theory TypedArithInClass
imports Main
begin

section "Syntax"

datatype trm =
    TTrue
  | TFalse
  | TIf trm trm trm
  | TZero
  | TSucc trm
  | TPred trm
  | TIsZero trm

inductive_set numeric_values :: "trm set" where
  zero_nv: "TZero \<in> numeric_values" |
  succ_nv: "nv \<in> numeric_values \<Longrightarrow> TSucc nv \<in> numeric_values"

thm numeric_values.cases
thm numeric_values.induct

definition vals :: "trm set" where
  "vals \<equiv> {TTrue, TFalse} \<union> numeric_values"
declare vals_def[simp]

section "Operational Semantics"

inductive reduction :: "trm \<Rightarrow> trm \<Rightarrow> bool" (infix "\<longrightarrow>" 80) where
  EIfTrue: "TIf TTrue t2 t3 \<longrightarrow> t2" |
  EIfFalse: "TIf TFalse t2 t3 \<longrightarrow> t3" |
  EIf: "t1 \<longrightarrow> t1' \<Longrightarrow> TIf t1 t2 t3 \<longrightarrow> TIf t1' t2 t3" |
  ESucc: "t1 \<longrightarrow> t1' \<Longrightarrow> TSucc t1 \<longrightarrow> TSucc t1'" |
  EPredZero: "TPred TZero \<longrightarrow> TZero" |
  EPredSucc: "nv \<in> numeric_values \<Longrightarrow> TPred (TSucc nv) \<longrightarrow> nv" |
  EPred: "t1 \<longrightarrow> t1' \<Longrightarrow> TPred t1 \<longrightarrow> TPred t1'" |
  EIsZero: "t1 \<longrightarrow> t1' \<Longrightarrow> TIsZero t1 \<longrightarrow> TIsZero t1'" |
  EIsZeroZero: "TIsZero TZero \<longrightarrow> TTrue" |
  EIsZeroSucc: "nv \<in> numeric_values \<Longrightarrow> TIsZero (TSucc nv) \<longrightarrow> TFalse"
  
section "Type System"

section "Meta Theory"

inductive_cases zero_case: "TZero \<longrightarrow> x" and
  iftru_case: "TIf TTrue t2 t3 \<longrightarrow> t" and
  succ_case: "TSucc nv \<longrightarrow> x"

theorem numeric_value_not_reducible:
  fixes nv::trm
  assumes nv: "nv \<in> numeric_values"
  and r: "nv \<longrightarrow> x"
  shows "False"
  using nv r apply (induct arbitrary: x rule: numeric_values.induct)
  apply (erule zero_case)
  apply (erule succ_case)
  apply blast
  done
  
lemma reduction_smaller:
  fixes t::trm assumes r: "t \<longrightarrow> t'"
  shows "size t' < size t"
  using r by (induct rule: reduction.induct) auto



end