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# Homework 6.3

This is the task corresponding to homework 6.3.

## Resources

### Definitions File

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

end```

### Template File

```theory Submission imports Defs begin

type_synonym ety = "ty option"
type_synonym etyenv = "vname \<Rightarrow> ety"

inductive infer_aty :: "etyenv \<Rightarrow> aexp \<Rightarrow> ty \<Rightarrow> bool"

inductive infer_bty :: "etyenv \<Rightarrow> bexp \<Rightarrow> bool"

definition is_inst :: "tyenv \<Rightarrow> etyenv \<Rightarrow> bool"
where "is_inst \<Gamma> e\<Gamma> \<equiv> \<forall>x \<tau>. e\<Gamma> x = Some \<tau> \<longrightarrow> \<Gamma> x = \<tau>"

theorem ainfer: assumes "infer_aty e\<Gamma> a \<tau>" and "is_inst \<Gamma> e\<Gamma>" shows "atyping \<Gamma> a \<tau>"
sorry

theorem binfer: assumes "infer_bty e\<Gamma> b" and "is_inst \<Gamma> e\<Gamma>" shows "btyping \<Gamma> b"
sorry

definition combine :: "etyenv \<Rightarrow> etyenv \<Rightarrow> etyenv \<Rightarrow> bool" where
"combine e\<Gamma>\<^sub>1 e\<Gamma>\<^sub>2 e\<Gamma> \<equiv> e\<Gamma> = e\<Gamma>\<^sub>1 ++ e\<Gamma>\<^sub>2  \<and>
(\<forall>x \<tau>\<^sub>1 \<tau>\<^sub>2. e\<Gamma>\<^sub>1 x = Some \<tau>\<^sub>1 \<and> e\<Gamma>\<^sub>2 x = Some \<tau>\<^sub>2 \<longrightarrow> \<tau>\<^sub>1 = \<tau>\<^sub>2)"

inductive infer_cty :: "etyenv \<Rightarrow> com \<Rightarrow> etyenv \<Rightarrow> bool"

abbreviation "test_c \<equiv>
''x''::=Ic 0;;
(IF Less (V ''x'') (Ic 2) THEN SKIP ELSE ''y'' ::= Rc 1.0);;
''y'' ::= Plus (V ''y'') (Rc 3.1)"

lemma "\<exists>e\<Gamma>'. infer_cty (\<lambda>_. None) test_c e\<Gamma>'"
apply (rule exI)
apply (rule infer_cty.intros)
apply simp
apply (rule infer_cty.intros)
apply (rule infer_cty.intros)
apply simp
apply (rule infer_aty.intros)  \<comment>\<open>and so on ...\<close>
sorry

theorem infer_typing: assumes "infer_cty e\<Gamma> c e\<Gamma>'" and "is_inst \<Gamma> e\<Gamma>'" shows "ctyping \<Gamma> c"
sorry

end```

### Check File

```theory Check imports Submission begin

theorem ainfer: assumes "infer_aty e\<Gamma> a \<tau>" and "is_inst \<Gamma> e\<Gamma>" shows "atyping \<Gamma> a \<tau>"
using assms by (rule Submission.ainfer)

theorem binfer: assumes "infer_bty e\<Gamma> b" and "is_inst \<Gamma> e\<Gamma>" shows "btyping \<Gamma> b"
using assms by (rule Submission.binfer)

theorem infer_typing: assumes "infer_cty e\<Gamma> c e\<Gamma>'" and "is_inst \<Gamma> e\<Gamma>'" shows "ctyping \<Gamma> c"
using assms by (rule Submission.infer_typing)

end```

Terms and Conditions