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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
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
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