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### Definitions File

### Template File

### Check File

theory Defs imports "HOL-IMP.AExp" begin declare [[names_short]] datatype bexp' = V vname | And "bexp'" "bexp'" | Not "bexp'" | TT | FF type_synonym assignment = "vname \<Rightarrow> bool" fun has_const where "has_const TT = True" | "has_const FF = True" | "has_const (Not a) = has_const a" | "has_const (And a b) \<longleftrightarrow> has_const a \<or> has_const b" | "has_const _ = False" definition "simplified \<phi> \<longleftrightarrow> \<phi> = TT \<or> \<phi> = FF \<or> \<not> has_const \<phi>" consts sat :: "bexp' \<Rightarrow> assignment \<Rightarrow> bool" consts models :: "bexp' \<Rightarrow> assignment set" consts simplify :: "bexp' \<Rightarrow> bexp'" end

theory Submission imports Defs begin fun sat :: "bexp' \<Rightarrow> assignment \<Rightarrow> bool" where "sat _ = undefined" fun models :: "bexp' \<Rightarrow> assignment set" where "models (V x) = {\<sigma>. \<sigma> x}" | "models TT = UNIV" | "models _ = undefined" theorem sat_iff_model: "sat \<phi> \<sigma> \<longleftrightarrow> \<sigma> \<in> models \<phi>" sorry fun simplify :: "bexp' \<Rightarrow> bexp'" where "simplify f = f" value "simplify (And (Not FF) (V ''x'')) = V ''x''" theorem simplify_simplified: "simplified (simplify \<phi>)" sorry theorem simplify_models: "models (simplify \<phi>) = models \<phi>" sorry end

theory Check imports Submission begin theorem sat_iff_model: "sat \<phi> \<sigma> \<longleftrightarrow> \<sigma> \<in> models \<phi>" by (rule Submission.sat_iff_model) theorem simplify_simplified: "simplified (simplify \<phi>)" by (rule Submission.simplify_simplified) theorem simplify_models: "models (simplify \<phi>) = models \<phi>" by (rule Submission.simplify_models) end

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