List.ma: added function nth (with default value in case of failure)

Assembly: a toy example on proving properties of assembly programs.
 It should be moved somewhere else. It shows HUGE problems with
 simplification and display of (very very large) terms.

diff --git a/matita/library/assembly/assembly.ma b/matita/library/assembly/assembly.ma
new file mode 100644 (file)
index 0000000..6cdf409
--- /dev/null
+++ b/matita/library/assembly/assembly.ma
@@ -0,0 +1,274 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/".
+
+include "nat/times.ma".
+include "nat/compare.ma".
+include "list/list.ma".
+
+alias id "OO" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
+alias id "SS" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
+
+let rec matita_nat_of_coq_nat n ≝
+ match n with
+  [ OO ⇒ O
+  | SS y ⇒ S (matita_nat_of_coq_nat y)
+  ].
+coercion cic:/matita/assembly/matita_nat_of_coq_nat.con.
+alias num (instance 0) = "natural number".
+
+definition byte ≝ nat.
+definition addr ≝ nat.
+
+inductive opcode: Type ≝
+   ADDd: opcode  (* 3 clock, 171 *)
+ | BEQ: opcode   (* 3, 55 *)
+ | BRA: opcode   (* 3, 48 *)
+ | DECd: opcode  (* 5, 58 *)
+ | LDAi: opcode  (* 2, 166 *)
+ | LDAd: opcode  (* 3, 182 *)
+ | STAd: opcode. (* 3, 183 *)
+
+let rec cycles_of_opcode op ≝
+ match op with
+  [ ADDd ⇒ matita_nat_of_coq_nat 3
+  | BEQ ⇒ matita_nat_of_coq_nat 3
+  | BRA ⇒ matita_nat_of_coq_nat 3
+  | DECd ⇒ matita_nat_of_coq_nat 5
+  | LDAi ⇒ matita_nat_of_coq_nat 2
+  | LDAd ⇒ matita_nat_of_coq_nat 3
+  | STAd ⇒ matita_nat_of_coq_nat 3
+  ].
+
+inductive cartesian_product (A,B: Type) : Type ≝
+ couple: ∀a:A.∀b:B. cartesian_product A B.
+
+definition opcodemap ≝
+ [ couple ? ? ADDd 171;
+   couple ? ? BEQ 55;
+   couple ? ? BRA 48;
+   couple ? ? DECd 58;
+   couple ? ? LDAi 166;
+   couple ? ? LDAd 182;
+   couple ? ? STAd 183 ].
+
+definition opcode_of_byte ≝
+ λb.
+  let rec aux l ≝
+   match l with
+    [ nil ⇒ ADDd
+    | cons c tl ⇒
+       match c with
+        [ couple op n ⇒
+           match eqb (matita_nat_of_coq_nat n) b with
+            [ true ⇒ op
+            | false ⇒ aux tl
+            ]]]
+  in
+   aux opcodemap.
+
+definition magic_of_opcode ≝
+ λop1.
+  match op1 with
+   [ ADDd ⇒ 0
+   | BEQ ⇒  1 
+   | BRA ⇒  2
+   | DECd ⇒ 3
+   | LDAi ⇒ 4
+   | LDAd ⇒ 5
+   | STAd ⇒ 6 ].
+   
+definition opcodeeqb ≝
+ λop1,op2. eqb (magic_of_opcode op1) (magic_of_opcode op2).
+
+definition byte_of_opcode : opcode → byte ≝
+ λop.
+  let rec aux l ≝
+   match l with
+    [ nil ⇒ matita_nat_of_coq_nat 0
+    | cons c tl ⇒
+       match c with
+        [ couple op' n ⇒
+           match opcodeeqb op op' with
+            [ true ⇒ matita_nat_of_coq_nat n
+            | false ⇒ aux tl
+            ]]]
+  in
+   aux opcodemap.
+
+notation "hvbox(# break a)"
+  non associative with precedence 80
+for @{ 'byte_of_opcode $a }.
+interpretation "byte_of_opcode" 'byte_of_opcode a =
+ (cic:/matita/assembly/byte_of_opcode.con a).
+
+definition byte_of_coq_nat : nat → byte ≝ matita_nat_of_coq_nat.
+ 
+notation "hvbox(@ break a)"
+  non associative with precedence 80
+for @{ 'matita_nat_of_coq_nat $a }.
+interpretation "matita_nat_of_coq_nat" 'matita_nat_of_coq_nat a =
+ (cic:/matita/assembly/byte_of_coq_nat.con a).
+
+definition mult_source : list byte ≝
+  [#LDAi; @0;
+   #STAd; @18; (* 18 = locazione $12 *)
+   #LDAd; @17; (* 17 = locazione $11 *)
+   #BEQ;  @14;
+   #LDAd; @18;
+   #DECd; @17;
+   #ADDd; @16; (* 16 = locazione $10 *)
+   #STAd; @18;
+   #LDAd; @17;
+   #BRA;  @246; (* 246 = -10 *)
+   #LDAd; @18].
+   
+definition mult_source' : list byte.
+
+record status : Type ≝ {
+  acc : byte;
+  pc  : addr;
+  spc : addr;
+  zf  : bool;
+  cf  : bool;
+  mem : addr → byte;
+  clk : nat
+}.
+
+definition mult_status : status ≝
+ mk_status @0 @0 @0 false false (λa:addr. nth ? mult_source @0 a) 0.
+
+definition tick ≝
+ λs:status.
+  (* fetch *)
+  let opc ≝ opcode_of_byte (mem s (pc s)) in
+  let op1 ≝ mem s (S (pc s)) in
+  let op2 ≝ mem s (S (S (pc s))) in
+  let clk' ≝ cycles_of_opcode opc in
+  match eqb (S (clk s)) clk' with
+   [ false ⇒
+       mk_status
+        (acc s) (pc s) (spc s) (zf s) (cf s) (mem s) (S (clk s))
+   | true ⇒
+      match opc with
+       [ ADDd ⇒ s
+       | BEQ ⇒
+          mk_status
+           (acc s)
+           (match zf s with
+             [ true ⇒ op1 + pc s   (* signed!!! *)
+             | false ⇒ 2 + pc s
+             ])
+           (spc s)
+           (zf s)
+           (cf s)
+           (mem s)
+           0
+       | BRA ⇒ s
+       | DECd ⇒ s
+       | LDAi ⇒ s
+       | LDAd ⇒ s
+       | STAd ⇒ s
+       ]
+   ].
+
+let rec execute s n on n ≝
+ match n with
+  [ O ⇒ s
+  | S n' ⇒ execute (tick s) n'
+  ].
+  
+lemma foo: ∀s,n. execute s (S n) = execute (tick s) n.
+ intros; reflexivity.
+qed.
+
+lemma goo: execute mult_status (matita_nat_of_coq_nat 4) = mult_status.
+ simplify;
+ change with (tick (tick (tick mult_status)) = mult_status);
+ change with (tick (tick mult_status) = mult_status);
+ change with (tick mult_status = mult_status);
+ change with (mult_status = mult_status);
+ reflexivity.
+qed.
+
+(*
+ unfold mult_status; 
+ simplify;
+ whd in ⊢ (? ?
+(?
+ (?
+  (?
+   match match ? % in nat return ? with [O\rArr ?|S\rArr ?] in bool return ?
+    with 
+   [true\rArr ?|false\rArr ?]))) ?);
+ simplify;
+ whd in ⊢ (? ?
+(?
+ (?
+  (?
+   match % in opcode return ? with 
+   [ADDd\rArr ?
+   |BEQ\rArr ?
+   |BRA\rArr ?
+   |DECd\rArr ?
+   |LDAi\rArr ?
+   |LDAd\rArr ?
+   |STAd\rArr ?]))) ?);
+ simplify;
+ whd in \vdash (? ?
+(?
+ (?
+  match match % in nat return ? with [O\rArr ?|S\rArr ?] in bool return ? with 
+  [true\rArr ?|false\rArr ?])) ?);
+ simplify;
+ whd in \vdash (? ?
+(?
+ (?
+  match % in opcode return ? with 
+  [ADDd\rArr ?
+  |BEQ\rArr ?
+  |BRA\rArr ?
+  |DECd\rArr ?
+  |LDAi\rArr ?
+  |LDAd\rArr ?
+  |STAd\rArr ?])) ?);
+ simplify;
+ whd in \vdash (? ? (? match % in bool return ? with [true\rArr ?|false\rArr ?]) ?);
+ simplify;
+ whd in \vdash (? ?
+(?
+ match % in opcode return ? with 
+ [ADDd\rArr ?
+ |BEQ\rArr ?
+ |BRA\rArr ?
+ |DECd\rArr ?
+ |LDAi\rArr ?
+ |LDAd\rArr ?
+ |STAd\rArr ?]) ?);
+ simplify;
+ whd in \vdash (? ? match % in bool return ? with [true\rArr ?|false\rArr ?] ?);
+ simplify;
+ whd in \vdash (? ?
+match % in opcode return ? with 
+[ADDd\rArr ?
+|BEQ\rArr ?
+|BRA\rArr ?
+|DECd\rArr ?
+|LDAi\rArr ?
+|LDAd\rArr ?
+|STAd\rArr ?] ?);
+ simplify;
+ reflexivity.
+*)
\ No newline at end of file

diff --git a/matita/library/list/list.ma b/matita/library/list/list.ma
index f214ff3efa11b6b2559bfa302f2d5dfb4174495d..f832d502002f3c9a59855a13acea0ac966fa69bc 100644 (file)
--- a/matita/library/list/list.ma
+++ b/matita/library/list/list.ma
@@ -133,3 +133,13 @@ simplify.
 reflexivity.
 qed.
 *)
+
+let rec nth (A:Type) l d n on n ≝
+ match n with
+  [ O ⇒
+     match l with
+      [ nil ⇒ d
+      | cons (x : A) _ ⇒ x
+      ]
+  | S n' ⇒ nth A (tail ? l) d n'
+  ].