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# Homework 8

This is the task corresponding to homework 8.

## Resources

### Definitions File

```theory Defs
imports "HOL-IMP.Com"
begin

datatype
com = SKIP
| Assign vname aexp       ("_ ::= _" [1000, 61] 61)
| Seq   com  com          ("_;;/ _"  [60, 61] 60)
| If     bexp com com     ("(IF _/ THEN _/ ELSE _)"  [0, 0, 61] 61)
| While  bexp com         ("(WHILE _/ DO _)"  [0, 61] 61)
| Or com com              ("_ OR _" [57,58] 59)
| ASSUME bexp
| Loop com                ("(LOOP _)"  [61] 61)

type_synonym com_den = "(state \<times> state) set"

consts big_step :: "com \<times> state \<Rightarrow> state \<Rightarrow> bool"

consts D :: "com \<Rightarrow> com_den"

end```

### Template File

```theory Submission
imports Defs
begin

inductive
big_step :: "com \<times> state \<Rightarrow> state \<Rightarrow> bool" (infix "\<Rightarrow>" 55)
where
Skip:    "(SKIP,s) \<Rightarrow> s" |
Assign:  "(x ::= a,s) \<Rightarrow> s(x := aval a s)" |
Seq:    "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> s\<^sub>2; (c\<^sub>2,s\<^sub>2) \<Rightarrow> s\<^sub>3 \<rbrakk> \<Longrightarrow> (c\<^sub>1;;c\<^sub>2, s\<^sub>1) \<Rightarrow> s\<^sub>3" |
IfTrue:  "\<lbrakk> bval b s;  (c\<^sub>1,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t" |
IfFalse: "\<lbrakk> \<not>bval b s;  (c\<^sub>2,s) \<Rightarrow> t \<rbrakk> \<Longrightarrow> (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t" |
WhileFalse: "\<not>bval b s \<Longrightarrow> (WHILE b DO c,s) \<Rightarrow> s" |
WhileTrue:  "\<lbrakk> bval b s\<^sub>1;  (c,s\<^sub>1) \<Rightarrow> s\<^sub>2;  (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3 \<rbrakk> \<Longrightarrow> (WHILE b DO c, s\<^sub>1) \<Rightarrow> s\<^sub>3" |
OrLeft: "\<lbrakk> (c\<^sub>1,s) \<Rightarrow> s' \<rbrakk> \<Longrightarrow> (c\<^sub>1 OR c\<^sub>2,s) \<Rightarrow> s'" |
OrRight: "\<lbrakk> (c\<^sub>2,s) \<Rightarrow> s' \<rbrakk> \<Longrightarrow> (c\<^sub>1 OR c\<^sub>2,s) \<Rightarrow> s'" |
Assume: "bval b s \<Longrightarrow> (ASSUME b, s) \<Rightarrow> s" |
declare big_step.intros [intro]
lemmas big_step_induct = big_step.induct[split_format(complete)]

inductive_cases skipE[elim!]: "(SKIP,s) \<Rightarrow> t"
inductive_cases AssignE[elim!]: "(x ::= a,s) \<Rightarrow> t"
inductive_cases SeqE[elim!]: "(c1;;c2,s1) \<Rightarrow> s3"
inductive_cases OrE: "(c1 OR c2,s1) \<Rightarrow> s3"
inductive_cases IfE[elim!]: "(IF b THEN c1 ELSE c2,s) \<Rightarrow> t"
inductive_cases WhileE[elim]: "(WHILE b DO c,s) \<Rightarrow> t"

type_synonym com_den = "(state \<times> state) set"

fun D :: "com \<Rightarrow> com_den" where
"D SKIP   = Id" |
"D (x ::= a) = {(s,t). t = s(x := aval a s)}" |
"D (c1;;c2)  = D(c1) O D(c2)" |
"D (IF b THEN c1 ELSE c2)
= {(s,t). if bval b s then (s,t) \<in> D c1 else (s,t) \<in> D c2}" |
"D (WHILE b DO c) = lfp (W (bval b) (D c))"
| "D _ = undefined"

theorem denotational_is_big_step:
"(s,t) \<in> D(c)  =  ((c,s) \<Rightarrow> t)"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem denotational_is_big_step: "(s,t) \<in> D(c)  =  ((c,s) \<Rightarrow> t)"
by (rule Submission.denotational_is_big_step)

end```

Terms and Conditions