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Homework 7

This is the task corresponding to homework 7.

Resources

Definitions File

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

declare [[names_short]]
declare [[syntax_ambiguity_warning=false]]

consts invar :: "vname \<Rightarrow> com \<Rightarrow> bool"

consts incr :: "vname \<Rightarrow> com \<Rightarrow> bool"

consts terminates :: "com \<Rightarrow> bool"

end```

Template File

```theory Submission
imports Defs
begin

fun invar :: "vname \<Rightarrow> com \<Rightarrow> bool"  where
"invar _ = undefined"

fun incr :: "vname \<Rightarrow> com \<Rightarrow> bool"  where
"incr _ = undefined"

value "incr ''x'' (''x'' ::= Plus (V ''x'') (N 0);; ''x'' ::= Plus (V ''x'') (N 2)) = True"
value "incr ''y'' (
WHILE Less (V ''y'') (N 2)
DO
''y'' ::= Plus (V ''y'') (N 1);;
''x''::=Plus (V ''x'') (N (-1))
) = False"

lemma incr_less: "(c,s) \<Rightarrow> t \<Longrightarrow> incr x c \<Longrightarrow> s x < t x"
sorry

fun terminates :: "com \<Rightarrow> bool"  where
"terminates _ = undefined"

lemma term_w:
assumes step: "\<And>s. \<exists>t. (c,s) \<Rightarrow> t"
and incr: "incr x c"
shows "\<exists>t. (WHILE Less (V x) (N k) DO c, s) \<Rightarrow> t"
sorry

theorem term_big_step: "terminates c \<Longrightarrow> \<exists>t. (c,s) \<Rightarrow> t"
sorry

end```

Check File

```theory Check
imports Submission
begin

lemma incr_less: "(c,s) \<Rightarrow> t \<Longrightarrow> incr x c \<Longrightarrow> s x < t x"
by (rule Submission.incr_less)

lemma term_w: "(\<And>s. \<exists>t. (c,s) \<Rightarrow> t) \<Longrightarrow> (incr x c) \<Longrightarrow> \<exists>t. (WHILE Less (V x) (N k) DO c, s) \<Rightarrow> t"
by (rule Submission.term_w)

theorem term_big_step: "terminates c \<Longrightarrow> \<exists>t. (c,s) \<Rightarrow> t"
by (rule Submission.term_big_step)

end```

Terms and Conditions