# Theory Def_Init_Big

theory Def_Init_Big
imports Def_Init_Exp Def_Init
```(* Author: Tobias Nipkow *)

theory Def_Init_Big
imports Def_Init_Exp Def_Init
begin

subsection "Initialization-Sensitive Big Step Semantics"

inductive
big_step :: "(com × state option) ⇒ state option ⇒ bool" (infix "⇒" 55)
where
None: "(c,None) ⇒ None" |
Skip: "(SKIP,s) ⇒ s" |
AssignNone: "aval a s = None ⟹ (x ::= a, Some s) ⇒ None" |
Assign: "aval a s = Some i ⟹ (x ::= a, Some s) ⇒ Some(s(x := Some i))" |
Seq:    "(c⇩1,s⇩1) ⇒ s⇩2 ⟹ (c⇩2,s⇩2) ⇒ s⇩3 ⟹ (c⇩1;;c⇩2,s⇩1) ⇒ s⇩3" |

IfNone:  "bval b s = None ⟹ (IF b THEN c⇩1 ELSE c⇩2,Some s) ⇒ None" |
IfTrue:  "⟦ bval b s = Some True;  (c⇩1,Some s) ⇒ s' ⟧ ⟹
(IF b THEN c⇩1 ELSE c⇩2,Some s) ⇒ s'" |
IfFalse: "⟦ bval b s = Some False;  (c⇩2,Some s) ⇒ s' ⟧ ⟹
(IF b THEN c⇩1 ELSE c⇩2,Some s) ⇒ s'" |

WhileNone: "bval b s = None ⟹ (WHILE b DO c,Some s) ⇒ None" |
WhileFalse: "bval b s = Some False ⟹ (WHILE b DO c,Some s) ⇒ Some s" |
WhileTrue:
"⟦ bval b s = Some True;  (c,Some s) ⇒ s';  (WHILE b DO c,s') ⇒ s'' ⟧ ⟹
(WHILE b DO c,Some s) ⇒ s''"

lemmas big_step_induct = big_step.induct[split_format(complete)]

subsection "Soundness wrt Big Steps"

text{* Note the special form of the induction because one of the arguments
of the inductive predicate is not a variable but the term @{term"Some s"}: *}

theorem Sound:
"⟦ (c,Some s) ⇒ s';  D A c A';  A ⊆ dom s ⟧
⟹ ∃ t. s' = Some t ∧ A' ⊆ dom t"
proof (induction c "Some s" s' arbitrary: s A A' rule:big_step_induct)
case AssignNone thus ?case
by auto (metis aval_Some option.simps(3) subset_trans)
next
case Seq thus ?case by auto metis
next
case IfTrue thus ?case by auto blast
next
case IfFalse thus ?case by auto blast
next
case IfNone thus ?case
by auto (metis bval_Some option.simps(3) order_trans)
next
case WhileNone thus ?case
by auto (metis bval_Some option.simps(3) order_trans)
next
case (WhileTrue b s c s' s'')
from `D A (WHILE b DO c) A'` obtain A' where "D A c A'" by blast
then obtain t' where "s' = Some t'" "A ⊆ dom t'"
by (metis D_incr WhileTrue(3,7) subset_trans)
from WhileTrue(5)[OF this(1) WhileTrue(6) this(2)] show ?case .
qed auto

corollary sound: "⟦  D (dom s) c A';  (c,Some s) ⇒ s' ⟧ ⟹ s' ≠ None"
by (metis Sound not_Some_eq subset_refl)

end

```