Theory Bool

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theory Bool
imports CTT

(*  Title:      CTT/bool
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
*)


header {* The two-element type (booleans and conditionals) *}

theory Bool
imports CTT
begin


definition
Bool :: "t" where
"Bool == T+T"


definition
true :: "i" where
"true == inl(tt)"


definition
false :: "i" where
"false == inr(tt)"


definition
cond :: "[i,i,i]=>i" where
"cond(a,b,c) == when(a, %u. b, %u. c)"


lemmas bool_defs = Bool_def true_def false_def cond_def


subsection {* Derivation of rules for the type Bool *}

(*formation rule*)
lemma boolF: "Bool type"
apply (unfold bool_defs)
apply (tactic "typechk_tac []")
done


(*introduction rules for true, false*)

lemma boolI_true: "true : Bool"
apply (unfold bool_defs)
apply (tactic "typechk_tac []")
done

lemma boolI_false: "false : Bool"
apply (unfold bool_defs)
apply (tactic "typechk_tac []")
done

(*elimination rule: typing of cond*)
lemma boolE:
"[| p:Bool; a : C(true); b : C(false) |] ==> cond(p,a,b) : C(p)"

apply (unfold bool_defs)
apply (tactic "typechk_tac []")
apply (erule_tac [!] TE)
apply (tactic "typechk_tac []")
done

lemma boolEL:
"[| p = q : Bool; a = c : C(true); b = d : C(false) |]
==> cond(p,a,b) = cond(q,c,d) : C(p)"

apply (unfold bool_defs)
apply (rule PlusEL)
apply (erule asm_rl refl_elem [THEN TEL])+
done

(*computation rules for true, false*)

lemma boolC_true:
"[| a : C(true); b : C(false) |] ==> cond(true,a,b) = a : C(true)"

apply (unfold bool_defs)
apply (rule comp_rls)
apply (tactic "typechk_tac []")
apply (erule_tac [!] TE)
apply (tactic "typechk_tac []")
done

lemma boolC_false:
"[| a : C(true); b : C(false) |] ==> cond(false,a,b) = b : C(false)"

apply (unfold bool_defs)
apply (rule comp_rls)
apply (tactic "typechk_tac []")
apply (erule_tac [!] TE)
apply (tactic "typechk_tac []")
done

end