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# Homework 9

This is the task corresponding to homework 9.

## Resources

### Definitions File

```theory Defs
imports "HOL-IMP.BExp"
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)
| CONTINUE
inductive
big_step :: "com \<times> state \<Rightarrow> bool \<times> state \<Rightarrow> bool" (infix "\<Rightarrow>" 55)
where
Skip: "(SKIP,s) \<Rightarrow> (False,s)" |
Assign: "(x ::= a,s) \<Rightarrow> (False,s(x := aval a s))" |
Seq1: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> (False,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" |
Seq2: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> (True,s\<^sub>2) \<rbrakk> \<Longrightarrow> (c\<^sub>1;;c\<^sub>2, s\<^sub>1) \<Rightarrow> (True,s\<^sub>2)" |
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> (False,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" | \<comment> \<open>We can simply reset the continue flag in a while loop\<close>
Continue: "(CONTINUE,s) \<Rightarrow> (True,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 ContinueE[elim!]: "(CONTINUE,s) \<Rightarrow> t"
inductive_cases AssignE[elim!]: "(x ::= a,s) \<Rightarrow> t"
inductive_cases SeqE[elim!]: "(c1;;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"

abbreviation state_subst :: "state \<Rightarrow> aexp \<Rightarrow> vname \<Rightarrow> state"
("_[_'/_]" [1000,0,0] 999)
where "s[a/x] == s(x := aval a s)"

type_synonym assn = "state \<Rightarrow> bool"

definition
hoare_valid :: "assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool" ("\<Turnstile> {(1_)}/ (_)/ {(1_)}" 50) where
"\<Turnstile> {P}c{Q} = (\<forall>s f t. P s \<and> (c,s) \<Rightarrow> (f, t)  \<longrightarrow>  Q t)"

definition
hoare_valid\<^sub>c :: "assn \<Rightarrow> assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool" ("\<Turnstile>\<^sub>c{(1_)}/ {(1_)}/ (_)/ {(1_)}" 50) where
"\<Turnstile>\<^sub>c{I} {P}c{Q} = (\<forall>s f t. P s \<and> (c,s) \<Rightarrow> (f, t)  \<longrightarrow> (if f then I t else Q t))"

consts hoare :: "assn \<Rightarrow> assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool"

consts wp :: "com \<Rightarrow> assn \<Rightarrow> assn \<Rightarrow> assn"

end```

### Template File

```theory Submission
imports Defs
begin

inductive
hoare :: "assn \<Rightarrow> assn \<Rightarrow> com \<Rightarrow> assn \<Rightarrow> bool" ("\<turnstile>{(1_)}/ ({(1_)}/ (_)/ {(1_)})" 50)
where
Skip: "\<turnstile>{I} {P} SKIP {P}"  |
Assign:  "\<turnstile>{I} {\<lambda>s. P(s[a/x])} x::=a {P}"  |
Seq: "\<lbrakk> \<turnstile>{I} {P} c\<^sub>1 {Q};  \<turnstile>{I} {Q} c\<^sub>2 {R} \<rbrakk>
\<Longrightarrow> \<turnstile>{I} {P} c\<^sub>1;;c\<^sub>2 {R}"  |
If: "\<lbrakk> \<turnstile>{I} {\<lambda>s. P s \<and> bval b s} c\<^sub>1 {Q};  \<turnstile>{I} {\<lambda>s. P s \<and> \<not> bval b s} c\<^sub>2 {Q} \<rbrakk>
\<Longrightarrow> \<turnstile>{I} {P} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"  |
conseq: "\<lbrakk> \<forall>s. P' s \<longrightarrow> P s;  \<turnstile>{I} {P} c {Q};  \<forall>s. Q s \<longrightarrow> Q' s \<rbrakk>
\<Longrightarrow> \<turnstile>{I} {P'} c {Q'}" |
theorem hoare_sound: "\<turnstile>{I} {P}c{Q} \<Longrightarrow> \<Turnstile>\<^sub>c{I} {P}c{Q}"
sorry

definition wp :: "com \<Rightarrow> assn \<Rightarrow> assn \<Rightarrow> assn"  where
"wp _ = undefined"

lemma hoare_complete: assumes "\<Turnstile> {P}c{Q}"
shows "\<turnstile>{Q} {P}c{Q}"
sorry

theorem hoare_sound_complete: "\<turnstile>{Q} {P}c{Q} \<longleftrightarrow> \<Turnstile> {P}c{Q}"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem hoare_sound: "\<turnstile>{I} {P}c{Q} \<Longrightarrow> \<Turnstile>\<^sub>c{I} {P}c{Q}"
by (rule Submission.hoare_sound)

lemma hoare_complete: "(\<Turnstile> {P}c{Q}) \<Longrightarrow> \<turnstile>{Q} {P}c{Q}"
by (rule Submission.hoare_complete)

theorem hoare_sound_complete: "\<turnstile>{Q} {P}c{Q} \<longleftrightarrow> \<Turnstile> {P}c{Q}"
by (rule Submission.hoare_sound_complete)

end```

Terms and Conditions