# Theory Type

```(*  Title:      CCL/Type.thy
Author:     Martin Coen
*)

section ‹Types in CCL are defined as sets of terms›

theory Type
imports Term
begin

definition Subtype :: "['a set, 'a ⇒ o] ⇒ 'a set"
where "Subtype(A, P) == {x. x:A ∧ P(x)}"

syntax
"_Subtype" :: "[idt, 'a set, o] ⇒ 'a set"  ("(1{_: _ ./ _})")
translations
"{x: A. B}" == "CONST Subtype(A, λx. B)"

definition Unit :: "i set"
where "Unit == {x. x=one}"

definition Bool :: "i set"
where "Bool == {x. x=true | x=false}"

definition Plus :: "[i set, i set] ⇒ i set"  (infixr "+" 55)
where "A+B == {x. (EX a:A. x=inl(a)) | (EX b:B. x=inr(b))}"

definition Pi :: "[i set, i ⇒ i set] ⇒ i set"
where "Pi(A,B) == {x. EX b. x=lam x. b(x) ∧ (ALL x:A. b(x):B(x))}"

definition Sigma :: "[i set, i ⇒ i set] ⇒ i set"
where "Sigma(A,B) == {x. EX a:A. EX b:B(a).x=<a,b>}"

syntax
"_Pi" :: "[idt, i set, i set] ⇒ i set"  ("(3PROD _:_./ _)" [0,0,60] 60)
"_Sigma" :: "[idt, i set, i set] ⇒ i set"  ("(3SUM _:_./ _)" [0,0,60] 60)
"_arrow" :: "[i set, i set] ⇒ i set"  ("(_ ->/ _)"  [54, 53] 53)
"_star"  :: "[i set, i set] ⇒ i set"  ("(_ */ _)" [56, 55] 55)
translations
"PROD x:A. B" ⇀ "CONST Pi(A, λx. B)"
"A -> B" ⇀ "CONST Pi(A, λ_. B)"
"SUM x:A. B" ⇀ "CONST Sigma(A, λx. B)"
"A * B" ⇀ "CONST Sigma(A, λ_. B)"
print_translation ‹
[(\<^const_syntax>‹Pi›,
fn _ => Syntax_Trans.dependent_tr' (\<^syntax_const>‹_Pi›, \<^syntax_const>‹_arrow›)),
(\<^const_syntax>‹Sigma›,
fn _ => Syntax_Trans.dependent_tr' (\<^syntax_const>‹_Sigma›, \<^syntax_const>‹_star›))]
›

definition Nat :: "i set"
where "Nat == lfp(λX. Unit + X)"

definition List :: "i set ⇒ i set"
where "List(A) == lfp(λX. Unit + A*X)"

definition Lists :: "i set ⇒ i set"
where "Lists(A) == gfp(λX. Unit + A*X)"

definition ILists :: "i set ⇒ i set"
where "ILists(A) == gfp(λX.{} + A*X)"

definition TAll :: "(i set ⇒ i set) ⇒ i set"  (binder "TALL " 55)
where "TALL X. B(X) == Inter({X. EX Y. X=B(Y)})"

definition TEx :: "(i set ⇒ i set) ⇒ i set"  (binder "TEX " 55)
where "TEX X. B(X) == Union({X. EX Y. X=B(Y)})"

definition Lift :: "i set ⇒ i set"  ("(3[_])")
where "[A] == A Un {bot}"

definition SPLIT :: "[i, [i, i] ⇒ i set] ⇒ i set"
where "SPLIT(p,B) == Union({A. EX x y. p=<x,y> ∧ A=B(x,y)})"

lemmas simp_type_defs =
Subtype_def Unit_def Bool_def Plus_def Sigma_def Pi_def Lift_def TAll_def TEx_def
and ind_type_defs = Nat_def List_def
and simp_data_defs = one_def inl_def inr_def
and ind_data_defs = zero_def succ_def nil_def cons_def

lemma subsetXH: "A <= B ⟷ (ALL x. x:A ⟶ x:B)"
by blast

subsection ‹Exhaustion Rules›

lemma EmptyXH: "⋀a. a : {} ⟷ False"
and SubtypeXH: "⋀a A P. a : {x:A. P(x)} ⟷ (a:A ∧ P(a))"
and UnitXH: "⋀a. a : Unit          ⟷ a=one"
and BoolXH: "⋀a. a : Bool          ⟷ a=true | a=false"
and PlusXH: "⋀a A B. a : A+B           ⟷ (EX x:A. a=inl(x)) | (EX x:B. a=inr(x))"
and PiXH: "⋀a A B. a : PROD x:A. B(x) ⟷ (EX b. a=lam x. b(x) ∧ (ALL x:A. b(x):B(x)))"
and SgXH: "⋀a A B. a : SUM x:A. B(x)  ⟷ (EX x:A. EX y:B(x).a=<x,y>)"
unfolding simp_type_defs by blast+

lemmas XHs = EmptyXH SubtypeXH UnitXH BoolXH PlusXH PiXH SgXH

lemma LiftXH: "a : [A] ⟷ (a=bot | a:A)"
and TallXH: "a : TALL X. B(X) ⟷ (ALL X. a:B(X))"
and TexXH: "a : TEX X. B(X) ⟷ (EX X. a:B(X))"
unfolding simp_type_defs by blast+

ML ‹ML_Thms.bind_thms ("case_rls", XH_to_Es @{thms XHs})›

subsection ‹Canonical Type Rules›

lemma oneT: "one : Unit"
and trueT: "true : Bool"
and falseT: "false : Bool"
and lamT: "⋀b B. (⋀x. x:A ⟹ b(x):B(x)) ⟹ lam x. b(x) : Pi(A,B)"
and pairT: "⋀b B. ⟦a:A; b:B(a)⟧ ⟹ <a,b>:Sigma(A,B)"
and inlT: "a:A ⟹ inl(a) : A+B"
and inrT: "b:B ⟹ inr(b) : A+B"
by (blast intro: XHs [THEN iffD2])+

lemmas canTs = oneT trueT falseT pairT lamT inlT inrT

subsection ‹Non-Canonical Type Rules›

lemma lem: "⟦a:B(u); u = v⟧ ⟹ a : B(v)"
by blast

ML ‹
fun mk_ncanT_tac top_crls crls =
SUBPROOF (fn {context = ctxt, prems = major :: prems, ...} =>
resolve_tac ctxt ([major] RL top_crls) 1 THEN
REPEAT_SOME (eresolve_tac ctxt (crls @ @{thms exE bexE conjE disjE})) THEN
ALLGOALS (asm_simp_tac ctxt) THEN
ALLGOALS (assume_tac ctxt ORELSE' resolve_tac ctxt (prems RL [@{thm lem}])
ORELSE' eresolve_tac ctxt @{thms bspec}) THEN
›

method_setup ncanT = ‹
Scan.succeed (SIMPLE_METHOD' o mk_ncanT_tac @{thms case_rls} @{thms case_rls})
›

lemma ifT: "⟦b:Bool; b=true ⟹ t:A(true); b=false ⟹ u:A(false)⟧ ⟹ if b then t else u : A(b)"
by ncanT

lemma applyT: "⟦f : Pi(A,B); a:A⟧ ⟹ f ` a : B(a)"
by ncanT

lemma splitT: "⟦p:Sigma(A,B); ⋀x y. ⟦x:A; y:B(x); p=<x,y>⟧ ⟹ c(x,y):C(<x,y>)⟧ ⟹ split(p,c):C(p)"
by ncanT

lemma whenT:
"⟦p:A+B;
⋀x. ⟦x:A; p=inl(x)⟧ ⟹ a(x):C(inl(x));
⋀y. ⟦y:B;  p=inr(y)⟧ ⟹ b(y):C(inr(y))⟧ ⟹ when(p,a,b) : C(p)"
by ncanT

lemmas ncanTs = ifT applyT splitT whenT

subsection ‹Subtypes›

lemma SubtypeD1: "a : Subtype(A, P) ⟹ a : A"
and SubtypeD2: "a : Subtype(A, P) ⟹ P(a)"

lemma SubtypeI: "⟦a:A; P(a)⟧ ⟹ a : {x:A. P(x)}"

lemma SubtypeE: "⟦a : {x:A. P(x)}; ⟦a:A; P(a)⟧ ⟹ Q⟧ ⟹ Q"

subsection ‹Monotonicity›

lemma idM: "mono (λX. X)"
apply (rule monoI)
apply assumption
done

lemma constM: "mono(λX. A)"
apply (rule monoI)
apply (rule subset_refl)
done

lemma "mono(λX. A(X)) ⟹ mono(λX.[A(X)])"
apply (rule subsetI [THEN monoI])
apply (drule LiftXH [THEN iffD1])
apply (erule disjE)
apply (erule disjI1 [THEN LiftXH [THEN iffD2]])
apply (rule disjI2 [THEN LiftXH [THEN iffD2]])
apply (drule (1) monoD)
apply blast
done

lemma SgM:
"⟦mono(λX. A(X)); ⋀x X. x:A(X) ⟹ mono(λX. B(X,x))⟧ ⟹
mono(λX. Sigma(A(X),B(X)))"
by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
dest!: monoD [THEN subsetD])

lemma PiM: "(⋀x. x:A ⟹ mono(λX. B(X,x))) ⟹ mono(λX. Pi(A,B(X)))"
by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
dest!: monoD [THEN subsetD])

lemma PlusM: "⟦mono(λX. A(X)); mono(λX. B(X))⟧ ⟹ mono(λX. A(X)+B(X))"
by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
dest!: monoD [THEN subsetD])

subsection ‹Recursive types›

subsubsection ‹Conversion Rules for Fixed Points via monotonicity and Tarski›

lemma NatM: "mono(λX. Unit+X)"
apply (rule PlusM constM idM)+
done

lemma def_NatB: "Nat = Unit + Nat"
apply (rule def_lfp_Tarski [OF Nat_def])
apply (rule NatM)
done

lemma ListM: "mono(λX.(Unit+Sigma(A,λy. X)))"
apply (rule PlusM SgM constM idM)+
done

lemma def_ListB: "List(A) = Unit + A * List(A)"
apply (rule def_lfp_Tarski [OF List_def])
apply (rule ListM)
done

lemma def_ListsB: "Lists(A) = Unit + A * Lists(A)"
apply (rule def_gfp_Tarski [OF Lists_def])
apply (rule ListM)
done

lemma IListsM: "mono(λX.({} + Sigma(A,λy. X)))"
apply (rule PlusM SgM constM idM)+
done

lemma def_IListsB: "ILists(A) = {} + A * ILists(A)"
apply (rule def_gfp_Tarski [OF ILists_def])
apply (rule IListsM)
done

lemmas ind_type_eqs = def_NatB def_ListB def_ListsB def_IListsB

subsection ‹Exhaustion Rules›

lemma NatXH: "a : Nat ⟷ (a=zero | (EX x:Nat. a=succ(x)))"
and ListXH: "a : List(A) ⟷ (a=[] | (EX x:A. EX xs:List(A).a=x\$xs))"
and ListsXH: "a : Lists(A) ⟷ (a=[] | (EX x:A. EX xs:Lists(A).a=x\$xs))"
and IListsXH: "a : ILists(A) ⟷ (EX x:A. EX xs:ILists(A).a=x\$xs)"
unfolding ind_data_defs
by (rule ind_type_eqs [THEN XHlemma1], blast intro!: canTs elim!: case_rls)+

lemmas iXHs = NatXH ListXH

ML ‹ML_Thms.bind_thms ("icase_rls", XH_to_Es @{thms iXHs})›

subsection ‹Type Rules›

lemma zeroT: "zero : Nat"
and succT: "n:Nat ⟹ succ(n) : Nat"
and nilT: "[] : List(A)"
and consT: "⟦h:A; t:List(A)⟧ ⟹ h\$t : List(A)"
by (blast intro: iXHs [THEN iffD2])+

lemmas icanTs = zeroT succT nilT consT

method_setup incanT = ‹
Scan.succeed (SIMPLE_METHOD' o mk_ncanT_tac @{thms icase_rls} @{thms case_rls})
›

lemma ncaseT: "⟦n:Nat; n=zero ⟹ b:C(zero); ⋀x. ⟦x:Nat; n=succ(x)⟧ ⟹ c(x):C(succ(x))⟧
⟹ ncase(n,b,c) : C(n)"
by incanT

lemma lcaseT: "⟦l:List(A); l = [] ⟹ b:C([]); ⋀h t. ⟦h:A; t:List(A); l=h\$t⟧ ⟹ c(h,t):C(h\$t)⟧
⟹ lcase(l,b,c) : C(l)"
by incanT

lemmas incanTs = ncaseT lcaseT

subsection ‹Induction Rules›

lemmas ind_Ms = NatM ListM

lemma Nat_ind: "⟦n:Nat; P(zero); ⋀x. ⟦x:Nat; P(x)⟧ ⟹ P(succ(x))⟧ ⟹ P(n)"
apply (unfold ind_data_defs)
apply (erule def_induct [OF Nat_def _ NatM])
apply (blast intro: canTs elim!: case_rls)
done

lemma List_ind: "⟦l:List(A); P([]); ⋀x xs. ⟦x:A; xs:List(A); P(xs)⟧ ⟹ P(x\$xs)⟧ ⟹ P(l)"
apply (unfold ind_data_defs)
apply (erule def_induct [OF List_def _ ListM])
apply (blast intro: canTs elim!: case_rls)
done

lemmas inds = Nat_ind List_ind

subsection ‹Primitive Recursive Rules›

lemma nrecT: "⟦n:Nat; b:C(zero); ⋀x g. ⟦x:Nat; g:C(x)⟧ ⟹ c(x,g):C(succ(x))⟧
⟹ nrec(n,b,c) : C(n)"
by (erule Nat_ind) auto

lemma lrecT: "⟦l:List(A); b:C([]); ⋀x xs g. ⟦x:A; xs:List(A); g:C(xs)⟧ ⟹ c(x,xs,g):C(x\$xs) ⟧
⟹ lrec(l,b,c) : C(l)"
by (erule List_ind) auto

lemmas precTs = nrecT lrecT

subsection ‹Theorem proving›

lemma SgE2: "⟦<a,b> : Sigma(A,B); ⟦a:A; b:B(a)⟧ ⟹ P⟧ ⟹ P"
unfolding SgXH by blast

(* General theorem proving ignores non-canonical term-formers,             *)
(*         - intro rules are type rules for canonical terms                *)
(*         - elim rules are case rules (no non-canonical terms appear)     *)

ML ‹ML_Thms.bind_thms ("XHEs", XH_to_Es @{thms XHs})›

lemmas [intro!] = SubtypeI canTs icanTs
and [elim!] = SubtypeE XHEs

subsection ‹Infinite Data Types›

lemma lfp_subset_gfp: "mono(f) ⟹ lfp(f) <= gfp(f)"
apply (rule lfp_lowerbound [THEN subset_trans])
apply (erule gfp_lemma3)
apply (rule subset_refl)
done

lemma gfpI:
assumes "a:A"
and "⋀x X. ⟦x:A; ALL y:A. t(y):X⟧ ⟹ t(x) : B(X)"
shows "t(a) : gfp(B)"
apply (rule coinduct)
apply (rule_tac P = "λx. EX y:A. x=t (y)" in CollectI)
apply (blast intro!: assms)+
done

lemma def_gfpI: "⟦C == gfp(B); a:A; ⋀x X. ⟦x:A; ALL y:A. t(y):X⟧ ⟹ t(x) : B(X)⟧ ⟹ t(a) : C"
apply unfold
apply (erule gfpI)
apply blast
done

(* EG *)
lemma "letrec g x be zero\$g(x) in g(bot) : Lists(Nat)"
apply (rule refl [THEN UnitXH [THEN iffD2], THEN Lists_def [THEN def_gfpI]])
apply (subst letrecB)
apply (unfold cons_def)
apply blast
done

subsection ‹Lemmas and tactics for using the rule ‹coinduct3› on ‹[=› and ‹=››

lemma lfpI: "⟦mono(f); a : f(lfp(f))⟧ ⟹ a : lfp(f)"
apply (erule lfp_Tarski [THEN ssubst])
apply assumption
done

lemma ssubst_single: "⟦a = a'; a' : A⟧ ⟹ a : A"
by simp

lemma ssubst_pair: "⟦a = a'; b = b'; <a',b'> : A⟧ ⟹ <a,b> : A"
by simp

ML ‹
val coinduct3_tac = SUBPROOF (fn {context = ctxt, prems = mono :: prems, ...} =>
fast_tac (ctxt addIs (mono RS @{thm coinduct3_mono_lemma} RS @{thm lfpI}) :: prems) 1);
›

method_setup coinduct3 = ‹Scan.succeed (SIMPLE_METHOD' o coinduct3_tac)›

lemma ci3_RI: "⟦mono(Agen); a : R⟧ ⟹ a : lfp(λx. Agen(x) Un R Un A)"
by coinduct3

lemma ci3_AgenI: "⟦mono(Agen); a : Agen(lfp(λx. Agen(x) Un R Un A))⟧ ⟹
a : lfp(λx. Agen(x) Un R Un A)"
by coinduct3

lemma ci3_AI: "⟦mono(Agen); a : A⟧ ⟹ a : lfp(λx. Agen(x) Un R Un A)"
by coinduct3

ML ‹
fun genIs_tac ctxt genXH gen_mono =
resolve_tac ctxt [genXH RS @{thm iffD2}] THEN'
simp_tac ctxt THEN'
TRY o fast_tac
(ctxt addIs [genXH RS @{thm iffD2}, gen_mono RS @{thm coinduct3_mono_lemma} RS @{thm lfpI}])
›

method_setup genIs = ‹
Attrib.thm -- Attrib.thm >>
(fn (genXH, gen_mono) => fn ctxt => SIMPLE_METHOD' (genIs_tac ctxt genXH gen_mono))
›

subsection ‹POgen›

lemma PO_refl: "<a,a> : PO"
by (rule po_refl [THEN PO_iff [THEN iffD1]])

lemma POgenIs:
"<true,true> : POgen(R)"
"<false,false> : POgen(R)"
"⟦<a,a'> : R; <b,b'> : R⟧ ⟹ <<a,b>,<a',b'>> : POgen(R)"
"⋀b b'. (⋀x. <b(x),b'(x)> : R) ⟹ <lam x. b(x),lam x. b'(x)> : POgen(R)"
"<one,one> : POgen(R)"
"<a,a'> : lfp(λx. POgen(x) Un R Un PO) ⟹
<inl(a),inl(a')> : POgen(lfp(λx. POgen(x) Un R Un PO))"
"<b,b'> : lfp(λx. POgen(x) Un R Un PO) ⟹
<inr(b),inr(b')> : POgen(lfp(λx. POgen(x) Un R Un PO))"
"<zero,zero> : POgen(lfp(λx. POgen(x) Un R Un PO))"
"<n,n'> : lfp(λx. POgen(x) Un R Un PO) ⟹
<succ(n),succ(n')> : POgen(lfp(λx. POgen(x) Un R Un PO))"
"<[],[]> : POgen(lfp(λx. POgen(x) Un R Un PO))"
"⟦<h,h'> : lfp(λx. POgen(x) Un R Un PO);  <t,t'> : lfp(λx. POgen(x) Un R Un PO)⟧
⟹ <h\$t,h'\$t'> : POgen(lfp(λx. POgen(x) Un R Un PO))"
unfolding data_defs by (genIs POgenXH POgen_mono)+

ML ‹
fun POgen_tac ctxt (rla, rlb) i =
SELECT_GOAL (safe_tac ctxt) i THEN
resolve_tac ctxt [rlb RS (rla RS @{thm ssubst_pair})] i THEN
(REPEAT (resolve_tac ctxt
(@{thms POgenIs} @ [@{thm PO_refl} RS (@{thm POgen_mono} RS @{thm ci3_AI})] @
(@{thms POgenIs} RL [@{thm POgen_mono} RS @{thm ci3_AgenI}]) @
[@{thm POgen_mono} RS @{thm ci3_RI}]) i))
›

subsection ‹EQgen›

lemma EQ_refl: "<a,a> : EQ"
by (rule refl [THEN EQ_iff [THEN iffD1]])

lemma EQgenIs:
"<true,true> : EQgen(R)"
"<false,false> : EQgen(R)"
"⟦<a,a'> : R; <b,b'> : R⟧ ⟹ <<a,b>,<a',b'>> : EQgen(R)"
"⋀b b'. (⋀x. <b(x),b'(x)> : R) ⟹ <lam x. b(x),lam x. b'(x)> : EQgen(R)"
"<one,one> : EQgen(R)"
"<a,a'> : lfp(λx. EQgen(x) Un R Un EQ) ⟹
<inl(a),inl(a')> : EQgen(lfp(λx. EQgen(x) Un R Un EQ))"
"<b,b'> : lfp(λx. EQgen(x) Un R Un EQ) ⟹
<inr(b),inr(b')> : EQgen(lfp(λx. EQgen(x) Un R Un EQ))"
"<zero,zero> : EQgen(lfp(λx. EQgen(x) Un R Un EQ))"
"<n,n'> : lfp(λx. EQgen(x) Un R Un EQ) ⟹
<succ(n),succ(n')> : EQgen(lfp(λx. EQgen(x) Un R Un EQ))"
"<[],[]> : EQgen(lfp(λx. EQgen(x) Un R Un EQ))"
"⟦<h,h'> : lfp(λx. EQgen(x) Un R Un EQ); <t,t'> : lfp(λx. EQgen(x) Un R Un EQ)⟧
⟹ <h\$t,h'\$t'> : EQgen(lfp(λx. EQgen(x) Un R Un EQ))"
unfolding data_defs by (genIs EQgenXH EQgen_mono)+

ML ‹
fun EQgen_raw_tac ctxt i =
(REPEAT (resolve_tac ctxt (@{thms EQgenIs} @
[@{thm EQ_refl} RS (@{thm EQgen_mono} RS @{thm ci3_AI})] @
(@{thms EQgenIs} RL [@{thm EQgen_mono} RS @{thm ci3_AgenI}]) @
[@{thm EQgen_mono} RS @{thm ci3_RI}]) i))

(* Goals of the form R <= EQgen(R) - rewrite elements <a,b> : EQgen(R) using rews and *)
(* then reduce this to a goal <a',b'> : R (hopefully?)                                *)
(*      rews are rewrite rules that would cause looping in the simpifier              *)

fun EQgen_tac ctxt rews i =
SELECT_GOAL
(TRY (safe_tac ctxt) THEN
resolve_tac ctxt ((rews @ [@{thm refl}]) RL ((rews @ [@{thm refl}]) RL [@{thm ssubst_pair}])) i THEN
ALLGOALS (simp_tac ctxt) THEN
ALLGOALS (EQgen_raw_tac ctxt)) i
›

method_setup EQgen = ‹
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (EQgen_tac ctxt ths))
›

end
```