | S a â False ]);
+ nrewrite > H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma eq_to_eqnat : ân1,n2:nat.n1 = n2 â eq_nat n1 n2 = true.
+ #n1;
+ nelim n1;
+ ##[ ##1: #n2;
+ nelim n2;
+ nnormalize;
+ ##[ ##1: #H; napply refl_eq
+ ##| ##2: #n3; #H; #H1; ndestruct (*nelim (nat_destruct_0_S ? H1)*)
+ ##]
+ ##| ##2: #n2; #H; #n3; #H1;
+ ncases n3 in H1:(%) ⢠%;
+ nnormalize;
+ ##[ ##1: #H1; ndestruct (*nelim (nat_destruct_S_0 ? H1)*)
+ ##| ##2: #n4; #H1;
+ napply (H n4 (nat_destruct_S_S ⦠H1))
+ ##]
+ ##]
+nqed.
+
+nlemma neqnat_to_neq : ân1,n2:nat.(eq_nat n1 n2 = false â n1 â n2).
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_nat n1 n2 = true) â¦);
+ ##[ ##1: napply (eq_to_eqnat n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue ⦠H)
+ ##]
+nqed.
+
+nlemma eqnat_to_eq : ân1,n2:nat.(eq_nat n1 n2 = true â n1 = n2).
+ #n1;
+ nelim n1;
+ ##[ ##1: #n2;
+ nelim n2;
+ nnormalize;
+ ##[ ##1: #H; napply refl_eq
+ ##| ##2: #n3; #H; #H1; ndestruct (*napply (bool_destruct ⦠(O = S n3) H1)*)
+ ##]
+ ##| ##2: #n2; #H; #n3; #H1;
+ ncases n3 in H1:(%) ⢠%;
+ nnormalize;
+ ##[ ##1: #H1; ndestruct (*napply (bool_destruct ⦠(S n2 = O) H1)*)
+ ##| ##2: #n4; #H1;
+ nrewrite > (H n4 H1);
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqnat : ân1,n2:nat.n1 â n2 â eq_nat n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_nat n1 n2));
+ napply (not_to_not (eq_nat n1 n2 = true) (n1 = n2) ? H);
+ napply (eqnat_to_eq n1 n2).
+nqed.
+
+nlemma decidable_nat : âx,y:nat.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_nat x y = true) (eq_nat x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x â y) (eqnat_to_eq ⦠H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x â y) (neqnat_to_neq ⦠H))
+ ##]
+nqed.
+
+nlemma symmetric_eqnat : symmetricT nat bool eq_nat.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_nat n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqnat n1 n2 H);
+ napply (symmetric_eq ? (eq_nat n2 n1) false);
+ napply (neq_to_neqnat n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma Sn_p_n_to_S_npn : ân1,n2.(S n1) + n2 = S (n1 + n2).
+ #n1;
+ nelim n1;
+ ##[ ##1: nnormalize; #n2; napply refl_eq
+ ##| ##2: #n; #H; #n2; nrewrite > (H n2);
+ ncases n in H:(%) ⢠%;
+ ##[ ##1: nnormalize; #H; napply refl_eq
+ ##| ##2: #n3; nnormalize; #H; napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma n_p_Sn_to_S_npn : ân1,n2.n1 + (S n2) = S (n1 + n2).
+ #n1;
+ nelim n1;
+ ##[ ##1: nnormalize; #n2; napply refl_eq
+ ##| ##2: #n; #H; #n2;
+ nrewrite > (Sn_p_n_to_S_npn n (S n2));
+ nrewrite > (H n2);
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma Opn_to_n : ân.O + n = n.
+ #n; nnormalize; napply refl_eq.
+nqed.
+
+nlemma npO_to_n : ân.n + O = n.
+ #n;
+ nelim n;
+ ##[ ##1: nnormalize; napply refl_eq
+ ##| ##2: #n1; #H;
+ nrewrite > (Sn_p_n_to_S_npn n1 O);
+ nrewrite > H;
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma symmetric_plusnat : symmetricT nat nat plus.
+ #n1;
+ nelim n1;
+ ##[ ##1: #n2; nrewrite > (npO_to_n n2); nnormalize; napply refl_eq
+ ##| ##2: #n2; #H; #n3;
+ nrewrite > (Sn_p_n_to_S_npn n2 n3);
+ nrewrite > (n_p_Sn_to_S_npn n3 n2);
+ nrewrite > (H n3);
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma nat_is_comparable : comparable.
+ napply (mk_comparable nat);
+ ##[ napply O
+ ##| napply (λx.false)
+ ##| napply eq_nat
+ ##| napply eqnat_to_eq
+ ##| napply eq_to_eqnat
+ ##| napply neqnat_to_neq
+ ##| napply neq_to_neqnat
+ ##| napply decidable_nat
+ ##| napply symmetric_eqnat
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr nat_is_comparable â¡ nat.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/nelist.ma b/helm/software/matita/contribs/ng_assembly2/common/nelist.ma
new file mode 100644
index 000000000..4c00777cb
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/nelist.ma
@@ -0,0 +1,599 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/list.ma".
+
+(* ************** *)
+(* NON-EMPTY LIST *)
+(* ************** *)
+
+(* lista non vuota *)
+ninductive ne_list (A:Type) : Type â
+ | ne_nil: A â ne_list A
+ | ne_cons: A â ne_list A â ne_list A.
+
+(* append *)
+nlet rec ne_append (A:Type) (l1,l2:ne_list A) on l1 â
+ match l1 with
+ [ ne_nil hd â ne_cons A hd l2
+ | ne_cons hd tl â ne_cons A hd (ne_append A tl l2) ].
+
+notation "hvbox(hd break §§ tl)"
+ right associative with precedence 46
+ for @{'ne_cons $hd $tl}.
+
+(* \laquo \raquo *)
+notation "« list0 x sep ; break £ y break »"
+ non associative with precedence 90
+ for ${fold right @{'ne_nil $y } rec acc @{'ne_cons $x $acc}}.
+
+notation "hvbox(l1 break & l2)"
+ right associative with precedence 47
+ for @{'ne_append $l1 $l2 }.
+
+interpretation "ne_nil" 'ne_nil hd = (ne_nil ? hd).
+interpretation "ne_cons" 'ne_cons hd tl = (ne_cons ? hd tl).
+interpretation "ne_append" 'ne_append l1 l2 = (ne_append ? l1 l2).
+
+nlemma nelist_destruct_nil_nil : âT.âx1,x2:T.ne_nil T x1 = ne_nil T x2 â x1 = x2.
+ #T; #x1; #x2; #H;
+ nchange with (match ne_nil T x2 with [ ne_cons _ _ â False | ne_nil a â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma nelist_destruct_cons_cons_1 : âT.âx1,x2:T.ây1,y2:ne_list T.ne_cons T x1 y1 = ne_cons T x2 y2 â x1 = x2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match ne_cons T x2 y2 with [ ne_nil _ â False | ne_cons a _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma nelist_destruct_cons_cons_2 : âT.âx1,x2:T.ây1,y2:ne_list T.ne_cons T x1 y1 = ne_cons T x2 y2 â y1 = y2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match ne_cons T x2 y2 with [ ne_nil _ â False | ne_cons _ b â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma nelist_destruct_cons_nil : âT.âx1,x2:T.ây1:ne_list T.ne_cons T x1 y1 = ne_nil T x2 â False.
+ #T; #x1; #x2; #y1; #H;
+ nchange with (match ne_cons T x1 y1 with [ ne_nil _ â True | ne_cons a b â False ]);
+ nrewrite > H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma nelist_destruct_nil_cons : âT.âx1,x2:T.ây2:ne_list T.ne_nil T x1 = ne_cons T x2 y2 â False.
+ #T; #x1; #x2; #y2; #H;
+ nchange with (match ne_cons T x2 y2 with [ ne_nil _ â True | ne_cons a b â False ]);
+ nrewrite < H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma associative_nelist : âT.associative (ne_list T) (ne_append T).
+ #T; #x; #y; #z;
+ nelim x;
+ nnormalize;
+ ##[ ##1: #hh; napply refl_eq
+ ##| ##2: #hh; #tt; #H;
+ nrewrite > H;
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma necons_append_commute : âT:Type.âl1,l2:ne_list T.âa:T.(a §§ (l1 & l2)) = ((a §§ l1) & l2).
+ #T; #l1; #l2; #a;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma append_necons_commute : âT:Type.âa:T.âl,l1:ne_list T.(l & (a §§ l1)) = (l & «£a») & l1.
+ #T; #a; #l; #l1;
+ nrewrite > (associative_nelist T l «£a» l1);
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+(* listlen *)
+nlet rec len_neList (T:Type) (nl:ne_list T) on nl â
+ match nl with [ ne_nil _ â (S O) | ne_cons _ t â S (len_neList T t) ].
+
+(* reverse *)
+nlet rec reverse_neList (T:Type) (nl:ne_list T) on nl â
+ match nl with
+ [ ne_nil h â ne_nil T h
+ | ne_cons h t â (reverse_neList T t)&(ne_nil T h)
+ ].
+
+(* getFirst *)
+ndefinition get_first_neList â
+λT:Type.λnl:ne_list T.match nl with
+ [ ne_nil h â h
+ | ne_cons h _ â h ].
+
+(* getLast *)
+ndefinition get_last_neList â
+λT:Type.λnl:ne_list T.match reverse_neList T nl with
+ [ ne_nil h â h
+ | ne_cons h _ â h ].
+
+(* cutFirst *)
+ndefinition cut_first_neList â
+λT:Type.λnl:ne_list T.match nl with
+ [ ne_nil h â ne_nil T h
+ | ne_cons _ t â t ].
+
+(* cutLast *)
+ndefinition cut_last_neList â
+λT:Type.λnl:ne_list T.match reverse_neList T nl with
+ [ ne_nil h â ne_nil T h
+ | ne_cons _ t â reverse_neList T t ].
+
+(* apply f *)
+nlet rec apply_f_neList (T1,T2:Type) (nl:ne_list T1) (f:T1 â T2) on nl â
+match nl with
+ [ ne_nil h â ne_nil T2 (f h)
+ | ne_cons h t â ne_cons T2 (f h) (apply_f_neList T1 T2 t f) ].
+
+(* fold right *)
+nlet rec fold_right_neList (T1,T2:Type) (f:T1 â T2 â T2) (acc:T2) (nl:ne_list T1) on nl â
+ match nl with
+ [ ne_nil h â f h acc
+ | ne_cons h t â f h (fold_right_neList T1 T2 f acc t)
+ ].
+
+(* double fold right *)
+nlemma fold_right_neList2_aux1 :
+âT.âh,h',t'.len_neList T «£h» = len_neList T (h'§§t') â False.
+ #T; #h; #h'; #t';
+ nnormalize;
+ ncases t';
+ nnormalize;
+ ##[ ##1: #x; #H; ndestruct (*nelim (nat_destruct_0_S ? (nat_destruct_S_S ⦠H))*)
+ ##| ##2: #x; #l; #H; ndestruct (*nelim (nat_destruct_0_S ? (nat_destruct_S_S ⦠H))*)
+ ##]
+nqed.
+
+nlemma fold_right_neList2_aux2 :
+âT.âh,h',t.len_neList T (h§§t) = len_neList T «£h'» â False.
+ #T; #h; #h'; #t;
+ nnormalize;
+ ncases t;
+ nnormalize;
+ ##[ ##1: #x; #H; ndestruct (*nelim (nat_destruct_S_0 ? (nat_destruct_S_S ⦠H))*)
+ ##| ##2: #x; #l; #H; ndestruct (*nelim (nat_destruct_S_0 ? (nat_destruct_S_S ⦠H))*)
+ ##]
+nqed.
+
+nlemma fold_right_neList2_aux3 :
+âT.âh,h',t,t'.len_neList T (h§§t) = len_neList T (h'§§t') â len_neList T t = len_neList T t'.
+ #T; #h; #h'; #t; #t';
+ nelim t;
+ nelim t';
+ ##[ ##1: nnormalize; #x; #y; #H; napply refl_eq
+ ##| ##2: #a; #l'; #H; #x; #H1;
+ nchange in H1:(%) with ((S (len_neList T «£x»)) = (S (len_neList T (a§§l'))));
+ nrewrite > (nat_destruct_S_S ⦠H1);
+ napply refl_eq
+ ##| ##3: #x; #a; #l'; #H; #H1;
+ nchange in H1:(%) with ((S (len_neList T (a§§l')))= (S (len_neList T «£x»)));
+ nrewrite > (nat_destruct_S_S ⦠H1);
+ napply refl_eq
+ ##| ##4: #a; #l; #H; #a1; #l1; #H1; #H2;
+ nchange in H2:(%) with ((S (len_neList T (a1§§l1))) = (S (len_neList T (a§§l))));
+ nrewrite > (nat_destruct_S_S ⦠H2);
+ napply refl_eq
+ ##]
+nqed.
+
+nlet rec fold_right_neList2 (T1,T2:Type) (f:T1 â T1 â T2 â T2) (acc:T2) (l1:ne_list T1) on l1 â
+ match l1
+ return λl1.Î l2.len_neList T1 l1 = len_neList T1 l2 â T2
+ with
+ [ ne_nil h â λl2.match l2 return λl2.len_neList T1 «£h» = len_neList T1 l2 â T2 with
+ [ ne_nil h' â λp:len_neList T1 «£h» = len_neList T1 «£h'».
+ f h h' acc
+ | ne_cons h' t' â λp:len_neList T1 «£h» = len_neList T1 (h'§§t').
+ False_rect_Type0 ? (fold_right_neList2_aux1 T1 h h' t' p)
+ ]
+ | ne_cons h t â λl2.match l2 return λl2.len_neList T1 (h§§t) = len_neList T1 l2 â T2 with
+ [ ne_nil h' â λp:len_neList T1 (h§§t) = len_neList T1 «£h'».
+ False_rect_Type0 ? (fold_right_neList2_aux2 T1 h h' t p)
+ | ne_cons h' t' â λp:len_neList T1 (h§§t) = len_neList T1 (h'§§t').
+ f h h' (fold_right_neList2 T1 T2 f acc t t' (fold_right_neList2_aux3 T1 h h' t t' p))
+ ]
+ ].
+
+nlet rec bfold_right_neList2 (T1:Type) (f:T1 â T1 â bool) (l1,l2:ne_list T1) on l1 â
+ match l1 with
+ [ ne_nil h â match l2 with
+ [ ne_nil h' â f h h' | ne_cons h' t' â false ]
+ | ne_cons h t â match l2 with
+ [ ne_nil h' â false | ne_cons h' t' â (f h h') â (bfold_right_neList2 T1 f t t')
+ ]
+ ].
+
+(* nth elem *)
+nlet rec nth_neList (T:Type) (nl:ne_list T) (n:nat) on nl â
+ match nl with
+ [ ne_nil h â match n with
+ [ O â Some ? h | S _ â None ? ]
+ | ne_cons h t â match n with
+ [ O â Some ? h | S n' â nth_neList T t n' ]
+ ].
+
+(* abs elem *)
+ndefinition abs_neList_aux1 : âT:Type.âh:T.ân.((len_neList T («£h»)) > (S n)) = true â False.
+ #T; #h; #n; nnormalize; #H; ndestruct (*napply (bool_destruct ⦠H)*). nqed.
+
+ndefinition abs_neList_aux2 : âT:Type.âh:T.ât:ne_list T.ân.((len_neList T (h§§t)) > (S n) = true) â ((len_neList T t) > n) = true.
+ #T; #h; #t; #n; nnormalize; #H; napply H. nqed.
+
+nlet rec abs_neList (T:Type) (l:ne_list T) on l â
+ match l
+ return λl.Î n.(((len_neList T l) > n) = true) â T
+ with
+ [ ne_nil h â λn.
+ match n
+ return λn.(((len_neList T (ne_nil T h)) > n) = true) â T
+ with
+ [ O â λp:(((len_neList T (ne_nil T h)) > O) = true).h
+ | S n' â λp:(((len_neList T (ne_nil T h)) > (S n')) = true).
+ False_rect_Type0 ? (abs_neList_aux1 T h n' p)
+ ]
+ | ne_cons h t â λn.
+ match n with
+ [ O â λp:(((len_neList T (ne_cons T h t)) > O) = true).h
+ | S n' â λp:(((len_neList T (ne_cons T h t)) > (S n')) = true).
+ abs_neList T t n' (abs_neList_aux2 T h t n' p)
+ ]
+ ].
+
+(* esempio: abs_neList ? « 1; 2; 3 £ 4 » 0 (refl_eq â¦) = 1. *)
+
+(* conversione *)
+nlet rec neList_to_list (T:Type) (nl:ne_list T) on nl : list T â
+ match nl with [ ne_nil h â [h] | ne_cons h t â [h]@(neList_to_list T t) ].
+
+nlemma symmetric_lennelist : âT.âl1,l2:ne_list T.len_neList T l1 = len_neList T l2 â len_neList T l2 = len_neList T l1.
+ #T; #l1;
+ nelim l1;
+ ##[ ##1: #h; #l2; ncases l2; nnormalize;
+ ##[ ##1: #H; #H1; napply refl_eq
+ ##| ##2: #h; #t; #H; nrewrite > H; napply refl_eq
+ ##]
+ ##| ##2: #h; #l2; ncases l2; nnormalize;
+ ##[ ##1: #h1; #H; #l; #H1; nrewrite < H1; napply refl_eq
+ ##| ##2: #h; #l; #H; #l3; #H1; nrewrite < H1; napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma symmetric_foldrightnelist2_aux :
+âT1,T2:Type.âf:T1 â T1 â T2 â T2.
+ (âx,y,z.f x y z = f y x z) â
+ (âacc:T2.âl1,l2:ne_list T1.
+ âH1:len_neList T1 l1 = len_neList T1 l2.âH2:len_neList T1 l2 = len_neList T1 l1.
+ fold_right_neList2 T1 T2 f acc l1 l2 H1 = fold_right_neList2 T1 T2 f acc l2 l1 H2).
+ #T1; #T2; #f; #H; #acc; #l1;
+ nelim l1;
+ ##[ ##1: #h; #l2; ncases l2;
+ ##[ ##1: #h1; nnormalize; #H1; #H2; nrewrite > (H h h1 acc); napply refl_eq
+ ##| ##2: #h1; #l; ncases l;
+ ##[ ##1: #h3; #H1; #H2;
+ nchange in H1:(%) with ((S O) = (S (S O)));
+ (* !!! ndestruct: si inceppa su un'ipotesi che non e' H1 *)
+ nelim (nat_destruct_0_S ? (nat_destruct_S_S ⦠H1))
+ ##| ##2: #h3; #l3; #H1; #H2;
+ nchange in H1:(%) with ((S O) = (S (S (len_neList ? l3))));
+ (* !!! ndestruct: si inceppa su un'ipotesi che non e' H1 *)
+ nelim (nat_destruct_0_S ? (nat_destruct_S_S ⦠H1))
+ ##]
+ ##]
+ ##| ##2: #h3; #l3; #H1; #l2; ncases l2;
+ ##[ ##1: #h4; ncases l3;
+ ##[ ##1: #h5; #H2; #H3;
+ nchange in H2:(%) with ((S (S O)) = (S O));
+ (* !!! ndestruct: si inceppa su un'ipotesi che non e' H1 *)
+ nelim (nat_destruct_S_0 ? (nat_destruct_S_S ⦠H2))
+ ##| ##2: #h5; #l5; #H2; #H3;
+ nchange in H2:(%) with ((S (S (len_neList ? l5))) = (S O));
+ (* !!! ndestruct: si inceppa su un'ipotesi che non e' H1 *)
+ nelim (nat_destruct_S_0 ? (nat_destruct_S_S ⦠H2))
+ ##]
+ ##| ##2: #h4; #l4; #H2; #H3;
+ nchange in H2:(%) with ((S (len_neList ? l3)) = (S (len_neList ? l4)));
+ nchange in H3:(%) with ((S (len_neList ? l4)) = (S (len_neList ? l3)));
+ nchange with ((f h3 h4 (fold_right_neList2 T1 T2 f acc l3 l4 (fold_right_neList2_aux3 T1 h3 h4 l3 l4 ?))) =
+ (f h4 h3 (fold_right_neList2 T1 T2 f acc l4 l3 (fold_right_neList2_aux3 T1 h4 h3 l4 l3 ?))));
+ nrewrite < (H1 l4 (fold_right_neList2_aux3 T1 h3 h4 l3 l4 H2) (fold_right_neList2_aux3 T1 h4 h3 l4 l3 H3));
+ nrewrite > (H h3 h4 (fold_right_neList2 T1 T2 f acc l3 l4 ?));
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma symmetric_foldrightnelist2 :
+âT1,T2:Type.âf:T1 â T1 â T2 â T2.
+ (âx,y,z.f x y z = f y x z) â
+ (âacc:T2.âl1,l2:ne_list T1.âH:len_neList T1 l1 = len_neList T1 l2.
+ fold_right_neList2 T1 T2 f acc l1 l2 H = fold_right_neList2 T1 T2 f acc l2 l1 (symmetric_lennelist T1 l1 l2 H)).
+ #T1; #T2; #f; #H; #acc; #l1; #l2; #H1;
+ nrewrite > (symmetric_foldrightnelist2_aux T1 T2 f H acc l1 l2 H1 (symmetric_lennelist T1 l1 l2 H1));
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_bfoldrightnelist2 :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.f x y = f y x) â
+ (âl1,l2:ne_list T1.
+ bfold_right_neList2 T1 f l1 l2 = bfold_right_neList2 T1 f l2 l1).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: #hh2; nnormalize; nrewrite > (H hh1 hh2); napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize; napply refl_eq
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #hh2; nnormalize; napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize;
+ nrewrite > (H1 ll2);
+ nrewrite > (H hh1 hh2);
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma bfoldrightnelist2_to_eq :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.(f x y = true â x = y)) â
+ (âl1,l2:ne_list T1.
+ (bfold_right_neList2 T1 f l1 l2 = true â l1 = l2)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: #hh2; #H1; nnormalize in H1:(%); nrewrite > (H hh1 hh2 H1); napply refl_eq
+ ##| ##2: #hh2; #ll2; nnormalize; #H1; ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #hh2; nnormalize; #H2; ndestruct (*napply (bool_destruct ⦠H2)*)
+ ##| ##2: #hh2; #ll2; #H2;
+ nchange in H2:(%) with (((f hh1 hh2)â(bfold_right_neList2 T f ll1 ll2)) = true);
+ nrewrite > (H hh1 hh2 (andb_true_true_l ⦠H2));
+ nrewrite > (H1 ll2 (andb_true_true_r ⦠H2));
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma eq_to_bfoldrightnelist2 :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.(x = y â f x y = true)) â
+ (âl1,l2:ne_list T1.
+ (l1 = l2 â bfold_right_neList2 T1 f l1 l2 = true)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: #hh2; #H1; nnormalize;
+ nrewrite > (H hh1 hh2 (nelist_destruct_nil_nil ⦠H1));
+ napply refl_eq
+ ##| ##2: #hh2; #ll2; #H1;
+ (* !!! ndestruct: assert false *)
+ nelim (nelist_destruct_nil_cons ???? H1)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #hh2; #H2;
+ (* !!! ndestruct: assert false *)
+ nelim (nelist_destruct_cons_nil ???? H2)
+ ##| ##2: #hh2; #ll2; #H2; nnormalize;
+ nrewrite > (nelist_destruct_cons_cons_1 ⦠H2);
+ nrewrite > (H hh2 hh2 (refl_eq â¦));
+ nnormalize;
+ nrewrite > (H1 ll2 (nelist_destruct_cons_cons_2 ⦠H2));
+ napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma bfoldrightnelist2_to_lennelist :
+âT.âf:T â T â bool.
+ (âl1,l2:ne_list T.bfold_right_neList2 T f l1 l2 = true â len_neList T l1 = len_neList T l2).
+ #T; #f; #l1;
+ nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: nnormalize; #hh2; #H; napply refl_eq
+ ##| ##2: nnormalize; #hh2; #tt2; #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+ ##| ##2: #hh1; #tt1; #H; #l2; ncases l2;
+ ##[ ##1: nnormalize; #hh2; #H1; ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##| ##2: #hh2; #tt2; #H1; nnormalize;
+ nrewrite > (H tt2 ?);
+ ##[ ##1: napply refl_eq
+ ##| ##2: nchange in H1:(%) with ((? â (bfold_right_neList2 T f tt1 tt2)) = true);
+ napply (andb_true_true_r ⦠H1)
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma decidable_nelist :
+âT.(âx,y:T.decidable (x = y)) â
+ (âx,y:ne_list T.decidable (x = y)).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #hh1; #y; ncases y;
+ ##[ ##1: #hh2; nnormalize; napply (or2_elim (hh1 = hh2) (hh1 â hh2) ? (H hh1 hh2));
+ ##[ ##1: #H1; nrewrite > H1; napply (or2_intro1 (? = ?) (? â ?) (refl_eq â¦))
+ ##| ##2: #H1; napply (or2_intro2 (? = ?) (? â ?) ?); nnormalize;
+ #H2; napply (H1 (nelist_destruct_nil_nil T ⦠H2))
+ ##]
+ ##| ##2: #hh2; #tt2; nnormalize; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H1;
+ (* !!! ndestruct: assert false *)
+ napply (nelist_destruct_nil_cons T ⦠H1)
+ ##]
+ ##| ##2: #hh1; #tt1; #H1; #y; ncases y;
+ ##[ ##1: #hh1; nnormalize; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (nelist_destruct_cons_nil T ⦠H2)
+ ##| ##2: #hh2; #tt2; nnormalize; napply (or2_elim (hh1 = hh2) (hh1 â hh2) ? (H â¦));
+ ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H3; napply (H2 (nelist_destruct_cons_cons_1 T ⦠H3))
+ ##| ##1: #H2; napply (or2_elim (tt1 = tt2) (tt1 â tt2) ? (H1 tt2));
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H4; napply (H3 (nelist_destruct_cons_cons_2 T ⦠H4))
+ ##| ##1: #H3; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H2; nrewrite > H3; napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma nbfoldrightnelist2_to_neq :
+âT1:Type.âf:T1 â T1 â bool.
+ (âx,y.(f x y = false â x â y)) â
+ (âl1,l2:ne_list T1.
+ (bfold_right_neList2 T1 f l1 l2 = false â l1 â l2)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: #hh2; nnormalize; #H1; #H2; napply (H hh1 hh2 H1 (nelist_destruct_nil_nil T ⦠H2))
+ ##| ##2: #hh2; #ll2; #H1; nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (nelist_destruct_nil_cons T ⦠H2)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #hh2; #H2; nnormalize; #H3;
+ (* !!! ndestruct: assert false *)
+ napply (nelist_destruct_cons_nil T ⦠H3)
+ ##| ##2: #hh2; #ll2; #H2; nnormalize; #H3;
+ nchange in H2:(%) with (((f hh1 hh2)â(bfold_right_neList2 T f ll1 ll2)) = false);
+ napply (H1 ll2 ? (nelist_destruct_cons_cons_2 T ⦠H3));
+ napply (or2_elim ??? (andb_false2 ⦠H2) );
+ ##[ ##1: #H4; napply (absurd (hh1 = hh2) â¦);
+ ##[ ##1: nrewrite > (nelist_destruct_cons_cons_1 T ⦠H3); napply refl_eq
+ ##| ##2: napply (H ⦠H4)
+ ##]
+ ##| ##2: #H4; napply H4
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma nelist_destruct :
+âT.(âx,y:T.decidable (x = y)) â
+ (âh1,h2:T.âl1,l2:ne_list T.
+ (h1§§l1) â (h2§§l2) â h1 â h2 ⨠l1 â l2).
+ #T; #H; #h1; #h2; #l1; nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: #hh2; #H1; napply (or2_elim (h1 = h2) (h1 â h2) ? (H â¦) â¦);
+ ##[ ##2: #H2; napply (or2_intro1 (h1 â h2) («£hh1» â «£hh2») H2)
+ ##| ##1: #H2; nrewrite > H2 in H1:(%); #H1;
+ napply (or2_elim (hh1 = hh2) (hh1 â hh2) ? (H â¦) â¦);
+ ##[ ##2: #H3; napply (or2_intro2 (h2 â h2) («£hh1» â «£hh2») ?);
+ nnormalize; #H4; napply (H3 (nelist_destruct_nil_nil T ⦠H4))
+ ##| ##1: #H3; nrewrite > H3 in H1:(%); #H1; nelim (H1 (refl_eq â¦))
+ ##]
+ ##]
+ ##| ##2: #hh2; #ll2; #H1; napply (or2_intro2 (h1 â h2) («£hh1» â (hh2§§ll2)) â¦);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (nelist_destruct_nil_cons T ⦠H2)
+ ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+ ##[ ##1: #hh2; #H2; napply (or2_intro2 (h1 â h2) ((hh1§§ll1) â «£hh2») â¦);
+ nnormalize; #H3;
+ (* !!! ndestruct: assert false *)
+ napply (nelist_destruct_cons_nil T ⦠H3)
+ ##| ##2: #hh2; #ll2; #H2;
+ napply (or2_elim (h1 = h2) (h1 â h2) ? (H h1 h2) â¦);
+ ##[ ##2: #H3; napply (or2_intro1 (h1 â h2) ((hh1§§ll1) â (hh2§§ll2)) H3)
+ ##| ##1: #H3; napply (or2_intro2 (h1 â h2) ((hh1§§ll1) â (hh2§§ll2) â¦));
+ nrewrite > H3 in H2:(%); #H2;
+ nnormalize; #H4; nrewrite > (nelist_destruct_cons_cons_1 T ⦠H4) in H2:(%); #H2;
+ nrewrite > (nelist_destruct_cons_cons_2 T ⦠H4) in H2:(%); #H2;
+ napply (H2 (refl_eq â¦))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_nbfoldrightnelist2 :
+âT:Type.âf:T â T â bool.
+ (âx,y:T.decidable (x = y)) â
+ (âx,y.(x â y â f x y = false)) â
+ (âl1,l2:ne_list T.
+ (l1 â l2 â bfold_right_neList2 T f l1 l2 = false)).
+ #T; #f; #H; #H1; #l1;
+ nelim l1;
+ ##[ ##1: #hh1; #l2; ncases l2;
+ ##[ ##1: #hh2; nnormalize; #H2; napply (H1 hh1 hh2 ?);
+ nnormalize; #H3; nrewrite > H3 in H2:(%); #H2; napply (H2 (refl_eq â¦))
+ ##| ##2: #hh2; #ll2; nnormalize; #H2; napply refl_eq
+ ##]
+ ##| ##2: #hh1; #ll1; #H2; #l2; ncases l2;
+ ##[ ##1: #hh2; nnormalize; #H3; napply refl_eq
+ ##| ##2: #hh2; #ll2; #H3;
+ nchange with (((f hh1 hh2)â(bfold_right_neList2 T f ll1 ll2)) = false);
+ napply (or2_elim (hh1 â hh2) (ll1 â ll2) ? (nelist_destruct T H ⦠H3) â¦);
+ ##[ ##1: #H4; nrewrite > (H1 hh1 hh2 H4); nnormalize; napply refl_eq
+ ##| ##2: #H4; nrewrite > (H2 ll2 H4);
+ nrewrite > (symmetric_andbool (f hh1 hh2) false);
+ nnormalize; napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma nelist_is_comparable : comparable â comparable.
+ #T; napply (mk_comparable (ne_list T));
+ ##[ napply (ne_nil ? (zeroc T))
+ ##| napply (λx.false)
+ ##| napply (bfold_right_neList2 T (eqc T))
+ ##| napply (bfoldrightnelist2_to_eq ⦠(eqc T));
+ napply (eqc_to_eq T)
+ ##| napply (eq_to_bfoldrightnelist2 ⦠(eqc T));
+ napply (eq_to_eqc T)
+ ##| napply (nbfoldrightnelist2_to_neq ⦠(eqc T));
+ napply (neqc_to_neq T)
+ ##| napply (neq_to_nbfoldrightnelist2 ⦠(eqc T));
+ ##[ napply (decidable_c T)
+ ##| napply (neq_to_neqc T)
+ ##]
+ ##| napply decidable_nelist;
+ napply (decidable_c T)
+ ##| napply symmetric_bfoldrightnelist2;
+ napply (symmetric_eqc T)
+ ##]
+nqed.
+
+unification hint 0 â S: comparable;
+ T â (carr S),
+ X â (nelist_is_comparable S)
+ (*********************************************) â¢
+ carr X â¡ ne_list T.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/option.ma b/helm/software/matita/contribs/ng_assembly2/common/option.ma
new file mode 100644
index 000000000..51fc5ce13
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/option.ma
@@ -0,0 +1,202 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "common/option_base.ma".
+include "num/bool_lemmas.ma".
+
+(* ****** *)
+(* OPTION *)
+(* ****** *)
+
+nlemma option_destruct_some_some : âT.âx1,x2:T.Some T x1 = Some T x2 â x1 = x2.
+ #T; #x1; #x2; #H;
+ nchange with (match Some T x2 with [ None â False | Some a â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma option_destruct_some_none : âT.âx:T.Some T x = None T â False.
+ #T; #x; #H;
+ nchange with (match Some T x with [ None â True | Some a â False ]);
+ nrewrite > H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma option_destruct_none_some : âT.âx:T.None T = Some T x â False.
+ #T; #x; #H;
+ nchange with (match Some T x with [ None â True | Some a â False ]);
+ nrewrite < H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma symmetric_eqoption :
+âT:Type.âf:T â T â bool.
+ (symmetricT T bool f) â
+ (âop1,op2:option T.
+ (eq_option T f op1 op2 = eq_option T f op2 op1)).
+ #T; #f; #H;
+ #op1; #op2; nelim op1; nelim op2;
+ nnormalize;
+ ##[ ##1: napply refl_eq
+ ##| ##2,3: #H; napply refl_eq
+ ##| ##4: #a; #a0;
+ nrewrite > (H a0 a);
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqoption :
+âT.âf:T â T â bool.
+ (âx1,x2:T.x1 = x2 â f x1 x2 = true) â
+ (âop1,op2:option T.
+ (op1 = op2 â eq_option T f op1 op2 = true)).
+ #T; #f; #H;
+ #op1; #op2; nelim op1; nelim op2;
+ nnormalize;
+ ##[ ##1: #H1; napply refl_eq
+ ##| ##2: #a; #H1;
+ (* !!! ndestruct: assert false *)
+ nelim (option_destruct_none_some ?? H1)
+ ##| ##3: #a; #H1;
+ (* !!! ndestruct: assert false *)
+ nelim (option_destruct_some_none ?? H1)
+ ##| ##4: #a; #a0; #H1;
+ nrewrite > (H ⦠(option_destruct_some_some ⦠H1));
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma eqoption_to_eq :
+âT.âf:T â T â bool.
+ (âx1,x2:T.f x1 x2 = true â x1 = x2) â
+ (âop1,op2:option T.
+ (eq_option T f op1 op2 = true â op1 = op2)).
+ #T; #f; #H;
+ #op1; #op2; nelim op1; nelim op2;
+ nnormalize;
+ ##[ ##1: #H1; napply refl_eq
+ ##| ##2,3: #a; #H1; ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##| ##4: #a; #a0; #H1;
+ nrewrite > (H ⦠H1);
+ napply refl_eq
+ ##]
+nqed.
+
+nlemma decidable_option :
+âT.(Î x,y:T.decidable (x = y)) â
+ (âx,y:option T.decidable (x = y)).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro1 (? = ?) (? â ?) (refl_eq â¦))
+ ##| ##2: #yy; nnormalize; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H1;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_none_some T ⦠H1)
+ ##]
+ ##| ##2: #xx; #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_some_none T ⦠H2)
+ ##| ##2: #yy; nnormalize; napply (or2_elim (xx = yy) (xx â yy) ? (H â¦));
+ ##[ ##2: #H1; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H2;
+ napply (H1 (option_destruct_some_some T ⦠H2))
+ ##| ##1: #H1; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H1; napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqoption :
+âT.âf:T â T â bool.
+ (âx1,x2:T.x1 â x2 â f x1 x2 = false) â
+ (âop1,op2:option T.
+ (op1 â op2 â eq_option T f op1 op2 = false)).
+ #T; #f; #H; #op1; nelim op1;
+ ##[ ##1: #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1; nelim (H1 (refl_eq â¦))
+ ##| ##2: #yy; nnormalize; #H1; napply refl_eq
+ ##]
+ ##| ##2: #xx; #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1; napply refl_eq
+ ##| ##2: #yy; nnormalize; #H1; napply (H xx yy â¦);
+ nnormalize; #H2; nrewrite > H2 in H1:(%); #H1;
+ napply (H1 (refl_eq â¦))
+ ##]
+ ##]
+nqed.
+
+nlemma neqoption_to_neq :
+âT.âf:T â T â bool.
+ (âx1,x2:T.f x1 x2 = false â x1 â x2) â
+ (âop1,op2:option T.
+ (eq_option T f op1 op2 = false â op1 â op2)).
+ #T; #f; #H; #op1; nelim op1;
+ ##[ ##1: #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1;
+ ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##| ##2: #yy; nnormalize; #H1; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_none_some T ⦠H2)
+ ##]
+ ##| ##2: #xx; #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_some_none T ⦠H2)
+ ##| ##2: #yy; nnormalize; #H1; #H2; napply (H xx yy H1 ?);
+ napply (option_destruct_some_some T ⦠H2)
+ ##]
+ ##]
+nqed.
+
+nlemma option_is_comparable :
+ comparable â comparable.
+ #T; napply (mk_comparable (option T));
+ ##[ napply (None ?)
+ ##| napply (λx.false)
+ ##| napply (eq_option ⦠(eqc T))
+ ##| napply (eqoption_to_eq ⦠(eqc T));
+ napply (eqc_to_eq T)
+ ##| napply (eq_to_eqoption ⦠(eqc T));
+ napply (eq_to_eqc T)
+ ##| napply (neqoption_to_neq ⦠(eqc T));
+ napply (neqc_to_neq T)
+ ##| napply (neq_to_neqoption ⦠(eqc T));
+ napply (neq_to_neqc T)
+ ##| napply decidable_option;
+ napply (decidable_c T)
+ ##| napply symmetric_eqoption;
+ napply (symmetric_eqc T)
+ ##]
+nqed.
+
+unification hint 0 â S: comparable;
+ T â (carr S),
+ X â (option_is_comparable S)
+ (*********************************************) â¢
+ carr X â¡ option T.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/option_base.ma b/helm/software/matita/contribs/ng_assembly2/common/option_base.ma
new file mode 100644
index 000000000..62da65900
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/option_base.ma
@@ -0,0 +1,43 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/bool.ma".
+
+(* ****** *)
+(* OPTION *)
+(* ****** *)
+
+ninductive option (A:Type) : Type â
+ None : option A
+| Some : A â option A.
+
+ndefinition eq_option â
+λT.λf:T â T â bool.λop1,op2:option T.
+ match op1 with
+ [ None â match op2 with [ None â true | Some _ â false ]
+ | Some x1 â match op2 with [ None â false | Some x2 â f x1 x2 ]
+ ].
+
+(* option map = match ... with [ None â None ? | Some .. â .. ] *)
+ndefinition opt_map â
+λT1,T2:Type.λt:option T1.λf:T1 â option T2.
+ match t with [ None â None ? | Some x â (f x) ].
diff --git a/helm/software/matita/contribs/ng_assembly2/common/prod.ma b/helm/software/matita/contribs/ng_assembly2/common/prod.ma
new file mode 100644
index 000000000..38852d1a9
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/prod.ma
@@ -0,0 +1,1118 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "common/prod_base.ma".
+include "num/bool_lemmas.ma".
+
+(* ********* *)
+(* TUPLE x 2 *)
+(* ********* *)
+
+nlemma pair_destruct_1 :
+âT1,T2.âx1,x2:T1.ây1,y2:T2.
+ pair T1 T2 x1 y1 = pair T1 T2 x2 y2 â x1 = x2.
+ #T1; #T2; #x1; #x2; #y1; #y2; #H;
+ nchange with (match pair T1 T2 x2 y2 with [ pair a _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma pair_destruct_2 :
+âT1,T2.âx1,x2:T1.ây1,y2:T2.
+ pair T1 T2 x1 y1 = pair T1 T2 x2 y2 â y1 = y2.
+ #T1; #T2; #x1; #x2; #y1; #y2; #H;
+ nchange with (match pair T1 T2 x2 y2 with [ pair _ b â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqpair :
+âT1,T2:Type.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.
+ (symmetricT T1 bool f1) â
+ (symmetricT T2 bool f2) â
+ (âp1,p2:ProdT T1 T2.
+ (eq_pair T1 T2 f1 f2 p1 p2 = eq_pair T1 T2 f1 f2 p2 p1)).
+ #T1; #T2; #f1; #f2; #H; #H1;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqpair :
+âT1,T2.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.
+ (âx,y:T1.x = y â f1 x y = true) â
+ (âx,y:T2.x = y â f2 x y = true) â
+ (âp1,p2:ProdT T1 T2.
+ (p1 = p2 â eq_pair T1 T2 f1 f2 p1 p2 = true)).
+ #T1; #T2; #f1; #f2; #H1; #H2;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2; #H;
+ nnormalize;
+ nrewrite > (H1 ⦠(pair_destruct_1 ⦠H));
+ nnormalize;
+ nrewrite > (H2 ⦠(pair_destruct_2 ⦠H));
+ napply refl_eq.
+nqed.
+
+nlemma eqpair_to_eq :
+âT1,T2.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.
+ (âx,y:T1.f1 x y = true â x = y) â
+ (âx,y:T2.f2 x y = true â x = y) â
+ (âp1,p2:ProdT T1 T2.
+ (eq_pair T1 T2 f1 f2 p1 p2 = true â p1 = p2)).
+ #T1; #T2; #f1; #f2; #H1; #H2;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2; #H;
+ nnormalize in H:(%);
+ nletin K â (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ #H3;
+ ##[ ##2: ndestruct (*napply (bool_destruct ⦠H3)*) ##]
+ #H4;
+ nrewrite > (H4 (refl_eq â¦));
+ nrewrite > (H2 y1 y2 H3);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_pair :
+âT1,T2.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:ProdT T1 T2.decidable (x = y)).
+ #T1; #T2; #H; #H1;
+ #x; nelim x; #xx1; #xx2;
+ #y; nelim y; #yy1; #yy2;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 â yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H3; napply (H2 (pair_destruct_1 T1 T2 ⦠H3))
+ ##| ##1: #H2; napply (or2_elim (xx2 = yy2) (xx2 â yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H4; napply (H3 (pair_destruct_2 T1 T2 ⦠H4))
+ ##| ##1: #H3; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H2; nrewrite > H3; napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma neqpair_to_neq :
+âT1,T2.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.
+ (âx,y:T1.f1 x y = false â x â y) â
+ (âx,y:T2.f2 x y = false â x â y) â
+ (âp1,p2:ProdT T1 T2.
+ (eq_pair T1 T2 f1 f2 p1 p2 = false â p1 â p2)).
+ #T1; #T2; #f1; #f2; #H1; #H2;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2;
+ nchange with ((((f1 x1 x2) â (f2 y1 y2)) = false) â ?); #H;
+ nnormalize; #H3;
+ napply (or2_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ? (andb_false2 ⦠H) ?);
+ ##[ ##1: #H4; napply (H1 x1 x2 H4); napply (pair_destruct_1 T1 T2 ⦠H3)
+ ##| ##2: #H4; napply (H2 y1 y2 H4); napply (pair_destruct_2 T1 T2 ⦠H3)
+ ##]
+nqed.
+
+nlemma pair_destruct :
+âT1,T2.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx1,x2:T1.ây1,y2:T2.
+ (pair T1 T2 x1 y1) â (pair T1 T2 x2 y2) â x1 â x2 ⨠y1 â y2).
+ #T1; #T2; #H1; #H2; #x1; #x2; #y1; #y2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 â x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H3; napply (or2_intro1 ⦠H3)
+ ##| ##1: #H3; napply (or2_elim (y1 = y2) (y1 â y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H4; napply (or2_intro2 ⦠H4)
+ ##| ##1: #H4; nrewrite > H3 in H:(%);
+ nrewrite > H4; #H; nelim (H (refl_eq â¦))
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqpair :
+âT1,T2.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T1.x â y â f1 x y = false) â
+ (âx,y:T2.x â y â f2 x y = false) â
+ (âp1,p2:ProdT T1 T2.
+ (p1 â p2 â eq_pair T1 T2 f1 f2 p1 p2 = false)).
+ #T1; #T2; #f1; #f2; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2; #H;
+ nchange with (((f1 x1 x2) â (f2 y1 y2)) = false);
+ napply (or2_elim (x1 â x2) (y1 â y2) ? (pair_destruct T1 T2 H1 H2 ⦠H) ?);
+ ##[ ##2: #H5; nrewrite > (H4 ⦠H5); nrewrite > (andb_false2_2 (f1 x1 x2)); napply refl_eq
+ ##| ##1: #H5; nrewrite > (H3 ⦠H5); nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma pair_is_comparable :
+ comparable â comparable â comparable.
+ #T1; #T2; napply (mk_comparable (ProdT T1 T2));
+ ##[ napply (pair ⦠(zeroc T1) (zeroc T2))
+ ##| napply (λx.false)
+ ##| napply (eq_pair ⦠(eqc T1) (eqc T2))
+ ##| napply (eqpair_to_eq ⦠(eqc T1) (eqc T2));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##]
+ ##| napply (eq_to_eqpair ⦠(eqc T1) (eqc T2));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##]
+ ##| napply (neqpair_to_neq ⦠(eqc T1) (eqc T2));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##]
+ ##| napply (neq_to_neqpair ⦠(eqc T1) (eqc T2));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##]
+ ##| napply decidable_pair;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##]
+ ##| napply symmetric_eqpair;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 â S1: comparable, S2: comparable;
+ T1 â (carr S1), T2 â (carr S2),
+ X â (pair_is_comparable S1 S2)
+ (*********************************************) â¢
+ carr X â¡ ProdT T1 T2.
+
+(* ********* *)
+(* TUPLE x 3 *)
+(* ********* *)
+
+nlemma triple_destruct_1 :
+âT1,T2,T3.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.
+ triple T1 T2 T3 x1 y1 z1 = triple T1 T2 T3 x2 y2 z2 â x1 = x2.
+ #T1; #T2; #T3; #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match triple T1 T2 T3 x2 y2 z2 with [ triple a _ _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma triple_destruct_2 :
+âT1,T2,T3.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.
+ triple T1 T2 T3 x1 y1 z1 = triple T1 T2 T3 x2 y2 z2 â y1 = y2.
+ #T1; #T2; #T3; #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match triple T1 T2 T3 x2 y2 z2 with [ triple _ b _ â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma triple_destruct_3 :
+âT1,T2,T3.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.
+ triple T1 T2 T3 x1 y1 z1 = triple T1 T2 T3 x2 y2 z2 â z1 = z2.
+ #T1; #T2; #T3; #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match triple T1 T2 T3 x2 y2 z2 with [ triple _ _ c â z1 = c ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqtriple :
+âT1,T2,T3:Type.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.
+ (symmetricT T1 bool f1) â
+ (symmetricT T2 bool f2) â
+ (symmetricT T3 bool f3) â
+ (âp1,p2:Prod3T T1 T2 T3.
+ (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = eq_triple T1 T2 T3 f1 f2 f3 p2 p1)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H; #H1; #H2;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2);
+ ncases (f2 y2 y1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H2 z1 z2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqtriple :
+âT1,T2,T3.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.
+ (âx1,x2:T1.x1 = x2 â f1 x1 x2 = true) â
+ (ây1,y2:T2.y1 = y2 â f2 y1 y2 = true) â
+ (âz1,z2:T3.z1 = z2 â f3 z1 z2 = true) â
+ (âp1,p2:Prod3T T1 T2 T3.
+ (p1 = p2 â eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = true)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2; #H;
+ nnormalize;
+ nrewrite > (H1 ⦠(triple_destruct_1 ⦠H));
+ nnormalize;
+ nrewrite > (H2 ⦠(triple_destruct_2 ⦠H));
+ nnormalize;
+ nrewrite > (H3 ⦠(triple_destruct_3 ⦠H));
+ napply refl_eq.
+nqed.
+
+nlemma eqtriple_to_eq :
+âT1,T2,T3.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.
+ (âx1,x2:T1.f1 x1 x2 = true â x1 = x2) â
+ (ây1,y2:T2.f2 y1 y2 = true â y1 = y2) â
+ (âz1,z2:T3.f3 z1 z2 = true â z1 = z2) â
+ (âp1,p2:Prod3T T1 T2 T3.
+ (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = true â p1 = p2)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2; #H;
+ nnormalize in H:(%);
+ nletin K â (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ ##[ ##2: #H4; ndestruct (*napply (bool_destruct ⦠H4)*) ##]
+ nletin K1 â (H2 y1 y2);
+ ncases (f2 y1 y2) in K1:(%) ⢠%;
+ nnormalize;
+ ##[ ##2: #H4; #H5; ndestruct (*napply (bool_destruct ⦠H5)*) ##]
+ #H4; #H5; #H6;
+ nrewrite > (H4 (refl_eq â¦));
+ nrewrite > (H6 (refl_eq â¦));
+ nrewrite > (H3 z1 z2 H5);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_triple :
+âT1,T2,T3.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:Prod3T T1 T2 T3.decidable (x = y)).
+ #T1; #T2; #T3; #H; #H1; #H2;
+ #x; nelim x; #xx1; #xx2; #xx3;
+ #y; nelim y; #yy1; #yy2; #yy3;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 â yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H4; napply (H3 (triple_destruct_1 T1 T2 T3 ⦠H4))
+ ##| ##1: #H3; napply (or2_elim (xx2 = yy2) (xx2 â yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H4; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H5; napply (H4 (triple_destruct_2 T1 T2 T3 ⦠H5))
+ ##| ##1: #H4; napply (or2_elim (xx3 = yy3) (xx3 â yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##2: #H5; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H6; napply (H5 (triple_destruct_3 T1 T2 T3 ⦠H6))
+ ##| ##1: #H5; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H3;
+ nrewrite > H4;
+ nrewrite > H5;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqtriple_to_neq :
+âT1,T2,T3.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.
+ (âx,y:T1.f1 x y = false â x â y) â
+ (âx,y:T2.f2 x y = false â x â y) â
+ (âx,y:T3.f3 x y = false â x â y) â
+ (âp1,p2:Prod3T T1 T2 T3.
+ (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = false â p1 â p2)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2;
+ nchange with ((((f1 x1 x2) â (f2 y1 y2) â (f3 z1 z2)) = false) â ?); #H;
+ nnormalize; #H4;
+ napply (or3_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ((f3 z1 z2) = false) ? (andb_false3 ⦠H) ?);
+ ##[ ##1: #H5; napply (H1 x1 x2 H5); napply (triple_destruct_1 T1 T2 T3 ⦠H4)
+ ##| ##2: #H5; napply (H2 y1 y2 H5); napply (triple_destruct_2 T1 T2 T3 ⦠H4)
+ ##| ##3: #H5; napply (H3 z1 z2 H5); napply (triple_destruct_3 T1 T2 T3 ⦠H4)
+ ##]
+nqed.
+
+nlemma triple_destruct :
+âT1,T2,T3.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.
+ (triple T1 T2 T3 x1 y1 z1) â (triple T1 T2 T3 x2 y2 z2) â
+ Or3 (x1 â x2) (y1 â y2) (z1 â z2)).
+ #T1; #T2; #T3; #H1; #H2; #H3; #x1; #x2; #y1; #y2; #z1; #z2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 â x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H4; napply (or3_intro1 ⦠H4)
+ ##| ##1: #H4; napply (or2_elim (y1 = y2) (y1 â y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H5; napply (or3_intro2 ⦠H5)
+ ##| ##1: #H5; napply (or2_elim (z1 = z2) (z1 â z2) ? (H3 z1 z2) ?);
+ ##[ ##2: #H6; napply (or3_intro3 ⦠H6)
+ ##| ##1: #H6; nrewrite > H4 in H:(%);
+ nrewrite > H5;
+ nrewrite > H6; #H; nelim (H (refl_eq â¦))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqtriple :
+âT1,T2,T3.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T1.x â y â f1 x y = false) â
+ (âx,y:T2.x â y â f2 x y = false) â
+ (âx,y:T3.x â y â f3 x y = false) â
+ (âp1,p2:Prod3T T1 T2 T3.
+ (p1 â p2 â eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = false)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3; #H4; #H5; #H6;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2; #H;
+ nchange with (((f1 x1 x2) â (f2 y1 y2) â (f3 z1 z2)) = false);
+ napply (or3_elim (x1 â x2) (y1 â y2) (z1 â z2) ? (triple_destruct T1 T2 T3 H1 H2 H3 ⦠H) ?);
+ ##[ ##1: #H7; nrewrite > (H4 ⦠H7); nrewrite > (andb_false3_1 (f2 y1 y2) (f3 z1 z2)); napply refl_eq
+ ##| ##2: #H7; nrewrite > (H5 ⦠H7); nrewrite > (andb_false3_2 (f1 x1 x2) (f3 z1 z2)); napply refl_eq
+ ##| ##3: #H7; nrewrite > (H6 ⦠H7); nrewrite > (andb_false3_3 (f1 x1 x2) (f2 y1 y2)); napply refl_eq
+ ##]
+nqed.
+
+nlemma triple_is_comparable :
+ comparable â comparable â comparable â comparable.
+ #T1; #T2; #T3; napply (mk_comparable (Prod3T T1 T2 T3));
+ ##[ napply (triple ⦠(zeroc T1) (zeroc T2) (zeroc T3))
+ ##| napply (λx.false)
+ ##| napply (eq_triple ⦠(eqc T1) (eqc T2) (eqc T3))
+ ##| napply (eqtriple_to_eq ⦠(eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##| napply (eqc_to_eq T3)
+ ##]
+ ##| napply (eq_to_eqtriple ⦠(eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##| napply (eq_to_eqc T3)
+ ##]
+ ##| napply (neqtriple_to_neq ⦠(eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##| napply (neqc_to_neq T3)
+ ##]
+ ##| napply (neq_to_neqtriple ⦠(eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##| napply (neq_to_neqc T3)
+ ##]
+ ##| napply decidable_triple;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##]
+ ##| napply symmetric_eqtriple;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##| napply (symmetric_eqc T3)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 â S1: comparable, S2: comparable, S3: comparable;
+ T1 â (carr S1), T2 â (carr S2), T3 â (carr S3),
+ X â (triple_is_comparable S1 S2 S3)
+ (*********************************************) â¢
+ carr X â¡ Prod3T T1 T2 T3.
+
+(* ********* *)
+(* TUPLE x 4 *)
+(* ********* *)
+
+nlemma quadruple_destruct_1 :
+âT1,T2,T3,T4.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 â x1 = x2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple a _ _ _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quadruple_destruct_2 :
+âT1,T2,T3,T4.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 â y1 = y2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple _ b _ _ â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quadruple_destruct_3 :
+âT1,T2,T3,T4.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 â z1 = z2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple _ _ c _ â z1 = c ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quadruple_destruct_4 :
+âT1,T2,T3,T4.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 â v1 = v2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple _ _ _ d â v1 = d ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqquadruple :
+âT1,T2,T3,T4:Type.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.
+ (symmetricT T1 bool f1) â
+ (symmetricT T2 bool f2) â
+ (symmetricT T3 bool f3) â
+ (symmetricT T4 bool f4) â
+ (âp1,p2:Prod4T T1 T2 T3 T4.
+ (eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p2 p1)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2);
+ ncases (f2 y2 y1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H2 z1 z2);
+ ncases (f3 z2 z1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H3 v1 v2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqquadruple :
+âT1,T2,T3,T4.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.
+ (âx,y:T1.x = y â f1 x y = true) â
+ (âx,y:T2.x = y â f2 x y = true) â
+ (âx,y:T3.x = y â f3 x y = true) â
+ (âx,y:T4.x = y â f4 x y = true) â
+ (âp1,p2:Prod4T T1 T2 T3 T4.
+ (p1 = p2 â eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = true)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #H;
+ nnormalize;
+ nrewrite > (H1 ⦠(quadruple_destruct_1 ⦠H));
+ nnormalize;
+ nrewrite > (H2 ⦠(quadruple_destruct_2 ⦠H));
+ nnormalize;
+ nrewrite > (H3 ⦠(quadruple_destruct_3 ⦠H));
+ nnormalize;
+ nrewrite > (H4 ⦠(quadruple_destruct_4 ⦠H));
+ napply refl_eq.
+nqed.
+
+nlemma eqquadruple_to_eq :
+âT1,T2,T3,T4.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.
+ (âx,y:T1.f1 x y = true â x = y) â
+ (âx,y:T2.f2 x y = true â x = y) â
+ (âx,y:T3.f3 x y = true â x = y) â
+ (âx,y:T4.f4 x y = true â x = y) â
+ (âp1,p2:Prod4T T1 T2 T3 T4.
+ (eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = true â p1 = p2)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #H;
+ nnormalize in H:(%);
+ nletin K â (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ ##[ ##2: #H5; ndestruct (*napply (bool_destruct ⦠H5)*) ##]
+ nletin K1 â (H2 y1 y2);
+ ncases (f2 y1 y2) in K1:(%) ⢠%;
+ nnormalize;
+ ##[ ##2: #H5; #H6; ndestruct (*napply (bool_destruct ⦠H6)*) ##]
+ nletin K2 â (H3 z1 z2);
+ ncases (f3 z1 z2) in K2:(%) ⢠%;
+ nnormalize;
+ ##[ ##2: #H5; #H6; #H7; ndestruct (*napply (bool_destruct ⦠H7)*) ##]
+ #H5; #H6; #H7; #H8;
+ nrewrite > (H5 (refl_eq â¦));
+ nrewrite > (H6 (refl_eq â¦));
+ nrewrite > (H8 (refl_eq â¦));
+ nrewrite > (H4 v1 v2 H7);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_quadruple :
+âT1,T2,T3,T4.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T4.decidable (x = y)) â
+ (âx,y:Prod4T T1 T2 T3 T4.decidable (x = y)).
+ #T1; #T2; #T3; #T4; #H; #H1; #H2; #H3;
+ #x; nelim x; #xx1; #xx2; #xx3; #xx4;
+ #y; nelim y; #yy1; #yy2; #yy3; #yy4;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 â yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H4; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H5; napply (H4 (quadruple_destruct_1 T1 T2 T3 T4 ⦠H5))
+ ##| ##1: #H4; napply (or2_elim (xx2 = yy2) (xx2 â yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H5; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H6; napply (H5 (quadruple_destruct_2 T1 T2 T3 T4 ⦠H6))
+ ##| ##1: #H5; napply (or2_elim (xx3 = yy3) (xx3 â yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##2: #H6; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H7; napply (H6 (quadruple_destruct_3 T1 T2 T3 T4 ⦠H7))
+ ##| ##1: #H6; napply (or2_elim (xx4 = yy4) (xx4 â yy4) ? (H3 xx4 yy4) ?);
+ ##[ ##2: #H7; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H8; napply (H7 (quadruple_destruct_4 T1 T2 T3 T4 ⦠H8))
+ ##| ##1: #H7; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H4;
+ nrewrite > H5;
+ nrewrite > H6;
+ nrewrite > H7;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqquadruple_to_neq :
+âT1,T2,T3,T4.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.
+ (âx,y:T1.f1 x y = false â x â y) â
+ (âx,y:T2.f2 x y = false â x â y) â
+ (âx,y:T3.f3 x y = false â x â y) â
+ (âx,y:T4.f4 x y = false â x â y) â
+ (âp1,p2:Prod4T T1 T2 T3 T4.
+ (eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = false â p1 â p2)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2;
+ nchange with ((((f1 x1 x2) â (f2 y1 y2) â (f3 z1 z2) â (f4 v1 v2)) = false) â ?); #H;
+ nnormalize; #H5;
+ napply (or4_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ((f3 z1 z2) = false) ((f4 v1 v2) = false) ? (andb_false4 ⦠H) ?);
+ ##[ ##1: #H6; napply (H1 x1 x2 H6); napply (quadruple_destruct_1 T1 T2 T3 T4 ⦠H5)
+ ##| ##2: #H6; napply (H2 y1 y2 H6); napply (quadruple_destruct_2 T1 T2 T3 T4 ⦠H5)
+ ##| ##3: #H6; napply (H3 z1 z2 H6); napply (quadruple_destruct_3 T1 T2 T3 T4 ⦠H5)
+ ##| ##4: #H6; napply (H4 v1 v2 H6); napply (quadruple_destruct_4 T1 T2 T3 T4 ⦠H5)
+ ##]
+nqed.
+
+nlemma quadruple_destruct :
+âT1,T2,T3,T4.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T4.decidable (x = y)) â
+ (âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.
+ (quadruple T1 T2 T3 T4 x1 y1 z1 v1) â (quadruple T1 T2 T3 T4 x2 y2 z2 v2) â
+ Or4 (x1 â x2) (y1 â y2) (z1 â z2) (v1 â v2)).
+ #T1; #T2; #T3; #T4; #H1; #H2; #H3; #H4;
+ #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 â x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H5; napply (or4_intro1 ⦠H5)
+ ##| ##1: #H5; napply (or2_elim (y1 = y2) (y1 â y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H6; napply (or4_intro2 ⦠H6)
+ ##| ##1: #H6; napply (or2_elim (z1 = z2) (z1 â z2) ? (H3 z1 z2) ?);
+ ##[ ##2: #H7; napply (or4_intro3 ⦠H7)
+ ##| ##1: #H7; napply (or2_elim (v1 = v2) (v1 â v2) ? (H4 v1 v2) ?);
+ ##[ ##2: #H8; napply (or4_intro4 ⦠H8)
+ ##| ##1: #H8; nrewrite > H5 in H:(%);
+ nrewrite > H6;
+ nrewrite > H7;
+ nrewrite > H8; #H; nelim (H (refl_eq â¦))
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqquadruple :
+âT1,T2,T3,T4.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T4.decidable (x = y)) â
+ (âx,y:T1.x â y â f1 x y = false) â
+ (âx,y:T2.x â y â f2 x y = false) â
+ (âx,y:T3.x â y â f3 x y = false) â
+ (âx,y:T4.x â y â f4 x y = false) â
+ (âp1,p2:Prod4T T1 T2 T3 T4.
+ (p1 â p2 â eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = false)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #H;
+ nchange with (((f1 x1 x2) â (f2 y1 y2) â (f3 z1 z2) â (f4 v1 v2)) = false);
+ napply (or4_elim (x1 â x2) (y1 â y2) (z1 â z2) (v1 â v2) ? (quadruple_destruct T1 T2 T3 T4 H1 H2 H3 H4 ⦠H) ?);
+ ##[ ##1: #H9; nrewrite > (H5 ⦠H9); nrewrite > (andb_false4_1 (f2 y1 y2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##| ##2: #H9; nrewrite > (H6 ⦠H9); nrewrite > (andb_false4_2 (f1 x1 x2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##| ##3: #H9; nrewrite > (H7 ⦠H9); nrewrite > (andb_false4_3 (f1 x1 x2) (f2 y1 y2) (f4 v1 v2)); napply refl_eq
+ ##| ##4: #H9; nrewrite > (H8 ⦠H9); nrewrite > (andb_false4_4 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2)); napply refl_eq
+ ##]
+nqed.
+
+nlemma quadruple_is_comparable :
+ comparable â comparable â comparable â comparable â comparable.
+ #T1; #T2; #T3; #T4; napply (mk_comparable (Prod4T T1 T2 T3 T4));
+ ##[ napply (quadruple ⦠(zeroc T1) (zeroc T2) (zeroc T3) (zeroc T4))
+ ##| napply (λx.false)
+ ##| napply (eq_quadruple ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4))
+ ##| napply (eqquadruple_to_eq ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##| napply (eqc_to_eq T3)
+ ##| napply (eqc_to_eq T4)
+ ##]
+ ##| napply (eq_to_eqquadruple ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##| napply (eq_to_eqc T3)
+ ##| napply (eq_to_eqc T4)
+ ##]
+ ##| napply (neqquadruple_to_neq ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##| napply (neqc_to_neq T3)
+ ##| napply (neqc_to_neq T4)
+ ##]
+ ##| napply (neq_to_neqquadruple ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##| napply (neq_to_neqc T3)
+ ##| napply (neq_to_neqc T4)
+ ##]
+ ##| napply decidable_quadruple;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##]
+ ##| napply symmetric_eqquadruple;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##| napply (symmetric_eqc T3)
+ ##| napply (symmetric_eqc T4)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 â S1: comparable, S2: comparable, S3: comparable, S4: comparable;
+ T1 â (carr S1), T2 â (carr S2), T3 â (carr S3), T4 â (carr S4),
+ X â (quadruple_is_comparable S1 S2 S3 S4)
+ (*********************************************) â¢
+ carr X â¡ Prod4T T1 T2 T3 T4.
+
+(* ********* *)
+(* TUPLE x 5 *)
+(* ********* *)
+
+nlemma quintuple_destruct_1 :
+âT1,T2,T3,T4,T5.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.âw1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 â x1 = x2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple a _ _ _ _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_2 :
+âT1,T2,T3,T4,T5.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.âw1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 â y1 = y2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ b _ _ _ â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_3 :
+âT1,T2,T3,T4,T5.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.âw1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 â z1 = z2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ _ c _ _ â z1 = c ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_4 :
+âT1,T2,T3,T4,T5.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.âw1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 â v1 = v2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ _ _ d _ â v1 = d ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_5 :
+âT1,T2,T3,T4,T5.âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.âw1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 â w1 = w2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ _ _ _ e â w1 = e ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqquintuple :
+âT1,T2,T3,T4,T5:Type.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.âf5:T5 â T5 â bool.
+ (symmetricT T1 bool f1) â
+ (symmetricT T2 bool f2) â
+ (symmetricT T3 bool f3) â
+ (symmetricT T4 bool f4) â
+ (symmetricT T5 bool f5) â
+ (âp1,p2:Prod5T T1 T2 T3 T4 T5.
+ (eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p2 p1)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2);
+ ncases (f2 y2 y1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H2 z1 z2);
+ ncases (f3 z2 z1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H3 v1 v2);
+ ncases (f4 v2 v1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H4 w1 w2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqquintuple :
+âT1,T2,T3,T4,T5.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.âf5:T5 â T5 â bool.
+ (âx,y:T1.x = y â f1 x y = true) â
+ (âx,y:T2.x = y â f2 x y = true) â
+ (âx,y:T3.x = y â f3 x y = true) â
+ (âx,y:T4.x = y â f4 x y = true) â
+ (âx,y:T5.x = y â f5 x y = true) â
+ (âp1,p2:Prod5T T1 T2 T3 T4 T5.
+ (p1 = p2 â eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = true)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2; #H;
+ nnormalize;
+ nrewrite > (H1 ⦠(quintuple_destruct_1 ⦠H));
+ nnormalize;
+ nrewrite > (H2 ⦠(quintuple_destruct_2 ⦠H));
+ nnormalize;
+ nrewrite > (H3 ⦠(quintuple_destruct_3 ⦠H));
+ nnormalize;
+ nrewrite > (H4 ⦠(quintuple_destruct_4 ⦠H));
+ nnormalize;
+ nrewrite > (H5 ⦠(quintuple_destruct_5 ⦠H));
+ napply refl_eq.
+nqed.
+
+nlemma eqquintuple_to_eq :
+âT1,T2,T3,T4,T5.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.âf5:T5 â T5 â bool.
+ (âx,y:T1.f1 x y = true â x = y) â
+ (âx,y:T2.f2 x y = true â x = y) â
+ (âx,y:T3.f3 x y = true â x = y) â
+ (âx,y:T4.f4 x y = true â x = y) â
+ (âx,y:T5.f5 x y = true â x = y) â
+ (âp1,p2:Prod5T T1 T2 T3 T4 T5.
+ (eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = true â p1 = p2)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2; #H;
+ nnormalize in H:(%);
+ nletin K â (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ ##[ ##2: #H6; ndestruct (*napply (bool_destruct ⦠H6)*) ##]
+ nletin K1 â (H2 y1 y2);
+ ncases (f2 y1 y2) in K1:(%) ⢠%;
+ nnormalize;
+ ##[ ##2: #H6; #H7; ndestruct (*napply (bool_destruct ⦠H7)*) ##]
+ nletin K2 â (H3 z1 z2);
+ ncases (f3 z1 z2) in K2:(%) ⢠%;
+ nnormalize;
+ ##[ ##2: #H6; #H7; #H8; ndestruct (*napply (bool_destruct ⦠H8)*) ##]
+ nletin K3 â (H4 v1 v2);
+ ncases (f4 v1 v2) in K3:(%) ⢠%;
+ nnormalize;
+ ##[ ##2: #H6; #H7; #H8; #H9; ndestruct (*napply (bool_destruct ⦠H9)*) ##]
+ #H6; #H7; #H8; #H9; #H10;
+ nrewrite > (H6 (refl_eq â¦));
+ nrewrite > (H7 (refl_eq â¦));
+ nrewrite > (H8 (refl_eq â¦));
+ nrewrite > (H10 (refl_eq â¦));
+ nrewrite > (H5 w1 w2 H9);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_quintuple :
+âT1,T2,T3,T4,T5.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T4.decidable (x = y)) â
+ (âx,y:T5.decidable (x = y)) â
+ (âx,y:Prod5T T1 T2 T3 T4 T5.decidable (x = y)).
+ #T1; #T2; #T3; #T4; #T5; #H; #H1; #H2; #H3; #H4;
+ #x; nelim x; #xx1; #xx2; #xx3; #xx4; #xx5;
+ #y; nelim y; #yy1; #yy2; #yy3; #yy4; #yy5;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 â yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H5; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H6; napply (H5 (quintuple_destruct_1 T1 T2 T3 T4 T5 ⦠H6))
+ ##| ##1: #H5; napply (or2_elim (xx2 = yy2) (xx2 â yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H6; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H7; napply (H6 (quintuple_destruct_2 T1 T2 T3 T4 T5 ⦠H7))
+ ##| ##1: #H6; napply (or2_elim (xx3 = yy3) (xx3 â yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##2: #H7; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H8; napply (H7 (quintuple_destruct_3 T1 T2 T3 T4 T5 ⦠H8))
+ ##| ##1: #H7; napply (or2_elim (xx4 = yy4) (xx4 â yy4) ? (H3 xx4 yy4) ?);
+ ##[ ##2: #H8; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H9; napply (H8 (quintuple_destruct_4 T1 T2 T3 T4 T5 ⦠H9))
+ ##| ##1: #H8; napply (or2_elim (xx5 = yy5) (xx5 â yy5) ? (H4 xx5 yy5) ?);
+ ##[ ##2: #H9; napply (or2_intro2 (? = ?) (? â ?) ?);
+ nnormalize; #H10; napply (H9 (quintuple_destruct_5 T1 T2 T3 T4 T5 ⦠H10))
+ ##| ##1: #H9; napply (or2_intro1 (? = ?) (? â ?) ?);
+ nrewrite > H5;
+ nrewrite > H6;
+ nrewrite > H7;
+ nrewrite > H8;
+ nrewrite > H9;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqquintuple_to_neq :
+âT1,T2,T3,T4,T5.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.âf5:T5 â T5 â bool.
+ (âx,y:T1.f1 x y = false â x â y) â
+ (âx,y:T2.f2 x y = false â x â y) â
+ (âx,y:T3.f3 x y = false â x â y) â
+ (âx,y:T4.f4 x y = false â x â y) â
+ (âx,y:T5.f5 x y = false â x â y) â
+ (âp1,p2:Prod5T T1 T2 T3 T4 T5.
+ (eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = false â p1 â p2)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2;
+ nchange with ((((f1 x1 x2) â (f2 y1 y2) â (f3 z1 z2) â (f4 v1 v2) â (f5 w1 w2)) = false) â ?); #H;
+ nnormalize; #H6;
+ napply (or5_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ((f3 z1 z2) = false) ((f4 v1 v2) = false) ((f5 w1 w2) = false) ? (andb_false5 ⦠H) ?);
+ ##[ ##1: #H7; napply (H1 x1 x2 H7); napply (quintuple_destruct_1 T1 T2 T3 T4 T5 ⦠H6)
+ ##| ##2: #H7; napply (H2 y1 y2 H7); napply (quintuple_destruct_2 T1 T2 T3 T4 T5 ⦠H6)
+ ##| ##3: #H7; napply (H3 z1 z2 H7); napply (quintuple_destruct_3 T1 T2 T3 T4 T5 ⦠H6)
+ ##| ##4: #H7; napply (H4 v1 v2 H7); napply (quintuple_destruct_4 T1 T2 T3 T4 T5 ⦠H6)
+ ##| ##5: #H7; napply (H5 w1 w2 H7); napply (quintuple_destruct_5 T1 T2 T3 T4 T5 ⦠H6)
+ ##]
+nqed.
+
+nlemma quintuple_destruct :
+âT1,T2,T3,T4,T5.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T4.decidable (x = y)) â
+ (âx,y:T5.decidable (x = y)) â
+ (âx1,x2:T1.ây1,y2:T2.âz1,z2:T3.âv1,v2:T4.âw1,w2:T5.
+ (quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1) â (quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2) â
+ Or5 (x1 â x2) (y1 â y2) (z1 â z2) (v1 â v2) (w1 â w2)).
+ #T1; #T2; #T3; #T4; #T5; #H1; #H2; #H3; #H4; #H5;
+ #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 â x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H6; napply (or5_intro1 ⦠H6)
+ ##| ##1: #H6; napply (or2_elim (y1 = y2) (y1 â y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H7; napply (or5_intro2 ⦠H7)
+ ##| ##1: #H7; napply (or2_elim (z1 = z2) (z1 â z2) ? (H3 z1 z2) ?);
+ ##[ ##2: #H8; napply (or5_intro3 ⦠H8)
+ ##| ##1: #H8; napply (or2_elim (v1 = v2) (v1 â v2) ? (H4 v1 v2) ?);
+ ##[ ##2: #H9; napply (or5_intro4 ⦠H9)
+ ##| ##1: #H9; napply (or2_elim (w1 = w2) (w1 â w2) ? (H5 w1 w2) ?);
+ ##[ ##2: #H10; napply (or5_intro5 ⦠H10)
+ ##| ##1: #H10; nrewrite > H6 in H:(%);
+ nrewrite > H7;
+ nrewrite > H8;
+ nrewrite > H9;
+ nrewrite > H10; #H; nelim (H (refl_eq â¦))
+ ##]
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqquintuple :
+âT1,T2,T3,T4,T5.
+âf1:T1 â T1 â bool.âf2:T2 â T2 â bool.âf3:T3 â T3 â bool.âf4:T4 â T4 â bool.âf5:T5 â T5 â bool.
+ (âx,y:T1.decidable (x = y)) â
+ (âx,y:T2.decidable (x = y)) â
+ (âx,y:T3.decidable (x = y)) â
+ (âx,y:T4.decidable (x = y)) â
+ (âx,y:T5.decidable (x = y)) â
+ (âx,y:T1.x â y â f1 x y = false) â
+ (âx,y:T2.x â y â f2 x y = false) â
+ (âx,y:T3.x â y â f3 x y = false) â
+ (âx,y:T4.x â y â f4 x y = false) â
+ (âx,y:T5.x â y â f5 x y = false) â
+ (âp1,p2:Prod5T T1 T2 T3 T4 T5.
+ (p1 â p2 â eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = false)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5;
+ #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8; #H9; #H10;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2; #H;
+ nchange with (((f1 x1 x2) â (f2 y1 y2) â (f3 z1 z2) â (f4 v1 v2) â (f5 w1 w2)) = false);
+ napply (or5_elim (x1 â x2) (y1 â y2) (z1 â z2) (v1 â v2) (w1 â w2) ? (quintuple_destruct T1 T2 T3 T4 T5 H1 H2 H3 H4 H5 ⦠H) ?);
+ ##[ ##1: #H11; nrewrite > (H6 ⦠H11); nrewrite > (andb_false5_1 (f2 y1 y2) (f3 z1 z2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##2: #H11; nrewrite > (H7 ⦠H11); nrewrite > (andb_false5_2 (f1 x1 x2) (f3 z1 z2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##3: #H11; nrewrite > (H8 ⦠H11); nrewrite > (andb_false5_3 (f1 x1 x2) (f2 y1 y2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##4: #H11; nrewrite > (H9 ⦠H11); nrewrite > (andb_false5_4 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2) (f5 w1 w2)); napply refl_eq
+ ##| ##5: #H11; nrewrite > (H10 ⦠H11); nrewrite > (andb_false5_5 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##]
+nqed.
+
+nlemma quintuple_is_comparable :
+ comparable â comparable â comparable â comparable â comparable â comparable.
+ #T1; #T2; #T3; #T4; #T5; napply (mk_comparable (Prod5T T1 T2 T3 T4 T5));
+ ##[ napply (quintuple ⦠(zeroc T1) (zeroc T2) (zeroc T3) (zeroc T4) (zeroc T5))
+ ##| napply (λx.false)
+ ##| napply (eq_quintuple ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5))
+ ##| napply (eqquintuple_to_eq ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##| napply (eqc_to_eq T3)
+ ##| napply (eqc_to_eq T4)
+ ##| napply (eqc_to_eq T5)
+ ##]
+ ##| napply (eq_to_eqquintuple ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##| napply (eq_to_eqc T3)
+ ##| napply (eq_to_eqc T4)
+ ##| napply (eq_to_eqc T5)
+ ##]
+ ##| napply (neqquintuple_to_neq ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##| napply (neqc_to_neq T3)
+ ##| napply (neqc_to_neq T4)
+ ##| napply (neqc_to_neq T5)
+ ##]
+ ##| napply (neq_to_neqquintuple ⦠(eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##| napply (decidable_c T5)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##| napply (neq_to_neqc T3)
+ ##| napply (neq_to_neqc T4)
+ ##| napply (neq_to_neqc T5)
+ ##]
+ ##| napply decidable_quintuple;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##| napply (decidable_c T5)
+ ##]
+ ##| napply symmetric_eqquintuple;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##| napply (symmetric_eqc T3)
+ ##| napply (symmetric_eqc T4)
+ ##| napply (symmetric_eqc T5)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 â S1: comparable, S2: comparable, S3: comparable, S4: comparable, S5: comparable;
+ T1 â (carr S1), T2 â (carr S2), T3 â (carr S3), T4 â (carr S4), T5 â (carr S5),
+ X â (quintuple_is_comparable S1 S2 S3 S4 S5)
+ (*********************************************) â¢
+ carr X â¡ Prod5T T1 T2 T3 T4 T5.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/prod_base.ma b/helm/software/matita/contribs/ng_assembly2/common/prod_base.ma
new file mode 100644
index 000000000..1080533d1
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/prod_base.ma
@@ -0,0 +1,115 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/bool.ma".
+
+(* ***** *)
+(* TUPLE *)
+(* ***** *)
+
+ninductive ProdT (T1:Type) (T2:Type) : Type â
+ pair : T1 â T2 â ProdT T1 T2.
+
+ndefinition fst â
+λT1,T2:Type.λp:ProdT T1 T2.match p with [ pair x _ â x ].
+
+ndefinition snd â
+λT1,T2:Type.λp:ProdT T1 T2.match p with [ pair _ x â x ].
+
+ndefinition eq_pair â
+λT1,T2:Type.
+λf1:T1 â T1 â bool.λf2:T2 â T2 â bool.
+λp1,p2:ProdT T1 T2.
+ (f1 (fst ⦠p1) (fst ⦠p2)) â
+ (f2 (snd ⦠p1) (snd ⦠p2)).
+
+ninductive Prod3T (T1:Type) (T2:Type) (T3:Type) : Type â
+triple : T1 â T2 â T3 â Prod3T T1 T2 T3.
+
+ndefinition fst3T â
+λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ triple x _ _ â x ].
+
+ndefinition snd3T â
+λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ triple _ x _ â x ].
+
+ndefinition thd3T â
+λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ triple _ _ x â x ].
+
+ndefinition eq_triple â
+λT1,T2,T3:Type.
+λf1:T1 â T1 â bool.λf2:T2 â T2 â bool.λf3:T3 â T3 â bool.
+λp1,p2:Prod3T T1 T2 T3.
+ (f1 (fst3T ⦠p1) (fst3T ⦠p2)) â
+ (f2 (snd3T ⦠p1) (snd3T ⦠p2)) â
+ (f3 (thd3T ⦠p1) (thd3T ⦠p2)).
+
+ninductive Prod4T (T1:Type) (T2:Type) (T3:Type) (T4:Type) : Type â
+quadruple : T1 â T2 â T3 â T4 â Prod4T T1 T2 T3 T4.
+
+ndefinition fst4T â
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadruple x _ _ _ â x ].
+
+ndefinition snd4T â
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadruple _ x _ _ â x ].
+
+ndefinition thd4T â
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadruple _ _ x _ â x ].
+
+ndefinition fth4T â
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadruple _ _ _ x â x ].
+
+ndefinition eq_quadruple â
+λT1,T2,T3,T4:Type.
+λf1:T1 â T1 â bool.λf2:T2 â T2 â bool.λf3:T3 â T3 â bool.λf4:T4 â T4 â bool.
+λp1,p2:Prod4T T1 T2 T3 T4.
+ (f1 (fst4T ⦠p1) (fst4T ⦠p2)) â
+ (f2 (snd4T ⦠p1) (snd4T ⦠p2)) â
+ (f3 (thd4T ⦠p1) (thd4T ⦠p2)) â
+ (f4 (fth4T ⦠p1) (fth4T ⦠p2)).
+
+ninductive Prod5T (T1:Type) (T2:Type) (T3:Type) (T4:Type) (T5:Type) : Type â
+quintuple : T1 â T2 â T3 â T4 â T5 â Prod5T T1 T2 T3 T4 T5.
+
+ndefinition fst5T â
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintuple x _ _ _ _ â x ].
+
+ndefinition snd5T â
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintuple _ x _ _ _ â x ].
+
+ndefinition thd5T â
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintuple _ _ x _ _ â x ].
+
+ndefinition fth5T â
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintuple _ _ _ x _ â x ].
+
+ndefinition fft5T â
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintuple _ _ _ _ x â x ].
+
+ndefinition eq_quintuple â
+λT1,T2,T3,T4,T5:Type.
+λf1:T1 â T1 â bool.λf2:T2 â T2 â bool.λf3:T3 â T3 â bool.λf4:T4 â T4 â bool.λf5:T5 â T5 â bool.
+λp1,p2:Prod5T T1 T2 T3 T4 T5.
+ (f1 (fst5T ⦠p1) (fst5T ⦠p2)) â
+ (f2 (snd5T ⦠p1) (snd5T ⦠p2)) â
+ (f3 (thd5T ⦠p1) (thd5T ⦠p2)) â
+ (f4 (fth5T ⦠p1) (fth5T ⦠p2)) â
+ (f5 (fft5T ⦠p1) (fft5T ⦠p2)).
diff --git a/helm/software/matita/contribs/ng_assembly2/common/pts.ma b/helm/software/matita/contribs/ng_assembly2/common/pts.ma
new file mode 100644
index 000000000..428772bbc
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/pts.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+universe constraint Type[0] < Type[1].
+universe constraint Type[1] < Type[2].
+universe constraint Type[2] < Type[3].
+universe constraint Type[3] < Type[4].
diff --git a/helm/software/matita/contribs/ng_assembly2/common/sigma.ma b/helm/software/matita/contribs/ng_assembly2/common/sigma.ma
new file mode 100755
index 000000000..529ca7ada
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/sigma.ma
@@ -0,0 +1,52 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/theory.ma".
+
+(* coppia dipendente *)
+
+ninductive sigma (A:Type) (P:A â Type) : Type â
+ sigma_intro: âx:A.P x â sigma A P.
+
+notation < "hvbox(\Sigma ident i opt (: tx) break . p)"
+ right associative with precedence 20
+for @{ 'Sigma ${default
+ @{\lambda ${ident i} : $tx. $p}
+ @{\lambda ${ident i} . $p}}}.
+
+notation > "\Sigma list1 ident x sep , opt (: T). term 19 Px"
+ with precedence 20
+ for ${ default
+ @{ ${ fold right @{$Px} rec acc @{'Sigma (λ${ident x}:$T.$acc)} } }
+ @{ ${ fold right @{$Px} rec acc @{'Sigma (λ${ident x}.$acc)} } }
+ }.
+
+notation "\ll term 19 a, break term 19 b \gg"
+with precedence 90 for @{'dependent_pair (λx:?.? x) $a $b}.
+interpretation "dependent pair" 'dependent_pair \eta.c a b = (sigma_intro ? c a b).
+
+interpretation "sigma" 'Sigma \eta.x = (sigma ? x).
+
+ndefinition sigmaFst â
+λT:Type.λf:T â Type.λs:sigma T f.match s with [ sigma_intro x _ â x ].
+ndefinition sigmaSnd â
+λT:Type.λf:T â Type.λs:sigma T f.match s return λs.f (sigmaFst ?? s) with [ sigma_intro _ x â x ].
diff --git a/helm/software/matita/contribs/ng_assembly2/common/string.ma b/helm/software/matita/contribs/ng_assembly2/common/string.ma
new file mode 100644
index 000000000..00b70cfc4
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/string.ma
@@ -0,0 +1,183 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/ascii.ma".
+include "common/list.ma".
+
+(* ******* *)
+(* STRINGA *)
+(* ******* *)
+
+(* tipo pubblico *)
+ndefinition aux_str_type â list ascii.
+
+unification hint 0 â ⢠carr (list_is_comparable ascii_is_comparable) â¡ list ascii.
+unification hint 0 â ⢠carr (list_is_comparable ascii_is_comparable) â¡ aux_str_type.
+
+(* ************ *)
+(* STRINGA + ID *)
+(* ************ *)
+
+(* tipo pubblico *)
+nrecord strId : Type â
+ {
+ str_elem: aux_str_type;
+ id_elem: nat
+ }.
+
+(* confronto *)
+ndefinition eq_strId â
+λsid,sid':strId.
+ (eqc ? (str_elem sid) (str_elem sid'))â
+ (eqc ? (id_elem sid) (id_elem sid')).
+
+nlemma strid_destruct_1 : âx1,x2,y1,y2.mk_strId x1 y1 = mk_strId x2 y2 â x1 = x2.
+ #x1; #x2; #y1; #y2; #H;
+ nchange with (match mk_strId x2 y2 with [ mk_strId a _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma strid_destruct_2 : âx1,x2,y1,y2.mk_strId x1 y1 = mk_strId x2 y2 â y1 = y2.
+ #x1; #x2; #y1; #y2; #H;
+ nchange with (match mk_strId x2 y2 with [ mk_strId _ b â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqstrid : symmetricT strId bool eq_strId.
+ #si1; #si2;
+ nchange with (
+ ((eqc ? (str_elem si1) (str_elem si2))â(eqc ? (id_elem si1) (id_elem si2))) =
+ ((eqc ? (str_elem si2) (str_elem si1))â(eqc ? (id_elem si2) (id_elem si1))));
+ nrewrite > (symmetric_eqc ? (str_elem si1) (str_elem si2));
+ nrewrite > (symmetric_eqc ? (id_elem si1) (id_elem si2));
+ napply refl_eq.
+nqed.
+
+nlemma eqstrid_to_eq : âs,s'.eq_strId s s' = true â s = s'.
+ #si1; #si2;
+ nelim si1;
+ #l1; #n1;
+ nelim si2;
+ #l2; #n2; #H;
+ nchange in H:(%) with (((eqc ? l1 l2)â(eqc ? n1 n2)) = true);
+ nrewrite > (eqc_to_eq ? l1 l2 (andb_true_true_l ⦠H));
+ nrewrite > (eqc_to_eq ? n1 n2 (andb_true_true_r ⦠H));
+ napply refl_eq.
+nqed.
+
+nlemma eq_to_eqstrid : âs,s'.s = s' â eq_strId s s' = true.
+ #si1; #si2;
+ nelim si1;
+ #l1; #n1;
+ nelim si2;
+ #l2; #n2; #H;
+ nchange with (((eqc ? l1 l2)â(eqc ? n1 n2)) = true);
+ nrewrite > (strid_destruct_1 ⦠H);
+ nrewrite > (strid_destruct_2 ⦠H);
+ nrewrite > (eq_to_eqc ? l2 l2 (refl_eq â¦));
+ nrewrite > (eq_to_eqc ? n2 n2 (refl_eq â¦));
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma decidable_strid_aux1 : âs1,n1,s2,n2.s1 â s2 â (mk_strId s1 n1) â (mk_strId s2 n2).
+ #s1; #n1; #s2; #n2;
+ nnormalize; #H; #H1;
+ napply (H (strid_destruct_1 ⦠H1)).
+nqed.
+
+nlemma decidable_strid_aux2 : âs1,n1,s2,n2.n1 â n2 â (mk_strId s1 n1) â (mk_strId s2 n2).
+ #s1; #n1; #s2; #n2;
+ nnormalize; #H; #H1;
+ napply (H (strid_destruct_2 ⦠H1)).
+nqed.
+
+nlemma decidable_strid : âx,y:strId.decidable (x = y).
+ #x; nelim x; #s1; #n1;
+ #y; nelim y; #s2; #n2;
+ nnormalize;
+ napply (or2_elim (s1 = s2) (s1 â s2) ? (decidable_c ? s1 s2) â¦);
+ ##[ ##2: #H; napply (or2_intro2 ⦠(decidable_strid_aux1 ⦠H))
+ ##| ##1: #H; napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_c ? n1 n2) â¦);
+ ##[ ##2: #H1; napply (or2_intro2 ⦠(decidable_strid_aux2 ⦠H1))
+ ##| ##1: #H1; nrewrite > H; nrewrite > H1;
+ napply (or2_intro1 ⦠(refl_eq ? (mk_strId s2 n2)))
+ ##]
+ ##]
+nqed.
+
+nlemma neqstrid_to_neq : âsid1,sid2:strId.(eq_strId sid1 sid2 = false) â (sid1 â sid2).
+ #sid1; nelim sid1; #s1; #n1;
+ #sid2; nelim sid2; #s2; #n2;
+ nchange with ((((eqc ? s1 s2) â (eqc ? n1 n2)) = false) â ?);
+ #H;
+ napply (or2_elim ((eqc ? s1 s2) = false) ((eqc ? n1 n2) = false) ? (andb_false2 ⦠H) â¦);
+ ##[ ##1: #H1; napply (decidable_strid_aux1 ⦠(neqc_to_neq ⦠H1))
+ ##| ##2: #H1; napply (decidable_strid_aux2 ⦠(neqc_to_neq ⦠H1))
+ ##]
+nqed.
+
+nlemma strid_destruct : âs1,s2,n1,n2.(mk_strId s1 n1) â (mk_strId s2 n2) â s1 â s2 ⨠n1 â n2.
+ #s1; #s2; #n1; #n2;
+ nnormalize; #H;
+ napply (or2_elim (s1 = s2) (s1 â s2) ? (decidable_c ? s1 s2) â¦);
+ ##[ ##2: #H1; napply (or2_intro1 ⦠H1)
+ ##| ##1: #H1; napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_c ? n1 n2) â¦);
+ ##[ ##2: #H2; napply (or2_intro2 ⦠H2)
+ ##| ##1: #H2; nrewrite > H1 in H:(%);
+ nrewrite > H2;
+ #H; nelim (H (refl_eq â¦))
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqstrid : âsid1,sid2.sid1 â sid2 â eq_strId sid1 sid2 = false.
+ #sid1; nelim sid1; #s1; #n1;
+ #sid2; nelim sid2; #s2; #n2;
+ #H; nchange with (((eqc ? s1 s2) â (eqc ? n1 n2)) = false);
+ napply (or2_elim (s1 â s2) (n1 â n2) ? (strid_destruct ⦠H) â¦);
+ ##[ ##1: #H1; nrewrite > (neq_to_neqc ⦠H1); nnormalize; napply refl_eq
+ ##| ##2: #H1; nrewrite > (neq_to_neqc ⦠H1);
+ nrewrite > (symmetric_andbool (eqc ? s1 s2) false);
+ nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma strid_is_comparable : comparable.
+ napply (mk_comparable strId);
+ ##[ napply (mk_strId (nil ?) O)
+ ##| napply (λx.false)
+ ##| napply eq_strId
+ ##| napply eqstrid_to_eq
+ ##| napply eq_to_eqstrid
+ ##| napply neqstrid_to_neq
+ ##| napply neq_to_neqstrid
+ ##| napply decidable_strid
+ ##| napply symmetric_eqstrid
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr strid_is_comparable â¡ strId.
diff --git a/helm/software/matita/contribs/ng_assembly2/common/theory.ma b/helm/software/matita/contribs/ng_assembly2/common/theory.ma
new file mode 100644
index 000000000..da7812069
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/common/theory.ma
@@ -0,0 +1,608 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/pts.ma".
+
+(* ********************************** *)
+(* SOTTOINSIEME MINIMALE DELLA TEORIA *)
+(* ********************************** *)
+
+(* logic/connectives.ma *)
+
+ninductive True: Prop â
+ I : True.
+
+ninductive False: Prop â.
+
+ndefinition Not: Prop â Prop â
+λA.(A â False).
+
+interpretation "logical not" 'not x = (Not x).
+
+nlemma absurd : âA,C:Prop.A â ¬A â C.
+ #A; #C; #H;
+ nnormalize;
+ #H1;
+ nelim (H1 H).
+nqed.
+
+nlemma not_to_not : âA,B:Prop. (A â B) â ((¬B) â (¬A)).
+ #A; #B; #H;
+ nnormalize;
+ #H1; #H2;
+ nelim (H1 (H H2)).
+nqed.
+
+nlemma prop_to_nnprop : âP.P â ¬¬P.
+ #P; nnormalize; #H; #H1;
+ napply (H1 H).
+nqed.
+
+ninductive And2 (A,B:Prop) : Prop â
+ conj2 : A â B â (And2 A B).
+
+interpretation "logical and" 'and x y = (And2 x y).
+
+nlemma proj2_1: âA,B:Prop.A ⧠B â A.
+ #A; #B; #H;
+ napply (And2_ind A B ⦠H);
+ #H1; #H2;
+ napply H1.
+nqed.
+
+nlemma proj2_2: âA,B:Prop.A ⧠B â B.
+ #A; #B; #H;
+ napply (And2_ind A B ⦠H);
+ #H1; #H2;
+ napply H2.
+nqed.
+
+ninductive And3 (A,B,C:Prop) : Prop â
+ conj3 : A â B â C â (And3 A B C).
+
+nlemma proj3_1: âA,B,C:Prop.And3 A B C â A.
+ #A; #B; #C; #H;
+ napply (And3_ind A B C ⦠H);
+ #H1; #H2; #H3;
+ napply H1.
+nqed.
+
+nlemma proj3_2: âA,B,C:Prop.And3 A B C â B.
+ #A; #B; #C; #H;
+ napply (And3_ind A B C ⦠H);
+ #H1; #H2; #H3;
+ napply H2.
+nqed.
+
+nlemma proj3_3: âA,B,C:Prop.And3 A B C â C.
+ #A; #B; #C; #H;
+ napply (And3_ind A B C ⦠H);
+ #H1; #H2; #H3;
+ napply H3.
+nqed.
+
+ninductive And4 (A,B,C,D:Prop) : Prop â
+ conj4 : A â B â C â D â (And4 A B C D).
+
+nlemma proj4_1: âA,B,C,D:Prop.And4 A B C D â A.
+ #A; #B; #C; #D; #H;
+ napply (And4_ind A B C D ⦠H);
+ #H1; #H2; #H3; #H4;
+ napply H1.
+nqed.
+
+nlemma proj4_2: âA,B,C,D:Prop.And4 A B C D â B.
+ #A; #B; #C; #D; #H;
+ napply (And4_ind A B C D ⦠H);
+ #H1; #H2; #H3; #H4;
+ napply H2.
+nqed.
+
+nlemma proj4_3: âA,B,C,D:Prop.And4 A B C D â C.
+ #A; #B; #C; #D; #H;
+ napply (And4_ind A B C D ⦠H);
+ #H1; #H2; #H3; #H4;
+ napply H3.
+nqed.
+
+nlemma proj4_4: âA,B,C,D:Prop.And4 A B C D â D.
+ #A; #B; #C; #D; #H;
+ napply (And4_ind A B C D ⦠H);
+ #H1; #H2; #H3; #H4;
+ napply H4.
+nqed.
+
+ninductive And5 (A,B,C,D,E:Prop) : Prop â
+ conj5 : A â B â C â D â E â (And5 A B C D E).
+
+nlemma proj5_1: âA,B,C,D,E:Prop.And5 A B C D E â A.
+ #A; #B; #C; #D; #E; #H;
+ napply (And5_ind A B C D E ⦠H);
+ #H1; #H2; #H3; #H4; #H5;
+ napply H1.
+nqed.
+
+nlemma proj5_2: âA,B,C,D,E:Prop.And5 A B C D E â B.
+ #A; #B; #C; #D; #E; #H;
+ napply (And5_ind A B C D E ⦠H);
+ #H1; #H2; #H3; #H4; #H5;
+ napply H2.
+nqed.
+
+nlemma proj5_3: âA,B,C,D,E:Prop.And5 A B C D E â C.
+ #A; #B; #C; #D; #E; #H;
+ napply (And5_ind A B C D E ⦠H);
+ #H1; #H2; #H3; #H4; #H5;
+ napply H3.
+nqed.
+
+nlemma proj5_4: âA,B,C,D,E:Prop.And5 A B C D E â D.
+ #A; #B; #C; #D; #E; #H;
+ napply (And5_ind A B C D E ⦠H);
+ #H1; #H2; #H3; #H4; #H5;
+ napply H4.
+nqed.
+
+nlemma proj5_5: âA,B,C,D,E:Prop.And5 A B C D E â E.
+ #A; #B; #C; #D; #E; #H;
+ napply (And5_ind A B C D E ⦠H);
+ #H1; #H2; #H3; #H4; #H5;
+ napply H5.
+nqed.
+
+ninductive Or2 (A,B:Prop) : Prop â
+ or2_intro1 : A â (Or2 A B)
+| or2_intro2 : B â (Or2 A B).
+
+interpretation "logical or" 'or x y = (Or2 x y).
+
+ndefinition decidable â λA:Prop.A ⨠(¬A).
+
+nlemma or2_elim
+ : âP1,P2,Q:Prop.Or2 P1 P2 â âf1:P1 â Q.âf2:P2 â Q.Q.
+ #P1; #P2; #Q; #H; #f1; #f2;
+ napply (Or2_ind P1 P2 ? f1 f2 ?);
+ napply H.
+nqed.
+
+nlemma symmetric_or2 : âP1,P2.Or2 P1 P2 â Or2 P2 P1.
+ #P1; #P2; #H;
+ napply (or2_elim P1 P2 ? H);
+ ##[ ##1: #H1; napply (or2_intro2 P2 P1 H1)
+ ##| ##2: #H1; napply (or2_intro1 P2 P1 H1)
+ ##]
+nqed.
+
+ninductive Or3 (A,B,C:Prop) : Prop â
+ or3_intro1 : A â (Or3 A B C)
+| or3_intro2 : B â (Or3 A B C)
+| or3_intro3 : C â (Or3 A B C).
+
+nlemma or3_elim
+ : âP1,P2,P3,Q:Prop.Or3 P1 P2 P3 â âf1:P1 â Q.âf2:P2 â Q.âf3:P3 â Q.Q.
+ #P1; #P2; #P3; #Q; #H; #f1; #f2; #f3;
+ napply (Or3_ind P1 P2 P3 ? f1 f2 f3 ?);
+ napply H.
+nqed.
+
+nlemma symmetric_or3_12 : âP1,P2,P3:Prop.Or3 P1 P2 P3 â Or3 P2 P1 P3.
+ #P1; #P2; #P3; #H;
+ napply (or3_elim P1 P2 P3 ? H);
+ ##[ ##1: #H1; napply (or3_intro2 P2 P1 P3 H1)
+ ##| ##2: #H1; napply (or3_intro1 P2 P1 P3 H1)
+ ##| ##3: #H1; napply (or3_intro3 P2 P1 P3 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or3_13 : âP1,P2,P3:Prop.Or3 P1 P2 P3 â Or3 P3 P2 P1.
+ #P1; #P2; #P3; #H;
+ napply (or3_elim P1 P2 P3 ? H);
+ ##[ ##1: #H1; napply (or3_intro3 P3 P2 P1 H1)
+ ##| ##2: #H1; napply (or3_intro2 P3 P2 P1 H1)
+ ##| ##3: #H1; napply (or3_intro1 P3 P2 P1 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or3_23 : âP1,P2,P3:Prop.Or3 P1 P2 P3 â Or3 P1 P3 P2.
+ #P1; #P2; #P3; #H;
+ napply (or3_elim P1 P2 P3 ? H);
+ ##[ ##1: #H1; napply (or3_intro1 P1 P3 P2 H1)
+ ##| ##2: #H1; napply (or3_intro3 P1 P3 P2 H1)
+ ##| ##3: #H1; napply (or3_intro2 P1 P3 P2 H1)
+ ##]
+nqed.
+
+ninductive Or4 (A,B,C,D:Prop) : Prop â
+ or4_intro1 : A â (Or4 A B C D)
+| or4_intro2 : B â (Or4 A B C D)
+| or4_intro3 : C â (Or4 A B C D)
+| or4_intro4 : D â (Or4 A B C D).
+
+nlemma or4_elim
+ : âP1,P2,P3,P4,Q:Prop.Or4 P1 P2 P3 P4 â âf1:P1 â Q.âf2:P2 â Q.
+ âf3:P3 â Q.âf4:P4 â Q.Q.
+ #P1; #P2; #P3; #P4; #Q; #H; #f1; #f2; #f3; #f4;
+ napply (Or4_ind P1 P2 P3 P4 ? f1 f2 f3 f4 ?);
+ napply H.
+nqed.
+
+nlemma symmetric_or4_12 : âP1,P2,P3,P4:Prop.Or4 P1 P2 P3 P4 â Or4 P2 P1 P3 P4.
+ #P1; #P2; #P3; #P4; #H;
+ napply (or4_elim P1 P2 P3 P4 ? H);
+ ##[ ##1: #H1; napply (or4_intro2 P2 P1 P3 P4 H1)
+ ##| ##2: #H1; napply (or4_intro1 P2 P1 P3 P4 H1)
+ ##| ##3: #H1; napply (or4_intro3 P2 P1 P3 P4 H1)
+ ##| ##4: #H1; napply (or4_intro4 P2 P1 P3 P4 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or4_13 : âP1,P2,P3,P4:Prop.Or4 P1 P2 P3 P4 â Or4 P3 P2 P1 P4.
+ #P1; #P2; #P3; #P4; #H;
+ napply (or4_elim P1 P2 P3 P4 ? H);
+ ##[ ##1: #H1; napply (or4_intro3 P3 P2 P1 P4 H1)
+ ##| ##2: #H1; napply (or4_intro2 P3 P2 P1 P4 H1)
+ ##| ##3: #H1; napply (or4_intro1 P3 P2 P1 P4 H1)
+ ##| ##4: #H1; napply (or4_intro4 P3 P2 P1 P4 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or4_14 : âP1,P2,P3,P4:Prop.Or4 P1 P2 P3 P4 â Or4 P4 P2 P3 P1.
+ #P1; #P2; #P3; #P4; #H;
+ napply (or4_elim P1 P2 P3 P4 ? H);
+ ##[ ##1: #H1; napply (or4_intro4 P4 P2 P3 P1 H1)
+ ##| ##2: #H1; napply (or4_intro2 P4 P2 P3 P1 H1)
+ ##| ##3: #H1; napply (or4_intro3 P4 P2 P3 P1 H1)
+ ##| ##4: #H1; napply (or4_intro1 P4 P2 P3 P1 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or4_23 : âP1,P2,P3,P4:Prop.Or4 P1 P2 P3 P4 â Or4 P1 P3 P2 P4.
+ #P1; #P2; #P3; #P4; #H;
+ napply (or4_elim P1 P2 P3 P4 ? H);
+ ##[ ##1: #H1; napply (or4_intro1 P1 P3 P2 P4 H1)
+ ##| ##2: #H1; napply (or4_intro3 P1 P3 P2 P4 H1)
+ ##| ##3: #H1; napply (or4_intro2 P1 P3 P2 P4 H1)
+ ##| ##4: #H1; napply (or4_intro4 P1 P3 P2 P4 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or4_24 : âP1,P2,P3,P4:Prop.Or4 P1 P2 P3 P4 â Or4 P1 P4 P3 P2.
+ #P1; #P2; #P3; #P4; #H;
+ napply (or4_elim P1 P2 P3 P4 ? H);
+ ##[ ##1: #H1; napply (or4_intro1 P1 P4 P3 P2 H1)
+ ##| ##2: #H1; napply (or4_intro4 P1 P4 P3 P2 H1)
+ ##| ##3: #H1; napply (or4_intro3 P1 P4 P3 P2 H1)
+ ##| ##4: #H1; napply (or4_intro2 P1 P4 P3 P2 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or4_34 : âP1,P2,P3,P4:Prop.Or4 P1 P2 P3 P4 â Or4 P1 P2 P4 P3.
+ #P1; #P2; #P3; #P4; #H;
+ napply (or4_elim P1 P2 P3 P4 ? H);
+ ##[ ##1: #H1; napply (or4_intro1 P1 P2 P4 P3 H1)
+ ##| ##2: #H1; napply (or4_intro2 P1 P2 P4 P3 H1)
+ ##| ##3: #H1; napply (or4_intro4 P1 P2 P4 P3 H1)
+ ##| ##4: #H1; napply (or4_intro3 P1 P2 P4 P3 H1)
+ ##]
+nqed.
+
+ninductive Or5 (A,B,C,D,E:Prop) : Prop â
+ or5_intro1 : A â (Or5 A B C D E)
+| or5_intro2 : B â (Or5 A B C D E)
+| or5_intro3 : C â (Or5 A B C D E)
+| or5_intro4 : D â (Or5 A B C D E)
+| or5_intro5 : E â (Or5 A B C D E).
+
+nlemma or5_elim
+ : âP1,P2,P3,P4,P5,Q:Prop.Or5 P1 P2 P3 P4 P5 â âf1:P1 â Q.âf2:P2 â Q.
+ âf3:P3 â Q.âf4:P4 â Q.âf5:P5 â Q.Q.
+ #P1; #P2; #P3; #P4; #P5; #Q; #H; #f1; #f2; #f3; #f4; #f5;
+ napply (Or5_ind P1 P2 P3 P4 P5 ? f1 f2 f3 f4 f5 ?);
+ napply H.
+nqed.
+
+nlemma symmetric_or5_12 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P2 P1 P3 P4 P5.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro2 P2 P1 P3 P4 P5 H1)
+ ##| ##2: #H1; napply (or5_intro1 P2 P1 P3 P4 P5 H1)
+ ##| ##3: #H1; napply (or5_intro3 P2 P1 P3 P4 P5 H1)
+ ##| ##4: #H1; napply (or5_intro4 P2 P1 P3 P4 P5 H1)
+ ##| ##5: #H1; napply (or5_intro5 P2 P1 P3 P4 P5 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_13 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P3 P2 P1 P4 P5.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro3 P3 P2 P1 P4 P5 H1)
+ ##| ##2: #H1; napply (or5_intro2 P3 P2 P1 P4 P5 H1)
+ ##| ##3: #H1; napply (or5_intro1 P3 P2 P1 P4 P5 H1)
+ ##| ##4: #H1; napply (or5_intro4 P3 P2 P1 P4 P5 H1)
+ ##| ##5: #H1; napply (or5_intro5 P3 P2 P1 P4 P5 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_14 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P4 P2 P3 P1 P5.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro4 P4 P2 P3 P1 P5 H1)
+ ##| ##2: #H1; napply (or5_intro2 P4 P2 P3 P1 P5 H1)
+ ##| ##3: #H1; napply (or5_intro3 P4 P2 P3 P1 P5 H1)
+ ##| ##4: #H1; napply (or5_intro1 P4 P2 P3 P1 P5 H1)
+ ##| ##5: #H1; napply (or5_intro5 P4 P2 P3 P1 P5 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_15 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P5 P2 P3 P4 P1.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro5 P5 P2 P3 P4 P1 H1)
+ ##| ##2: #H1; napply (or5_intro2 P5 P2 P3 P4 P1 H1)
+ ##| ##3: #H1; napply (or5_intro3 P5 P2 P3 P4 P1 H1)
+ ##| ##4: #H1; napply (or5_intro4 P5 P2 P3 P4 P1 H1)
+ ##| ##5: #H1; napply (or5_intro1 P5 P2 P3 P4 P1 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_23 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P1 P3 P2 P4 P5.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro1 P1 P3 P2 P4 P5 H1)
+ ##| ##2: #H1; napply (or5_intro3 P1 P3 P2 P4 P5 H1)
+ ##| ##3: #H1; napply (or5_intro2 P1 P3 P2 P4 P5 H1)
+ ##| ##4: #H1; napply (or5_intro4 P1 P3 P2 P4 P5 H1)
+ ##| ##5: #H1; napply (or5_intro5 P1 P3 P2 P4 P5 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_24 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P1 P4 P3 P2 P5.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro1 P1 P4 P3 P2 P5 H1)
+ ##| ##2: #H1; napply (or5_intro4 P1 P4 P3 P2 P5 H1)
+ ##| ##3: #H1; napply (or5_intro3 P1 P4 P3 P2 P5 H1)
+ ##| ##4: #H1; napply (or5_intro2 P1 P4 P3 P2 P5 H1)
+ ##| ##5: #H1; napply (or5_intro5 P1 P4 P3 P2 P5 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_25 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P1 P5 P3 P4 P2.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro1 P1 P5 P3 P4 P2 H1)
+ ##| ##2: #H1; napply (or5_intro5 P1 P5 P3 P4 P2 H1)
+ ##| ##3: #H1; napply (or5_intro3 P1 P5 P3 P4 P2 H1)
+ ##| ##4: #H1; napply (or5_intro4 P1 P5 P3 P4 P2 H1)
+ ##| ##5: #H1; napply (or5_intro2 P1 P5 P3 P4 P2 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_34 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P1 P2 P4 P3 P5.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro1 P1 P2 P4 P3 P5 H1)
+ ##| ##2: #H1; napply (or5_intro2 P1 P2 P4 P3 P5 H1)
+ ##| ##3: #H1; napply (or5_intro4 P1 P2 P4 P3 P5 H1)
+ ##| ##4: #H1; napply (or5_intro3 P1 P2 P4 P3 P5 H1)
+ ##| ##5: #H1; napply (or5_intro5 P1 P2 P4 P3 P5 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_35 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P1 P2 P5 P4 P3.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro1 P1 P2 P5 P4 P3 H1)
+ ##| ##2: #H1; napply (or5_intro2 P1 P2 P5 P4 P3 H1)
+ ##| ##3: #H1; napply (or5_intro5 P1 P2 P5 P4 P3 H1)
+ ##| ##4: #H1; napply (or5_intro4 P1 P2 P5 P4 P3 H1)
+ ##| ##5: #H1; napply (or5_intro3 P1 P2 P5 P4 P3 H1)
+ ##]
+nqed.
+
+nlemma symmetric_or5_45 : âP1,P2,P3,P4,P5:Prop.Or5 P1 P2 P3 P4 P5 â Or5 P1 P2 P3 P5 P4.
+ #P1; #P2; #P3; #P4; #P5; #H;
+ napply (or5_elim P1 P2 P3 P4 P5 ? H);
+ ##[ ##1: #H1; napply (or5_intro1 P1 P2 P3 P5 P4 H1)
+ ##| ##2: #H1; napply (or5_intro2 P1 P2 P3 P5 P4 H1)
+ ##| ##3: #H1; napply (or5_intro3 P1 P2 P3 P5 P4 H1)
+ ##| ##4: #H1; napply (or5_intro5 P1 P2 P3 P5 P4 H1)
+ ##| ##5: #H1; napply (or5_intro4 P1 P2 P3 P5 P4 H1)
+ ##]
+nqed.
+
+ninductive ex (A:Type) (Q:A â Prop) : Prop â
+ ex_intro: âx:A.Q x â ex A Q.
+
+interpretation "exists" 'exists x = (ex ? x).
+
+ninductive ex2 (A:Type) (Q,R:A â Prop) : Prop â
+ ex_intro2: âx:A.Q x â R x â ex2 A Q R.
+
+(* higher_order_defs/relations *)
+
+ndefinition relation : Type â Type â
+λA.A â A â Prop.
+
+ndefinition reflexive : âA:Type.âR:relation A.Prop â
+λA.λR.âx:A.R x x.
+
+ndefinition symmetric : âA:Type.âR:relation A.Prop â
+λA.λR.âx,y:A.R x y â R y x.
+
+ndefinition transitive : âA:Type.âR:relation A.Prop â
+λA.λR.âx,y,z:A.R x y â R y z â R x z.
+
+ndefinition irreflexive : âA:Type.âR:relation A.Prop â
+λA.λR.âx:A.¬ (R x x).
+
+ndefinition cotransitive : âA:Type.âR:relation A.Prop â
+λA.λR.âx,y:A.R x y â âz:A. R x z ⨠R z y.
+
+ndefinition tight_apart : âA:Type.âeq,ap:relation A.Prop â
+λA.λeq,ap.âx,y:A. (¬ (ap x y) â eq x y) ⧠(eq x y â ¬ (ap x y)).
+
+ndefinition antisymmetric : âA:Type.âR:relation A.Prop â
+λA.λR.âx,y:A.R x y â ¬ (R y x).
+
+(* logic/equality.ma *)
+
+ninductive eq (A:Type) (x:A) : A â Prop â
+ refl_eq : eq A x x.
+
+ndefinition refl â refl_eq.
+
+interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
+
+interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
+
+nlemma eq_f : âT1,T2:Type.âx,y:T1.âf:T1 â T2.x = y â (f x) = (f y).
+ #T1; #T2; #x; #y; #f; #H;
+ nrewrite < H;
+ napply refl_eq.
+nqed.
+
+nlemma eq_f2 : âT1,T2,T3:Type.âx1,y1:T1.âx2,y2:T2.âf:T1 â T2 â T3.x1 = y1 â x2 = y2 â f x1 x2 = f y1 y2.
+ #T1; #T2; #T3; #x1; #y1; #x2; #y2; #f; #H1; #H2;
+ nrewrite < H1;
+ nrewrite < H2;
+ napply refl_eq.
+nqed.
+
+nlemma neqf_to_neq : âT1,T2:Type.âx,y:T1.âf:T1 â T2.((f x) â (f y)) â x â y.
+ #T1; #T2; #x; #y; #f;
+ nnormalize; #H; #H1;
+ napply (H (eq_f ⦠H1)).
+nqed.
+
+nlemma symmetric_eq: âA:Type. symmetric A (eq A).
+ #A;
+ nnormalize;
+ #x; #y; #H;
+ nrewrite < H;
+ napply refl_eq.
+nqed.
+
+nlemma eq_ind_r: âA:Type[0].âx:A.âP:A â Prop.P x â ây:A.y=x â P y.
+ #A; #x; #P; #H; #y; #H1;
+ nrewrite < (symmetric_eq ⦠H1);
+ napply H.
+nqed.
+
+ndefinition R0 â λT:Type[0].λt:T.t.
+
+ndefinition R1 â eq_rect_Type0.
+
+ndefinition R2 :
+ âT0:Type[0].
+ âa0:T0.
+ âT1:âx0:T0. a0=x0 â Type[0].
+ âa1:T1 a0 (refl_eq ? a0).
+ âT2:âx0:T0. âp0:a0=x0. âx1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 â Type[0].
+ âa2:T2 a0 (refl_eq ? a0) a1 (refl_eq ? a1).
+ âb0:T0.
+ âe0:a0 = b0.
+ âb1: T1 b0 e0.
+ âe1:R1 ?? T1 a1 ? e0 = b1.
+ T2 b0 e0 b1 e1.
+ #T0;#a0;#T1;#a1;#T2;#a2;#b0;#e0;#b1;#e1;
+ napply (eq_rect_Type0 ????? e1);
+ napply (R1 ?? ? ?? e0);
+ napply a2;
+nqed.
+
+ndefinition R3 :
+ âT0:Type[0].
+ âa0:T0.
+ âT1:âx0:T0. a0=x0 â Type[0].
+ âa1:T1 a0 (refl_eq ? a0).
+ âT2:âx0:T0. âp0:a0=x0. âx1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 â Type[0].
+ âa2:T2 a0 (refl_eq ? a0) a1 (refl_eq ? a1).
+ âT3:âx0:T0. âp0:a0=x0. âx1:T1 x0 p0.âp1:R1 ?? T1 a1 ? p0 = x1.
+ âx2:T2 x0 p0 x1 p1.R2 ???? T2 a2 x0 p0 ? p1 = x2 â Type[0].
+ âa3:T3 a0 (refl_eq ? a0) a1 (refl_eq ? a1) a2 (refl_eq ? a2).
+ âb0:T0.
+ âe0:a0 = b0.
+ âb1: T1 b0 e0.
+ âe1:R1 ?? T1 a1 ? e0 = b1.
+ âb2: T2 b0 e0 b1 e1.
+ âe2:R2 ???? T2 a2 b0 e0 ? e1 = b2.
+ T3 b0 e0 b1 e1 b2 e2.
+ #T0;#a0;#T1;#a1;#T2;#a2;#T3;#a3;#b0;#e0;#b1;#e1;#b2;#e2;
+ napply (eq_rect_Type0 ????? e2);
+ napply (R2 ?? ? ???? e0 ? e1);
+ napply a3;
+nqed.
+
+ndefinition R4 :
+ âT0:Type[0].
+ âa0:T0.
+ âT1:âx0:T0. eq T0 a0 x0 â Type[0].
+ âa1:T1 a0 (refl_eq T0 a0).
+ âT2:âx0:T0. âp0:eq (T0 â¦) a0 x0. âx1:T1 x0 p0.eq (T1 â¦) (R1 T0 a0 T1 a1 x0 p0) x1 â Type[0].
+ âa2:T2 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1).
+ âT3:âx0:T0. âp0:eq (T0 â¦) a0 x0. âx1:T1 x0 p0.âp1:eq (T1 â¦) (R1 T0 a0 T1 a1 x0 p0) x1.
+ âx2:T2 x0 p0 x1 p1.eq (T2 â¦) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 â Type[0].
+ âa3:T3 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1)
+ a2 (refl_eq (T2 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1)) a2).
+ âT4:âx0:T0. âp0:eq (T0 â¦) a0 x0. âx1:T1 x0 p0.âp1:eq (T1 â¦) (R1 T0 a0 T1 a1 x0 p0) x1.
+ âx2:T2 x0 p0 x1 p1.âp2:eq (T2 â¦) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
+ âx3:T3 x0 p0 x1 p1 x2 p2.âp3:eq (T3 â¦) (R3 T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3.
+ Type[0].
+ âa4:T4 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1)
+ a2 (refl_eq (T2 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1)) a2)
+ a3 (refl_eq (T3 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1)
+ a2 (refl_eq (T2 a0 (refl_eq T0 a0) a1 (refl_eq (T1 a0 (refl_eq T0 a0)) a1)) a2))
+ a3).
+ âb0:T0.
+ âe0:eq (T0 â¦) a0 b0.
+ âb1: T1 b0 e0.
+ âe1:eq (T1 â¦) (R1 T0 a0 T1 a1 b0 e0) b1.
+ âb2: T2 b0 e0 b1 e1.
+ âe2:eq (T2 â¦) (R2 T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
+ âb3: T3 b0 e0 b1 e1 b2 e2.
+ âe3:eq (T3 â¦) (R3 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
+ T4 b0 e0 b1 e1 b2 e2 b3 e3.
+ #T0;#a0;#T1;#a1;#T2;#a2;#T3;#a3;#T4;#a4;#b0;#e0;#b1;#e1;#b2;#e2;#b3;#e3;
+ napply (eq_rect_Type0 ????? e3);
+ napply (R3 ????????? e0 ? e1 ? e2);
+ napply a4;
+nqed.
+
+naxiom streicherK : âT:Type[0].ât:T.âP:t = t â Type[2].P (refl ? t) â âp.P p.
+
+nlemma symmetric_neq : âT:Type.âx,y:T.x â y â y â x.
+ #T; #x; #y;
+ nnormalize;
+ #H; #H1;
+ nrewrite > H1 in H:(%); #H;
+ napply (H (refl_eq â¦)).
+nqed.
+
+ndefinition relationT : Type â Type â Type â
+λA,T:Type.A â A â T.
+
+ndefinition symmetricT: âA,T:Type.âR:relationT A T.Prop â
+λA,T.λR.âx,y:A.R x y = R y x.
+
+ndefinition associative : âA:Type.âR:relationT A A.Prop â
+λA.λR.âx,y,z:A.R (R x y) z = R x (R y z).
diff --git a/helm/software/matita/contribs/ng_assembly2/depends b/helm/software/matita/contribs/ng_assembly2/depends
new file mode 100644
index 000000000..cadd99a6a
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/depends
@@ -0,0 +1,27 @@
+common/nat.ma common/comp.ma num/bool_lemmas.ma
+num/word32.ma num/word16.ma
+common/ascii_base.ma num/bool.ma
+num/bitrigesim.ma common/comp.ma num/bool_lemmas.ma
+common/pts.ma
+common/prod.ma common/comp.ma common/prod_base.ma num/bool_lemmas.ma
+common/prod_base.ma num/bool.ma
+universe/universe.ma common/nelist.ma common/prod.ma
+num/bool_lemmas.ma num/bool.ma
+common/comp.ma common/hints_declaration.ma num/bool.ma
+num/exadecim.ma num/bool_lemmas.ma num/comp_ext.ma num/oct.ma
+num/word24.ma num/byte8.ma
+common/theory.ma common/pts.ma
+common/sigma.ma common/theory.ma
+num/oct.ma common/comp.ma num/bool_lemmas.ma
+common/hints_declaration.ma common/pts.ma
+num/byte8.ma num/bitrigesim.ma num/comp_num.ma num/exadecim.ma
+common/option.ma common/comp.ma common/option_base.ma num/bool_lemmas.ma
+num/bool.ma common/theory.ma
+common/option_base.ma num/bool.ma
+num/word16.ma common/nat.ma num/byte8.ma
+common/string.ma common/ascii.ma common/list.ma
+common/ascii.ma common/ascii_base.ma common/comp.ma num/bool_lemmas.ma
+common/list.ma common/comp.ma common/nat.ma common/option.ma
+num/comp_num.ma num/bool_lemmas.ma num/comp_ext.ma
+num/comp_ext.ma common/comp.ma common/prod.ma
+common/nelist.ma common/list.ma
diff --git a/helm/software/matita/contribs/ng_assembly2/num/bitrigesim.ma b/helm/software/matita/contribs/ng_assembly2/num/bitrigesim.ma
new file mode 100755
index 000000000..82c14bdc2
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/bitrigesim.ma
@@ -0,0 +1,262 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "num/bool_lemmas.ma".
+
+(* ************* *)
+(* BITRIGESIMALI *)
+(* ************* *)
+
+ninductive bitrigesim : Type â
+ t00: bitrigesim
+| t01: bitrigesim
+| t02: bitrigesim
+| t03: bitrigesim
+| t04: bitrigesim
+| t05: bitrigesim
+| t06: bitrigesim
+| t07: bitrigesim
+| t08: bitrigesim
+| t09: bitrigesim
+| t0A: bitrigesim
+| t0B: bitrigesim
+| t0C: bitrigesim
+| t0D: bitrigesim
+| t0E: bitrigesim
+| t0F: bitrigesim
+| t10: bitrigesim
+| t11: bitrigesim
+| t12: bitrigesim
+| t13: bitrigesim
+| t14: bitrigesim
+| t15: bitrigesim
+| t16: bitrigesim
+| t17: bitrigesim
+| t18: bitrigesim
+| t19: bitrigesim
+| t1A: bitrigesim
+| t1B: bitrigesim
+| t1C: bitrigesim
+| t1D: bitrigesim
+| t1E: bitrigesim
+| t1F: bitrigesim.
+
+(* operatore = *)
+ndefinition eq_bit â
+λt1,t2:bitrigesim.
+ match t1 with
+ [ t00 â match t2 with [ t00 â true | _ â false ] | t01 â match t2 with [ t01 â true | _ â false ]
+ | t02 â match t2 with [ t02 â true | _ â false ] | t03 â match t2 with [ t03 â true | _ â false ]
+ | t04 â match t2 with [ t04 â true | _ â false ] | t05 â match t2 with [ t05 â true | _ â false ]
+ | t06 â match t2 with [ t06 â true | _ â false ] | t07 â match t2 with [ t07 â true | _ â false ]
+ | t08 â match t2 with [ t08 â true | _ â false ] | t09 â match t2 with [ t09 â true | _ â false ]
+ | t0A â match t2 with [ t0A â true | _ â false ] | t0B â match t2 with [ t0B â true | _ â false ]
+ | t0C â match t2 with [ t0C â true | _ â false ] | t0D â match t2 with [ t0D â true | _ â false ]
+ | t0E â match t2 with [ t0E â true | _ â false ] | t0F â match t2 with [ t0F â true | _ â false ]
+ | t10 â match t2 with [ t10 â true | _ â false ] | t11 â match t2 with [ t11 â true | _ â false ]
+ | t12 â match t2 with [ t12 â true | _ â false ] | t13 â match t2 with [ t13 â true | _ â false ]
+ | t14 â match t2 with [ t14 â true | _ â false ] | t15 â match t2 with [ t15 â true | _ â false ]
+ | t16 â match t2 with [ t16 â true | _ â false ] | t17 â match t2 with [ t17 â true | _ â false ]
+ | t18 â match t2 with [ t18 â true | _ â false ] | t19 â match t2 with [ t19 â true | _ â false ]
+ | t1A â match t2 with [ t1A â true | _ â false ] | t1B â match t2 with [ t1B â true | _ â false ]
+ | t1C â match t2 with [ t1C â true | _ â false ] | t1D â match t2 with [ t1D â true | _ â false ]
+ | t1E â match t2 with [ t1E â true | _ â false ] | t1F â match t2 with [ t1F â true | _ â false ]
+ ].
+
+(* iteratore sui bitrigesimali *)
+ndefinition forall_bit â λP.
+ P t00 â P t01 â P t02 â P t03 â P t04 â P t05 â P t06 â P t07 â
+ P t08 â P t09 â P t0A â P t0B â P t0C â P t0D â P t0E â P t0F â
+ P t10 â P t11 â P t12 â P t13 â P t14 â P t15 â P t16 â P t17 â
+ P t18 â P t19 â P t1A â P t1B â P t1C â P t1D â P t1E â P t1F.
+
+(* operatore successore *)
+ndefinition succ_bit â
+λn.match n with
+ [ t00 â t01 | t01 â t02 | t02 â t03 | t03 â t04 | t04 â t05 | t05 â t06 | t06 â t07 | t07 â t08
+ | t08 â t09 | t09 â t0A | t0A â t0B | t0B â t0C | t0C â t0D | t0D â t0E | t0E â t0F | t0F â t10
+ | t10 â t11 | t11 â t12 | t12 â t13 | t13 â t14 | t14 â t15 | t15 â t16 | t16 â t17 | t17 â t18
+ | t18 â t19 | t19 â t1A | t1A â t1B | t1B â t1C | t1C â t1D | t1D â t1E | t1E â t1F | t1F â t00
+ ].
+
+(* bitrigesimali ricorsivi *)
+ninductive rec_bitrigesim : bitrigesim â Type â
+ bi_O : rec_bitrigesim t00
+| bi_S : ân.rec_bitrigesim n â rec_bitrigesim (succ_bit n).
+
+(* bitrigesimali â bitrigesimali ricorsivi *)
+ndefinition bit_to_recbit â
+λn.match n return λx.rec_bitrigesim x with
+ [ t00 â bi_O
+ | t01 â bi_S ? bi_O
+ | t02 â bi_S ? (bi_S ? bi_O)
+ | t03 â bi_S ? (bi_S ? (bi_S ? bi_O))
+ | t04 â bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))
+ | t05 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))
+ | t06 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))
+ | t07 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))
+ | t08 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ bi_O)))))))
+ | t09 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? bi_O))))))))
+ | t0A â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? bi_O)))))))))
+ | t0B â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))
+ | t0C â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))))))))
+ | t0D â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))
+ | t0E â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))))))))))
+ | t0F â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))
+ | t10 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ bi_O)))))))))))))))
+ | t11 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? bi_O))))))))))))))))
+ | t12 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? bi_O)))))))))))))))))
+ | t13 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))))))
+ | t14 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))))))))))))))))
+ | t15 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))))))))
+ | t16 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))))))))))))))))))
+ | t17 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))))))))))
+ | t18 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ bi_O)))))))))))))))))))))))
+ | t19 â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? bi_O))))))))))))))))))))))))
+ | t1A â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? bi_O)))))))))))))))))))))))))
+ | t1B â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))))))))))))))
+ | t1C â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))))))))))))))))))))))))
+ | t1D â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))))))))))))))))
+ | t1E â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O)))))))))))))))))))))))))))))
+ | t1F â bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ?
+ (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? (bi_S ? bi_O))))))))))))))))))))))))))))))
+ ].
+
+ndefinition bitrigesim_destruct_aux â
+Î t1,t2:bitrigesim.Î P:Prop.t1 = t2 â
+ match eq_bit t1 t2 with [ true â P â P | false â P ].
+
+ndefinition bitrigesim_destruct : bitrigesim_destruct_aux.
+ #t1; #t2; #P; #H;
+ nrewrite < H;
+ nelim t1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+nlemma eq_to_eqbit : ân1,n2.n1 = n2 â eq_bit n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqbit_to_neq : ân1,n2.eq_bit n1 n2 = false â n1 â n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_bit n1 n2 = true) â¦);
+ ##[ ##1: napply (eq_to_eqbit n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue ⦠H)
+ ##]
+nqed.
+
+(* !!! per brevita... *)
+naxiom eqbit_to_eq : ât1,t2.eq_bit t1 t2 = true â t1 = t2.
+
+nlemma neq_to_neqbit : ân1,n2.n1 â n2 â eq_bit n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_bit n1 n2));
+ napply (not_to_not (eq_bit n1 n2 = true) (n1 = n2) ? H);
+ napply (eqbit_to_eq n1 n2).
+nqed.
+
+nlemma decidable_bit : âx,y:bitrigesim.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_bit x y = true) (eq_bit x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x â y) (eqbit_to_eq ⦠H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x â y) (neqbit_to_neq ⦠H))
+ ##]
+nqed.
+
+nlemma symmetric_eqbit : symmetricT bitrigesim bool eq_bit.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_bit n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqbit n1 n2 H);
+ napply (symmetric_eq ? (eq_bit n2 n1) false);
+ napply (neq_to_neqbit n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma bitrigesim_is_comparable : comparable.
+ @ bitrigesim
+ ##[ napply t00
+ ##| napply forall_bit
+ ##| napply eq_bit
+ ##| napply eqbit_to_eq
+ ##| napply eq_to_eqbit
+ ##| napply neqbit_to_neq
+ ##| napply neq_to_neqbit
+ ##| napply decidable_bit
+ ##| napply symmetric_eqbit
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr bitrigesim_is_comparable â¡ bitrigesim.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/bool.ma b/helm/software/matita/contribs/ng_assembly2/num/bool.ma
new file mode 100755
index 000000000..8092713f2
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/bool.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/theory.ma".
+
+(* ******** *)
+(* BOOLEANI *)
+(* ******** *)
+
+ninductive bool : Type â
+ true : bool
+| false : bool.
+
+(* operatori booleani *)
+
+ndefinition eq_bool â
+λb1,b2:bool.match b1 with
+ [ true â match b2 with [ true â true | false â false ]
+ | false â match b2 with [ true â false | false â true ]
+ ].
+
+ndefinition not_bool â
+λb:bool.match b with [ true â false | false â true ].
+
+ndefinition and_bool â
+λb1,b2:bool.match b1 with
+ [ true â b2 | false â false ].
+
+ndefinition or_bool â
+λb1,b2:bool.match b1 with
+ [ true â true | false â b2 ].
+
+ndefinition xor_bool â
+λb1,b2:bool.match b1 with
+ [ true â not_bool b2
+ | false â b2 ].
+
+(* \ominus *)
+notation "hvbox(â a)" non associative with precedence 36
+ for @{ 'not_bool $a }.
+interpretation "not_bool" 'not_bool x = (not_bool x).
+
+(* \otimes *)
+notation "hvbox(a break â b)" left associative with precedence 35
+ for @{ 'and_bool $a $b }.
+interpretation "and_bool" 'and_bool x y = (and_bool x y).
+
+(* \oplus *)
+notation "hvbox(a break â b)" left associative with precedence 34
+ for @{ 'or_bool $a $b }.
+interpretation "or_bool" 'or_bool x y = (or_bool x y).
+
+(* \odot *)
+notation "hvbox(a break â b)" left associative with precedence 33
+ for @{ 'xor_bool $a $b }.
+interpretation "xor_bool" 'xor_bool x y = (xor_bool x y).
+
+ndefinition boolRelation : Type â Type â
+λA:Type.A â A â bool.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/bool_lemmas.ma b/helm/software/matita/contribs/ng_assembly2/num/bool_lemmas.ma
new file mode 100755
index 000000000..660dd7e67
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/bool_lemmas.ma
@@ -0,0 +1,322 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/bool.ma".
+
+(* ******** *)
+(* BOOLEANI *)
+(* ******** *)
+
+ndefinition bool_destruct_aux â
+Î b1,b2:bool.Î P:Prop.b1 = b2 â
+ match eq_bool b1 b2 with [ true â P â P | false â P ].
+
+ndefinition bool_destruct : bool_destruct_aux.
+ #b1; #b2; #P; #H;
+ nrewrite < H;
+ nelim b1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+nlemma symmetric_eqbool : symmetricT bool bool eq_bool.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_andbool : symmetricT bool bool and_bool.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_andbool : âb1,b2,b3.((b1 â b2) â b3) = (b1 â (b2 â b3)).
+ #b1; #b2; #b3;
+ nelim b1;
+ nelim b2;
+ nelim b3;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_orbool : symmetricT bool bool or_bool.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_orbool : âb1,b2,b3.((b1 â b2) â b3) = (b1 â (b2 â b3)).
+ #b1; #b2; #b3;
+ nelim b1;
+ nelim b2;
+ nelim b3;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_xorbool : symmetricT bool bool xor_bool.
+ #b1; #b2;
+ nelim b1;
+ nelim b2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma associative_xorbool : âb1,b2,b3.((b1 â b2) â b3) = (b1 â (b2 â b3)).
+ #b1; #b2; #b3;
+ nelim b1;
+ nelim b2;
+ nelim b3;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma eqbool_to_eq : âb1,b2:bool.(eq_bool b1 b2 = true) â (b1 = b2).
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,4: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma eq_to_eqbool : âb1,b2.b1 = b2 â eq_bool b1 b2 = true.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,4: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma decidable_bool : âx,y:bool.decidable (x = y).
+ #x; #y;
+ nnormalize;
+ nelim x;
+ nelim y;
+ ##[ ##1,4: napply (or2_intro1 (? = ?) (? â ?) â¦); napply refl_eq
+ ##| ##*: napply (or2_intro2 (? = ?) (? â ?) â¦);
+ nnormalize; #H;
+ napply False_ind;
+ ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma decidable_bexpr : âx.(x = true) ⨠(x = false).
+ #x; ncases x;
+ ##[ ##1: napply (or2_intro1 (true = true) (true = false) (refl_eq â¦))
+ ##| ##2: napply (or2_intro2 (false = true) (false = false) (refl_eq â¦))
+ ##]
+nqed.
+
+nlemma neqbool_to_neq : âb1,b2:bool.(eq_bool b1 b2 = false) â (b1 â b2).
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,4: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##| ##*: #H; #H1; ndestruct (*napply (bool_destruct ⦠H1)*)
+ ##]
+nqed.
+
+nlemma neq_to_neqbool : âb1,b2.b1 â b2 â eq_bool b1 b2 = false.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,4: #H; nelim (H (refl_eq â¦))
+ ##| ##*: #H; napply refl_eq
+ ##]
+nqed.
+
+nlemma eqfalse_to_neqtrue : âx.x = false â x â 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 : âx.x = true â x â 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 : âx.x â false â x = true.
+ #x; nelim x;
+ ##[ ##1: #H; napply refl_eq
+ ##| ##2: nnormalize; #H; nelim (H (refl_eq â¦))
+ ##]
+nqed.
+
+nlemma neqtrue_to_eqfalse : âx.x â true â x = false.
+ #x; nelim x;
+ ##[ ##1: nnormalize; #H; nelim (H (refl_eq â¦))
+ ##| ##2: #H; napply refl_eq
+ ##]
+nqed.
+
+nlemma andb_true_true_l: âb1,b2.(b1 â b2) = true â b1 = true.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,2: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma andb_true_true_r: âb1,b2.(b1 â b2) = true â b2 = true.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1,3: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma andb_false2
+ : âb1,b2.(b1 â b2) = false â
+ (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
+ : âb1,b2,b3.(b1 â b2 â b3) = false â
+ 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
+ : âb1,b2,b3,b4.(b1 â b2 â b3 â b4) = false â
+ 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
+ : âb1,b2,b3,b4,b5.(b1 â b2 â b3 â b4 â b5) = false â
+ 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 : âb.(false â b) = false.
+ #b; nnormalize; napply refl_eq. nqed.
+nlemma andb_false2_2 : âb.(b â false) = false.
+ #b; nelim b; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false3_1 : âb1,b2.(false â b1 â b2) = false.
+ #b1; #b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false3_2 : âb1,b2.(b1 â false â b2) = false.
+ #b1; #b2; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false3_3 : âb1,b2.(b1 â b2 â false) = false.
+ #b1; #b2; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false4_1 : âb1,b2,b3.(false â b1 â b2 â b3) = false.
+ #b1; #b2; #b3; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_2 : âb1,b2,b3.(b1 â false â b2 â b3) = false.
+ #b1; #b2; #b3; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_3 : âb1,b2,b3.(b1 â b2 â false â b3) = false.
+ #b1; #b2; #b3; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_4 : âb1,b2,b3.(b1 â b2 â b3 â false) = false.
+ #b1; #b2; #b3; nelim b1; nelim b2; nelim b3; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false5_1 : âb1,b2,b3,b4.(false â b1 â b2 â b3 â b4) = false.
+ #b1; #b2; #b3; #b4; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_2 : âb1,b2,b3,b4.(b1 â false â b2 â b3 â b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_3 : âb1,b2,b3,b4.(b1 â b2 â false â b3 â b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_4 : âb1,b2,b3,b4.(b1 â b2 â b3 â false â b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nelim b3; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_5 : âb1,b2,b3,b4.(b1 â b2 â b3 â b4 â false) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nelim b3; nelim b4; nnormalize; napply refl_eq. nqed.
+
+nlemma orb_false_false_l : âb1,b2:bool.(b1 â b2) = false â b1 = false.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##4: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma orb_false_false_r : âb1,b2:bool.(b1 â b2) = false â b2 = false.
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##4: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/byte8.ma b/helm/software/matita/contribs/ng_assembly2/num/byte8.ma
new file mode 100755
index 000000000..c45d71c6e
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/byte8.ma
@@ -0,0 +1,337 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/comp_num.ma".
+include "num/exadecim.ma".
+include "num/bitrigesim.ma".
+
+(* **** *)
+(* BYTE *)
+(* **** *)
+
+ndefinition byte8 â comp_num exadecim.
+ndefinition mk_byte8 â λe1,e2.mk_comp_num exadecim e1 e2.
+
+(* \langle \rangle *)
+notation "â©x,yâª" non associative with precedence 80
+ for @{ mk_comp_num exadecim $x $y }.
+
+ndefinition byte8_is_comparable_ext â cn_is_comparable_ext exadecim_is_comparable_ext.
+unification hint 0 â ⢠carr (comp_base byte8_is_comparable_ext) â¡ comp_num exadecim.
+unification hint 0 â ⢠carr (comp_base byte8_is_comparable_ext) â¡ byte8.
+
+(* operatore estensione unsigned *)
+ndefinition extu_b8 â λe2.â©zeroc ?,e2âª.
+
+(* operatore estensione signed *)
+ndefinition exts_b8 â
+λe2.â©(match getMSBc ? e2 with
+ [ true â predc ? (zeroc ?) | false â zeroc ? ]),e2âª.
+
+(* operatore moltiplicazione senza segno: e*e=[0x00,0xE1] *)
+ndefinition mulu_ex â
+λe1,e2:exadecim.match e1 with
+ [ x0 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x0⪠| x2 â â©x0,x0⪠| x3 â â©x0,x0âª
+ | x4 â â©x0,x0⪠| x5 â â©x0,x0⪠| x6 â â©x0,x0⪠| x7 â â©x0,x0âª
+ | x8 â â©x0,x0⪠| x9 â â©x0,x0⪠| xA â â©x0,x0⪠| xB â â©x0,x0âª
+ | xC â â©x0,x0⪠| xD â â©x0,x0⪠| xE â â©x0,x0⪠| xF â â©x0,x0⪠]
+ | x1 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x1⪠| x2 â â©x0,x2⪠| x3 â â©x0,x3âª
+ | x4 â â©x0,x4⪠| x5 â â©x0,x5⪠| x6 â â©x0,x6⪠| x7 â â©x0,x7âª
+ | x8 â â©x0,x8⪠| x9 â â©x0,x9⪠| xA â â©x0,xA⪠| xB â â©x0,xBâª
+ | xC â â©x0,xC⪠| xD â â©x0,xD⪠| xE â â©x0,xE⪠| xF â â©x0,xF⪠]
+ | x2 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x2⪠| x2 â â©x0,x4⪠| x3 â â©x0,x6âª
+ | x4 â â©x0,x8⪠| x5 â â©x0,xA⪠| x6 â â©x0,xC⪠| x7 â â©x0,xEâª
+ | x8 â â©x1,x0⪠| x9 â â©x1,x2⪠| xA â â©x1,x4⪠| xB â â©x1,x6âª
+ | xC â â©x1,x8⪠| xD â â©x1,xA⪠| xE â â©x1,xC⪠| xF â â©x1,xE⪠]
+ | x3 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x3⪠| x2 â â©x0,x6⪠| x3 â â©x0,x9âª
+ | x4 â â©x0,xC⪠| x5 â â©x0,xF⪠| x6 â â©x1,x2⪠| x7 â â©x1,x5âª
+ | x8 â â©x1,x8⪠| x9 â â©x1,xB⪠| xA â â©x1,xE⪠| xB â â©x2,x1âª
+ | xC â â©x2,x4⪠| xD â â©x2,x7⪠| xE â â©x2,xA⪠| xF â â©x2,xD⪠]
+ | x4 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x4⪠| x2 â â©x0,x8⪠| x3 â â©x0,xCâª
+ | x4 â â©x1,x0⪠| x5 â â©x1,x4⪠| x6 â â©x1,x8⪠| x7 â â©x1,xCâª
+ | x8 â â©x2,x0⪠| x9 â â©x2,x4⪠| xA â â©x2,x8⪠| xB â â©x2,xCâª
+ | xC â â©x3,x0⪠| xD â â©x3,x4⪠| xE â â©x3,x8⪠| xF â â©x3,xC⪠]
+ | x5 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x5⪠| x2 â â©x0,xA⪠| x3 â â©x0,xFâª
+ | x4 â â©x1,x4⪠| x5 â â©x1,x9⪠| x6 â â©x1,xE⪠| x7 â â©x2,x3âª
+ | x8 â â©x2,x8⪠| x9 â â©x2,xD⪠| xA â â©x3,x2⪠| xB â â©x3,x7âª
+ | xC â â©x3,xC⪠| xD â â©x4,x1⪠| xE â â©x4,x6⪠| xF â â©x4,xB⪠]
+ | x6 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x6⪠| x2 â â©x0,xC⪠| x3 â â©x1,x2âª
+ | x4 â â©x1,x8⪠| x5 â â©x1,xE⪠| x6 â â©x2,x4⪠| x7 â â©x2,xAâª
+ | x8 â â©x3,x0⪠| x9 â â©x3,x6⪠| xA â â©x3,xC⪠| xB â â©x4,x2âª
+ | xC â â©x4,x8⪠| xD â â©x4,xE⪠| xE â â©x5,x4⪠| xF â â©x5,xA⪠]
+ | x7 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x7⪠| x2 â â©x0,xE⪠| x3 â â©x1,x5âª
+ | x4 â â©x1,xC⪠| x5 â â©x2,x3⪠| x6 â â©x2,xA⪠| x7 â â©x3,x1âª
+ | x8 â â©x3,x8⪠| x9 â â©x3,xF⪠| xA â â©x4,x6⪠| xB â â©x4,xDâª
+ | xC â â©x5,x4⪠| xD â â©x5,xB⪠| xE â â©x6,x2⪠| xF â â©x6,x9⪠]
+ | x8 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x8⪠| x2 â â©x1,x0⪠| x3 â â©x1,x8âª
+ | x4 â â©x2,x0⪠| x5 â â©x2,x8⪠| x6 â â©x3,x0⪠| x7 â â©x3,x8âª
+ | x8 â â©x4,x0⪠| x9 â â©x4,x8⪠| xA â â©x5,x0⪠| xB â â©x5,x8âª
+ | xC â â©x6,x0⪠| xD â â©x6,x8⪠| xE â â©x7,x0⪠| xF â â©x7,x8⪠]
+ | x9 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x9⪠| x2 â â©x1,x2⪠| x3 â â©x1,xBâª
+ | x4 â â©x2,x4⪠| x5 â â©x2,xD⪠| x6 â â©x3,x6⪠| x7 â â©x3,xFâª
+ | x8 â â©x4,x8⪠| x9 â â©x5,x1⪠| xA â â©x5,xA⪠| xB â â©x6,x3âª
+ | xC â â©x6,xC⪠| xD â â©x7,x5⪠| xE â â©x7,xE⪠| xF â â©x8,x7⪠]
+ | xA â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,xA⪠| x2 â â©x1,x4⪠| x3 â â©x1,xEâª
+ | x4 â â©x2,x8⪠| x5 â â©x3,x2⪠| x6 â â©x3,xC⪠| x7 â â©x4,x6âª
+ | x8 â â©x5,x0⪠| x9 â â©x5,xA⪠| xA â â©x6,x4⪠| xB â â©x6,xEâª
+ | xC â â©x7,x8⪠| xD â â©x8,x2⪠| xE â â©x8,xC⪠| xF â â©x9,x6⪠]
+ | xB â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,xB⪠| x2 â â©x1,x6⪠| x3 â â©x2,x1âª
+ | x4 â â©x2,xC⪠| x5 â â©x3,x7⪠| x6 â â©x4,x2⪠| x7 â â©x4,xDâª
+ | x8 â â©x5,x8⪠| x9 â â©x6,x3⪠| xA â â©x6,xE⪠| xB â â©x7,x9âª
+ | xC â â©x8,x4⪠| xD â â©x8,xF⪠| xE â â©x9,xA⪠| xF â â©xA,x5⪠]
+ | xC â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,xC⪠| x2 â â©x1,x8⪠| x3 â â©x2,x4âª
+ | x4 â â©x3,x0⪠| x5 â â©x3,xC⪠| x6 â â©x4,x8⪠| x7 â â©x5,x4âª
+ | x8 â â©x6,x0⪠| x9 â â©x6,xC⪠| xA â â©x7,x8⪠| xB â â©x8,x4âª
+ | xC â â©x9,x0⪠| xD â â©x9,xC⪠| xE â â©xA,x8⪠| xF â â©xB,x4⪠]
+ | xD â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,xD⪠| x2 â â©x1,xA⪠| x3 â â©x2,x7âª
+ | x4 â â©x3,x4⪠| x5 â â©x4,x1⪠| x6 â â©x4,xE⪠| x7 â â©x5,xBâª
+ | x8 â â©x6,x8⪠| x9 â â©x7,x5⪠| xA â â©x8,x2⪠| xB â â©x8,xFâª
+ | xC â â©x9,xC⪠| xD â â©xA,x9⪠| xE â â©xB,x6⪠| xF â â©xC,x3⪠]
+ | xE â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,xE⪠| x2 â â©x1,xC⪠| x3 â â©x2,xAâª
+ | x4 â â©x3,x8⪠| x5 â â©x4,x6⪠| x6 â â©x5,x4⪠| x7 â â©x6,x2âª
+ | x8 â â©x7,x0⪠| x9 â â©x7,xE⪠| xA â â©x8,xC⪠| xB â â©x9,xAâª
+ | xC â â©xA,x8⪠| xD â â©xB,x6⪠| xE â â©xC,x4⪠| xF â â©xD,x2⪠]
+ | xF â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,xF⪠| x2 â â©x1,xE⪠| x3 â â©x2,xDâª
+ | x4 â â©x3,xC⪠| x5 â â©x4,xB⪠| x6 â â©x5,xA⪠| x7 â â©x6,x9âª
+ | x8 â â©x7,x8⪠| x9 â â©x8,x7⪠| xA â â©x9,x6⪠| xB â â©xA,x5âª
+ | xC â â©xB,x4⪠| xD â â©xC,x3⪠| xE â â©xD,x2⪠| xF â â©xE,x1⪠]
+ ].
+
+(* operatore moltiplicazione con segno *)
+ndefinition muls_ex â
+λe1,e2:exadecim.match e1 with
+ [ x0 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x0⪠| x2 â â©x0,x0⪠| x3 â â©x0,x0âª
+ | x4 â â©x0,x0⪠| x5 â â©x0,x0⪠| x6 â â©x0,x0⪠| x7 â â©x0,x0âª
+ | x8 â â©x0,x0⪠| x9 â â©x0,x0⪠| xA â â©x0,x0⪠| xB â â©x0,x0âª
+ | xC â â©x0,x0⪠| xD â â©x0,x0⪠| xE â â©x0,x0⪠| xF â â©x0,x0⪠]
+ | x1 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x1⪠| x2 â â©x0,x2⪠| x3 â â©x0,x3âª
+ | x4 â â©x0,x4⪠| x5 â â©x0,x5⪠| x6 â â©x0,x6⪠| x7 â â©x0,x7âª
+ | x8 â â©xF,x8⪠| x9 â â©xF,x9⪠| xA â â©xF,xA⪠| xB â â©xF,xBâª
+ | xC â â©xF,xC⪠| xD â â©xF,xD⪠| xE â â©xF,xE⪠| xF â â©xF,xF⪠]
+ | x2 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x2⪠| x2 â â©x0,x4⪠| x3 â â©x0,x6âª
+ | x4 â â©x0,x8⪠| x5 â â©x0,xA⪠| x6 â â©x0,xC⪠| x7 â â©x0,xEâª
+ | x8 â â©xF,x0⪠| x9 â â©xF,x2⪠| xA â â©xF,x4⪠| xB â â©xF,x6âª
+ | xC â â©xF,x8⪠| xD â â©xF,xA⪠| xE â â©xF,xC⪠| xF â â©xF,xE⪠]
+ | x3 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x3⪠| x2 â â©x0,x6⪠| x3 â â©x0,x9âª
+ | x4 â â©x0,xC⪠| x5 â â©x0,xF⪠| x6 â â©x1,x2⪠| x7 â â©x1,x5âª
+ | x8 â â©xE,x8⪠| x9 â â©xE,xB⪠| xA â â©xE,xE⪠| xB â â©xF,x1âª
+ | xC â â©xF,x4⪠| xD â â©xF,x7⪠| xE â â©xF,xA⪠| xF â â©xF,xD⪠]
+ | x4 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x4⪠| x2 â â©x0,x8⪠| x3 â â©x0,xCâª
+ | x4 â â©x1,x0⪠| x5 â â©x1,x4⪠| x6 â â©x1,x8⪠| x7 â â©x1,xCâª
+ | x8 â â©xE,x0⪠| x9 â â©xE,x4⪠| xA â â©xE,x8⪠| xB â â©xE,xCâª
+ | xC â â©xF,x0⪠| xD â â©xF,x4⪠| xE â â©xF,x8⪠| xF â â©xF,xC⪠]
+ | x5 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x5⪠| x2 â â©x0,xA⪠| x3 â â©x0,xFâª
+ | x4 â â©x1,x4⪠| x5 â â©x1,x9⪠| x6 â â©x1,xE⪠| x7 â â©x2,x3âª
+ | x8 â â©xD,x8⪠| x9 â â©xD,xD⪠| xA â â©xE,x2⪠| xB â â©xE,x7âª
+ | xC â â©xE,xC⪠| xD â â©xF,x1⪠| xE â â©xF,x6⪠| xF â â©xF,xB⪠]
+ | x6 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x6⪠| x2 â â©x0,xC⪠| x3 â â©x1,x2âª
+ | x4 â â©x1,x8⪠| x5 â â©x1,xE⪠| x6 â â©x2,x4⪠| x7 â â©x2,xAâª
+ | x8 â â©xD,x0⪠| x9 â â©xD,x6⪠| xA â â©xD,xC⪠| xB â â©xE,x2âª
+ | xC â â©xE,x8⪠| xD â â©xE,xE⪠| xE â â©xF,x4⪠| xF â â©xF,xA⪠]
+ | x7 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©x0,x7⪠| x2 â â©x0,xE⪠| x3 â â©x1,x5âª
+ | x4 â â©x1,xC⪠| x5 â â©x2,x3⪠| x6 â â©x2,xA⪠| x7 â â©x3,x1âª
+ | x8 â â©xC,x8⪠| x9 â â©xC,xF⪠| xA â â©xD,x6⪠| xB â â©xD,xDâª
+ | xC â â©xE,x4⪠| xD â â©xE,xB⪠| xE â â©xF,x2⪠| xF â â©xF,x9⪠]
+ | x8 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,x8⪠| x2 â â©xF,x0⪠| x3 â â©xE,x8âª
+ | x4 â â©xE,x0⪠| x5 â â©xD,x8⪠| x6 â â©xD,x0⪠| x7 â â©xC,x8âª
+ | x8 â â©x4,x0⪠| x9 â â©x3,x8⪠| xA â â©x3,x0⪠| xB â â©x2,x8âª
+ | xC â â©x2,x0⪠| xD â â©x1,x8⪠| xE â â©x1,x0⪠| xF â â©x0,x8⪠]
+ | x9 â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,x9⪠| x2 â â©xF,x2⪠| x3 â â©xE,xBâª
+ | x4 â â©xE,x4⪠| x5 â â©xD,xD⪠| x6 â â©xD,x6⪠| x7 â â©xC,xFâª
+ | x8 â â©x3,x8⪠| x9 â â©x3,x1⪠| xA â â©x2,xA⪠| xB â â©x2,x3âª
+ | xC â â©x1,xC⪠| xD â â©x1,x5⪠| xE â â©x0,xE⪠| xF â â©x0,x7⪠]
+ | xA â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,xA⪠| x2 â â©xF,x4⪠| x3 â â©xE,xEâª
+ | x4 â â©xE,x8⪠| x5 â â©xE,x2⪠| x6 â â©xD,xC⪠| x7 â â©xD,x6âª
+ | x8 â â©x3,x0⪠| x9 â â©x2,xA⪠| xA â â©x2,x4⪠| xB â â©x1,xEâª
+ | xC â â©x1,x8⪠| xD â â©x1,x2⪠| xE â â©x0,xC⪠| xF â â©x0,x6⪠]
+ | xB â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,xB⪠| x2 â â©xF,x6⪠| x3 â â©xF,x1âª
+ | x4 â â©xE,xC⪠| x5 â â©xE,x7⪠| x6 â â©xE,x2⪠| x7 â â©xD,xDâª
+ | x8 â â©x2,x8⪠| x9 â â©x2,x3⪠| xA â â©x1,xE⪠| xB â â©x1,x9âª
+ | xC â â©x1,x4⪠| xD â â©x0,xF⪠| xE â â©x0,xA⪠| xF â â©x0,x5⪠]
+ | xC â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,xC⪠| x2 â â©xF,x8⪠| x3 â â©xF,x4âª
+ | x4 â â©xF,x0⪠| x5 â â©xE,xC⪠| x6 â â©xE,x8⪠| x7 â â©xE,x4âª
+ | x8 â â©x2,x0⪠| x9 â â©x1,xC⪠| xA â â©x1,x8⪠| xB â â©x1,x4âª
+ | xC â â©x1,x0⪠| xD â â©x0,xC⪠| xE â â©x0,x8⪠| xF â â©x0,x4⪠]
+ | xD â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,xD⪠| x2 â â©xF,xA⪠| x3 â â©xF,x7âª
+ | x4 â â©xF,x4⪠| x5 â â©xF,x1⪠| x6 â â©xE,xE⪠| x7 â â©xE,xBâª
+ | x8 â â©x1,x8⪠| x9 â â©x1,x5⪠| xA â â©x1,x2⪠| xB â â©x0,xFâª
+ | xC â â©x0,xC⪠| xD â â©x0,x9⪠| xE â â©x0,x6⪠| xF â â©x0,x3⪠]
+ | xE â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,xE⪠| x2 â â©xF,xC⪠| x3 â â©xF,xAâª
+ | x4 â â©xF,x8⪠| x5 â â©xF,x6⪠| x6 â â©xF,x4⪠| x7 â â©xF,x2âª
+ | x8 â â©x1,x0⪠| x9 â â©x0,xE⪠| xA â â©x0,xC⪠| xB â â©x0,xAâª
+ | xC â â©x0,x8⪠| xD â â©x0,x6⪠| xE â â©x0,x4⪠| xF â â©x0,x2⪠]
+ | xF â match e2 with
+ [ x0 â â©x0,x0⪠| x1 â â©xF,xF⪠| x2 â â©xF,xE⪠| x3 â â©xF,xDâª
+ | x4 â â©xF,xC⪠| x5 â â©xF,xB⪠| x6 â â©xF,xA⪠| x7 â â©xF,x9âª
+ | x8 â â©x0,x8⪠| x9 â â©x0,x7⪠| xA â â©x0,x6⪠| xB â â©x0,x5âª
+ | xC â â©x0,x4⪠| xD â â©x0,x3⪠| xE â â©x0,x2⪠| xF â â©x0,x1⪠]
+ ].
+
+(* correzione per somma su BCD *)
+(* input: halfcarry,carry,X(BCD+BCD) *)
+(* output: X',carry' *)
+ndefinition daa_b8 â
+λh,c:bool.λX:byte8.
+ match ltc ? X â©x9,xA⪠with
+ (* [X:0x00-0x99] *)
+ (* c' = c *)
+ (* X' = [(b16l X):0x0-0x9] X + [h=1 ? 0x06 : 0x00] + [c=1 ? 0x60 : 0x00]
+ [(b16l X):0xA-0xF] X + 0x06 + [c=1 ? 0x60 : 0x00] *)
+ [ true â
+ let X' â match (ltc ? (cnL ? X) xA) â (âh) with
+ [ true â X
+ | false â plusc_d_d ? X â©x0,x6⪠] in
+ let X'' â match c with
+ [ true â plusc_d_d ? X' â©x6,x0âª
+ | false â X' ] in
+ pair ⦠c X''
+ (* [X:0x9A-0xFF] *)
+ (* c' = 1 *)
+ (* X' = [X:0x9A-0xFF]
+ [(b16l X):0x0-0x9] X + [h=1 ? 0x06 : 0x00] + 0x60
+ [(b16l X):0xA-0xF] X + 0x6 + 0x60 *)
+ | false â
+ let X' â match (ltc ? (cnL ? X) xA) â (âh) with
+ [ true â X
+ | false â plusc_d_d ? X â©x0,x6⪠] in
+ let X'' â plusc_d_d ? X' â©x6,x0⪠in
+ pair ⦠true X''
+ ].
+
+(* byte ricorsivi *)
+ninductive rec_byte8 : byte8 â Type â
+ b8_O : rec_byte8 (zeroc ?)
+| b8_S : ân.rec_byte8 n â rec_byte8 (succc ? n).
+
+(* byte â byte ricorsivi *)
+ndefinition b8_to_recb8_aux1 : Î n.rec_byte8 â©n,x0⪠â rec_byte8 â©succc ? n,x0⪠â
+λn.λrecb:rec_byte8 â©n,x0âª.
+ b8_S â©n,xF⪠(b8_S â©n,xE⪠(b8_S â©n,xD⪠(b8_S â©n,xC⪠(
+ b8_S â©n,xB⪠(b8_S â©n,xA⪠(b8_S â©n,x9⪠(b8_S â©n,x8⪠(
+ b8_S â©n,x7⪠(b8_S â©n,x6⪠(b8_S â©n,x5⪠(b8_S â©n,x4⪠(
+ b8_S â©n,x3⪠(b8_S â©n,x2⪠(b8_S â©n,x1⪠(b8_S â©n,x0⪠recb))))))))))))))).
+
+(* ... cifra esadecimale superiore *)
+nlet rec b8_to_recb8_aux2 (n:exadecim) (r:rec_exadecim n) on r â
+ match r return λx.λy:rec_exadecim x.rec_byte8 â©x,x0⪠with
+ [ ex_O â b8_O
+ | ex_S t n' â b8_to_recb8_aux1 ? (b8_to_recb8_aux2 t n')
+ ].
+
+(* ... cifra esadecimale inferiore *)
+ndefinition b8_to_recb8_aux3 : Î n1,n2.rec_byte8 â©n1,x0⪠â rec_byte8 â©n1,n2⪠â
+λn1,n2.λrecb:rec_byte8 â©n1,x0âª.
+ match n2 return λx.rec_byte8 â©n1,x⪠with
+ [ x0 â recb
+ | x1 â b8_S â©n1,x0⪠recb
+ | x2 â b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)
+ | x3 â b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb))
+ | x4 â b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)))
+ | x5 â b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(
+ b8_S â©n1,x0⪠recb))))
+ | x6 â b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(
+ b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)))))
+ | x7 â b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(
+ b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb))))))
+ | x8 â b8_S â©n1,x7⪠(b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(
+ b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)))))))
+ | x9 â b8_S â©n1,x8⪠(b8_S â©n1,x7⪠(b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(
+ b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(
+ b8_S â©n1,x0⪠recb))))))))
+ | xA â b8_S â©n1,x9⪠(b8_S â©n1,x8⪠(b8_S â©n1,x7⪠(b8_S â©n1,x6⪠(
+ b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(
+ b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)))))))))
+ | xB â b8_S â©n1,xA⪠(b8_S â©n1,x9⪠(b8_S â©n1,x8⪠(b8_S â©n1,x7⪠(
+ b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(
+ b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb))))))))))
+ | xC â b8_S â©n1,xB⪠(b8_S â©n1,xA⪠(b8_S â©n1,x9⪠(b8_S â©n1,x8⪠(
+ b8_S â©n1,x7⪠(b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(
+ b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)))))))))))
+ | xD â b8_S â©n1,xC⪠(b8_S â©n1,xB⪠(b8_S â©n1,xA⪠(b8_S â©n1,x9⪠(
+ b8_S â©n1,x8⪠(b8_S â©n1,x7⪠(b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(
+ b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(
+ b8_S â©n1,x0⪠recb))))))))))))
+ | xE â b8_S â©n1,xD⪠(b8_S â©n1,xC⪠(b8_S â©n1,xB⪠(b8_S â©n1,xA⪠(
+ b8_S â©n1,x9⪠(b8_S â©n1,x8⪠(b8_S â©n1,x7⪠(b8_S â©n1,x6⪠(
+ b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(b8_S â©n1,x2⪠(
+ b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb)))))))))))))
+ | xF â b8_S â©n1,xE⪠(b8_S â©n1,xD⪠(b8_S â©n1,xC⪠(b8_S â©n1,xB⪠(
+ b8_S â©n1,xA⪠(b8_S â©n1,x9⪠(b8_S â©n1,x8⪠(b8_S â©n1,x7⪠(
+ b8_S â©n1,x6⪠(b8_S â©n1,x5⪠(b8_S â©n1,x4⪠(b8_S â©n1,x3⪠(
+ b8_S â©n1,x2⪠(b8_S â©n1,x1⪠(b8_S â©n1,x0⪠recb))))))))))))))
+ ].
+
+(*
+nlemma b8_to_recb8 : Î b.rec_byte8 b.
+ #b; nletin K â (b8_to_recb8_aux3
+ (b8h b) (b8l b) (b8_to_recb8_aux2 (b8h b) (ex_to_recex (b8h b))));
+ ncases b in K; #e1; #e2; #K; napply K;
+nqed.
+*)
+
+ndefinition b8_to_recb8 : Î b.rec_byte8 b â
+λb.match b with
+ [ mk_comp_num h l â b8_to_recb8_aux3 h l (b8_to_recb8_aux2 h (ex_to_recex h)) ].
+
+(* ottali â esadecimali *)
+ndefinition b8_of_bit â
+λn.match n with
+ [ t00 â â©x0,x0⪠| t01 â â©x0,x1⪠| t02 â â©x0,x2⪠| t03 â â©x0,x3âª
+ | t04 â â©x0,x4⪠| t05 â â©x0,x5⪠| t06 â â©x0,x6⪠| t07 â â©x0,x7âª
+ | t08 â â©x0,x8⪠| t09 â â©x0,x9⪠| t0A â â©x0,xA⪠| t0B â â©x0,xBâª
+ | t0C â â©x0,xC⪠| t0D â â©x0,xD⪠| t0E â â©x0,xE⪠| t0F â â©x0,xFâª
+ | t10 â â©x1,x0⪠| t11 â â©x1,x1⪠| t12 â â©x1,x2⪠| t13 â â©x1,x3âª
+ | t14 â â©x1,x4⪠| t15 â â©x1,x5⪠| t16 â â©x1,x6⪠| t17 â â©x1,x7âª
+ | t18 â â©x1,x8⪠| t19 â â©x1,x9⪠| t1A â â©x1,xA⪠| t1B â â©x1,xBâª
+ | t1C â â©x1,xC⪠| t1D â â©x1,xD⪠| t1E â â©x1,xE⪠| t1F â â©x1,xFâª
+ ].
diff --git a/helm/software/matita/contribs/ng_assembly2/num/comp_ext.ma b/helm/software/matita/contribs/ng_assembly2/num/comp_ext.ma
new file mode 100755
index 000000000..58df52a1e
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/comp_ext.ma
@@ -0,0 +1,62 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "common/prod.ma".
+
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+
+nrecord comparable_ext : Type[1] â
+ {
+ comp_base : comparable;
+ ltc : (carr comp_base) â (carr comp_base) â bool;
+ lec : (carr comp_base) â (carr comp_base) â bool;
+ gtc : (carr comp_base) â (carr comp_base) â bool;
+ gec : (carr comp_base) â (carr comp_base) â bool;
+ andc : (carr comp_base) â (carr comp_base) â (carr comp_base);
+ orc : (carr comp_base) â (carr comp_base) â (carr comp_base);
+ xorc : (carr comp_base) â (carr comp_base) â (carr comp_base);
+ getMSBc : (carr comp_base) â bool;
+ setMSBc : (carr comp_base) â (carr comp_base);
+ clrMSBc : (carr comp_base) â (carr comp_base);
+ getLSBc : (carr comp_base) â bool;
+ setLSBc : (carr comp_base) â (carr comp_base);
+ clrLSBc : (carr comp_base) â (carr comp_base);
+ rcrc : bool â (carr comp_base) â ProdT bool (carr comp_base);
+ shrc : (carr comp_base) â ProdT bool (carr comp_base);
+ rorc : (carr comp_base) â (carr comp_base);
+ rclc : bool â (carr comp_base) â ProdT bool (carr comp_base);
+ shlc : (carr comp_base) â ProdT bool (carr comp_base);
+ rolc : (carr comp_base) â (carr comp_base);
+ notc : (carr comp_base) â (carr comp_base);
+ plusc_dc_dc : bool â (carr comp_base) â (carr comp_base) â ProdT bool (carr comp_base);
+ plusc_d_dc : (carr comp_base) â (carr comp_base) â ProdT bool (carr comp_base);
+ plusc_dc_d : bool â (carr comp_base) â (carr comp_base) â (carr comp_base);
+ plusc_d_d : (carr comp_base) â (carr comp_base) â (carr comp_base);
+ plusc_dc_c : bool â (carr comp_base) â (carr comp_base) â bool;
+ plusc_d_c : (carr comp_base) â (carr comp_base) â bool;
+ predc : (carr comp_base) â (carr comp_base);
+ succc : (carr comp_base) â (carr comp_base);
+ complc : (carr comp_base) â (carr comp_base);
+ absc : (carr comp_base) â (carr comp_base);
+ inrangec : (carr comp_base) â (carr comp_base) â (carr comp_base) â bool
+ }.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/comp_num.ma b/helm/software/matita/contribs/ng_assembly2/num/comp_num.ma
new file mode 100755
index 000000000..a062a5624
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/comp_num.ma
@@ -0,0 +1,354 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/comp_ext.ma".
+include "num/bool_lemmas.ma".
+
+(* ******** *)
+(* COMPOSTI *)
+(* ******** *)
+
+nrecord comp_num (T:Type) : Type â
+ {
+ cnH: T;
+ cnL: T
+ }.
+
+(* operatore = *)
+ndefinition eq_cn â
+λT.λfeq:T â T â bool.
+λcn1,cn2:comp_num T.
+ (feq (cnH ? cn1) (cnH ? cn2)) â (feq (cnL ? cn1) (cnL ? cn2)).
+
+nlemma cn_destruct_1 :
+âT.âx1,x2,y1,y2:T.
+ mk_comp_num T x1 y1 = mk_comp_num T x2 y2 â x1 = x2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match mk_comp_num ? x2 y2 with [ mk_comp_num a _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma cn_destruct_2 :
+âT.âx1,x2,y1,y2:T.
+ mk_comp_num T x1 y1 = mk_comp_num T x2 y2 â y1 = y2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match mk_comp_num ? x2 y2 with [ mk_comp_num _ b â y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqcn :
+âT.âfeq:T â T â bool.
+ (symmetricT T bool feq) â
+ (symmetricT (comp_num T) bool (eq_cn T feq)).
+ #T; #feq; #H;
+ #b1; nelim b1; #e1; #e2;
+ #b2; nelim b2; #e3; #e4;
+ nchange with (((feq e1 e3)â(feq e2 e4)) = ((feq e3 e1)â(feq e4 e2)));
+ nrewrite > (H e1 e3);
+ nrewrite > (H e2 e4);
+ napply refl_eq.
+nqed.
+
+nlemma eqcn_to_eq :
+âT.âfeq:T â T â bool.
+ (âx,y:T.(feq x y = true) â (x = y)) â
+ (âb1,b2:comp_num T.
+ ((eq_cn T feq b1 b2 = true) â (b1 = b2))).
+ #T; #feq; #H; #b1; #b2;
+ nelim b1; #e1; #e2;
+ nelim b2; #e3; #e4;
+ nchange in ⢠(% â ?) with (((feq e1 e3)â(feq e2 e4)) = true);
+ #H1;
+ nrewrite < (H ⦠(andb_true_true_l ⦠H1));
+ nrewrite < (H ⦠(andb_true_true_r ⦠H1));
+ napply refl_eq.
+nqed.
+
+nlemma eq_to_eqcn :
+âT.âfeq:T â T â bool.
+ (âx,y:T.(x = y) â (feq x y = true)) â
+ (âb1,b2:comp_num T.
+ ((b1 = b2) â (eq_cn T feq b1 b2 = true))).
+ #T; #feq; #H; #b1; #b2;
+ nelim b1; #e1; #e2;
+ nelim b2; #e3; #e4;
+ #H1;
+ nrewrite < (cn_destruct_1 ⦠H1);
+ nrewrite < (cn_destruct_2 ⦠H1);
+ nchange with (((feq e1 e1)â(feq e2 e2)) = true);
+ nrewrite > (H e1 e1 (refl_eq â¦));
+ nrewrite > (H e2 e2 (refl_eq â¦));
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma decidable_cn_aux1 :
+âT.âe1,e2,e3,e4:T.e1 â e3 â (mk_comp_num T e1 e2) â (mk_comp_num T e3 e4).
+ #T; #e1; #e2; #e3; #e4;
+ nnormalize; #H; #H1;
+ napply (H (cn_destruct_1 ⦠H1)).
+nqed.
+
+nlemma decidable_cn_aux2 :
+âT.âe1,e2,e3,e4:T.e2 â e4 â (mk_comp_num T e1 e2) â (mk_comp_num T e3 e4).
+ #T; #e1; #e2; #e3; #e4;
+ nnormalize; #H; #H1;
+ napply (H (cn_destruct_2 ⦠H1)).
+nqed.
+
+nlemma decidable_cn :
+âT.(âx,y:T.decidable (x = y)) â
+ (âb1,b2:comp_num T.
+ (decidable (b1 = b2))).
+ #T; #H;
+ #b1; nelim b1; #e1; #e2;
+ #b2; nelim b2; #e3; #e4;
+ nnormalize;
+ napply (or2_elim (e1 = e3) (e1 â e3) ? (H e1 e3) â¦);
+ ##[ ##2: #H1; napply (or2_intro2 ⦠(decidable_cn_aux1 T e1 e2 e3 e4 H1))
+ ##| ##1: #H1; napply (or2_elim (e2 = e4) (e2 â e4) ? (H e2 e4) â¦);
+ ##[ ##2: #H2; napply (or2_intro2 ⦠(decidable_cn_aux2 T e1 e2 e3 e4 H2))
+ ##| ##1: #H2; nrewrite > H1; nrewrite > H2;
+ napply (or2_intro1 ⦠(refl_eq ? (mk_comp_num T e3 e4)))
+ ##]
+ ##]
+nqed.
+
+nlemma neqcn_to_neq :
+âT.âfeq:T â T â bool.
+ (âx,y:T.(feq x y = false) â (x â y)) â
+ (âb1,b2:comp_num T.
+ ((eq_cn T feq b1 b2 = false) â (b1 â b2))).
+ #T; #feq; #H; #b1; #b2;
+ nelim b1; #e1; #e2;
+ nelim b2; #e3; #e4;
+ nchange with ((((feq e1 e3) â (feq e2 e4)) = false) â ?);
+ #H1;
+ napply (or2_elim ((feq e1 e3) = false) ((feq e2 e4) = false) ? (andb_false2 ⦠H1) â¦);
+ ##[ ##1: #H2; napply (decidable_cn_aux1 ⦠(H ⦠H2))
+ ##| ##2: #H2; napply (decidable_cn_aux2 ⦠(H ⦠H2))
+ ##]
+nqed.
+
+nlemma cn_destruct :
+âT.(âx,y:T.decidable (x = y)) â
+ (âe1,e2,e3,e4:T.
+ ((mk_comp_num T e1 e2) â (mk_comp_num T e3 e4)) â
+ ((e1 â e3) ⨠(e2 â e4))).
+ #T; #H; #e1; #e2; #e3; #e4;
+ nnormalize; #H1;
+ napply (or2_elim (e1 = e3) (e1 â e3) ? (H e1 e3) â¦);
+ ##[ ##2: #H2; napply (or2_intro1 ⦠H2)
+ ##| ##1: #H2; napply (or2_elim (e2 = e4) (e2 â e4) ? (H e2 e4) â¦);
+ ##[ ##2: #H3; napply (or2_intro2 ⦠H3)
+ ##| ##1: #H3; nrewrite > H2 in H1:(%);
+ nrewrite > H3;
+ #H1; nelim (H1 (refl_eq â¦))
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqcn :
+âT.âfeq:T â T â bool.
+ (âx,y:T.(x â y) â (feq x y = false)) â
+ (âx,y:T.decidable (x = y)) â
+ (âb1,b2:comp_num T.
+ ((b1 â b2) â (eq_cn T feq b1 b2 = false))).
+ #T; #feq; #H; #H1; #b1; #b2;
+ nelim b1; #e1; #e2;
+ nelim b2; #e3; #e4;
+ #H2; nchange with (((feq e1 e3) â (feq e2 e4)) = false);
+ napply (or2_elim (e1 â e3) (e2 â e4) ? (cn_destruct T H1 e1 e2 e3 e4 ⦠H2) â¦);
+ ##[ ##1: #H3; nrewrite > (H ⦠H3); nnormalize; napply refl_eq
+ ##| ##2: #H3; nrewrite > (H ⦠H3);
+ nrewrite > (symmetric_andbool (feq e1 e3) false);
+ nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma cn_is_comparable : comparable â comparable.
+ #T; @ (comp_num T)
+ (* zero *)
+ ##[ napply (mk_comp_num ? (zeroc ?) (zeroc ?))
+ (* forall *)
+ ##| napply (λP.forallc T
+ (λh.forallc T
+ (λl.P (mk_comp_num ? h l))))
+ (* eq *)
+ ##| napply (eq_cn ? (eqc T))
+ (* eqc_to_eq *)
+ ##| napply (eqcn_to_eq ⦠(eqc_to_eq T))
+ (* eq_to_eqc *)
+ ##| napply (eq_to_eqcn ⦠(eq_to_eqc T))
+ (* neqc_to_neq *)
+ ##| napply (neqcn_to_neq ⦠(neqc_to_neq T))
+ (* neq_to_neqc *)
+ ##| napply (neq_to_neqcn ⦠(neq_to_neqc T));
+ napply (decidable_c T)
+ (* decidable_c *)
+ ##| napply (decidable_cn ⦠(decidable_c T))
+ (* symmetric_eqc *)
+ ##| napply (symmetric_eqcn ⦠(symmetric_eqc T))
+ ##]
+nqed.
+
+nlemma cn_is_comparable_ext : comparable_ext â comparable_ext.
+ #T; nelim T; #c;
+ #ltc; #lec; #gtc; #gec; #andc; #orc; #xorc;
+ #getMSBc; #setMSBc; #clrMSBc; #getLSBc; #setLSBc; #clrLSBc;
+ #rcrc; #shrc; #rorc; #rclc; #shlc; #rolc; #notc;
+ #plusc_dc_dc; #plusc_d_dc; #plusc_dc_d; #plusc_d_d; #plusc_dc_c; #plusc_d_c;
+ #predc; #succc; #complc; #absc; #inrangec;
+ napply (mk_comparable_ext);
+ ##[ napply (cn_is_comparable c)
+ (* lt *)
+ ##| napply (λx,y.(ltc (cnH ? x) (cnH ? y)) â
+ (((eqc c) (cnH ? x) (cnH ? y)) â (ltc (cnL ? x) (cnL ? y))))
+ (* le *)
+ ##| napply (λx,y.(ltc (cnH ? x) (cnH ? y)) â
+ (((eqc c) (cnH ? x) (cnH ? y)) â (lec (cnL ? x) (cnL ? y))))
+ (* gt *)
+ ##| napply (λx,y.(gtc (cnH ? x) (cnH ? y)) â
+ (((eqc c) (cnH ? x) (cnH ? y)) â (gtc (cnL ? x) (cnL ? y))))
+ (* ge *)
+ ##| napply (λx,y.(gtc (cnH ? x) (cnH ? y)) â
+ (((eqc c) (cnH ? x) (cnH ? y)) â (gec (cnL ? x) (cnL ? y))))
+ (* and *)
+ ##| napply (λx,y.mk_comp_num ? (andc (cnH ? x) (cnH ? y))
+ (andc (cnL ? x) (cnL ? y)))
+ (* or *)
+ ##| napply (λx,y.mk_comp_num ? (orc (cnH ? x) (cnH ? y))
+ (orc (cnL ? x) (cnL ? y)))
+ (* xor *)
+ ##| napply (λx,y.mk_comp_num ? (xorc (cnH ? x) (cnH ? y))
+ (xorc (cnL ? x) (cnL ? y)))
+ (* getMSB *)
+ ##| napply (λx.getMSBc (cnH ? x))
+ (* setMSB *)
+ ##| napply (λx.mk_comp_num ? (setMSBc (cnH ? x)) (cnL ? x))
+ (* clrMSB *)
+ ##| napply (λx.mk_comp_num ? (clrMSBc (cnH ? x)) (cnL ? x))
+ (* getLSB *)
+ ##| napply (λx.getLSBc (cnL ? x))
+ (* setLSB *)
+ ##| napply (λx.mk_comp_num ? (cnH ? x) (setLSBc (cnL ? x)))
+ (* clrLSB *)
+ ##| napply (λx.mk_comp_num ? (cnH ? x) (clrLSBc (cnL ? x)))
+ (* rcr *)
+ ##| napply (λcy,x.match rcrc cy (cnH ? x) with
+ [ pair cy' cnh' â match rcrc cy' (cnL ? x) with
+ [ pair cy'' cnl' â pair ⦠cy'' (mk_comp_num ? cnh' cnl')]])
+ (* shr *)
+ ##| napply (λx.match shrc (cnH ? x) with
+ [ pair cy' cnh' â match rcrc cy' (cnL ? x) with
+ [ pair cy'' cnl' â pair ⦠cy'' (mk_comp_num ? cnh' cnl')]])
+ (* ror *)
+ ##| napply (λx.match shrc (cnH ? x) with
+ [ pair cy' cnh' â match rcrc cy' (cnL ? x) with
+ [ pair cy'' cnl' â mk_comp_num ?
+ (match cy'' with [ true â setMSBc
+ | false â λh.h ] cnh') cnl']])
+ (* rcl *)
+ ##| napply (λcy,x.match rclc cy (cnL ? x) with
+ [ pair cy' cnl' â match rclc cy' (cnH ? x) with
+ [ pair cy'' cnh' â pair ⦠cy'' (mk_comp_num ? cnh' cnl')]])
+ (* shl *)
+ ##| napply (λx.match shlc (cnL ? x) with
+ [ pair cy' cnl' â match rclc cy' (cnH ? x) with
+ [ pair cy'' cnh' â pair ⦠cy'' (mk_comp_num ? cnh' cnl')]])
+ (* rol *)
+ ##| napply (λx.match shlc (cnL ? x) with
+ [ pair cy' cnl' â match rclc cy' (cnH ? x) with
+ [ pair cy'' cnh' â mk_comp_num ?
+ cnh' (match cy'' with [ true â setLSBc
+ | false â λh.h ] cnl')]])
+ (* not *)
+ ##| napply (λx.mk_comp_num ? (notc (cnH ? x)) (notc (cnL ? x)))
+ (* plus_dc_dc *)
+ ##| napply (λcy,x,y.match plusc_dc_dc cy (cnL ? x) (cnL ? y) with
+ [ pair cy' cnl' â match plusc_dc_dc cy' (cnH ? x) (cnH ? y) with
+ [ pair cy'' cnh' â pair ⦠cy'' (mk_comp_num ? cnh' cnl')]])
+ (* plus_d_dc *)
+ ##| napply (λx,y.match plusc_d_dc (cnL ? x) (cnL ? y) with
+ [ pair cy' cnl' â match plusc_dc_dc cy' (cnH ? x) (cnH ? y) with
+ [ pair cy'' cnh' â pair ⦠cy'' (mk_comp_num ? cnh' cnl')]])
+ (* plus_dc_d *)
+ ##| napply (λcy,x,y.match plusc_dc_dc cy (cnL ? x) (cnL ? y) with
+ [ pair cy' cnl' â mk_comp_num ? (plusc_dc_d cy' (cnH ? x) (cnH ? y)) cnl'])
+ (* plus_d_d *)
+ ##| napply (λx,y.match plusc_d_dc (cnL ? x) (cnL ? y) with
+ [ pair cy' cnl' â mk_comp_num ? (plusc_dc_d cy' (cnH ? x) (cnH ? y)) cnl'])
+ (* plus_dc_c *)
+ ##| napply (λcy,x,y.plusc_dc_c (plusc_dc_c cy (cnL ? x) (cnL ? y))
+ (cnH ? x) (cnH ? y))
+ (* plus_d_c *)
+ ##| napply (λx,y.plusc_dc_c (plusc_d_c (cnL ? x) (cnL ? y))
+ (cnH ? x) (cnH ? y))
+ (* pred *)
+ ##| napply (λx.match (eqc c) (zeroc c) (cnL ? x) with
+ [ true â mk_comp_num ? (predc (cnH ? x)) (predc (cnL ? x))
+ | false â mk_comp_num ? (cnH ? x) (predc (cnL ? x)) ])
+ (* succ *)
+ ##| napply (λx.match (eqc c) (predc (zeroc c)) (cnL ? x) with
+ [ true â mk_comp_num ? (succc (cnH ? x)) (succc (cnL ? x))
+ | false â mk_comp_num ? (cnH ? x) (succc (cnL ? x)) ])
+ (* compl *)
+ ##| napply (λx.(match (eqc c) (zeroc c) (cnL ? x) with
+ [ true â mk_comp_num ? (complc (cnH ? x)) (complc (cnL ? x))
+ | false â mk_comp_num ? (notc (cnH ? x)) (complc (cnL ? x)) ]))
+ (* abs *)
+ ##| napply (λx.match getMSBc (cnH ? x) with
+ [ true â match (eqc c) (zeroc c) (cnL ? x) with
+ [ true â mk_comp_num ? (complc (cnH ? x)) (complc (cnL ? x))
+ | false â mk_comp_num ? (notc (cnH ? x)) (complc (cnL ? x)) ]
+ | false â x ])
+ (* inrange *)
+ ##| napply (λx,inf,sup.
+ match (ltc (cnH ? inf) (cnH ? sup)) â
+ (((eqc c) (cnH ? inf) (cnH ? sup)) â (lec (cnL ? inf) (cnL ? sup))) with
+ [ true â and_bool | false â or_bool ]
+ ((ltc (cnH ? inf) (cnH ? x)) â
+ (((eqc c) (cnH ? inf) (cnH ? x)) â (lec (cnL ? inf) (cnL ? x))))
+ ((ltc (cnH ? x) (cnH ? sup)) â
+ (((eqc c) (cnH ? x) (cnH ? sup)) â (lec (cnL ? x) (cnL ? sup)))))
+ ##]
+nqed.
+
+unification hint 0 â S: comparable;
+ T â (carr S),
+ X â (cn_is_comparable S)
+ (*********************************************) â¢
+ carr X â¡ comp_num T.
+
+ndefinition zeroc â λx:comparable_ext.zeroc (comp_base x).
+ndefinition forallc â λx:comparable_ext.forallc (comp_base x).
+ndefinition eqc â λx:comparable_ext.eqc (comp_base x).
+ndefinition eqc_to_eq â λx:comparable_ext.eqc_to_eq (comp_base x).
+ndefinition eq_to_eqc â λx:comparable_ext.eq_to_eqc (comp_base x).
+ndefinition neqc_to_neq â λx:comparable_ext.neqc_to_neq (comp_base x).
+ndefinition neq_to_neqc â λx:comparable_ext.neq_to_neqc (comp_base x).
+ndefinition decidable_c â λx:comparable_ext.decidable_c (comp_base x).
+ndefinition symmetric_eqc â λx:comparable_ext.symmetric_eqc (comp_base x).
diff --git a/helm/software/matita/contribs/ng_assembly2/num/exadecim.ma b/helm/software/matita/contribs/ng_assembly2/num/exadecim.ma
new file mode 100755
index 000000000..72f81d0d2
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/exadecim.ma
@@ -0,0 +1,1583 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/comp_ext.ma".
+include "num/bool_lemmas.ma".
+include "num/oct.ma".
+
+(* *********** *)
+(* ESADECIMALI *)
+(* *********** *)
+
+ninductive exadecim : Type â
+ x0: exadecim
+| x1: exadecim
+| x2: exadecim
+| x3: exadecim
+| x4: exadecim
+| x5: exadecim
+| x6: exadecim
+| x7: exadecim
+| x8: exadecim
+| x9: exadecim
+| xA: exadecim
+| xB: exadecim
+| xC: exadecim
+| xD: exadecim
+| xE: exadecim
+| xF: exadecim.
+
+(* iteratore sugli esadecimali *)
+ndefinition forall_ex â λP.
+ P x0 â P x1 â P x2 â P x3 â P x4 â P x5 â P x6 â P x7 â
+ P x8 â P x9 â P xA â P xB â P xC â P xD â P xE â P xF.
+
+(* operatore = *)
+ndefinition eq_ex â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with [ x0 â true | _ â false ]
+ | x1 â match e2 with [ x1 â true | _ â false ]
+ | x2 â match e2 with [ x2 â true | _ â false ]
+ | x3 â match e2 with [ x3 â true | _ â false ]
+ | x4 â match e2 with [ x4 â true | _ â false ]
+ | x5 â match e2 with [ x5 â true | _ â false ]
+ | x6 â match e2 with [ x6 â true | _ â false ]
+ | x7 â match e2 with [ x7 â true | _ â false ]
+ | x8 â match e2 with [ x8 â true | _ â false ]
+ | x9 â match e2 with [ x9 â true | _ â false ]
+ | xA â match e2 with [ xA â true | _ â false ]
+ | xB â match e2 with [ xB â true | _ â false ]
+ | xC â match e2 with [ xC â true | _ â false ]
+ | xD â match e2 with [ xD â true | _ â false ]
+ | xE â match e2 with [ xE â true | _ â false ]
+ | xF â match e2 with [ xF â true | _ â false ]
+ ].
+
+(* operatore < *)
+ndefinition lt_ex â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â false | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x1 â match e2 with
+ [ x0 â false | x1 â false | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x2 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x3 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x4 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x5 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x6 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x7 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x8 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x9 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xA â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xB â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xC â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â true | xE â true | xF â true ]
+ | xD â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â true | xF â true ]
+ | xE â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â true ]
+ | xF â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ ].
+
+(* operatore ⤠*)
+ndefinition le_ex â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x1 â match e2 with
+ [ x0 â false | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x2 â match e2 with
+ [ x0 â false | x1 â false | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x3 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x4 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x5 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x6 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x7 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x8 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | x9 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xA â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xB â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xC â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â true | xD â true | xE â true | xF â true ]
+ | xD â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â true | xE â true | xF â true ]
+ | xE â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â true | xF â true ]
+ | xF â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â true ]
+ ].
+
+(* operatore > *)
+ndefinition gt_ex â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x1 â match e2 with
+ [ x0 â true | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x2 â match e2 with
+ [ x0 â true | x1 â true | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x3 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x4 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x5 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x6 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x7 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x8 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x9 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xA â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xB â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xC â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xD â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â false | xE â false | xF â false ]
+ | xE â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â false | xF â false ]
+ | xF â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â false ]
+ ].
+
+(* operatore ⥠*)
+ndefinition ge_ex â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â true | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x1 â match e2 with
+ [ x0 â true | x1 â true | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x2 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x3 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x4 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x5 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x6 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x7 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â false | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x8 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â false | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | x9 â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â false | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xA â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â false
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xB â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â false | xD â false | xE â false | xF â false ]
+ | xC â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â false | xE â false | xF â false ]
+ | xD â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â false | xF â false ]
+ | xE â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â false ]
+ | xF â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true
+ | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ]
+ ].
+
+(* operatore and *)
+ndefinition and_ex â
+λe1,e2:exadecim.match e1 with
+ [ x0 â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x0 | x3 â x0
+ | x4 â x0 | x5 â x0 | x6 â x0 | x7 â x0
+ | x8 â x0 | x9 â x0 | xA â x0 | xB â x0
+ | xC â x0 | xD â x0 | xE â x0 | xF â x0 ]
+ | x1 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x0 | x3 â x1
+ | x4 â x0 | x5 â x1 | x6 â x0 | x7 â x1
+ | x8 â x0 | x9 â x1 | xA â x0 | xB â x1
+ | xC â x0 | xD â x1 | xE â x0 | xF â x1 ]
+ | x2 â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x2 | x3 â x2
+ | x4 â x0 | x5 â x0 | x6 â x2 | x7 â x2
+ | x8 â x0 | x9 â x0 | xA â x2 | xB â x2
+ | xC â x0 | xD â x0 | xE â x2 | xF â x2 ]
+ | x3 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x0 | x5 â x1 | x6 â x2 | x7 â x3
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3
+ | xC â x0 | xD â x1 | xE â x2 | xF â x3 ]
+ | x4 â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x0 | x3 â x0
+ | x4 â x4 | x5 â x4 | x6 â x4 | x7 â x4
+ | x8 â x0 | x9 â x0 | xA â x0 | xB â x0
+ | xC â x4 | xD â x4 | xE â x4 | xF â x4 ]
+ | x5 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x0 | x3 â x1
+ | x4 â x4 | x5 â x5 | x6 â x4 | x7 â x5
+ | x8 â x0 | x9 â x1 | xA â x0 | xB â x1
+ | xC â x4 | xD â x5 | xE â x4 | xF â x5 ]
+ | x6 â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x2 | x3 â x2
+ | x4 â x4 | x5 â x4 | x6 â x6 | x7 â x6
+ | x8 â x0 | x9 â x0 | xA â x2 | xB â x2
+ | xC â x4 | xD â x4 | xE â x6 | xF â x6 ]
+ | x7 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3
+ | xC â x4 | xD â x5 | xE â x6 | xF â x7 ]
+ | x8 â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x0 | x3 â x0
+ | x4 â x0 | x5 â x0 | x6 â x0 | x7 â x0
+ | x8 â x8 | x9 â x8 | xA â x8 | xB â x8
+ | xC â x8 | xD â x8 | xE â x8 | xF â x8 ]
+ | x9 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x0 | x3 â x1
+ | x4 â x0 | x5 â x1 | x6 â x0 | x7 â x1
+ | x8 â x8 | x9 â x9 | xA â x8 | xB â x9
+ | xC â x8 | xD â x9 | xE â x8 | xF â x9 ]
+ | xA â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x2 | x3 â x2
+ | x4 â x0 | x5 â x0 | x6 â x2 | x7 â x2
+ | x8 â x8 | x9 â x8 | xA â xA | xB â xA
+ | xC â x8 | xD â x8 | xE â xA | xF â xA ]
+ | xB â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x0 | x5 â x1 | x6 â x2 | x7 â x3
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB
+ | xC â x8 | xD â x9 | xE â xA | xF â xB ]
+ | xC â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x0 | x3 â x0
+ | x4 â x4 | x5 â x4 | x6 â x4 | x7 â x4
+ | x8 â x8 | x9 â x8 | xA â x8 | xB â x8
+ | xC â xC | xD â xC | xE â xC | xF â xC ]
+ | xD â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x0 | x3 â x1
+ | x4 â x4 | x5 â x5 | x6 â x4 | x7 â x5
+ | x8 â x8 | x9 â x9 | xA â x8 | xB â x9
+ | xC â xC | xD â xD | xE â xC | xF â xD ]
+ | xE â match e2 with
+ [ x0 â x0 | x1 â x0 | x2 â x2 | x3 â x2
+ | x4 â x4 | x5 â x4 | x6 â x6 | x7 â x6
+ | x8 â x8 | x9 â x8 | xA â xA | xB â xA
+ | xC â xC | xD â xC | xE â xE | xF â xE ]
+ | xF â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB
+ | xC â xC | xD â xD | xE â xE | xF â xF ]
+ ].
+
+(* operatore or *)
+ndefinition or_ex â
+λe1,e2:exadecim.match e1 with
+ [ x0 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB
+ | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | x1 â match e2 with
+ [ x0 â x1 | x1 â x1 | x2 â x3 | x3 â x3
+ | x4 â x5 | x5 â x5 | x6 â x7 | x7 â x7
+ | x8 â x9 | x9 â x9 | xA â xB | xB â xB
+ | xC â xD | xD â xD | xE â xF | xF â xF ]
+ | x2 â match e2 with
+ [ x0 â x2 | x1 â x3 | x2 â x2 | x3 â x3
+ | x4 â x6 | x5 â x7 | x6 â x6 | x7 â x7
+ | x8 â xA | x9 â xB | xA â xA | xB â xB
+ | xC â xE | xD â xF | xE â xE | xF â xF ]
+ | x3 â match e2 with
+ [ x0 â x3 | x1 â x3 | x2 â x3 | x3 â x3
+ | x4 â x7 | x5 â x7 | x6 â x7 | x7 â x7
+ | x8 â xB | x9 â xB | xA â xB | xB â xB
+ | xC â xF | xD â xF | xE â xF | xF â xF ]
+ | x4 â match e2 with
+ [ x0 â x4 | x1 â x5 | x2 â x6 | x3 â x7
+ | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â xC | x9 â xD | xA â xE | xB â xF
+ | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | x5 â match e2 with
+ [ x0 â x5 | x1 â x5 | x2 â x7 | x3 â x7
+ | x4 â x5 | x5 â x5 | x6 â x7 | x7 â x7
+ | x8 â xD | x9 â xD | xA â xF | xB â xF
+ | xC â xD | xD â xD | xE â xF | xF â xF ]
+ | x6 â match e2 with
+ [ x0 â x6 | x1 â x7 | x2 â x6 | x3 â x7
+ | x4 â x6 | x5 â x7 | x6 â x6 | x7 â x7
+ | x8 â xE | x9 â xF | xA â xE | xB â xF
+ | xC â xE | xD â xF | xE â xE | xF â xF ]
+ | x7 â match e2 with
+ [ x0 â x7 | x1 â x7 | x2 â x7 | x3 â x7
+ | x4 â x7 | x5 â x7 | x6 â x7 | x7 â x7
+ | x8 â xF | x9 â xF | xA â xF | xB â xF
+ | xC â xF | xD â xF | xE â xF | xF â xF ]
+ | x8 â match e2 with
+ [ x0 â x8 | x1 â x9 | x2 â xA | x3 â xB
+ | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB
+ | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | x9 â match e2 with
+ [ x0 â x9 | x1 â x9 | x2 â xB | x3 â xB
+ | x4 â xD | x5 â xD | x6 â xF | x7 â xF
+ | x8 â x9 | x9 â x9 | xA â xB | xB â xB
+ | xC â xD | xD â xD | xE â xF | xF â xF ]
+ | xA â match e2 with
+ [ x0 â xA | x1 â xB | x2 â xA | x3 â xB
+ | x4 â xE | x5 â xF | x6 â xE | x7 â xF
+ | x8 â xA | x9 â xB | xA â xA | xB â xB
+ | xC â xE | xD â xF | xE â xE | xF â xF ]
+ | xB â match e2 with
+ [ x0 â xB | x1 â xB | x2 â xB | x3 â xB
+ | x4 â xF | x5 â xF | x6 â xF | x7 â xF
+ | x8 â xB | x9 â xB | xA â xB | xB â xB
+ | xC â xF | xD â xF | xE â xF | xF â xF ]
+ | xC â match e2 with
+ [ x0 â xC | x1 â xD | x2 â xE | x3 â xF
+ | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â xC | x9 â xD | xA â xE | xB â xF
+ | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | xD â match e2 with
+ [ x0 â xD | x1 â xD | x2 â xF | x3 â xF
+ | x4 â xD | x5 â xD | x6 â xF | x7 â xF
+ | x8 â xD | x9 â xD | xA â xF | xB â xF
+ | xC â xD | xD â xD | xE â xF | xF â xF ]
+ | xE â match e2 with
+ [ x0 â xE | x1 â xF | x2 â xE | x3 â xF
+ | x4 â xE | x5 â xF | x6 â xE | x7 â xF
+ | x8 â xE | x9 â xF | xA â xE | xB â xF
+ | xC â xE | xD â xF | xE â xE | xF â xF ]
+ | xF â match e2 with
+ [ x0 â xF | x1 â xF | x2 â xF | x3 â xF
+ | x4 â xF | x5 â xF | x6 â xF | x7 â xF
+ | x8 â xF | x9 â xF | xA â xF | xB â xF
+ | xC â xF | xD â xF | xE â xF | xF â xF ]
+ ].
+
+(* operatore xor *)
+ndefinition xor_ex â
+λe1,e2:exadecim.match e1 with
+ [ x0 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB
+ | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | x1 â match e2 with
+ [ x0 â x1 | x1 â x0 | x2 â x3 | x3 â x2
+ | x4 â x5 | x5 â x4 | x6 â x7 | x7 â x6
+ | x8 â x9 | x9 â x8 | xA â xB | xB â xA
+ | xC â xD | xD â xC | xE â xF | xF â xE ]
+ | x2 â match e2 with
+ [ x0 â x2 | x1 â x3 | x2 â x0 | x3 â x1
+ | x4 â x6 | x5 â x7 | x6 â x4 | x7 â x5
+ | x8 â xA | x9 â xB | xA â x8 | xB â x9
+ | xC â xE | xD â xF | xE â xC | xF â xD ]
+ | x3 â match e2 with
+ [ x0 â x3 | x1 â x2 | x2 â x1 | x3 â x0
+ | x4 â x7 | x5 â x6 | x6 â x5 | x7 â x4
+ | x8 â xB | x9 â xA | xA â x9 | xB â x8
+ | xC â xF | xD â xE | xE â xD | xF â xC ]
+ | x4 â match e2 with
+ [ x0 â x4 | x1 â x5 | x2 â x6 | x3 â x7
+ | x4 â x0 | x5 â x1 | x6 â x2 | x7 â x3
+ | x8 â xC | x9 â xD | xA â xE | xB â xF
+ | xC â x8 | xD â x9 | xE â xA | xF â xB ]
+ | x5 â match e2 with
+ [ x0 â x5 | x1 â x4 | x2 â x7 | x3 â x6
+ | x4 â x1 | x5 â x0 | x6 â x3 | x7 â x2
+ | x8 â xD | x9 â xC | xA â xF | xB â xE
+ | xC â x9 | xD â x8 | xE â xB | xF â xA ]
+ | x6 â match e2 with
+ [ x0 â x6 | x1 â x7 | x2 â x4 | x3 â x5
+ | x4 â x2 | x5 â x3 | x6 â x0 | x7 â x1
+ | x8 â xE | x9 â xF | xA â xC | xB â xD
+ | xC â xA | xD â xB | xE â x8 | xF â x9 ]
+ | x7 â match e2 with
+ [ x0 â x7 | x1 â x6 | x2 â x5 | x3 â x4
+ | x4 â x3 | x5 â x2 | x6 â x1 | x7 â x0
+ | x8 â xF | x9 â xE | xA â xD | xB â xC
+ | xC â xB | xD â xA | xE â x9 | xF â x8 ]
+ | x8 â match e2 with
+ [ x0 â x8 | x1 â x9 | x2 â xA | x3 â xB
+ | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3
+ | xC â x4 | xD â x5 | xE â x6 | xF â x7 ]
+ | x9 â match e2 with
+ [ x0 â x9 | x1 â x8 | x2 â xB | x3 â xA
+ | x4 â xD | x5 â xC | x6 â xF | x7 â xE
+ | x8 â x1 | x9 â x0 | xA â x3 | xB â x2
+ | xC â x5 | xD â x4 | xE â x7 | xF â x6 ]
+ | xA â match e2 with
+ [ x0 â xA | x1 â xB | x2 â x8 | x3 â x9
+ | x4 â xE | x5 â xF | x6 â xC | x7 â xD
+ | x8 â x2 | x9 â x3 | xA â x0 | xB â x1
+ | xC â x6 | xD â x7 | xE â x4 | xF â x5 ]
+ | xB â match e2 with
+ [ x0 â xB | x1 â xA | x2 â x9 | x3 â x8
+ | x4 â xF | x5 â xE | x6 â xD | x7 â xC
+ | x8 â x3 | x9 â x2 | xA â x1 | xB â x0
+ | xC â x7 | xD â x6 | xE â x5 | xF â x4 ]
+ | xC â match e2 with
+ [ x0 â xC | x1 â xD | x2 â xE | x3 â xF
+ | x4 â x8 | x5 â x9 | x6 â xA | x7 â xB
+ | x8 â x4 | x9 â x5 | xA â x6 | xB â x7
+ | xC â x0 | xD â x1 | xE â x2 | xF â x3 ]
+ | xD â match e2 with
+ [ x0 â xD | x1 â xC | x2 â xF | x3 â xE
+ | x4 â x9 | x5 â x8 | x6 â xB | x7 â xA
+ | x8 â x5 | x9 â x4 | xA â x7 | xB â x6
+ | xC â x1 | xD â x0 | xE â x3 | xF â x2 ]
+ | xE â match e2 with
+ [ x0 â xE | x1 â xF | x2 â xC | x3 â xD
+ | x4 â xA | x5 â xB | x6 â x8 | x7 â x9
+ | x8 â x6 | x9 â x7 | xA â x4 | xB â x5
+ | xC â x2 | xD â x3 | xE â x0 | xF â x1 ]
+ | xF â match e2 with
+ [ x0 â xF | x1 â xE | x2 â xD | x3 â xC
+ | x4 â xB | x5 â xA | x6 â x9 | x7 â x8
+ | x8 â x7 | x9 â x6 | xA â x5 | xB â x4
+ | xC â x3 | xD â x2 | xE â x1 | xF â x0 ]
+ ].
+
+(* operatore Most Significant Bit *)
+ndefinition getMSB_ex â
+λe:exadecim.match e with
+ [ x0 â false | x1 â false | x2 â false | x3 â false
+ | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â true | x9 â true | xA â true | xB â true
+ | xC â true | xD â true | xE â true | xF â true ].
+
+ndefinition setMSB_ex â
+λe:exadecim.match e with
+ [ x0 â x8 | x1 â x9 | x2 â xA | x3 â xB
+ | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB
+ | xC â xC | xD â xD | xE â xE | xF â xF ].
+
+ndefinition clrMSB_ex â
+λe:exadecim.match e with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3
+ | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3
+ | xC â x4 | xD â x5 | xE â x6 | xF â x7 ].
+
+(* operatore Least Significant Bit *)
+ndefinition getLSB_ex â
+λe:exadecim.match e with
+ [ x0 â false | x1 â true | x2 â false | x3 â true
+ | x4 â false | x5 â true | x6 â false | x7 â true
+ | x8 â false | x9 â true | xA â false | xB â true
+ | xC â false | xD â true | xE â false | xF â true ].
+
+ndefinition setLSB_ex â
+λe:exadecim.match e with
+ [ x0 â x1 | x1 â x1 | x2 â x3 | x3 â x3
+ | x4 â x5 | x5 â x5 | x6 â x7 | x7 â x7
+ | x8 â x9 | x9 â x9 | xA â xB | xB â xB
+ | xC â xD | xD â xD | xE â xF | xF â xF ].
+
+ndefinition clrLSB_ex â
+λe:exadecim.match e with
+ [ x0 â x0 | x1 â x0 | x2 â x2 | x3 â x2
+ | x4 â x4 | x5 â x4 | x6 â x6 | x7 â x6
+ | x8 â x8 | x9 â x8 | xA â xA | xB â xA
+ | xC â xC | xD â xC | xE â xE | xF â xE ].
+
+(* operatore rotazione destra con carry *)
+ndefinition rcr_ex â
+λc:bool.λe:exadecim.match c with
+ [ true â match e with
+ [ x0 â pair ⦠false x8 | x1 â pair ⦠true x8
+ | x2 â pair ⦠false x9 | x3 â pair ⦠true x9
+ | x4 â pair ⦠false xA | x5 â pair ⦠true xA
+ | x6 â pair ⦠false xB | x7 â pair ⦠true xB
+ | x8 â pair ⦠false xC | x9 â pair ⦠true xC
+ | xA â pair ⦠false xD | xB â pair ⦠true xD
+ | xC â pair ⦠false xE | xD â pair ⦠true xE
+ | xE â pair ⦠false xF | xF â pair ⦠true xF ]
+ | false â match e with
+ [ x0 â pair ⦠false x0 | x1 â pair ⦠true x0
+ | x2 â pair ⦠false x1 | x3 â pair ⦠true x1
+ | x4 â pair ⦠false x2 | x5 â pair ⦠true x2
+ | x6 â pair ⦠false x3 | x7 â pair ⦠true x3
+ | x8 â pair ⦠false x4 | x9 â pair ⦠true x4
+ | xA â pair ⦠false x5 | xB â pair ⦠true x5
+ | xC â pair ⦠false x6 | xD â pair ⦠true x6
+ | xE â pair ⦠false x7 | xF â pair ⦠true x7 ]
+ ].
+
+(* operatore shift destro *)
+ndefinition shr_ex â
+λe:exadecim.match e with
+ [ x0 â pair ⦠false x0 | x1 â pair ⦠true x0
+ | x2 â pair ⦠false x1 | x3 â pair ⦠true x1
+ | x4 â pair ⦠false x2 | x5 â pair ⦠true x2
+ | x6 â pair ⦠false x3 | x7 â pair ⦠true x3
+ | x8 â pair ⦠false x4 | x9 â pair ⦠true x4
+ | xA â pair ⦠false x5 | xB â pair ⦠true x5
+ | xC â pair ⦠false x6 | xD â pair ⦠true x6
+ | xE â pair ⦠false x7 | xF â pair ⦠true x7 ].
+
+(* operatore rotazione destra *)
+ndefinition ror_ex â
+λe:exadecim.match e with
+ [ x0 â x0 | x1 â x8 | x2 â x1 | x3 â x9
+ | x4 â x2 | x5 â xA | x6 â x3 | x7 â xB
+ | x8 â x4 | x9 â xC | xA â x5 | xB â xD
+ | xC â x6 | xD â xE | xE â x7 | xF â xF ].
+
+(* operatore rotazione sinistra con carry *)
+ndefinition rcl_ex â
+λc:bool.λe:exadecim.match c with
+ [ true â match e with
+ [ x0 â pair ⦠false x1 | x1 â pair ⦠false x3
+ | x2 â pair ⦠false x5 | x3 â pair ⦠false x7
+ | x4 â pair ⦠false x9 | x5 â pair ⦠false xB
+ | x6 â pair ⦠false xD | x7 â pair ⦠false xF
+ | x8 â pair ⦠true x1 | x9 â pair ⦠true x3
+ | xA â pair ⦠true x5 | xB â pair ⦠true x7
+ | xC â pair ⦠true x9 | xD â pair ⦠true xB
+ | xE â pair ⦠true xD | xF â pair ⦠true xF ]
+ | false â match e with
+ [ x0 â pair ⦠false x0 | x1 â pair ⦠false x2
+ | x2 â pair ⦠false x4 | x3 â pair ⦠false x6
+ | x4 â pair ⦠false x8 | x5 â pair ⦠false xA
+ | x6 â pair ⦠false xC | x7 â pair ⦠false xE
+ | x8 â pair ⦠true x0 | x9 â pair ⦠true x2
+ | xA â pair ⦠true x4 | xB â pair ⦠true x6
+ | xC â pair ⦠true x8 | xD â pair ⦠true xA
+ | xE â pair ⦠true xC | xF â pair ⦠true xE ]
+ ].
+
+(* operatore shift sinistro *)
+ndefinition shl_ex â
+λe:exadecim.match e with
+ [ x0 â pair ⦠false x0 | x1 â pair ⦠false x2
+ | x2 â pair ⦠false x4 | x3 â pair ⦠false x6
+ | x4 â pair ⦠false x8 | x5 â pair ⦠false xA
+ | x6 â pair ⦠false xC | x7 â pair ⦠false xE
+ | x8 â pair ⦠true x0 | x9 â pair ⦠true x2
+ | xA â pair ⦠true x4 | xB â pair ⦠true x6
+ | xC â pair ⦠true x8 | xD â pair ⦠true xA
+ | xE â pair ⦠true xC | xF â pair ⦠true xE ].
+
+(* operatore rotazione sinistra *)
+ndefinition rol_ex â
+λe:exadecim.match e with
+ [ x0 â x0 | x1 â x2 | x2 â x4 | x3 â x6
+ | x4 â x8 | x5 â xA | x6 â xC | x7 â xE
+ | x8 â x1 | x9 â x3 | xA â x5 | xB â x7
+ | xC â x9 | xD â xB | xE â xD | xF â xF ].
+
+(* operatore not/complemento a 1 *)
+ndefinition not_ex â
+λe:exadecim.match e with
+ [ x0 â xF | x1 â xE | x2 â xD | x3 â xC
+ | x4 â xB | x5 â xA | x6 â x9 | x7 â x8
+ | x8 â x7 | x9 â x6 | xA â x5 | xB â x4
+ | xC â x3 | xD â x2 | xE â x1 | xF â x0 ].
+
+(* operatore somma con data+carry â data+carry *)
+ndefinition plus_ex_dc_dc â
+λc:bool.λe1,e2:exadecim.
+ match c with
+ [ true â match e1 with
+ [ x0 â match e2 with
+ [ x0 â pair ⦠false x1 | x1 â pair ⦠false x2 | x2 â pair ⦠false x3 | x3 â pair ⦠false x4
+ | x4 â pair ⦠false x5 | x5 â pair ⦠false x6 | x6 â pair ⦠false x7 | x7 â pair ⦠false x8
+ | x8 â pair ⦠false x9 | x9 â pair ⦠false xA | xA â pair ⦠false xB | xB â pair ⦠false xC
+ | xC â pair ⦠false xD | xD â pair ⦠false xE | xE â pair ⦠false xF | xF â pair ⦠true x0 ]
+ | x1 â match e2 with
+ [ x0 â pair ⦠false x2 | x1 â pair ⦠false x3 | x2 â pair ⦠false x4 | x3 â pair ⦠false x5
+ | x4 â pair ⦠false x6 | x5 â pair ⦠false x7 | x6 â pair ⦠false x8 | x7 â pair ⦠false x9
+ | x8 â pair ⦠false xA | x9 â pair ⦠false xB | xA â pair ⦠false xC | xB â pair ⦠false xD
+ | xC â pair ⦠false xE | xD â pair ⦠false xF | xE â pair ⦠true x0 | xF â pair ⦠true x1 ]
+ | x2 â match e2 with
+ [ x0 â pair ⦠false x3 | x1 â pair ⦠false x4 | x2 â pair ⦠false x5 | x3 â pair ⦠false x6
+ | x4 â pair ⦠false x7 | x5 â pair ⦠false x8 | x6 â pair ⦠false x9 | x7 â pair ⦠false xA
+ | x8 â pair ⦠false xB | x9 â pair ⦠false xC | xA â pair ⦠false xD | xB â pair ⦠false xE
+ | xC â pair ⦠false xF | xD â pair ⦠true x0 | xE â pair ⦠true x1 | xF â pair ⦠true x2 ]
+ | x3 â match e2 with
+ [ x0 â pair ⦠false x4 | x1 â pair ⦠false x5 | x2 â pair ⦠false x6 | x3 â pair ⦠false x7
+ | x4 â pair ⦠false x8 | x5 â pair ⦠false x9 | x6 â pair ⦠false xA | x7 â pair ⦠false xB
+ | x8 â pair ⦠false xC | x9 â pair ⦠false xD | xA â pair ⦠false xE | xB â pair ⦠false xF
+ | xC â pair ⦠true x0 | xD â pair ⦠true x1 | xE â pair ⦠true x2 | xF â pair ⦠true x3 ]
+ | x4 â match e2 with
+ [ x0 â pair ⦠false x5 | x1 â pair ⦠false x6 | x2 â pair ⦠false x7 | x3 â pair ⦠false x8
+ | x4 â pair ⦠false x9 | x5 â pair ⦠false xA | x6 â pair ⦠false xB | x7 â pair ⦠false xC
+ | x8 â pair ⦠false xD | x9 â pair ⦠false xE | xA â pair ⦠false xF | xB â pair ⦠true x0
+ | xC â pair ⦠true x1 | xD â pair ⦠true x2 | xE â pair ⦠true x3 | xF â pair ⦠true x4 ]
+ | x5 â match e2 with
+ [ x0 â pair ⦠false x6 | x1 â pair ⦠false x7 | x2 â pair ⦠false x8 | x3 â pair ⦠false x9
+ | x4 â pair ⦠false xA | x5 â pair ⦠false xB | x6 â pair ⦠false xC | x7 â pair ⦠false xD
+ | x8 â pair ⦠false xE | x9 â pair ⦠false xF | xA â pair ⦠true x0 | xB â pair ⦠true x1
+ | xC â pair ⦠true x2 | xD â pair ⦠true x3 | xE â pair ⦠true x4 | xF â pair ⦠true x5 ]
+ | x6 â match e2 with
+ [ x0 â pair ⦠false x7 | x1 â pair ⦠false x8 | x2 â pair ⦠false x9 | x3 â pair ⦠false xA
+ | x4 â pair ⦠false xB | x5 â pair ⦠false xC | x6 â pair ⦠false xD | x7 â pair ⦠false xE
+ | x8 â pair ⦠false xF | x9 â pair ⦠true x0 | xA â pair ⦠true x1 | xB â pair ⦠true x2
+ | xC â pair ⦠true x3 | xD â pair ⦠true x4 | xE â pair ⦠true x5 | xF â pair ⦠true x6 ]
+ | x7 â match e2 with
+ [ x0 â pair ⦠false x8 | x1 â pair ⦠false x9 | x2 â pair ⦠false xA | x3 â pair ⦠false xB
+ | x4 â pair ⦠false xC | x5 â pair ⦠false xD | x6 â pair ⦠false xE | x7 â pair ⦠false xF
+ | x8 â pair ⦠true x0 | x9 â pair ⦠true x1 | xA â pair ⦠true x2 | xB â pair ⦠true x3
+ | xC â pair ⦠true x4 | xD â pair ⦠true x5 | xE â pair ⦠true x6 | xF â pair ⦠true x7 ]
+ | x8 â match e2 with
+ [ x0 â pair ⦠false x9 | x1 â pair ⦠false xA | x2 â pair ⦠false xB | x3 â pair ⦠false xC
+ | x4 â pair ⦠false xD | x5 â pair ⦠false xE | x6 â pair ⦠false xF | x7 â pair ⦠true x0
+ | x8 â pair ⦠true x1 | x9 â pair ⦠true x2 | xA â pair ⦠true x3 | xB â pair ⦠true x4
+ | xC â pair ⦠true x5 | xD â pair ⦠true x6 | xE â pair ⦠true x7 | xF â pair ⦠true x8 ]
+ | x9 â match e2 with
+ [ x0 â pair ⦠false xA | x1 â pair ⦠false xB | x2 â pair ⦠false xC | x3 â pair ⦠false xD
+ | x4 â pair ⦠false xE | x5 â pair ⦠false xF | x6 â pair ⦠true x0 | x7 â pair ⦠true x1
+ | x8 â pair ⦠true x2 | x9 â pair ⦠true x3 | xA â pair ⦠true x4 | xB â pair ⦠true x5
+ | xC â pair ⦠true x6 | xD â pair ⦠true x7 | xE â pair ⦠true x8 | xF â pair ⦠true x9 ]
+ | xA â match e2 with
+ [ x0 â pair ⦠false xB | x1 â pair ⦠false xC | x2 â pair ⦠false xD | x3 â pair ⦠false xE
+ | x4 â pair ⦠false xF | x5 â pair ⦠true x0 | x6 â pair ⦠true x1 | x7 â pair ⦠true x2
+ | x8 â pair ⦠true x3 | x9 â pair ⦠true x4 | xA â pair ⦠true x5 | xB â pair ⦠true x6
+ | xC â pair ⦠true x7 | xD â pair ⦠true x8 | xE â pair ⦠true x9 | xF â pair ⦠true xA ]
+ | xB â match e2 with
+ [ x0 â pair ⦠false xC | x1 â pair ⦠false xD | x2 â pair ⦠false xE | x3 â pair ⦠false xF
+ | x4 â pair ⦠true x0 | x5 â pair ⦠true x1 | x6 â pair ⦠true x2 | x7 â pair ⦠true x3
+ | x8 â pair ⦠true x4 | x9 â pair ⦠true x5 | xA â pair ⦠true x6 | xB â pair ⦠true x7
+ | xC â pair ⦠true x8 | xD â pair ⦠true x9 | xE â pair ⦠true xA | xF â pair ⦠true xB ]
+ | xC â match e2 with
+ [ x0 â pair ⦠false xD | x1 â pair ⦠false xE | x2 â pair ⦠false xF | x3 â pair ⦠true x0
+ | x4 â pair ⦠true x1 | x5 â pair ⦠true x2 | x6 â pair ⦠true x3 | x7 â pair ⦠true x4
+ | x8 â pair ⦠true x5 | x9 â pair ⦠true x6 | xA â pair ⦠true x7 | xB â pair ⦠true x8
+ | xC â pair ⦠true x9 | xD â pair ⦠true xA | xE â pair ⦠true xB | xF â pair ⦠true xC ]
+ | xD â match e2 with
+ [ x0 â pair ⦠false xE | x1 â pair ⦠false xF | x2 â pair ⦠true x0 | x3 â pair ⦠true x1
+ | x4 â pair ⦠true x2 | x5 â pair ⦠true x3 | x6 â pair ⦠true x4 | x7 â pair ⦠true x5
+ | x8 â pair ⦠true x6 | x9 â pair ⦠true x7 | xA â pair ⦠true x8 | xB â pair ⦠true x9
+ | xC â pair ⦠true xA | xD â pair ⦠true xB | xE â pair ⦠true xC | xF â pair ⦠true xD ]
+ | xE â match e2 with
+ [ x0 â pair ⦠false xF | x1 â pair ⦠true x0 | x2 â pair ⦠true x1 | x3 â pair ⦠true x2
+ | x4 â pair ⦠true x3 | x5 â pair ⦠true x4 | x6 â pair ⦠true x5 | x7 â pair ⦠true x6
+ | x8 â pair ⦠true x7 | x9 â pair ⦠true x8 | xA â pair ⦠true x9 | xB â pair ⦠true xA
+ | xC â pair ⦠true xB | xD â pair ⦠true xC | xE â pair ⦠true xD | xF â pair ⦠true xE ]
+ | xF â match e2 with
+ [ x0 â pair ⦠true x0 | x1 â pair ⦠true x1 | x2 â pair ⦠true x2 | x3 â pair ⦠true x3
+ | x4 â pair ⦠true x4 | x5 â pair ⦠true x5 | x6 â pair ⦠true x6 | x7 â pair ⦠true x7
+ | x8 â pair ⦠true x8 | x9 â pair ⦠true x9 | xA â pair ⦠true xA | xB â pair ⦠true xB
+ | xC â pair ⦠true xC | xD â pair ⦠true xD | xE â pair ⦠true xE | xF â pair ⦠true xF ]
+ ]
+ | false â match e1 with
+ [ x0 â match e2 with
+ [ x0 â pair ⦠false x0 | x1 â pair ⦠false x1 | x2 â pair ⦠false x2 | x3 â pair ⦠false x3
+ | x4 â pair ⦠false x4 | x5 â pair ⦠false x5 | x6 â pair ⦠false x6 | x7 â pair ⦠false x7
+ | x8 â pair ⦠false x8 | x9 â pair ⦠false x9 | xA â pair ⦠false xA | xB â pair ⦠false xB
+ | xC â pair ⦠false xC | xD â pair ⦠false xD | xE â pair ⦠false xE | xF â pair ⦠false xF ]
+ | x1 â match e2 with
+ [ x0 â pair ⦠false x1 | x1 â pair ⦠false x2 | x2 â pair ⦠false x3 | x3 â pair ⦠false x4
+ | x4 â pair ⦠false x5 | x5 â pair ⦠false x6 | x6 â pair ⦠false x7 | x7 â pair ⦠false x8
+ | x8 â pair ⦠false x9 | x9 â pair ⦠false xA | xA â pair ⦠false xB | xB â pair ⦠false xC
+ | xC â pair ⦠false xD | xD â pair ⦠false xE | xE â pair ⦠false xF | xF â pair ⦠true x0 ]
+ | x2 â match e2 with
+ [ x0 â pair ⦠false x2 | x1 â pair ⦠false x3 | x2 â pair ⦠false x4 | x3 â pair ⦠false x5
+ | x4 â pair ⦠false x6 | x5 â pair ⦠false x7 | x6 â pair ⦠false x8 | x7 â pair ⦠false x9
+ | x8 â pair ⦠false xA | x9 â pair ⦠false xB | xA â pair ⦠false xC | xB â pair ⦠false xD
+ | xC â pair ⦠false xE | xD â pair ⦠false xF | xE â pair ⦠true x0 | xF â pair ⦠true x1 ]
+ | x3 â match e2 with
+ [ x0 â pair ⦠false x3 | x1 â pair ⦠false x4 | x2 â pair ⦠false x5 | x3 â pair ⦠false x6
+ | x4 â pair ⦠false x7 | x5 â pair ⦠false x8 | x6 â pair ⦠false x9 | x7 â pair ⦠false xA
+ | x8 â pair ⦠false xB | x9 â pair ⦠false xC | xA â pair ⦠false xD | xB â pair ⦠false xE
+ | xC â pair ⦠false xF | xD â pair ⦠true x0 | xE â pair ⦠true x1 | xF â pair ⦠true x2 ]
+ | x4 â match e2 with
+ [ x0 â pair ⦠false x4 | x1 â pair ⦠false x5 | x2 â pair ⦠false x6 | x3 â pair ⦠false x7
+ | x4 â pair ⦠false x8 | x5 â pair ⦠false x9 | x6 â pair ⦠false xA | x7 â pair ⦠false xB
+ | x8 â pair ⦠false xC | x9 â pair ⦠false xD | xA â pair ⦠false xE | xB â pair ⦠false xF
+ | xC â pair ⦠true x0 | xD â pair ⦠true x1 | xE â pair ⦠true x2 | xF â pair ⦠true x3 ]
+ | x5 â match e2 with
+ [ x0 â pair ⦠false x5 | x1 â pair ⦠false x6 | x2 â pair ⦠false x7 | x3 â pair ⦠false x8
+ | x4 â pair ⦠false x9 | x5 â pair ⦠false xA | x6 â pair ⦠false xB | x7 â pair ⦠false xC
+ | x8 â pair ⦠false xD | x9 â pair ⦠false xE | xA â pair ⦠false xF | xB â pair ⦠true x0
+ | xC â pair ⦠true x1 | xD â pair ⦠true x2 | xE â pair ⦠true x3 | xF â pair ⦠true x4 ]
+ | x6 â match e2 with
+ [ x0 â pair ⦠false x6 | x1 â pair ⦠false x7 | x2 â pair ⦠false x8 | x3 â pair ⦠false x9
+ | x4 â pair ⦠false xA | x5 â pair ⦠false xB | x6 â pair ⦠false xC | x7 â pair ⦠false xD
+ | x8 â pair ⦠false xE | x9 â pair ⦠false xF | xA â pair ⦠true x0 | xB â pair ⦠true x1
+ | xC â pair ⦠true x2 | xD â pair ⦠true x3 | xE â pair ⦠true x4 | xF â pair ⦠true x5 ]
+ | x7 â match e2 with
+ [ x0 â pair ⦠false x7 | x1 â pair ⦠false x8 | x2 â pair ⦠false x9 | x3 â pair ⦠false xA
+ | x4 â pair ⦠false xB | x5 â pair ⦠false xC | x6 â pair ⦠false xD | x7 â pair ⦠false xE
+ | x8 â pair ⦠false xF | x9 â pair ⦠true x0 | xA â pair ⦠true x1 | xB â pair ⦠true x2
+ | xC â pair ⦠true x3 | xD â pair ⦠true x4 | xE â pair ⦠true x5 | xF â pair ⦠true x6 ]
+ | x8 â match e2 with
+ [ x0 â pair ⦠false x8 | x1 â pair ⦠false x9 | x2 â pair ⦠false xA | x3 â pair ⦠false xB
+ | x4 â pair ⦠false xC | x5 â pair ⦠false xD | x6 â pair ⦠false xE | x7 â pair ⦠false xF
+ | x8 â pair ⦠true x0 | x9 â pair ⦠true x1 | xA â pair ⦠true x2 | xB â pair ⦠true x3
+ | xC â pair ⦠true x4 | xD â pair ⦠true x5 | xE â pair ⦠true x6 | xF â pair ⦠true x7 ]
+ | x9 â match e2 with
+ [ x0 â pair ⦠false x9 | x1 â pair ⦠false xA | x2 â pair ⦠false xB | x3 â pair ⦠false xC
+ | x4 â pair ⦠false xD | x5 â pair ⦠false xE | x6 â pair ⦠false xF | x7 â pair ⦠true x0
+ | x8 â pair ⦠true x1 | x9 â pair ⦠true x2 | xA â pair ⦠true x3 | xB â pair ⦠true x4
+ | xC â pair ⦠true x5 | xD â pair ⦠true x6 | xE â pair ⦠true x7 | xF â pair ⦠true x8 ]
+ | xA â match e2 with
+ [ x0 â pair ⦠false xA | x1 â pair ⦠false xB | x2 â pair ⦠false xC | x3 â pair ⦠false xD
+ | x4 â pair ⦠false xE | x5 â pair ⦠false xF | x6 â pair ⦠true x0 | x7 â pair ⦠true x1
+ | x8 â pair ⦠true x2 | x9 â pair ⦠true x3 | xA â pair ⦠true x4 | xB â pair ⦠true x5
+ | xC â pair ⦠true x6 | xD â pair ⦠true x7 | xE â pair ⦠true x8 | xF â pair ⦠true x9 ]
+ | xB â match e2 with
+ [ x0 â pair ⦠false xB | x1 â pair ⦠false xC | x2 â pair ⦠false xD | x3 â pair ⦠false xE
+ | x4 â pair ⦠false xF | x5 â pair ⦠true x0 | x6 â pair ⦠true x1 | x7 â pair ⦠true x2
+ | x8 â pair ⦠true x3 | x9 â pair ⦠true x4 | xA â pair ⦠true x5 | xB â pair ⦠true x6
+ | xC â pair ⦠true x7 | xD â pair ⦠true x8 | xE â pair ⦠true x9 | xF â pair ⦠true xA ]
+ | xC â match e2 with
+ [ x0 â pair ⦠false xC | x1 â pair ⦠false xD | x2 â pair ⦠false xE | x3 â pair ⦠false xF
+ | x4 â pair ⦠true x0 | x5 â pair ⦠true x1 | x6 â pair ⦠true x2 | x7 â pair ⦠true x3
+ | x8 â pair ⦠true x4 | x9 â pair ⦠true x5 | xA â pair ⦠true x6 | xB â pair ⦠true x7
+ | xC â pair ⦠true x8 | xD â pair ⦠true x9 | xE â pair ⦠true xA | xF â pair ⦠true xB ]
+ | xD â match e2 with
+ [ x0 â pair ⦠false xD | x1 â pair ⦠false xE | x2 â pair ⦠false xF | x3 â pair ⦠true x0
+ | x4 â pair ⦠true x1 | x5 â pair ⦠true x2 | x6 â pair ⦠true x3 | x7 â pair ⦠true x4
+ | x8 â pair ⦠true x5 | x9 â pair ⦠true x6 | xA â pair ⦠true x7 | xB â pair ⦠true x8
+ | xC â pair ⦠true x9 | xD â pair ⦠true xA | xE â pair ⦠true xB | xF â pair ⦠true xC ]
+ | xE â match e2 with
+ [ x0 â pair ⦠false xE | x1 â pair ⦠false xF | x2 â pair ⦠true x0 | x3 â pair ⦠true x1
+ | x4 â pair ⦠true x2 | x5 â pair ⦠true x3 | x6 â pair ⦠true x4 | x7 â pair ⦠true x5
+ | x8 â pair ⦠true x6 | x9 â pair ⦠true x7 | xA â pair ⦠true x8 | xB â pair ⦠true x9
+ | xC â pair ⦠true xA | xD â pair ⦠true xB | xE â pair ⦠true xC | xF â pair ⦠true xD ]
+ | xF â match e2 with
+ [ x0 â pair ⦠false xF | x1 â pair ⦠true x0 | x2 â pair ⦠true x1 | x3 â pair ⦠true x2
+ | x4 â pair ⦠true x3 | x5 â pair ⦠true x4 | x6 â pair ⦠true x5 | x7 â pair ⦠true x6
+ | x8 â pair ⦠true x7 | x9 â pair ⦠true x8 | xA â pair ⦠true x9 | xB â pair ⦠true xA
+ | xC â pair ⦠true xB | xD â pair ⦠true xC | xE â pair ⦠true xD | xF â pair ⦠true xE ]
+ ]].
+
+(* operatore somma con data â data+carry *)
+ndefinition plus_ex_d_dc â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â pair ⦠false x0 | x1 â pair ⦠false x1 | x2 â pair ⦠false x2 | x3 â pair ⦠false x3
+ | x4 â pair ⦠false x4 | x5 â pair ⦠false x5 | x6 â pair ⦠false x6 | x7 â pair ⦠false x7
+ | x8 â pair ⦠false x8 | x9 â pair ⦠false x9 | xA â pair ⦠false xA | xB â pair ⦠false xB
+ | xC â pair ⦠false xC | xD â pair ⦠false xD | xE â pair ⦠false xE | xF â pair ⦠false xF ]
+ | x1 â match e2 with
+ [ x0 â pair ⦠false x1 | x1 â pair ⦠false x2 | x2 â pair ⦠false x3 | x3 â pair ⦠false x4
+ | x4 â pair ⦠false x5 | x5 â pair ⦠false x6 | x6 â pair ⦠false x7 | x7 â pair ⦠false x8
+ | x8 â pair ⦠false x9 | x9 â pair ⦠false xA | xA â pair ⦠false xB | xB â pair ⦠false xC
+ | xC â pair ⦠false xD | xD â pair ⦠false xE | xE â pair ⦠false xF | xF â pair ⦠true x0 ]
+ | x2 â match e2 with
+ [ x0 â pair ⦠false x2 | x1 â pair ⦠false x3 | x2 â pair ⦠false x4 | x3 â pair ⦠false x5
+ | x4 â pair ⦠false x6 | x5 â pair ⦠false x7 | x6 â pair ⦠false x8 | x7 â pair ⦠false x9
+ | x8 â pair ⦠false xA | x9 â pair ⦠false xB | xA â pair ⦠false xC | xB â pair ⦠false xD
+ | xC â pair ⦠false xE | xD â pair ⦠false xF | xE â pair ⦠true x0 | xF â pair ⦠true x1 ]
+ | x3 â match e2 with
+ [ x0 â pair ⦠false x3 | x1 â pair ⦠false x4 | x2 â pair ⦠false x5 | x3 â pair ⦠false x6
+ | x4 â pair ⦠false x7 | x5 â pair ⦠false x8 | x6 â pair ⦠false x9 | x7 â pair ⦠false xA
+ | x8 â pair ⦠false xB | x9 â pair ⦠false xC | xA â pair ⦠false xD | xB â pair ⦠false xE
+ | xC â pair ⦠false xF | xD â pair ⦠true x0 | xE â pair ⦠true x1 | xF â pair ⦠true x2 ]
+ | x4 â match e2 with
+ [ x0 â pair ⦠false x4 | x1 â pair ⦠false x5 | x2 â pair ⦠false x6 | x3 â pair ⦠false x7
+ | x4 â pair ⦠false x8 | x5 â pair ⦠false x9 | x6 â pair ⦠false xA | x7 â pair ⦠false xB
+ | x8 â pair ⦠false xC | x9 â pair ⦠false xD | xA â pair ⦠false xE | xB â pair ⦠false xF
+ | xC â pair ⦠true x0 | xD â pair ⦠true x1 | xE â pair ⦠true x2 | xF â pair ⦠true x3 ]
+ | x5 â match e2 with
+ [ x0 â pair ⦠false x5 | x1 â pair ⦠false x6 | x2 â pair ⦠false x7 | x3 â pair ⦠false x8
+ | x4 â pair ⦠false x9 | x5 â pair ⦠false xA | x6 â pair ⦠false xB | x7 â pair ⦠false xC
+ | x8 â pair ⦠false xD | x9 â pair ⦠false xE | xA â pair ⦠false xF | xB â pair ⦠true x0
+ | xC â pair ⦠true x1 | xD â pair ⦠true x2 | xE â pair ⦠true x3 | xF â pair ⦠true x4 ]
+ | x6 â match e2 with
+ [ x0 â pair ⦠false x6 | x1 â pair ⦠false x7 | x2 â pair ⦠false x8 | x3 â pair ⦠false x9
+ | x4 â pair ⦠false xA | x5 â pair ⦠false xB | x6 â pair ⦠false xC | x7 â pair ⦠false xD
+ | x8 â pair ⦠false xE | x9 â pair ⦠false xF | xA â pair ⦠true x0 | xB â pair ⦠true x1
+ | xC â pair ⦠true x2 | xD â pair ⦠true x3 | xE â pair ⦠true x4 | xF â pair ⦠true x5 ]
+ | x7 â match e2 with
+ [ x0 â pair ⦠false x7 | x1 â pair ⦠false x8 | x2 â pair ⦠false x9 | x3 â pair ⦠false xA
+ | x4 â pair ⦠false xB | x5 â pair ⦠false xC | x6 â pair ⦠false xD | x7 â pair ⦠false xE
+ | x8 â pair ⦠false xF | x9 â pair ⦠true x0 | xA â pair ⦠true x1 | xB â pair ⦠true x2
+ | xC â pair ⦠true x3 | xD â pair ⦠true x4 | xE â pair ⦠true x5 | xF â pair ⦠true x6 ]
+ | x8 â match e2 with
+ [ x0 â pair ⦠false x8 | x1 â pair ⦠false x9 | x2 â pair ⦠false xA | x3 â pair ⦠false xB
+ | x4 â pair ⦠false xC | x5 â pair ⦠false xD | x6 â pair ⦠false xE | x7 â pair ⦠false xF
+ | x8 â pair ⦠true x0 | x9 â pair ⦠true x1 | xA â pair ⦠true x2 | xB â pair ⦠true x3
+ | xC â pair ⦠true x4 | xD â pair ⦠true x5 | xE â pair ⦠true x6 | xF â pair ⦠true x7 ]
+ | x9 â match e2 with
+ [ x0 â pair ⦠false x9 | x1 â pair ⦠false xA | x2 â pair ⦠false xB | x3 â pair ⦠false xC
+ | x4 â pair ⦠false xD | x5 â pair ⦠false xE | x6 â pair ⦠false xF | x7 â pair ⦠true x0
+ | x8 â pair ⦠true x1 | x9 â pair ⦠true x2 | xA â pair ⦠true x3 | xB â pair ⦠true x4
+ | xC â pair ⦠true x5 | xD â pair ⦠true x6 | xE â pair ⦠true x7 | xF â pair ⦠true x8 ]
+ | xA â match e2 with
+ [ x0 â pair ⦠false xA | x1 â pair ⦠false xB | x2 â pair ⦠false xC | x3 â pair ⦠false xD
+ | x4 â pair ⦠false xE | x5 â pair ⦠false xF | x6 â pair ⦠true x0 | x7 â pair ⦠true x1
+ | x8 â pair ⦠true x2 | x9 â pair ⦠true x3 | xA â pair ⦠true x4 | xB â pair ⦠true x5
+ | xC â pair ⦠true x6 | xD â pair ⦠true x7 | xE â pair ⦠true x8 | xF â pair ⦠true x9 ]
+ | xB â match e2 with
+ [ x0 â pair ⦠false xB | x1 â pair ⦠false xC | x2 â pair ⦠false xD | x3 â pair ⦠false xE
+ | x4 â pair ⦠false xF | x5 â pair ⦠true x0 | x6 â pair ⦠true x1 | x7 â pair ⦠true x2
+ | x8 â pair ⦠true x3 | x9 â pair ⦠true x4 | xA â pair ⦠true x5 | xB â pair ⦠true x6
+ | xC â pair ⦠true x7 | xD â pair ⦠true x8 | xE â pair ⦠true x9 | xF â pair ⦠true xA ]
+ | xC â match e2 with
+ [ x0 â pair ⦠false xC | x1 â pair ⦠false xD | x2 â pair ⦠false xE | x3 â pair ⦠false xF
+ | x4 â pair ⦠true x0 | x5 â pair ⦠true x1 | x6 â pair ⦠true x2 | x7 â pair ⦠true x3
+ | x8 â pair ⦠true x4 | x9 â pair ⦠true x5 | xA â pair ⦠true x6 | xB â pair ⦠true x7
+ | xC â pair ⦠true x8 | xD â pair ⦠true x9 | xE â pair ⦠true xA | xF â pair ⦠true xB ]
+ | xD â match e2 with
+ [ x0 â pair ⦠false xD | x1 â pair ⦠false xE | x2 â pair ⦠false xF | x3 â pair ⦠true x0
+ | x4 â pair ⦠true x1 | x5 â pair ⦠true x2 | x6 â pair ⦠true x3 | x7 â pair ⦠true x4
+ | x8 â pair ⦠true x5 | x9 â pair ⦠true x6 | xA â pair ⦠true x7 | xB â pair ⦠true x8
+ | xC â pair ⦠true x9 | xD â pair ⦠true xA | xE â pair ⦠true xB | xF â pair ⦠true xC ]
+ | xE â match e2 with
+ [ x0 â pair ⦠false xE | x1 â pair ⦠false xF | x2 â pair ⦠true x0 | x3 â pair ⦠true x1
+ | x4 â pair ⦠true x2 | x5 â pair ⦠true x3 | x6 â pair ⦠true x4 | x7 â pair ⦠true x5
+ | x8 â pair ⦠true x6 | x9 â pair ⦠true x7 | xA â pair ⦠true x8 | xB â pair ⦠true x9
+ | xC â pair ⦠true xA | xD â pair ⦠true xB | xE â pair ⦠true xC | xF â pair ⦠true xD ]
+ | xF â match e2 with
+ [ x0 â pair ⦠false xF | x1 â pair ⦠true x0 | x2 â pair ⦠true x1 | x3 â pair ⦠true x2
+ | x4 â pair ⦠true x3 | x5 â pair ⦠true x4 | x6 â pair ⦠true x5 | x7 â pair ⦠true x6
+ | x8 â pair ⦠true x7 | x9 â pair ⦠true x8 | xA â pair ⦠true x9 | xB â pair ⦠true xA
+ | xC â pair ⦠true xB | xD â pair ⦠true xC | xE â pair ⦠true xD | xF â pair ⦠true xE ]
+ ].
+
+(* operatore somma con data+carry â data *)
+ndefinition plus_ex_dc_d â
+λc:bool.λe1,e2:exadecim.
+ match c with
+ [ true â match e1 with
+ [ x0 â match e2 with
+ [ x0 â x1 | x1 â x2 | x2 â x3 | x3 â x4 | x4 â x5 | x5 â x6 | x6 â x7 | x7 â x8
+ | x8 â x9 | x9 â xA | xA â xB | xB â xC | xC â xD | xD â xE | xE â xF | xF â x0 ]
+ | x1 â match e2 with
+ [ x0 â x2 | x1 â x3 | x2 â x4 | x3 â x5 | x4 â x6 | x5 â x7 | x6 â x8 | x7 â x9
+ | x8 â xA | x9 â xB | xA â xC | xB â xD | xC â xE | xD â xF | xE â x0 | xF â x1 ]
+ | x2 â match e2 with
+ [ x0 â x3 | x1 â x4 | x2 â x5 | x3 â x6 | x4 â x7 | x5 â x8 | x6 â x9 | x7 â xA
+ | x8 â xB | x9 â xC | xA â xD | xB â xE | xC â xF | xD â x0 | xE â x1 | xF â x2 ]
+ | x3 â match e2 with
+ [ x0 â x4 | x1 â x5 | x2 â x6 | x3 â x7 | x4 â x8 | x5 â x9 | x6 â xA | x7 â xB
+ | x8 â xC | x9 â xD | xA â xE | xB â xF | xC â x0 | xD â x1 | xE â x2 | xF â x3 ]
+ | x4 â match e2 with
+ [ x0 â x5 | x1 â x6 | x2 â x7 | x3 â x8 | x4 â x9 | x5 â xA | x6 â xB | x7 â xC
+ | x8 â xD | x9 â xE | xA â xF | xB â x0 | xC â x1 | xD â x2 | xE â x3 | xF â x4 ]
+ | x5 â match e2 with
+ [ x0 â x6 | x1 â x7 | x2 â x8 | x3 â x9 | x4 â xA | x5 â xB | x6 â xC | x7 â xD
+ | x8 â xE | x9 â xF | xA â x0 | xB â x1 | xC â x2 | xD â x3 | xE â x4 | xF â x5 ]
+ | x6 â match e2 with
+ [ x0 â x7 | x1 â x8 | x2 â x9 | x3 â xA | x4 â xB | x5 â xC | x6 â xD | x7 â xE
+ | x8 â xF | x9 â x0 | xA â x1 | xB â x2 | xC â x3 | xD â x4 | xE â x5 | xF â x6 ]
+ | x7 â match e2 with
+ [ x0 â x8 | x1 â x9 | x2 â xA | x3 â xB | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3 | xC â x4 | xD â x5 | xE â x6 | xF â x7 ]
+ | x8 â match e2 with
+ [ x0 â x9 | x1 â xA | x2 â xB | x3 â xC | x4 â xD | x5 â xE | x6 â xF | x7 â x0
+ | x8 â x1 | x9 â x2 | xA â x3 | xB â x4 | xC â x5 | xD â x6 | xE â x7 | xF â x8 ]
+ | x9 â match e2 with
+ [ x0 â xA | x1 â xB | x2 â xC | x3 â xD | x4 â xE | x5 â xF | x6 â x0 | x7 â x1
+ | x8 â x2 | x9 â x3 | xA â x4 | xB â x5 | xC â x6 | xD â x7 | xE â x8 | xF â x9 ]
+ | xA â match e2 with
+ [ x0 â xB | x1 â xC | x2 â xD | x3 â xE | x4 â xF | x5 â x0 | x6 â x1 | x7 â x2
+ | x8 â x3 | x9 â x4 | xA â x5 | xB â x6 | xC â x7 | xD â x8 | xE â x9 | xF â xA ]
+ | xB â match e2 with
+ [ x0 â xC | x1 â xD | x2 â xE | x3 â xF | x4 â x0 | x5 â x1 | x6 â x2 | x7 â x3
+ | x8 â x4 | x9 â x5 | xA â x6 | xB â x7 | xC â x8 | xD â x9 | xE â xA | xF â xB ]
+ | xC â match e2 with
+ [ x0 â xD | x1 â xE | x2 â xF | x3 â x0 | x4 â x1 | x5 â x2 | x6 â x3 | x7 â x4
+ | x8 â x5 | x9 â x6 | xA â x7 | xB â x8 | xC â x9 | xD â xA | xE â xB | xF â xC ]
+ | xD â match e2 with
+ [ x0 â xE | x1 â xF | x2 â x0 | x3 â x1 | x4 â x2 | x5 â x3 | x6 â x4 | x7 â x5
+ | x8 â x6 | x9 â x7 | xA â x8 | xB â x9 | xC â xA | xD â xB | xE â xC | xF â xD ]
+ | xE â match e2 with
+ [ x0 â xF | x1 â x0 | x2 â x1 | x3 â x2 | x4 â x3 | x5 â x4 | x6 â x5 | x7 â x6
+ | x8 â x7 | x9 â x8 | xA â x9 | xB â xA | xC â xB | xD â xC | xE â xD | xF â xE ]
+ | xF â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3 | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB | xC â xC | xD â xD | xE â xE | xF â xF ]
+ ]
+ | false â match e1 with
+ [ x0 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3 | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | x1 â match e2 with
+ [ x0 â x1 | x1 â x2 | x2 â x3 | x3 â x4 | x4 â x5 | x5 â x6 | x6 â x7 | x7 â x8
+ | x8 â x9 | x9 â xA | xA â xB | xB â xC | xC â xD | xD â xE | xE â xF | xF â x0 ]
+ | x2 â match e2 with
+ [ x0 â x2 | x1 â x3 | x2 â x4 | x3 â x5 | x4 â x6 | x5 â x7 | x6 â x8 | x7 â x9
+ | x8 â xA | x9 â xB | xA â xC | xB â xD | xC â xE | xD â xF | xE â x0 | xF â x1 ]
+ | x3 â match e2 with
+ [ x0 â x3 | x1 â x4 | x2 â x5 | x3 â x6 | x4 â x7 | x5 â x8 | x6 â x9 | x7 â xA
+ | x8 â xB | x9 â xC | xA â xD | xB â xE | xC â xF | xD â x0 | xE â x1 | xF â x2 ]
+ | x4 â match e2 with
+ [ x0 â x4 | x1 â x5 | x2 â x6 | x3 â x7 | x4 â x8 | x5 â x9 | x6 â xA | x7 â xB
+ | x8 â xC | x9 â xD | xA â xE | xB â xF | xC â x0 | xD â x1 | xE â x2 | xF â x3 ]
+ | x5 â match e2 with
+ [ x0 â x5 | x1 â x6 | x2 â x7 | x3 â x8 | x4 â x9 | x5 â xA | x6 â xB | x7 â xC
+ | x8 â xD | x9 â xE | xA â xF | xB â x0 | xC â x1 | xD â x2 | xE â x3 | xF â x4 ]
+ | x6 â match e2 with
+ [ x0 â x6 | x1 â x7 | x2 â x8 | x3 â x9 | x4 â xA | x5 â xB | x6 â xC | x7 â xD
+ | x8 â xE | x9 â xF | xA â x0 | xB â x1 | xC â x2 | xD â x3 | xE â x4 | xF â x5 ]
+ | x7 â match e2 with
+ [ x0 â x7 | x1 â x8 | x2 â x9 | x3 â xA | x4 â xB | x5 â xC | x6 â xD | x7 â xE
+ | x8 â xF | x9 â x0 | xA â x1 | xB â x2 | xC â x3 | xD â x4 | xE â x5 | xF â x6 ]
+ | x8 â match e2 with
+ [ x0 â x8 | x1 â x9 | x2 â xA | x3 â xB | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3 | xC â x4 | xD â x5 | xE â x6 | xF â x7 ]
+ | x9 â match e2 with
+ [ x0 â x9 | x1 â xA | x2 â xB | x3 â xC | x4 â xD | x5 â xE | x6 â xF | x7 â x0
+ | x8 â x1 | x9 â x2 | xA â x3 | xB â x4 | xC â x5 | xD â x6 | xE â x7 | xF â x8 ]
+ | xA â match e2 with
+ [ x0 â xA | x1 â xB | x2 â xC | x3 â xD | x4 â xE | x5 â xF | x6 â x0 | x7 â x1
+ | x8 â x2 | x9 â x3 | xA â x4 | xB â x5 | xC â x6 | xD â x7 | xE â x8 | xF â x9 ]
+ | xB â match e2 with
+ [ x0 â xB | x1 â xC | x2 â xD | x3 â xE | x4 â xF | x5 â x0 | x6 â x1 | x7 â x2
+ | x8 â x3 | x9 â x4 | xA â x5 | xB â x6 | xC â x7 | xD â x8 | xE â x9 | xF â xA ]
+ | xC â match e2 with
+ [ x0 â xC | x1 â xD | x2 â xE | x3 â xF | x4 â x0 | x5 â x1 | x6 â x2 | x7 â x3
+ | x8 â x4 | x9 â x5 | xA â x6 | xB â x7 | xC â x8 | xD â x9 | xE â xA | xF â xB ]
+ | xD â match e2 with
+ [ x0 â xD | x1 â xE | x2 â xF | x3 â x0 | x4 â x1 | x5 â x2 | x6 â x3 | x7 â x4
+ | x8 â x5 | x9 â x6 | xA â x7 | xB â x8 | xC â x9 | xD â xA | xE â xB | xF â xC ]
+ | xE â match e2 with
+ [ x0 â xE | x1 â xF | x2 â x0 | x3 â x1 | x4 â x2 | x5 â x3 | x6 â x4 | x7 â x5
+ | x8 â x6 | x9 â x7 | xA â x8 | xB â x9 | xC â xA | xD â xB | xE â xC | xF â xD ]
+ | xF â match e2 with
+ [ x0 â xF | x1 â x0 | x2 â x1 | x3 â x2 | x4 â x3 | x5 â x4 | x6 â x5 | x7 â x6
+ | x8 â x7 | x9 â x8 | xA â x9 | xB â xA | xC â xB | xD â xC | xE â xD | xF â xE ]
+ ]].
+
+(* operatore somma con data â data *)
+ndefinition plus_ex_d_d â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â x0 | x1 â x1 | x2 â x2 | x3 â x3 | x4 â x4 | x5 â x5 | x6 â x6 | x7 â x7
+ | x8 â x8 | x9 â x9 | xA â xA | xB â xB | xC â xC | xD â xD | xE â xE | xF â xF ]
+ | x1 â match e2 with
+ [ x0 â x1 | x1 â x2 | x2 â x3 | x3 â x4 | x4 â x5 | x5 â x6 | x6 â x7 | x7 â x8
+ | x8 â x9 | x9 â xA | xA â xB | xB â xC | xC â xD | xD â xE | xE â xF | xF â x0 ]
+ | x2 â match e2 with
+ [ x0 â x2 | x1 â x3 | x2 â x4 | x3 â x5 | x4 â x6 | x5 â x7 | x6 â x8 | x7 â x9
+ | x8 â xA | x9 â xB | xA â xC | xB â xD | xC â xE | xD â xF | xE â x0 | xF â x1 ]
+ | x3 â match e2 with
+ [ x0 â x3 | x1 â x4 | x2 â x5 | x3 â x6 | x4 â x7 | x5 â x8 | x6 â x9 | x7 â xA
+ | x8 â xB | x9 â xC | xA â xD | xB â xE | xC â xF | xD â x0 | xE â x1 | xF â x2 ]
+ | x4 â match e2 with
+ [ x0 â x4 | x1 â x5 | x2 â x6 | x3 â x7 | x4 â x8 | x5 â x9 | x6 â xA | x7 â xB
+ | x8 â xC | x9 â xD | xA â xE | xB â xF | xC â x0 | xD â x1 | xE â x2 | xF â x3 ]
+ | x5 â match e2 with
+ [ x0 â x5 | x1 â x6 | x2 â x7 | x3 â x8 | x4 â x9 | x5 â xA | x6 â xB | x7 â xC
+ | x8 â xD | x9 â xE | xA â xF | xB â x0 | xC â x1 | xD â x2 | xE â x3 | xF â x4 ]
+ | x6 â match e2 with
+ [ x0 â x6 | x1 â x7 | x2 â x8 | x3 â x9 | x4 â xA | x5 â xB | x6 â xC | x7 â xD
+ | x8 â xE | x9 â xF | xA â x0 | xB â x1 | xC â x2 | xD â x3 | xE â x4 | xF â x5 ]
+ | x7 â match e2 with
+ [ x0 â x7 | x1 â x8 | x2 â x9 | x3 â xA | x4 â xB | x5 â xC | x6 â xD | x7 â xE
+ | x8 â xF | x9 â x0 | xA â x1 | xB â x2 | xC â x3 | xD â x4 | xE â x5 | xF â x6 ]
+ | x8 â match e2 with
+ [ x0 â x8 | x1 â x9 | x2 â xA | x3 â xB | x4 â xC | x5 â xD | x6 â xE | x7 â xF
+ | x8 â x0 | x9 â x1 | xA â x2 | xB â x3 | xC â x4 | xD â x5 | xE â x6 | xF â x7 ]
+ | x9 â match e2 with
+ [ x0 â x9 | x1 â xA | x2 â xB | x3 â xC | x4 â xD | x5 â xE | x6 â xF | x7 â x0
+ | x8 â x1 | x9 â x2 | xA â x3 | xB â x4 | xC â x5 | xD â x6 | xE â x7 | xF â x8 ]
+ | xA â match e2 with
+ [ x0 â xA | x1 â xB | x2 â xC | x3 â xD | x4 â xE | x5 â xF | x6 â x0 | x7 â x1
+ | x8 â x2 | x9 â x3 | xA â x4 | xB â x5 | xC â x6 | xD â x7 | xE â x8 | xF â x9 ]
+ | xB â match e2 with
+ [ x0 â xB | x1 â xC | x2 â xD | x3 â xE | x4 â xF | x5 â x0 | x6 â x1 | x7 â x2
+ | x8 â x3 | x9 â x4 | xA â x5 | xB â x6 | xC â x7 | xD â x8 | xE â x9 | xF â xA ]
+ | xC â match e2 with
+ [ x0 â xC | x1 â xD | x2 â xE | x3 â xF | x4 â x0 | x5 â x1 | x6 â x2 | x7 â x3
+ | x8 â x4 | x9 â x5 | xA â x6 | xB â x7 | xC â x8 | xD â x9 | xE â xA | xF â xB ]
+ | xD â match e2 with
+ [ x0 â xD | x1 â xE | x2 â xF | x3 â x0 | x4 â x1 | x5 â x2 | x6 â x3 | x7 â x4
+ | x8 â x5 | x9 â x6 | xA â x7 | xB â x8 | xC â x9 | xD â xA | xE â xB | xF â xC ]
+ | xE â match e2 with
+ [ x0 â xE | x1 â xF | x2 â x0 | x3 â x1 | x4 â x2 | x5 â x3 | x6 â x4 | x7 â x5
+ | x8 â x6 | x9 â x7 | xA â x8 | xB â x9 | xC â xA | xD â xB | xE â xC | xF â xD ]
+ | xF â match e2 with
+ [ x0 â xF | x1 â x0 | x2 â x1 | x3 â x2 | x4 â x3 | x5 â x4 | x6 â x5 | x7 â x6
+ | x8 â x7 | x9 â x8 | xA â x9 | xB â xA | xC â xB | xD â xC | xE â xD | xF â xE ]
+ ].
+
+(* operatore somma con data+carry â carry *)
+ndefinition plus_ex_dc_c â
+λc:bool.λe1,e2:exadecim.
+ match c with
+ [ true â match e1 with
+ [ x0 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â false | xF â true ]
+ | x1 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â true | xF â true ]
+ | x2 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â true | xE â true | xF â true ]
+ | x3 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â true | xD â true | xE â true | xF â true ]
+ | x4 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x5 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x6 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x7 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x8 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x9 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xA â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xB â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xC â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xD â match e2 with
+ [ x0 â false | x1 â false | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xE â match e2 with
+ [ x0 â false | x1 â true | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xF â match e2 with
+ [ x0 â true | x1 â true | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ ]
+ | false â match e1 with
+ [ x0 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â false | xF â false ]
+ | x1 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â false | xF â true ]
+ | x2 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â true | xF â true ]
+ | x3 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â true | xE â true | xF â true ]
+ | x4 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â true | xD â true | xE â true | xF â true ]
+ | x5 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x6 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x7 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x8 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x9 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xA â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xB â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xC â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xD â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xE â match e2 with
+ [ x0 â false | x1 â false | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xF â match e2 with
+ [ x0 â false | x1 â true | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ ]].
+
+(* operatore somma con data â carry *)
+ndefinition plus_ex_d_c â
+λe1,e2:exadecim.
+ match e1 with
+ [ x0 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â false | xF â false ]
+ | x1 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â false | xF â true ]
+ | x2 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â false | xE â true | xF â true ]
+ | x3 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â false | xD â true | xE â true | xF â true ]
+ | x4 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â false | xC â true | xD â true | xE â true | xF â true ]
+ | x5 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â false | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x6 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â false | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x7 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â false | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x8 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â false
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | x9 â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â false | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xA â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â false | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xB â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â false | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xC â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â false | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xD â match e2 with
+ [ x0 â false | x1 â false | x2 â false | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xE â match e2 with
+ [ x0 â false | x1 â false | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ | xF â match e2 with
+ [ x0 â false | x1 â true | x2 â true | x3 â true | x4 â true | x5 â true | x6 â true | x7 â true
+ | x8 â true | x9 â true | xA â true | xB â true | xC â true | xD â true | xE â true | xF â true ]
+ ].
+
+(* operatore predecessore *)
+ndefinition pred_ex â
+λe:exadecim.
+ match e with
+ [ x0 â xF | x1 â x0 | x2 â x1 | x3 â x2
+ | x4 â x3 | x5 â x4 | x6 â x5 | x7 â x6
+ | x8 â x7 | x9 â x8 | xA â x9 | xB â xA
+ | xC â xB | xD â xC | xE â xD | xF â xE ].
+
+(* operatore successore *)
+ndefinition succ_ex â
+λe:exadecim.
+ match e with
+ [ x0 â x1 | x1 â x2 | x2 â x3 | x3 â x4
+ | x4 â x5 | x5 â x6 | x6 â x7 | x7 â x8
+ | x8 â x9 | x9 â xA | xA â xB | xB â xC
+ | xC â xD | xD â xE | xE â xF | xF â x0 ].
+
+(* operatore neg/complemento a 2 *)
+ndefinition compl_ex â
+λe:exadecim.match e with
+ [ x0 â x0 | x1 â xF | x2 â xE | x3 â xD
+ | x4 â xC | x5 â xB | x6 â xA | x7 â x9
+ | x8 â x8 | x9 â x7 | xA â x6 | xB â x5
+ | xC â x4 | xD â x3 | xE â x2 | xF â x1 ].
+
+(* operatore abs *)
+ndefinition abs_ex â
+λe:exadecim.match getMSB_ex e with
+ [ true â compl_ex e | false â e ].
+
+(* operatore x in [inf,sup] o in sup],[inf *)
+ndefinition inrange_ex â
+λx,inf,sup:exadecim.
+ match le_ex inf sup with
+ [ true â and_bool | false â or_bool ]
+ (le_ex inf x) (le_ex x sup).
+
+(* esadecimali ricorsivi *)
+ninductive rec_exadecim : exadecim â Type â
+ ex_O : rec_exadecim x0
+| ex_S : ân.rec_exadecim n â rec_exadecim (succ_ex n).
+
+(* esadecimali â esadecimali ricorsivi *)
+ndefinition ex_to_recex â
+λn.match n return λx.rec_exadecim x with
+ [ x0 â ex_O
+ | x1 â ex_S ? ex_O
+ | x2 â ex_S ? (ex_S ? ex_O)
+ | x3 â ex_S ? (ex_S ? (ex_S ? ex_O))
+ | x4 â ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O)))
+ | x5 â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O))))
+ | x6 â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O)))))
+ | x7 â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O))))))
+ | x8 â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O)))))))
+ | x9 â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O))))))))
+ | xA â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O)))))))))
+ | xB â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O))))))))))
+ | xC â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O)))))))))))
+ | xD â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O))))))))))))
+ | xE â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O)))))))))))))
+ | xF â ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? (ex_S ? ex_O))))))))))))))
+ ].
+
+(* ottali â esadecimali *)
+ndefinition ex_of_oct â
+λn.match n with
+ [ o0 â x0 | o1 â x1 | o2 â x2 | o3 â x3 | o4 â x4 | o5 â x5 | o6 â x6 | o7 â x7 ].
+
+(* esadecimali xNNNN â ottali *)
+ndefinition oct_of_exL â
+λn.match n with
+ [ x0 â o0 | x1 â o1 | x2 â o2 | x3 â o3 | x4 â o4 | x5 â o5 | x6 â o6 | x7 â o7
+ | x8 â o0 | x9 â o1 | xA â o2 | xB â o3 | xC â o4 | xD â o5 | xE â o6 | xF â o7 ].
+
+(* esadecimali NNNNx â ottali *)
+ndefinition oct_of_exH â
+λn.match n with
+ [ x0 â o0 | x1 â o0 | x2 â o1 | x3 â o1 | x4 â o2 | x5 â o2 | x6 â o3 | x7 â o3
+ | x8 â o4 | x9 â o4 | xA â o5 | xB â o5 | xC â o6 | xD â o6 | xE â o7 | xF â o7 ].
+
+ndefinition exadecim_destruct_aux â
+Î e1,e2.Î P:Prop.Î H:e1 = e2.
+ match eq_ex e1 e2 with [ true â P â P | false â P ].
+
+ndefinition exadecim_destruct : exadecim_destruct_aux.
+ #e1; #e2; #P; #H;
+ nrewrite < H;
+ nelim e1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+nlemma eq_to_eqex : ân1,n2.n1 = n2 â eq_ex n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqex_to_neq : ân1,n2.eq_ex n1 n2 = false â n1 â n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_ex n1 n2 = true) â¦);
+ ##[ ##1: napply (eq_to_eqex n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue ⦠H)
+ ##]
+nqed.
+
+nlemma eqex_to_eq : ân1,n2.eq_ex n1 n2 = true â n1 = n2.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
+ ##| ##*: #H; (*ndestruct lentissima...*) napply (bool_destruct ⦠H)
+ ##]
+nqed.
+
+nlemma neq_to_neqex : ân1,n2.n1 â n2 â eq_ex n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_ex n1 n2));
+ napply (not_to_not (eq_ex n1 n2 = true) (n1 = n2) ? H);
+ napply (eqex_to_eq n1 n2).
+nqed.
+
+nlemma decidable_ex : âx,y:exadecim.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_ex x y = true) (eq_ex x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x â y) (eqex_to_eq ⦠H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x â y) (neqex_to_neq ⦠H))
+ ##]
+nqed.
+
+nlemma symmetric_eqex : symmetricT exadecim bool eq_ex.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_ex n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqex n1 n2 H);
+ napply (symmetric_eq ? (eq_ex n2 n1) false);
+ napply (neq_to_neqex n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma exadecim_is_comparable : comparable.
+ @ exadecim
+ ##[ napply x0
+ ##| napply forall_ex
+ ##| napply eq_ex
+ ##| napply eqex_to_eq
+ ##| napply eq_to_eqex
+ ##| napply neqex_to_neq
+ ##| napply neq_to_neqex
+ ##| napply decidable_ex
+ ##| napply symmetric_eqex
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr exadecim_is_comparable â¡ exadecim.
+
+nlemma exadecim_is_comparable_ext : comparable_ext.
+ napply mk_comparable_ext;
+ ##[ napply exadecim_is_comparable
+ ##| napply lt_ex
+ ##| napply le_ex
+ ##| napply gt_ex
+ ##| napply ge_ex
+ ##| napply and_ex
+ ##| napply or_ex
+ ##| napply xor_ex
+ ##| napply getMSB_ex
+ ##| napply setMSB_ex
+ ##| napply clrMSB_ex
+ ##| napply getLSB_ex
+ ##| napply setLSB_ex
+ ##| napply clrLSB_ex
+ ##| napply rcr_ex
+ ##| napply shr_ex
+ ##| napply ror_ex
+ ##| napply rcl_ex
+ ##| napply shl_ex
+ ##| napply rol_ex
+ ##| napply not_ex
+ ##| napply plus_ex_dc_dc
+ ##| napply plus_ex_d_dc
+ ##| napply plus_ex_dc_d
+ ##| napply plus_ex_d_d
+ ##| napply plus_ex_dc_c
+ ##| napply plus_ex_d_c
+ ##| napply pred_ex
+ ##| napply succ_ex
+ ##| napply compl_ex
+ ##| napply abs_ex
+ ##| napply inrange_ex
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr (comp_base exadecim_is_comparable_ext) â¡ exadecim.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/oct.ma b/helm/software/matita/contribs/ng_assembly2/num/oct.ma
new file mode 100755
index 000000000..a8199dd53
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/oct.ma
@@ -0,0 +1,245 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "num/bool_lemmas.ma".
+
+(* ****** *)
+(* OTTALI *)
+(* ****** *)
+
+ninductive oct : Type â
+ o0: oct
+| o1: oct
+| o2: oct
+| o3: oct
+| o4: oct
+| o5: oct
+| o6: oct
+| o7: oct.
+
+(* iteratore sugli ottali *)
+ndefinition forall_oct â λP.
+ P o0 â P o1 â P o2 â P o3 â P o4 â P o5 â P o6 â P o7.
+
+(* operatore = *)
+ndefinition eq_oct â
+λn1,n2:oct.
+ match n1 with
+ [ o0 â match n2 with [ o0 â true | _ â false ]
+ | o1 â match n2 with [ o1 â true | _ â false ]
+ | o2 â match n2 with [ o2 â true | _ â false ]
+ | o3 â match n2 with [ o3 â true | _ â false ]
+ | o4 â match n2 with [ o4 â true | _ â false ]
+ | o5 â match n2 with [ o5 â true | _ â false ]
+ | o6 â match n2 with [ o6 â true | _ â false ]
+ | o7 â match n2 with [ o7 â true | _ â false ]
+ ].
+
+(* operatore and *)
+ndefinition and_oct â
+λe1,e2:oct.match e1 with
+ [ o0 â match e2 with
+ [ o0 â o0 | o1 â o0 | o2 â o0 | o3 â o0
+ | o4 â o0 | o5 â o0 | o6 â o0 | o7 â o0 ]
+ | o1 â match e2 with
+ [ o0 â o0 | o1 â o1 | o2 â o0 | o3 â o1
+ | o4 â o0 | o5 â o1 | o6 â o0 | o7 â o1 ]
+ | o2 â match e2 with
+ [ o0 â o0 | o1 â o0 | o2 â o2 | o3 â o2
+ | o4 â o0 | o5 â o0 | o6 â o2 | o7 â o2 ]
+ | o3 â match e2 with
+ [ o0 â o0 | o1 â o1 | o2 â o2 | o3 â o3
+ | o4 â o0 | o5 â o1 | o6 â o2 | o7 â o3 ]
+ | o4 â match e2 with
+ [ o0 â o0 | o1 â o0 | o2 â o0 | o3 â o0
+ | o4 â o4 | o5 â o4 | o6 â o4 | o7 â o4 ]
+ | o5 â match e2 with
+ [ o0 â o0 | o1 â o1 | o2 â o0 | o3 â o1
+ | o4 â o4 | o5 â o5 | o6 â o4 | o7 â o5 ]
+ | o6 â match e2 with
+ [ o0 â o0 | o1 â o0 | o2 â o2 | o3 â o2
+ | o4 â o4 | o5 â o4 | o6 â o6 | o7 â o6 ]
+ | o7 â match e2 with
+ [ o0 â o0 | o1 â o1 | o2 â o2 | o3 â o3
+ | o4 â o4 | o5 â o5 | o6 â o6 | o7 â o7 ]
+ ].
+
+(* operatore or *)
+ndefinition or_oct â
+λe1,e2:oct.match e1 with
+ [ o0 â match e2 with
+ [ o0 â o0 | o1 â o1 | o2 â o2 | o3 â o3
+ | o4 â o4 | o5 â o5 | o6 â o6 | o7 â o7 ]
+ | o1 â match e2 with
+ [ o0 â o1 | o1 â o1 | o2 â o3 | o3 â o3
+ | o4 â o5 | o5 â o5 | o6 â o7 | o7 â o7 ]
+ | o2 â match e2 with
+ [ o0 â o2 | o1 â o3 | o2 â o2 | o3 â o3
+ | o4 â o6 | o5 â o7 | o6 â o6 | o7 â o7 ]
+ | o3 â match e2 with
+ [ o0 â o3 | o1 â o3 | o2 â o3 | o3 â o3
+ | o4 â o7 | o5 â o7 | o6 â o7 | o7 â o7 ]
+ | o4 â match e2 with
+ [ o0 â o4 | o1 â o5 | o2 â o6 | o3 â o7
+ | o4 â o4 | o5 â o5 | o6 â o6 | o7 â o7 ]
+ | o5 â match e2 with
+ [ o0 â o5 | o1 â o5 | o2 â o7 | o3 â o7
+ | o4 â o5 | o5 â o5 | o6 â o7 | o7 â o7 ]
+ | o6 â match e2 with
+ [ o0 â o6 | o1 â o7 | o2 â o6 | o3 â o7
+ | o4 â o6 | o5 â o7 | o6 â o6 | o7 â o7 ]
+ | o7 â match e2 with
+ [ o0 â o7 | o1 â o7 | o2 â o7 | o3 â o7
+ | o4 â o7 | o5 â o7 | o6 â o7 | o7 â o7 ]
+ ].
+
+(* operatore xor *)
+ndefinition xor_oct â
+λe1,e2:oct.match e1 with
+ [ o0 â match e2 with
+ [ o0 â o0 | o1 â o1 | o2 â o2 | o3 â o3
+ | o4 â o4 | o5 â o5 | o6 â o6 | o7 â o7 ]
+ | o1 â match e2 with
+ [ o0 â o1 | o1 â o0 | o2 â o3 | o3 â o2
+ | o4 â o5 | o5 â o4 | o6 â o7 | o7 â o6 ]
+ | o2 â match e2 with
+ [ o0 â o2 | o1 â o3 | o2 â o0 | o3 â o1
+ | o4 â o6 | o5 â o7 | o6 â o4 | o7 â o5 ]
+ | o3 â match e2 with
+ [ o0 â o3 | o1 â o2 | o2 â o1 | o3 â o0
+ | o4 â o7 | o5 â o6 | o6 â o5 | o7 â o4 ]
+ | o4 â match e2 with
+ [ o0 â o4 | o1 â o5 | o2 â o6 | o3 â o7
+ | o4 â o0 | o5 â o1 | o6 â o2 | o7 â o3 ]
+ | o5 â match e2 with
+ [ o0 â o5 | o1 â o4 | o2 â o7 | o3 â o6
+ | o4 â o1 | o5 â o0 | o6 â o3 | o7 â o2 ]
+ | o6 â match e2 with
+ [ o0 â o6 | o1 â o7 | o2 â o4 | o3 â o5
+ | o4 â o2 | o5 â o3 | o6 â o0 | o7 â o1 ]
+ | o7 â match e2 with
+ [ o0 â o7 | o1 â o6 | o2 â o5 | o3 â o4
+ | o4 â o3 | o5 â o2 | o6 â o1 | o7 â o0 ]
+ ].
+
+(* operatore successore *)
+ndefinition succ_oct â
+λn.match n with
+ [ o0 â o1 | o1 â o2 | o2 â o3 | o3 â o4 | o4 â o5 | o5 â o6 | o6 â o7 | o7 â o0 ].
+
+(* ottali ricorsivi *)
+ninductive rec_oct : oct â Type â
+ oc_O : rec_oct o0
+| oc_S : ân.rec_oct n â rec_oct (succ_oct n).
+
+(* ottali â ottali ricorsivi *)
+ndefinition oct_to_recoct â
+λn.match n return λx.rec_oct x with
+ [ o0 â oc_O
+ | o1 â oc_S ? oc_O
+ | o2 â oc_S ? (oc_S ? oc_O)
+ | o3 â oc_S ? (oc_S ? (oc_S ? oc_O))
+ | o4 â oc_S ? (oc_S ? (oc_S ? (oc_S ? oc_O)))
+ | o5 â oc_S ? (oc_S ? (oc_S ? (oc_S ? (oc_S ? oc_O))))
+ | o6 â oc_S ? (oc_S ? (oc_S ? (oc_S ? (oc_S ? (oc_S ? oc_O)))))
+ | o7 â oc_S ? (oc_S ? (oc_S ? (oc_S ? (oc_S ? (oc_S ? (oc_S ? oc_O))))))
+ ].
+
+ndefinition oct_destruct_aux â
+Î n1,n2:oct.Î P:Prop.n1 = n2 â
+ match eq_oct n1 n2 with [ true â P â P | false â P ].
+
+ndefinition oct_destruct : oct_destruct_aux.
+ #n1; #n2; #P; #H;
+ nrewrite < H;
+ nelim n1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+nlemma eq_to_eqoct : ân1,n2.n1 = n2 â eq_oct n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqoct_to_neq : ân1,n2.eq_oct n1 n2 = false â n1 â n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_oct n1 n2 = true) â¦);
+ ##[ ##1: napply (eq_to_eqoct n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue ⦠H)
+ ##]
+nqed.
+
+nlemma eqoct_to_eq : ân1,n2.eq_oct n1 n2 = true â n1 = n2.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,10,19,28,37,46,55,64: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct ⦠H)*)
+ ##]
+nqed.
+
+nlemma neq_to_neqoct : ân1,n2.n1 â n2 â eq_oct n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_oct n1 n2));
+ napply (not_to_not (eq_oct n1 n2 = true) (n1 = n2) ? H);
+ napply (eqoct_to_eq n1 n2).
+nqed.
+
+nlemma decidable_oct : âx,y:oct.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_oct x y = true) (eq_oct x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x â y) (eqoct_to_eq ⦠H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x â y) (neqoct_to_neq ⦠H))
+ ##]
+nqed.
+
+nlemma symmetric_eqoct : symmetricT oct bool eq_oct.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_oct n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqoct n1 n2 H);
+ napply (symmetric_eq ? (eq_oct n2 n1) false);
+ napply (neq_to_neqoct n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+nlemma oct_is_comparable : comparable.
+ @ oct
+ ##[ napply o0
+ ##| napply forall_oct
+ ##| napply eq_oct
+ ##| napply eqoct_to_eq
+ ##| napply eq_to_eqoct
+ ##| napply neqoct_to_neq
+ ##| napply neq_to_neqoct
+ ##| napply decidable_oct
+ ##| napply symmetric_eqoct
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr oct_is_comparable â¡ oct.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/word16.ma b/helm/software/matita/contribs/ng_assembly2/num/word16.ma
new file mode 100755
index 000000000..c0d331a5a
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/word16.ma
@@ -0,0 +1,616 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/byte8.ma".
+include "common/nat.ma".
+
+(* **** *)
+(* WORD *)
+(* **** *)
+
+ndefinition word16 â comp_num byte8.
+ndefinition mk_word16 â λb1,b2.mk_comp_num byte8 b1 b2.
+
+(* \langle \rangle *)
+notation "â©x:yâª" non associative with precedence 80
+ for @{ mk_comp_num byte8 $x $y }.
+
+ndefinition word16_is_comparable_ext â cn_is_comparable_ext byte8_is_comparable_ext.
+unification hint 0 â ⢠carr (comp_base word16_is_comparable_ext) â¡ comp_num (comp_num exadecim).
+unification hint 0 â ⢠carr (comp_base word16_is_comparable_ext) â¡ comp_num byte8.
+unification hint 0 â ⢠carr (comp_base word16_is_comparable_ext) â¡ word16.
+
+(* operatore estensione unsigned *)
+ndefinition extu_w16 â λb2.â©zeroc ?:b2âª.
+ndefinition extu2_w16 â λe2.â©zeroc ?:â©zeroc ?,e2âªâª.
+
+(* operatore estensione signed *)
+ndefinition exts_w16 â
+λb2.â©(match getMSBc ? b2 with
+ [ true â predc ? (zeroc ?) | false â zeroc ? ]):b2âª.
+ndefinition exts2_w16 â
+λe2.(match getMSBc ? e2 with
+ [ true â â©predc ? (zeroc ?):â©predc ? (zeroc ?),e2âªâª
+ | false â â©zeroc ?:â©zeroc ?,e2âªâª ]).
+
+(* operatore moltiplicazione senza segno *)
+(* â©a1,a2⪠* â©b1,b2⪠= (a1*b1) x0 x0 + x0 (a1*b2) x0 + x0 (a2*b1) x0 + x0 x0 (a2*b2) *)
+ndefinition mulu_b8_aux â
+λw:word16.nat_it ? (rolc ?) w nat4.
+
+ndefinition mulu_b8 â
+λb1,b2:byte8.
+ plusc_d_d ? â©(mulu_ex (cnH ? b1) (cnH ? b2)):zeroc ?âª
+ (plusc_d_d ? (mulu_b8_aux (extu_w16 (mulu_ex (cnH ? b1) (cnL ? b2))))
+ (plusc_d_d ? (mulu_b8_aux (extu_w16 (mulu_ex (cnL ? b1) (cnH ? b2))))
+ (extu_w16 (mulu_ex (cnL ? b1) (cnL ? b2))))).
+
+(* operatore moltiplicazione con segno *)
+(* x * y = sgn(x) * sgn(y) * |x| * |y| *)
+ndefinition muls_b8 â
+λb1,b2:byte8.
+(* ++/-- â +, +-/-+ â - *)
+ match (getMSBc ? b1) â (getMSBc ? b2) with
+ (* +- -+ â - *)
+ [ true â complc ?
+ (* ++/-- â + *)
+ | false â λx.x ] (mulu_b8 (absc ? b1) (absc ? b2)).
+
+(* divisione senza segno (secondo la logica delle ALU): (quoziente resto) overflow *)
+nlet rec div_b8_aux (divd:word16) (divs:word16) (molt:byte8) (q:byte8) (n:nat) on n â
+ let w' â plusc_d_d ? divd (complc ? divs) in
+ match n with
+ [ O â match lec ? divs divd with
+ [ true â triple ⦠(orc ? molt q) (cnL ? w') (â (eqc ? (cnH ? w') (zeroc ?)))
+ | false â triple ⦠q (cnL ? divd) (â (eqc ? (cnH ? divd) (zeroc ?))) ]
+ | S n' â match lec ? divs divd with
+ [ true â div_b8_aux w' (rorc ? divs) (rorc ? molt) (orc ? molt q) n'
+ | false â div_b8_aux divd (rorc ? divs) (rorc ? molt) q n' ]].
+
+ndefinition div_b8 â
+λw:word16.λb:byte8.match eqc ? b (zeroc ?) with
+(* la combinazione n/0 e' illegale, segnala solo overflow senza dare risultato *)
+ [ true â triple ⦠â©xF,xF⪠(cnL ? w) true
+ | false â match eqc ? w (zeroc ?) with
+(* 0 diviso qualsiasi cosa diverso da 0 da' q=0 r=0 o=false *)
+ [ true â triple ⦠(zeroc ?) (zeroc ?) false
+(* 1) e' una divisione sensata che produrra' overflow/risultato *)
+(* 2) parametri: dividendo, divisore, moltiplicatore, quoziente, contatore *)
+(* 3) ad ogni ciclo il divisore e il moltiplicatore vengono scalati di 1 a dx *)
+(* 4) il moltiplicatore e' la quantita' aggiunta al quoziente se il divisore *)
+(* puo' essere sottratto al dividendo *)
+ | false â div_b8_aux w (nat_it ? (rolc ?) (extu_w16 b) nat7) â©x8,x0⪠(zeroc ?) nat7 ]].
+
+(* word16 ricorsive *)
+ninductive rec_word16 : word16 â Type â
+ w16_O : rec_word16 (zeroc ?)
+| w16_S : ân.rec_word16 n â rec_word16 (succc ? n).
+
+(* word16 â word16 ricorsive *)
+
+(* EX: ancora problema di tempi ???
+ndefinition w16_to_recw16_aux1_1 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x1,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (
+ w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? (w16_S ? recw
+ ))))))))))))))).
+*)
+
+ndefinition w16_to_recw16_aux1_1 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x1,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x0,xFâªâª (w16_S â©n:â©x0,xEâªâª (w16_S â©n:â©x0,xDâªâª (w16_S â©n:â©x0,xCâªâª (
+ w16_S â©n:â©x0,xBâªâª (w16_S â©n:â©x0,xAâªâª (w16_S â©n:â©x0,x9âªâª (w16_S â©n:â©x0,x8âªâª (
+ w16_S â©n:â©x0,x7âªâª (w16_S â©n:â©x0,x6âªâª (w16_S â©n:â©x0,x5âªâª (w16_S â©n:â©x0,x4âªâª (
+ w16_S â©n:â©x0,x3âªâª (w16_S â©n:â©x0,x2âªâª (w16_S â©n:â©x0,x1âªâª (w16_S â©n:â©x0,x0âªâª recw
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_2 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x2,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x1,xFâªâª (w16_S â©n:â©x1,xEâªâª (w16_S â©n:â©x1,xDâªâª (w16_S â©n:â©x1,xCâªâª (
+ w16_S â©n:â©x1,xBâªâª (w16_S â©n:â©x1,xAâªâª (w16_S â©n:â©x1,x9âªâª (w16_S â©n:â©x1,x8âªâª (
+ w16_S â©n:â©x1,x7âªâª (w16_S â©n:â©x1,x6âªâª (w16_S â©n:â©x1,x5âªâª (w16_S â©n:â©x1,x4âªâª (
+ w16_S â©n:â©x1,x3âªâª (w16_S â©n:â©x1,x2âªâª (w16_S â©n:â©x1,x1âªâª (w16_S â©n:â©x1,x0âªâª (w16_to_recw16_aux1_1 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_3 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x3,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x2,xFâªâª (w16_S â©n:â©x2,xEâªâª (w16_S â©n:â©x2,xDâªâª (w16_S â©n:â©x2,xCâªâª (
+ w16_S â©n:â©x2,xBâªâª (w16_S â©n:â©x2,xAâªâª (w16_S â©n:â©x2,x9âªâª (w16_S â©n:â©x2,x8âªâª (
+ w16_S â©n:â©x2,x7âªâª (w16_S â©n:â©x2,x6âªâª (w16_S â©n:â©x2,x5âªâª (w16_S â©n:â©x2,x4âªâª (
+ w16_S â©n:â©x2,x3âªâª (w16_S â©n:â©x2,x2âªâª (w16_S â©n:â©x2,x1âªâª (w16_S â©n:â©x2,x0âªâª (w16_to_recw16_aux1_2 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_4 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x4,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x3,xFâªâª (w16_S â©n:â©x3,xEâªâª (w16_S â©n:â©x3,xDâªâª (w16_S â©n:â©x3,xCâªâª (
+ w16_S â©n:â©x3,xBâªâª (w16_S â©n:â©x3,xAâªâª (w16_S â©n:â©x3,x9âªâª (w16_S â©n:â©x3,x8âªâª (
+ w16_S â©n:â©x3,x7âªâª (w16_S â©n:â©x3,x6âªâª (w16_S â©n:â©x3,x5âªâª (w16_S â©n:â©x3,x4âªâª (
+ w16_S â©n:â©x3,x3âªâª (w16_S â©n:â©x3,x2âªâª (w16_S â©n:â©x3,x1âªâª (w16_S â©n:â©x3,x0âªâª (w16_to_recw16_aux1_3 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_5 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x5,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x4,xFâªâª (w16_S â©n:â©x4,xEâªâª (w16_S â©n:â©x4,xDâªâª (w16_S â©n:â©x4,xCâªâª (
+ w16_S â©n:â©x4,xBâªâª (w16_S â©n:â©x4,xAâªâª (w16_S â©n:â©x4,x9âªâª (w16_S â©n:â©x4,x8âªâª (
+ w16_S â©n:â©x4,x7âªâª (w16_S â©n:â©x4,x6âªâª (w16_S â©n:â©x4,x5âªâª (w16_S â©n:â©x4,x4âªâª (
+ w16_S â©n:â©x4,x3âªâª (w16_S â©n:â©x4,x2âªâª (w16_S â©n:â©x4,x1âªâª (w16_S â©n:â©x4,x0âªâª (w16_to_recw16_aux1_4 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_6 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x6,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x5,xFâªâª (w16_S â©n:â©x5,xEâªâª (w16_S â©n:â©x5,xDâªâª (w16_S â©n:â©x5,xCâªâª (
+ w16_S â©n:â©x5,xBâªâª (w16_S â©n:â©x5,xAâªâª (w16_S â©n:â©x5,x9âªâª (w16_S â©n:â©x5,x8âªâª (
+ w16_S â©n:â©x5,x7âªâª (w16_S â©n:â©x5,x6âªâª (w16_S â©n:â©x5,x5âªâª (w16_S â©n:â©x5,x4âªâª (
+ w16_S â©n:â©x5,x3âªâª (w16_S â©n:â©x5,x2âªâª (w16_S â©n:â©x5,x1âªâª (w16_S â©n:â©x5,x0âªâª (w16_to_recw16_aux1_5 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_7 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x7,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x6,xFâªâª (w16_S â©n:â©x6,xEâªâª (w16_S â©n:â©x6,xDâªâª (w16_S â©n:â©x6,xCâªâª (
+ w16_S â©n:â©x6,xBâªâª (w16_S â©n:â©x6,xAâªâª (w16_S â©n:â©x6,x9âªâª (w16_S â©n:â©x6,x8âªâª (
+ w16_S â©n:â©x6,x7âªâª (w16_S â©n:â©x6,x6âªâª (w16_S â©n:â©x6,x5âªâª (w16_S â©n:â©x6,x4âªâª (
+ w16_S â©n:â©x6,x3âªâª (w16_S â©n:â©x6,x2âªâª (w16_S â©n:â©x6,x1âªâª (w16_S â©n:â©x6,x0âªâª (w16_to_recw16_aux1_6 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_8 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x8,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x7,xFâªâª (w16_S â©n:â©x7,xEâªâª (w16_S â©n:â©x7,xDâªâª (w16_S â©n:â©x7,xCâªâª (
+ w16_S â©n:â©x7,xBâªâª (w16_S â©n:â©x7,xAâªâª (w16_S â©n:â©x7,x9âªâª (w16_S â©n:â©x7,x8âªâª (
+ w16_S â©n:â©x7,x7âªâª (w16_S â©n:â©x7,x6âªâª (w16_S â©n:â©x7,x5âªâª (w16_S â©n:â©x7,x4âªâª (
+ w16_S â©n:â©x7,x3âªâª (w16_S â©n:â©x7,x2âªâª (w16_S â©n:â©x7,x1âªâª (w16_S â©n:â©x7,x0âªâª (w16_to_recw16_aux1_7 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_9 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©x9,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x8,xFâªâª (w16_S â©n:â©x8,xEâªâª (w16_S â©n:â©x8,xDâªâª (w16_S â©n:â©x8,xCâªâª (
+ w16_S â©n:â©x8,xBâªâª (w16_S â©n:â©x8,xAâªâª (w16_S â©n:â©x8,x9âªâª (w16_S â©n:â©x8,x8âªâª (
+ w16_S â©n:â©x8,x7âªâª (w16_S â©n:â©x8,x6âªâª (w16_S â©n:â©x8,x5âªâª (w16_S â©n:â©x8,x4âªâª (
+ w16_S â©n:â©x8,x3âªâª (w16_S â©n:â©x8,x2âªâª (w16_S â©n:â©x8,x1âªâª (w16_S â©n:â©x8,x0âªâª (w16_to_recw16_aux1_8 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_10 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©xA,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©x9,xFâªâª (w16_S â©n:â©x9,xEâªâª (w16_S â©n:â©x9,xDâªâª (w16_S â©n:â©x9,xCâªâª (
+ w16_S â©n:â©x9,xBâªâª (w16_S â©n:â©x9,xAâªâª (w16_S â©n:â©x9,x9âªâª (w16_S â©n:â©x9,x8âªâª (
+ w16_S â©n:â©x9,x7âªâª (w16_S â©n:â©x9,x6âªâª (w16_S â©n:â©x9,x5âªâª (w16_S â©n:â©x9,x4âªâª (
+ w16_S â©n:â©x9,x3âªâª (w16_S â©n:â©x9,x2âªâª (w16_S â©n:â©x9,x1âªâª (w16_S â©n:â©x9,x0âªâª (w16_to_recw16_aux1_9 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_11 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©xB,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©xA,xFâªâª (w16_S â©n:â©xA,xEâªâª (w16_S â©n:â©xA,xDâªâª (w16_S â©n:â©xA,xCâªâª (
+ w16_S â©n:â©xA,xBâªâª (w16_S â©n:â©xA,xAâªâª (w16_S â©n:â©xA,x9âªâª (w16_S â©n:â©xA,x8âªâª (
+ w16_S â©n:â©xA,x7âªâª (w16_S â©n:â©xA,x6âªâª (w16_S â©n:â©xA,x5âªâª (w16_S â©n:â©xA,x4âªâª (
+ w16_S â©n:â©xA,x3âªâª (w16_S â©n:â©xA,x2âªâª (w16_S â©n:â©xA,x1âªâª (w16_S â©n:â©xA,x0âªâª (w16_to_recw16_aux1_10 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_12 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©xC,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©xB,xFâªâª (w16_S â©n:â©xB,xEâªâª (w16_S â©n:â©xB,xDâªâª (w16_S â©n:â©xB,xCâªâª (
+ w16_S â©n:â©xB,xBâªâª (w16_S â©n:â©xB,xAâªâª (w16_S â©n:â©xB,x9âªâª (w16_S â©n:â©xB,x8âªâª (
+ w16_S â©n:â©xB,x7âªâª (w16_S â©n:â©xB,x6âªâª (w16_S â©n:â©xB,x5âªâª (w16_S â©n:â©xB,x4âªâª (
+ w16_S â©n:â©xB,x3âªâª (w16_S â©n:â©xB,x2âªâª (w16_S â©n:â©xB,x1âªâª (w16_S â©n:â©xB,x0âªâª (w16_to_recw16_aux1_11 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_13 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©xD,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©xC,xFâªâª (w16_S â©n:â©xC,xEâªâª (w16_S â©n:â©xC,xDâªâª (w16_S â©n:â©xC,xCâªâª (
+ w16_S â©n:â©xC,xBâªâª (w16_S â©n:â©xC,xAâªâª (w16_S â©n:â©xC,x9âªâª (w16_S â©n:â©xC,x8âªâª (
+ w16_S â©n:â©xC,x7âªâª (w16_S â©n:â©xC,x6âªâª (w16_S â©n:â©xC,x5âªâª (w16_S â©n:â©xC,x4âªâª (
+ w16_S â©n:â©xC,x3âªâª (w16_S â©n:â©xC,x2âªâª (w16_S â©n:â©xC,x1âªâª (w16_S â©n:â©xC,x0âªâª (w16_to_recw16_aux1_12 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_14 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©xE,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©xD,xFâªâª (w16_S â©n:â©xD,xEâªâª (w16_S â©n:â©xD,xDâªâª (w16_S â©n:â©xD,xCâªâª (
+ w16_S â©n:â©xD,xBâªâª (w16_S â©n:â©xD,xAâªâª (w16_S â©n:â©xD,x9âªâª (w16_S â©n:â©xD,x8âªâª (
+ w16_S â©n:â©xD,x7âªâª (w16_S â©n:â©xD,x6âªâª (w16_S â©n:â©xD,x5âªâª (w16_S â©n:â©xD,x4âªâª (
+ w16_S â©n:â©xD,x3âªâª (w16_S â©n:â©xD,x2âªâª (w16_S â©n:â©xD,x1âªâª (w16_S â©n:â©xD,x0âªâª (w16_to_recw16_aux1_13 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1_15 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©n:â©xF,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©xE,xFâªâª (w16_S â©n:â©xE,xEâªâª (w16_S â©n:â©xE,xDâªâª (w16_S â©n:â©xE,xCâªâª (
+ w16_S â©n:â©xE,xBâªâª (w16_S â©n:â©xE,xAâªâª (w16_S â©n:â©xE,x9âªâª (w16_S â©n:â©xE,x8âªâª (
+ w16_S â©n:â©xE,x7âªâª (w16_S â©n:â©xE,x6âªâª (w16_S â©n:â©xE,x5âªâª (w16_S â©n:â©xE,x4âªâª (
+ w16_S â©n:â©xE,x3âªâª (w16_S â©n:â©xE,x2âªâª (w16_S â©n:â©xE,x1âªâª (w16_S â©n:â©xE,x0âªâª (w16_to_recw16_aux1_14 ? recw)
+ ))))))))))))))).
+
+ndefinition w16_to_recw16_aux1 : Î n.rec_word16 â©n:â©x0,x0âªâª â rec_word16 â©(succc ? n):â©x0,x0âªâª â
+λn.λrecw:rec_word16 â©n:â©x0,x0âªâª.
+ w16_S â©n:â©xF,xFâªâª (w16_S â©n:â©xF,xEâªâª (w16_S â©n:â©xF,xDâªâª (w16_S â©n:â©xF,xCâªâª (
+ w16_S â©n:â©xF,xBâªâª (w16_S â©n:â©xF,xAâªâª (w16_S â©n:â©xF,x9âªâª (w16_S â©n:â©xF,x8âªâª (
+ w16_S â©n:â©xF,x7âªâª (w16_S â©n:â©xF,x6âªâª (w16_S â©n:â©xF,x5âªâª (w16_S â©n:â©xF,x4âªâª (
+ w16_S â©n:â©xF,x3âªâª (w16_S â©n:â©xF,x2âªâª (w16_S â©n:â©xF,x1âªâª (w16_S â©n:â©xF,x0âªâª (w16_to_recw16_aux1_15 ? recw)
+ ))))))))))))))).
+
+(* ... cifra byte superiore *)
+nlet rec w16_to_recw16_aux2 (n:byte8) (r:rec_byte8 n) on r â
+ match r return λx.λy:rec_byte8 x.rec_word16 â©x:â©x0,x0âªâª with
+ [ b8_O â w16_O
+ | b8_S t n' â w16_to_recw16_aux1 ? (w16_to_recw16_aux2 t n')
+ ].
+
+(* ... cifra esadecimale n.2 *)
+ndefinition w16_to_recw16_aux3 â
+λb,n.λrecw:rec_word16 â©b:â©x0,x0âªâª.
+ match n return λx.rec_word16 â©b:â©x,x0âªâª with
+ [ x0 â recw
+ | x1 â w16_to_recw16_aux1_1 ? recw
+ | x2 â w16_to_recw16_aux1_2 ? recw
+ | x3 â w16_to_recw16_aux1_3 ? recw
+ | x4 â w16_to_recw16_aux1_4 ? recw
+ | x5 â w16_to_recw16_aux1_5 ? recw
+ | x6 â w16_to_recw16_aux1_6 ? recw
+ | x7 â w16_to_recw16_aux1_7 ? recw
+ | x8 â w16_to_recw16_aux1_8 ? recw
+ | x9 â w16_to_recw16_aux1_9 ? recw
+ | xA â w16_to_recw16_aux1_10 ? recw
+ | xB â w16_to_recw16_aux1_11 ? recw
+ | xC â w16_to_recw16_aux1_12 ? recw
+ | xD â w16_to_recw16_aux1_13 ? recw
+ | xE â w16_to_recw16_aux1_14 ? recw
+ | xF â w16_to_recw16_aux1_15 ? recw
+ ].
+
+ndefinition w16_to_recw16_aux4_1 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x1âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x1âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? recw
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? recw
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? recw
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? recw
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? recw
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? recw
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? recw
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? recw
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? recw
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? recw
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? recw
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? recw
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? recw
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? recw
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? recw
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? recw
+ ].
+
+ndefinition w16_to_recw16_aux4_2 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x2âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x2âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_1 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_3 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x3âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x3âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_2 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_4 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x4âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x4âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_3 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_5 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x5âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x5âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_4 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_6 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x6âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x6âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_5 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_7 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x7âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x7âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_6 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_8 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x8âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x8âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_7 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_9 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,x9âªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,x9âªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_8 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_10 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,xAâªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,xAâªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_9 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_11 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,xBâªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,xBâªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_10 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_12 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,xCâªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,xCâªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_11 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_13 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,xDâªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,xDâªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_12 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_14 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,xEâªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,xEâªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_13 ⦠recw)
+ ].
+
+ndefinition w16_to_recw16_aux4_15 : Î n,e.rec_word16 â©n:â©e,x0âªâª â rec_word16 â©n:â©e,xFâªâª â
+λn,e.
+ match e return λx.rec_word16 â©n:â©x,x0âªâª â rec_word16 â©n:â©x,xFâªâª with
+ [ x0 â λrecw:rec_word16 â©n:â©x0,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x1 â λrecw:rec_word16 â©n:â©x1,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x2 â λrecw:rec_word16 â©n:â©x2,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x3 â λrecw:rec_word16 â©n:â©x3,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x4 â λrecw:rec_word16 â©n:â©x4,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x5 â λrecw:rec_word16 â©n:â©x5,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x6 â λrecw:rec_word16 â©n:â©x6,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x7 â λrecw:rec_word16 â©n:â©x7,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x8 â λrecw:rec_word16 â©n:â©x8,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | x9 â λrecw:rec_word16 â©n:â©x9,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | xA â λrecw:rec_word16 â©n:â©xA,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | xB â λrecw:rec_word16 â©n:â©xB,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | xC â λrecw:rec_word16 â©n:â©xC,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | xD â λrecw:rec_word16 â©n:â©xD,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | xE â λrecw:rec_word16 â©n:â©xE,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ | xF â λrecw:rec_word16 â©n:â©xF,x0âªâª.w16_S ? (w16_to_recw16_aux4_14 ⦠recw)
+ ].
+
+(* ... cifra esadecimale n.1 *)
+ndefinition w16_to_recw16_aux4 â
+λb,e,n.λrecw:rec_word16 â©b:â©e,x0âªâª.
+ match n return λx.rec_word16 â©b:â©e,xâªâª with
+ [ x0 â recw
+ | x1 â w16_to_recw16_aux4_1 ⦠recw
+ | x2 â w16_to_recw16_aux4_2 ⦠recw
+ | x3 â w16_to_recw16_aux4_3 ⦠recw
+ | x4 â w16_to_recw16_aux4_4 ⦠recw
+ | x5 â w16_to_recw16_aux4_5 ⦠recw
+ | x6 â w16_to_recw16_aux4_6 ⦠recw
+ | x7 â w16_to_recw16_aux4_7 ⦠recw
+ | x8 â w16_to_recw16_aux4_8 ⦠recw
+ | x9 â w16_to_recw16_aux4_9 ⦠recw
+ | xA â w16_to_recw16_aux4_10 ⦠recw
+ | xB â w16_to_recw16_aux4_11 ⦠recw
+ | xC â w16_to_recw16_aux4_12 ⦠recw
+ | xD â w16_to_recw16_aux4_13 ⦠recw
+ | xE â w16_to_recw16_aux4_14 ⦠recw
+ | xF â w16_to_recw16_aux4_15 ⦠recw
+ ].
+
+ndefinition w16_to_recw16 : Î w.rec_word16 w â
+λw.
+ match w with [ mk_comp_num h l â
+ match l with [ mk_comp_num lh ll â
+ w16_to_recw16_aux4 h lh ll (w16_to_recw16_aux3 h lh (w16_to_recw16_aux2 h (b8_to_recb8 h))) ]].
diff --git a/helm/software/matita/contribs/ng_assembly2/num/word24.ma b/helm/software/matita/contribs/ng_assembly2/num/word24.ma
new file mode 100755
index 000000000..f523e996e
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/word24.ma
@@ -0,0 +1,218 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/byte8.ma".
+
+(* ********* *)
+(* BYTE+WORD *)
+(* ********* *)
+
+nrecord word24 : Type â
+ {
+ w24x: byte8;
+ w24h: byte8;
+ w24l: byte8
+ }.
+
+(* \langle \rangle *)
+notation "â©x;y;zâª" non associative with precedence 80
+ for @{ 'mk_word24 $x $y $z }.
+interpretation "mk_word24" 'mk_word24 x y z = (mk_word24 x y z).
+
+(* operatore = *)
+ndefinition eq_w24 â
+λw1,w2.(eqc ? (w24x w1) (w24x w2)) â
+ (eqc ? (w24h w1) (w24h w2)) â
+ (eqc ? (w24l w1) (w24l w2)).
+
+nlemma word24_destruct_1 :
+âx1,x2,y1,y2,z1,z2.
+ mk_word24 x1 y1 z1 = mk_word24 x2 y2 z2 â x1 = x2.
+ #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match mk_word24 x2 y2 z2 with [ mk_word24 a _ _ â x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma word24_destruct_2 :
+âx1,x2,y1,y2,z1,z2.
+ mk_word24 x1 y1 z1 = mk_word24 x2 y2 z2 â y1 = y2.
+ #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match mk_word24 x2 y2 z2 with [ mk_word24 _ a _ â y1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma word24_destruct_3 :
+âx1,x2,y1,y2,z1,z2.
+ mk_word24 x1 y1 z1 = mk_word24 x2 y2 z2 â z1 = z2.
+ #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match mk_word24 x2 y2 z2 with [ mk_word24 _ _ a â z1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqw24 : symmetricT word24 bool eq_w24.
+ #b1; nelim b1; #e1; #e2; #e3;
+ #b2; nelim b2; #e4; #e5; #e6;
+ nchange with (((eqc ? e1 e4)â(eqc ? e2 e5)â(eqc ? e3 e6)) = ((eqc ? e4 e1)â(eqc ? e5 e2)â(eqc ? e6 e3)));
+ nrewrite > (symmetric_eqc ? e1 e4);
+ nrewrite > (symmetric_eqc ? e2 e5);
+ nrewrite > (symmetric_eqc ? e3 e6);
+ napply refl_eq.
+nqed.
+
+nlemma eqw24_to_eq : âb1,b2.(eq_w24 b1 b2 = true) â (b1 = b2).
+ #b1; nelim b1; #e1; #e2; #e3;
+ #b2; nelim b2; #e4; #e5; #e6;
+ nchange in ⢠(% â ?) with (((eqc ? e1 e4)â(eqc ? e2 e5)â(eqc ? e3 e6)) = true);
+ #H;
+ nrewrite < (eqc_to_eq ⦠(andb_true_true_r ⦠H));
+ nrewrite < (eqc_to_eq ⦠(andb_true_true_r ⦠(andb_true_true_l ⦠H)));
+ nrewrite < (eqc_to_eq ⦠(andb_true_true_l ⦠(andb_true_true_l ⦠H)));
+ napply refl_eq.
+nqed.
+
+nlemma eq_to_eqw24 : âb1,b2.(b1 = b2) â (eq_w24 b1 b2 = true).
+ #b1; nelim b1; #e1; #e2; #e3;
+ #b2; nelim b2; #e4; #e5; #e6;
+ #H;
+ nchange with (((eqc ? e1 e4)â(eqc ? e2 e5)â(eqc ? e3 e6)) = true);
+ nrewrite < (word24_destruct_1 ⦠H);
+ nrewrite < (word24_destruct_2 ⦠H);
+ nrewrite < (word24_destruct_3 ⦠H);
+ nrewrite > (eq_to_eqc ? e1 e1 (refl_eq â¦));
+ nrewrite > (eq_to_eqc ? e2 e2 (refl_eq â¦));
+ nrewrite > (eq_to_eqc ? e3 e3 (refl_eq â¦));
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma decidable_w24_aux1 :
+âe1,e2,e3,e4,e5,e6.e1 â e4 â (mk_word24 e1 e2 e3) â (mk_word24 e4 e5 e6).
+ #e1; #e2; #e3; #e4; #e5; #e6;
+ nnormalize; #H; #H1;
+ napply (H (word24_destruct_1 ⦠H1)).
+nqed.
+
+nlemma decidable_w24_aux2 :
+âe1,e2,e3,e4,e5,e6.e2 â e5 â (mk_word24 e1 e2 e3) â (mk_word24 e4 e5 e6).
+ #e1; #e2; #e3; #e4; #e5; #e6;
+ nnormalize; #H; #H1;
+ napply (H (word24_destruct_2 ⦠H1)).
+nqed.
+
+nlemma decidable_w24_aux3 :
+âe1,e2,e3,e4,e5,e6.e3 â e6 â (mk_word24 e1 e2 e3) â (mk_word24 e4 e5 e6).
+ #e1; #e2; #e3; #e4; #e5; #e6;
+ nnormalize; #H; #H1;
+ napply (H (word24_destruct_3 ⦠H1)).
+nqed.
+
+nlemma decidable_w24 : âb1,b2:word24.(decidable (b1 = b2)).
+ #b1; nelim b1; #e1; #e2; #e3;
+ #b2; nelim b2; #e4; #e5; #e6;
+ nnormalize;
+ napply (or2_elim (e1 = e4) (e1 â e4) ? (decidable_c ? e1 e4) â¦);
+ ##[ ##2: #H; napply (or2_intro2 ⦠(decidable_w24_aux1 e1 e2 e3 e4 e5 e6 H))
+ ##| ##1: #H; napply (or2_elim (e2 = e5) (e2 â e5) ? (decidable_c ? e2 e5) â¦);
+ ##[ ##2: #H1; napply (or2_intro2 ⦠(decidable_w24_aux2 e1 e2 e3 e4 e5 e6 H1))
+ ##| ##1: #H1; napply (or2_elim (e3 = e6) (e3 â e6) ? (decidable_c ? e3 e6) â¦);
+ ##[ ##2: #H2; napply (or2_intro2 ⦠(decidable_w24_aux3 e1 e2 e3 e4 e5 e6 H2))
+ ##| ##1: #H2; nrewrite > H; nrewrite > H1; nrewrite > H2;
+ napply (or2_intro1 ⦠(refl_eq ? (mk_word24 e4 e5 e6)))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqw24_to_neq : âb1,b2.(eq_w24 b1 b2 = false) â (b1 â b2).
+ #b1; nelim b1; #e1; #e2; #e3;
+ #b2; nelim b2; #e4; #e5; #e6;
+ #H;
+ nchange in H:(%) with (((eqc ? e1 e4)â(eqc ? e2 e5)â(eqc ? e3 e6)) = false);
+ #H1;
+ nrewrite > (word24_destruct_1 ⦠H1) in H:(%);
+ nrewrite > (word24_destruct_2 ⦠H1);
+ nrewrite > (word24_destruct_3 ⦠H1);
+ nrewrite > (eq_to_eqc ? e4 e4 (refl_eq â¦));
+ nrewrite > (eq_to_eqc ? e5 e5 (refl_eq â¦));
+ nrewrite > (eq_to_eqc ? e6 e6 (refl_eq â¦));
+ #H; ndestruct.
+nqed.
+
+nlemma word24_destruct :
+âe1,e2,e3,e4,e5,e6.
+ ((mk_word24 e1 e2 e3) â (mk_word24 e4 e5 e6)) â
+ (Or3 (e1 â e4) (e2 â e5) (e3 â e6)).
+ #e1; #e2; #e3; #e4; #e5; #e6;
+ nnormalize; #H;
+ napply (or2_elim (e1 = e4) (e1 â e4) ? (decidable_c ? e1 e4) â¦);
+ ##[ ##2: #H1; napply (or3_intro1 ⦠H1)
+ ##| ##1: #H1; napply (or2_elim (e2 = e5) (e2 â e5) ? (decidable_c ? e2 e5) â¦);
+ ##[ ##2: #H2; napply (or3_intro2 ⦠H2)
+ ##| ##1: #H2; napply (or2_elim (e3 = e6) (e3 â e6) ? (decidable_c ? e3 e6) â¦);
+ ##[ ##2: #H3; napply (or3_intro3 ⦠H3)
+ ##| ##1: #H3; nrewrite > H1 in H:(%);
+ nrewrite > H2; nrewrite > H3;
+ #H; nelim (H (refl_eq â¦))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqw24 : âb1,b2.((b1 â b2) â (eq_w24 b1 b2 = false)).
+ #b1; nelim b1; #e1; #e2; #e3;
+ #b2; nelim b2; #e4; #e5; #e6;
+ #H; nchange with (((eqc ? e1 e4)â(eqc ? e2 e5)â(eqc ? e3 e6)) = false);
+ napply (or3_elim (e1 â e4) (e2 â e5) (e3 â e6) ? (word24_destruct e1 e2 e3 e4 e5 e6 ⦠H) â¦);
+ ##[ ##1: #H1; nrewrite > (neq_to_neqc ? e1 e4 H1); nnormalize; napply refl_eq
+ ##| ##2: #H1; nrewrite > (neq_to_neqc ? e2 e5 H1);
+ nrewrite > (symmetric_andbool (eqc ? e1 e4) â¦);
+ nnormalize; napply refl_eq
+ ##| ##3: #H1; nrewrite > (neq_to_neqc ? e3 e6 H1);
+ nrewrite > (symmetric_andbool ((eqc ? e1 e4)â(eqc ? e2 e5)) â¦);
+ nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma word24_is_comparable : comparable.
+ @ word24
+ ##[ napply (mk_word24 (zeroc ?) (zeroc ?) (zeroc ?))
+ ##| napply (λP.forallc ?
+ (λx.forallc ?
+ (λh.forallc ?
+ (λl.P (mk_word24 x h l)))))
+ ##| napply eq_w24
+ ##| napply eqw24_to_eq
+ ##| napply eq_to_eqw24
+ ##| napply neqw24_to_neq
+ ##| napply neq_to_neqw24
+ ##| napply decidable_w24
+ ##| napply symmetric_eqw24
+ ##]
+nqed.
+
+unification hint 0 â ⢠carr word24_is_comparable â¡ word24.
diff --git a/helm/software/matita/contribs/ng_assembly2/num/word32.ma b/helm/software/matita/contribs/ng_assembly2/num/word32.ma
new file mode 100755
index 000000000..4fbbe96fc
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/num/word32.ma
@@ -0,0 +1,81 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/word16.ma".
+
+(* ***** *)
+(* DWORD *)
+(* ***** *)
+
+ndefinition word32 â comp_num word16.
+ndefinition mk_word32 â λw1,w2.mk_comp_num word16 w1 w2.
+
+(* \langle \rangle *)
+notation "â©x.yâª" non associative with precedence 80
+ for @{ mk_comp_num word16 $x $y }.
+
+ndefinition word32_is_comparable_ext â cn_is_comparable_ext word16_is_comparable_ext.
+unification hint 0 â ⢠carr (comp_base word32_is_comparable_ext) â¡ comp_num (comp_num (comp_num exadecim)).
+unification hint 0 â ⢠carr (comp_base word32_is_comparable_ext) â¡ comp_num (comp_num byte8).
+unification hint 0 â ⢠carr (comp_base word32_is_comparable_ext) â¡ comp_num word16.
+unification hint 0 â ⢠carr (comp_base word32_is_comparable_ext) â¡ word32.
+
+(* operatore estensione unsigned *)
+ndefinition extu_w32 â λw2.â©zeroc ?.w2âª.
+ndefinition extu2_w32 â λb2.â©zeroc ?.â©zeroc ?:b2âªâª.
+ndefinition extu3_w32 â λe2.â©zeroc ?.â©zeroc ?:â©zeroc ?,e2âªâªâª.
+
+(* operatore estensione signed *)
+ndefinition exts_w32 â
+λw2.â©(match getMSBc ? w2 with
+ [ true â predc ? (zeroc ?) | false â zeroc ? ]).w2âª.
+ndefinition exts2_w32 â
+λb2.(match getMSBc ? b2 with
+ [ true â â©predc ? (zeroc ?).â©predc ? (zeroc ?):b2âªâª
+ | false â â©zeroc ?.â©zeroc ?:b2âªâª ]).
+ndefinition exts3_w32 â
+λe2.(match getMSBc ? e2 with
+ [ true â â©predc ? (zeroc ?).â©predc ? (zeroc ?):â©predc ? (zeroc ?),e2âªâªâª
+ | false â â©zeroc ?.â©zeroc ?:â©zeroc ?,e2âªâªâª ]).
+
+(* operatore moltiplicazione senza segno *)
+(* â©a1,a2⪠* â©b1,b2⪠= (a1*b1) x0 x0 + x0 (a1*b2) x0 + x0 (a2*b1) x0 + x0 x0 (a2*b2) *)
+ndefinition mulu_w16_aux â
+λw:word32.nat_it ? (rolc ?) w nat8.
+
+ndefinition mulu_w16 â
+λw1,w2:word16.
+ plusc_d_d ? â©(mulu_b8 (cnH ? w1) (cnH ? w2)).zeroc ?âª
+ (plusc_d_d ? (mulu_w16_aux (extu_w32 (mulu_b8 (cnH ? w1) (cnL ? w2))))
+ (plusc_d_d ? (mulu_w16_aux (extu_w32 (mulu_b8 (cnL ? w1) (cnH ? w2))))
+ (extu_w32 (mulu_b8 (cnL ? w1) (cnL ? w2))))).
+
+(* operatore moltiplicazione con segno *)
+(* x * y = sgn(x) * sgn(y) * |x| * |y| *)
+ndefinition muls_w16 â
+λw1,w2:word16.
+(* ++/-- â +, +-/-+ â - *)
+ match (getMSBc ? w1) â (getMSBc ? w2) with
+ (* +- -+ â - *)
+ [ true â complc ?
+ (* ++/-- â + *)
+ | false â λx.x ] (mulu_w16 (absc ? w1) (absc ? w2)).
diff --git a/helm/software/matita/contribs/ng_assembly2/root b/helm/software/matita/contribs/ng_assembly2/root
new file mode 100644
index 000000000..e6f78ade0
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/root
@@ -0,0 +1 @@
+baseuri=cic:/matita
diff --git a/helm/software/matita/contribs/ng_assembly2/universe/universe.ma b/helm/software/matita/contribs/ng_assembly2/universe/universe.ma
new file mode 100755
index 000000000..24e5e5a18
--- /dev/null
+++ b/helm/software/matita/contribs/ng_assembly2/universe/universe.ma
@@ -0,0 +1,232 @@
+(**************************************************************************)
+(* ___ *)
+(* ||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: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/nelist.ma".
+include "common/prod.ma".
+
+nlet rec nmember_natList (elem:nat) (l:ne_list nat) on l â
+ match l with
+ [ ne_nil h â â(eqc ? elem h)
+ | ne_cons h t â match eqc ? elem h with
+ [ true â false | false â nmember_natList elem t ]
+ ].
+
+(* elem presente una ed una sola volta in l *)
+nlet rec member_natList (elem:nat) (l:ne_list nat) on l â
+ match l with
+ [ ne_nil h â eqc ? elem h
+ | ne_cons h t â match eqc ? elem h with
+ [ true â nmember_natList elem t | false â member_natList elem t ]
+ ].
+
+(* costruttore di un sottouniverso:
+ S_EL cioe' uno qualsiasi degli elementi del sottouniverso
+*)
+ninductive S_UN (l:ne_list nat) : Type â
+ S_EL : Î x:nat.((member_natList x l) = true) â S_UN l.
+
+ndefinition getelem : âl.âe:S_UN l.nat.
+ #l; #s; nelim s;
+ #u; #dim;
+ napply u.
+nqed.
+
+ndefinition eq_SUN â λl.λx,y:S_UN l.eq_nat (getelem ? x) (getelem ? y).
+
+ndefinition getdim : âl.âe:S_UN l.member_natList (getelem ? e) l = true.
+ #l; #s; nelim s;
+ #u; #dim;
+ napply dim.
+nqed.
+
+nlemma SUN_destruct_1
+ : âl.âe1,e2.âdim1,dim2.S_EL l e1 dim1 = S_EL l e2 dim2 â e1 = e2.
+ #l; #e1; #e2; #dim1; #dim2; #H;
+ nchange with (match S_EL l e2 dim2 with [ S_EL a _ â e1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+(* destruct universale *)
+ndefinition SUN_destruct : âl.âx,y:S_UN l.âP:Prop.x = y â match eq_SUN l x y with [ true â P â P | false â P ].
+ #l; #x; nelim x;
+ #u1; #dim1;
+ #y; nelim y;
+ #u2; #dim2;
+ #P;
+ nchange with (? â (match eq_nat u1 u2 with [ true â P â P | false â P ]));
+ #H;
+ nrewrite > (SUN_destruct_1 l ⦠H);
+ nrewrite > (eq_to_eqc ? u2 u2 (refl_eq â¦));
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+(* eq_to_eqxx universale *)
+nlemma eq_to_eqSUN : âl.âx,y:S_UN l.x = y â eq_SUN l x y = true.
+ #l; #x; nelim x;
+ #u1; #dim1;
+ #y; nelim y;
+ #u2; #dim2;
+ nchange with (? â eqc ? u1 u2 = true);
+ #H; napply (eq_to_eqc ? u1 u2);
+ napply (SUN_destruct_1 l ⦠H).
+nqed.
+
+(* neqxx_to_neq universale *)
+nlemma neqSUN_to_neq : âl.âx,y:S_UN l.eq_SUN l x y = false â x â y.
+ #l; #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_SUN l n1 n2 = true) â¦);
+ ##[ ##1: napply (eq_to_eqSUN l n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue ⦠H)
+ ##]
+nqed.
+
+(* eqxx_to_eq universale *)
+(* !!! evidente ma come si fa? *)
+naxiom eqSUN_to_eq_aux : âl,x,y.((getelem l x) = (getelem l y)) â x = y.
+
+nlemma eqSUN_to_eq : âl.âx,y:S_UN l.eq_SUN l x y = true â x = y.
+ #l; #x; #y;
+ nchange with ((eqc ? (getelem ? x) (getelem ? y) = true) â x = y);
+ #H; napply (eqSUN_to_eq_aux l x y (eqc_to_eq ⦠H)).
+nqed.
+
+(* neq_to_neqxx universale *)
+nlemma neq_to_neqSUN : âl.âx,y:S_UN l.x â y â eq_SUN l x y = false.
+ #l; #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_SUN l n1 n2));
+ napply (not_to_not (eq_SUN l n1 n2 = true) (n1 = n2) ? H);
+ napply (eqSUN_to_eq l n1 n2).
+nqed.
+
+(* decidibilita' universale *)
+nlemma decidable_SUN : âl.âx,y:S_UN l.decidable (x = y).
+ #l; #x; #y; nnormalize;
+ napply (or2_elim (eq_SUN l x y = true) (eq_SUN l x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x â y) (eqSUN_to_eq l ⦠H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x â y) (neqSUN_to_neq l ⦠H))
+ ##]
+nqed.
+
+(* simmetria di uguaglianza universale *)
+nlemma symmetric_eqSUN : âl.symmetricT (S_UN l) bool (eq_SUN l).
+ #l; #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 â n2) ? (decidable_SUN l n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqSUN l n1 n2 H);
+ napply (symmetric_eq ? (eq_SUN l n2 n1) false);
+ napply (neq_to_neqSUN l n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
+(* scheletro di funzione generica ad 1 argomento *)
+nlet rec farg1_auxT (T:Type) (l:ne_list nat) on l â
+ match l with
+ [ ne_nil _ â T
+ | ne_cons _ t â ProdT T (farg1_auxT T t)
+ ].
+
+nlemma farg1_auxDim : âh,t,x.eqc ? x h = false â member_natList x (h§§t) = true â member_natList x t = true.
+ #h; #t; #x; #H; #H1;
+ nnormalize in H1:(%);
+ nrewrite > H in H1:(%);
+ nnormalize;
+ napply (λx.x).
+nqed.
+
+nlet rec farg1 (T:Type) (l:ne_list nat) on l â
+ match l with
+ [ ne_nil h â λarg:farg1_auxT T «£h».λx:S_UN «£h».arg
+ | ne_cons h t â λarg:farg1_auxT T (h§§t).λx:S_UN (h§§t).
+ match eqc ? (getelem ? x) h
+ return λy.eqc ? (getelem ? x) h = y â ?
+ with
+ [ true â λp:(eqc ? (getelem ? x) h = true).fst ⦠arg
+ | false â λp:(eqc ? (getelem ? x) h = false).
+ farg1 T t
+ (snd ⦠arg)
+ (S_EL t (getelem ? x) (farg1_auxDim h t (getelem ? x) p (getdim ? x)))
+ ] (refl_eq ? (eqc ? (getelem ? x) h))
+ ].
+
+(* scheletro di funzione generica a 2 argomenti *)
+nlet rec farg2 (T:Type) (l,lfix:ne_list nat) on l â
+ match l with
+ [ ne_nil h â λarg:farg1_auxT (farg1_auxT T lfix) «£h».λx:S_UN «£h».farg1 T lfix arg
+ | ne_cons h t â λarg:farg1_auxT (farg1_auxT T lfix) (h§§t).λx:S_UN (h§§t).
+ match eqc ? (getelem ? x) h
+ return λy.eqc ? (getelem ? x) h = y â ?
+ with
+ [ true â λp:(eqc ? (getelem ? x) h = true).farg1 T lfix (fst ⦠arg)
+ | false â λp:(eqc ? (getelem ? x) h = false).
+ farg2 T t lfix
+ (snd ⦠arg)
+ (S_EL t (getelem ? x) (farg1_auxDim h t (getelem ? x) p (getdim ? x)))
+ ] (refl_eq ? (eqc ? (getelem ? x) h))
+ ].
+
+(* esempio0: universo ottale *)
+ndefinition oct0 â O.
+ndefinition oct1 â nat1.
+ndefinition oct2 â nat2.
+ndefinition oct3 â nat3.
+ndefinition oct4 â nat4.
+ndefinition oct5 â nat5.
+ndefinition oct6 â nat6.
+ndefinition oct7 â nat7.
+
+ndefinition oct_UN â « oct0 ; oct1 ; oct2 ; oct3 ; oct4 ; oct5 ; oct6 £ oct7 ».
+
+ndefinition uoct0 â S_EL oct_UN oct0 (refl_eq â¦).
+ndefinition uoct1 â S_EL oct_UN oct1 (refl_eq â¦).
+ndefinition uoct2 â S_EL oct_UN oct2 (refl_eq â¦).
+ndefinition uoct3 â S_EL oct_UN oct3 (refl_eq â¦).
+ndefinition uoct4 â S_EL oct_UN oct4 (refl_eq â¦).
+ndefinition uoct5 â S_EL oct_UN oct5 (refl_eq â¦).
+ndefinition uoct6 â S_EL oct_UN oct6 (refl_eq â¦).
+ndefinition uoct7 â S_EL oct_UN oct7 (refl_eq â¦).
+
+(* esempio1: NOT ottale *)
+ndefinition octNOT â
+ farg1 (S_UN oct_UN) oct_UN
+ (pair ⦠uoct7 (pair ⦠uoct6 (pair ⦠uoct5 (pair ⦠uoct4 (pair ⦠uoct3 (pair ⦠uoct2 (pair ⦠uoct1 uoct0))))))).
+
+(* esempio2: AND ottale *)
+ndefinition octAND0 â pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 uoct0)))))).
+ndefinition octAND1 â pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct0 uoct1)))))).
+ndefinition octAND2 â pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct2 (pair ⦠uoct2 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct2 uoct2)))))).
+ndefinition octAND3 â pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct2 (pair ⦠uoct3 (pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct2 uoct3)))))).
+ndefinition octAND4 â pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct4 (pair ⦠uoct4 (pair ⦠uoct4 uoct4)))))).
+ndefinition octAND5 â pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct4 (pair ⦠uoct5 (pair ⦠uoct4 uoct5)))))).
+ndefinition octAND6 â pair ⦠uoct0 (pair ⦠uoct0 (pair ⦠uoct2 (pair ⦠uoct2 (pair ⦠uoct4 (pair ⦠uoct4 (pair ⦠uoct6 uoct6)))))).
+ndefinition octAND7 â pair ⦠uoct0 (pair ⦠uoct1 (pair ⦠uoct2 (pair ⦠uoct3 (pair ⦠uoct4 (pair ⦠uoct5 (pair ⦠uoct6 uoct7)))))).
+
+ndefinition octAND â
+ farg2 (S_UN oct_UN) oct_UN oct_UN
+ (pair ⦠octAND0 (pair ⦠octAND1 (pair ⦠octAND2 (pair ⦠octAND3 (pair ⦠octAND4 (pair ⦠octAND5 (pair ⦠octAND6 octAND7))))))).
+
+(* ora e' possibile fare
+ octNOT uoctX
+ octAND uoctX uoctY
+*)