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theory CTT(* Title: CTT/CTT.thy ID: $Id: CTT.thy,v 1.18 2007/06/21 20:10:16 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) header {* Constructive Type Theory *} theory CTT imports Pure uses "~~/src/Provers/typedsimp.ML" ("rew.ML") begin typedecl i typedecl t typedecl o consts (*Types*) F :: "t" T :: "t" (*F is empty, T contains one element*) contr :: "i=>i" tt :: "i" (*Natural numbers*) N :: "t" succ :: "i=>i" rec :: "[i, i, [i,i]=>i] => i" (*Unions*) inl :: "i=>i" inr :: "i=>i" when :: "[i, i=>i, i=>i]=>i" (*General Sum and Binary Product*) Sum :: "[t, i=>t]=>t" fst :: "i=>i" snd :: "i=>i" split :: "[i, [i,i]=>i] =>i" (*General Product and Function Space*) Prod :: "[t, i=>t]=>t" (*Types*) Plus :: "[t,t]=>t" (infixr "+" 40) (*Equality type*) Eq :: "[t,i,i]=>t" eq :: "i" (*Judgements*) Type :: "t => prop" ("(_ type)" [10] 5) Eqtype :: "[t,t]=>prop" ("(_ =/ _)" [10,10] 5) Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5) Eqelem :: "[i,i,t]=>prop" ("(_ =/ _ :/ _)" [10,10,10] 5) Reduce :: "[i,i]=>prop" ("Reduce[_,_]") (*Types*) (*Functions*) lambda :: "(i => i) => i" (binder "lam " 10) app :: "[i,i]=>i" (infixl "`" 60) (*Natural numbers*) "0" :: "i" ("0") (*Pairing*) pair :: "[i,i]=>i" ("(1<_,/_>)") syntax "_PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10) "_SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10) translations "PROD x:A. B" == "Prod(A, %x. B)" "SUM x:A. B" == "Sum(A, %x. B)" abbreviation Arrow :: "[t,t]=>t" (infixr "-->" 30) where "A --> B == PROD _:A. B" abbreviation Times :: "[t,t]=>t" (infixr "*" 50) where "A * B == SUM _:A. B" notation (xsymbols) lambda (binder "λλ" 10) and Elem ("(_ /∈ _)" [10,10] 5) and Eqelem ("(2_ =/ _ ∈/ _)" [10,10,10] 5) and Arrow (infixr "-->" 30) and Times (infixr "×" 50) notation (HTML output) lambda (binder "λλ" 10) and Elem ("(_ /∈ _)" [10,10] 5) and Eqelem ("(2_ =/ _ ∈/ _)" [10,10,10] 5) and Times (infixr "×" 50) syntax (xsymbols) "_PROD" :: "[idt,t,t] => t" ("(3Π _∈_./ _)" 10) "_SUM" :: "[idt,t,t] => t" ("(3Σ _∈_./ _)" 10) syntax (HTML output) "_PROD" :: "[idt,t,t] => t" ("(3Π _∈_./ _)" 10) "_SUM" :: "[idt,t,t] => t" ("(3Σ _∈_./ _)" 10) axioms (*Reduction: a weaker notion than equality; a hack for simplification. Reduce[a,b] means either that a=b:A for some A or else that "a" and "b" are textually identical.*) (*does not verify a:A! Sound because only trans_red uses a Reduce premise No new theorems can be proved about the standard judgements.*) refl_red: "Reduce[a,a]" red_if_equal: "a = b : A ==> Reduce[a,b]" trans_red: "[| a = b : A; Reduce[b,c] |] ==> a = c : A" (*Reflexivity*) refl_type: "A type ==> A = A" refl_elem: "a : A ==> a = a : A" (*Symmetry*) sym_type: "A = B ==> B = A" sym_elem: "a = b : A ==> b = a : A" (*Transitivity*) trans_type: "[| A = B; B = C |] ==> A = C" trans_elem: "[| a = b : A; b = c : A |] ==> a = c : A" equal_types: "[| a : A; A = B |] ==> a : B" equal_typesL: "[| a = b : A; A = B |] ==> a = b : B" (*Substitution*) subst_type: "[| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type" subst_typeL: "[| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)" subst_elem: "[| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)" subst_elemL: "[| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)" (*The type N -- natural numbers*) NF: "N type" NI0: "0 : N" NI_succ: "a : N ==> succ(a) : N" NI_succL: "a = b : N ==> succ(a) = succ(b) : N" NE: "[| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==> rec(p, a, %u v. b(u,v)) : C(p)" NEL: "[| p = q : N; a = c : C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |] ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)" NC0: "[| a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==> rec(0, a, %u v. b(u,v)) = a : C(0)" NC_succ: "[| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==> rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))" (*The fourth Peano axiom. See page 91 of Martin-Lof's book*) zero_ne_succ: "[| a: N; 0 = succ(a) : N |] ==> 0: F" (*The Product of a family of types*) ProdF: "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type" ProdFL: "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> PROD x:A. B(x) = PROD x:C. D(x)" ProdI: "[| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)" ProdIL: "[| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==> lam x. b(x) = lam x. c(x) : PROD x:A. B(x)" ProdE: "[| p : PROD x:A. B(x); a : A |] ==> p`a : B(a)" ProdEL: "[| p=q: PROD x:A. B(x); a=b : A |] ==> p`a = q`b : B(a)" ProdC: "[| a : A; !!x. x:A ==> b(x) : B(x)|] ==> (lam x. b(x)) ` a = b(a) : B(a)" ProdC2: "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)" (*The Sum of a family of types*) SumF: "[| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type" SumFL: "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)" SumI: "[| a : A; b : B(a) |] ==> <a,b> : SUM x:A. B(x)" SumIL: "[| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)" SumE: "[| p: SUM x:A. B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |] ==> split(p, %x y. c(x,y)) : C(p)" SumEL: "[| p=q : SUM x:A. B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|] ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)" SumC: "[| a: A; b: B(a); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |] ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)" fst_def: "fst(a) == split(a, %x y. x)" snd_def: "snd(a) == split(a, %x y. y)" (*The sum of two types*) PlusF: "[| A type; B type |] ==> A+B type" PlusFL: "[| A = C; B = D |] ==> A+B = C+D" PlusI_inl: "[| a : A; B type |] ==> inl(a) : A+B" PlusI_inlL: "[| a = c : A; B type |] ==> inl(a) = inl(c) : A+B" PlusI_inr: "[| A type; b : B |] ==> inr(b) : A+B" PlusI_inrL: "[| A type; b = d : B |] ==> inr(b) = inr(d) : A+B" PlusE: "[| p: A+B; !!x. x:A ==> c(x): C(inl(x)); !!y. y:B ==> d(y): C(inr(y)) |] ==> when(p, %x. c(x), %y. d(y)) : C(p)" PlusEL: "[| p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x)); !!y. y: B ==> d(y) = f(y) : C(inr(y)) |] ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)" PlusC_inl: "[| a: A; !!x. x:A ==> c(x): C(inl(x)); !!y. y:B ==> d(y): C(inr(y)) |] ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))" PlusC_inr: "[| b: B; !!x. x:A ==> c(x): C(inl(x)); !!y. y:B ==> d(y): C(inr(y)) |] ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))" (*The type Eq*) EqF: "[| A type; a : A; b : A |] ==> Eq(A,a,b) type" EqFL: "[| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)" EqI: "a = b : A ==> eq : Eq(A,a,b)" EqE: "p : Eq(A,a,b) ==> a = b : A" (*By equality of types, can prove C(p) from C(eq), an elimination rule*) EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)" (*The type F*) FF: "F type" FE: "[| p: F; C type |] ==> contr(p) : C" FEL: "[| p = q : F; C type |] ==> contr(p) = contr(q) : C" (*The type T Martin-Lof's book (page 68) discusses elimination and computation. Elimination can be derived by computation and equality of types, but with an extra premise C(x) type x:T. Also computation can be derived from elimination. *) TF: "T type" TI: "tt : T" TE: "[| p : T; c : C(tt) |] ==> c : C(p)" TEL: "[| p = q : T; c = d : C(tt) |] ==> c = d : C(p)" TC: "p : T ==> p = tt : T" subsection "Tactics and derived rules for Constructive Type Theory" (*Formation rules*) lemmas form_rls = NF ProdF SumF PlusF EqF FF TF and formL_rls = ProdFL SumFL PlusFL EqFL (*Introduction rules OMITTED: EqI, because its premise is an eqelem, not an elem*) lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL (*Elimination rules OMITTED: EqE, because its conclusion is an eqelem, not an elem TE, because it does not involve a constructor *) lemmas elim_rls = NE ProdE SumE PlusE FE and elimL_rls = NEL ProdEL SumEL PlusEL FEL (*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *) lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr (*rules with conclusion a:A, an elem judgement*) lemmas element_rls = intr_rls elim_rls (*Definitions are (meta)equality axioms*) lemmas basic_defs = fst_def snd_def (*Compare with standard version: B is applied to UNSIMPLIFIED expression! *) lemma SumIL2: "[| c=a : A; d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)" apply (rule sym_elem) apply (rule SumIL) apply (rule_tac [!] sym_elem) apply assumption+ done lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL (*Exploit p:Prod(A,B) to create the assumption z:B(a). A more natural form of product elimination. *) lemma subst_prodE: assumes "p: Prod(A,B)" and "a: A" and "!!z. z: B(a) ==> c(z): C(z)" shows "c(p`a): C(p`a)" apply (rule prems ProdE)+ done subsection {* Tactics for type checking *} ML {* local fun is_rigid_elem (Const("CTT.Elem",_) $ a $ _) = not(is_Var (head_of a)) | is_rigid_elem (Const("CTT.Eqelem",_) $ a $ _ $ _) = not(is_Var (head_of a)) | is_rigid_elem (Const("CTT.Type",_) $ a) = not(is_Var (head_of a)) | is_rigid_elem _ = false in (*Try solving a:A or a=b:A by assumption provided a is rigid!*) val test_assume_tac = SUBGOAL(fn (prem,i) => if is_rigid_elem (Logic.strip_assums_concl prem) then assume_tac i else no_tac) fun ASSUME tf i = test_assume_tac i ORELSE tf i end; *} (*For simplification: type formation and checking, but no equalities between terms*) lemmas routine_rls = form_rls formL_rls refl_type element_rls ML {* local val routine_rls = thms "routine_rls"; val form_rls = thms "form_rls"; val intr_rls = thms "intr_rls"; val element_rls = thms "element_rls"; val equal_rls = form_rls @ element_rls @ thms "intrL_rls" @ thms "elimL_rls" @ thms "refl_elem" in fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4); (*Solve all subgoals "A type" using formation rules. *) val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac(form_rls) 1)); (*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) fun typechk_tac thms = let val tac = filt_resolve_tac (thms @ form_rls @ element_rls) 3 in REPEAT_FIRST (ASSUME tac) end (*Solve a:A (a flexible, A rigid) by introduction rules. Cannot use stringtrees (filt_resolve_tac) since goals like ?a:SUM(A,B) have a trivial head-string *) fun intr_tac thms = let val tac = filt_resolve_tac(thms@form_rls@intr_rls) 1 in REPEAT_FIRST (ASSUME tac) end (*Equality proving: solve a=b:A (where a is rigid) by long rules. *) fun equal_tac thms = REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3)) end *} subsection {* Simplification *} (*To simplify the type in a goal*) lemma replace_type: "[| B = A; a : A |] ==> a : B" apply (rule equal_types) apply (rule_tac [2] sym_type) apply assumption+ done (*Simplify the parameter of a unary type operator.*) lemma subst_eqtyparg: assumes 1: "a=c : A" and 2: "!!z. z:A ==> B(z) type" shows "B(a)=B(c)" apply (rule subst_typeL) apply (rule_tac [2] refl_type) apply (rule 1) apply (erule 2) done (*Simplification rules for Constructive Type Theory*) lemmas reduction_rls = comp_rls [THEN trans_elem] ML {* local val EqI = thm "EqI"; val EqE = thm "EqE"; val NE = thm "NE"; val FE = thm "FE"; val ProdI = thm "ProdI"; val SumI = thm "SumI"; val SumE = thm "SumE"; val PlusE = thm "PlusE"; val PlusI_inl = thm "PlusI_inl"; val PlusI_inr = thm "PlusI_inr"; val subst_prodE = thm "subst_prodE"; val intr_rls = thms "intr_rls"; in (*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. Uses other intro rules to avoid changing flexible goals.*) val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac(EqI::intr_rls) 1)) (** Tactics that instantiate CTT-rules. Vars in the given terms will be incremented! The (rtac EqE i) lets them apply to equality judgements. **) fun NE_tac (sp: string) i = TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] NE i fun SumE_tac (sp: string) i = TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] SumE i fun PlusE_tac (sp: string) i = TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] PlusE i (** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) (*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) fun add_mp_tac i = rtac subst_prodE i THEN assume_tac i THEN assume_tac i (*Finds P-->Q and P in the assumptions, replaces implication by Q *) fun mp_tac i = etac subst_prodE i THEN assume_tac i (*"safe" when regarded as predicate calculus rules*) val safe_brls = sort (make_ord lessb) [ (true,FE), (true,asm_rl), (false,ProdI), (true,SumE), (true,PlusE) ] val unsafe_brls = [ (false,PlusI_inl), (false,PlusI_inr), (false,SumI), (true,subst_prodE) ] (*0 subgoals vs 1 or more*) val (safe0_brls, safep_brls) = List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls fun safestep_tac thms i = form_tac ORELSE resolve_tac thms i ORELSE biresolve_tac safe0_brls i ORELSE mp_tac i ORELSE DETERM (biresolve_tac safep_brls i) fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i) fun step_tac thms = safestep_tac thms ORELSE' biresolve_tac unsafe_brls (*Fails unless it solves the goal!*) fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms) end *} use "rew.ML" subsection {* The elimination rules for fst/snd *} lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A" apply (unfold basic_defs) apply (erule SumE) apply assumption done (*The first premise must be p:Sum(A,B) !!*) lemma SumE_snd: assumes major: "p: Sum(A,B)" and "A type" and "!!x. x:A ==> B(x) type" shows "snd(p) : B(fst(p))" apply (unfold basic_defs) apply (rule major [THEN SumE]) apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) apply (tactic {* typechk_tac (thms "prems") *}) done end
lemma form_rls:
N type
[| A type; !!x. x ∈ A ==> B(x) type |] ==> Prod(A, B) type
[| A type; !!x. x ∈ A ==> B(x) type |] ==> Sum(A, B) type
[| A type; B type |] ==> A + B type
[| A type; a ∈ A; b ∈ A |] ==> Eq(A, a, b) type
F type
T type
and formL_rls:
[| A = C; !!x. x ∈ A ==> B(x) = D(x) |] ==> Prod(A, B) = Prod(C, D)
[| A = C; !!x. x ∈ A ==> B(x) = D(x) |] ==> Sum(A, B) = Sum(C, D)
[| A = C; B = D |] ==> A + B = C + D
[| A = B; a = c ∈ A; b = d ∈ A |] ==> Eq(A, a, b) = Eq(B, c, d)
lemma intr_rls:
0 ∈ N
a ∈ N ==> succ(a) ∈ N
[| A type; !!x. x ∈ A ==> b(x) ∈ B(x) |] ==> λλx. b(x) ∈ Prod(A, B)
[| a ∈ A; b ∈ B(a) |] ==> <a,b> ∈ Sum(A, B)
[| a ∈ A; B type |] ==> inl(a) ∈ A + B
[| A type; b ∈ B |] ==> inr(b) ∈ A + B
tt ∈ T
and intrL_rls:
a = b ∈ N ==> succ(a) = succ(b) ∈ N
[| A type; !!x. x ∈ A ==> b(x) = c(x) ∈ B(x) |]
==> λλx. b(x) = λλx. c(x) ∈ Prod(A, B)
[| a = c ∈ A; b = d ∈ B(a) |] ==> <a,b> = <c,d> ∈ Sum(A, B)
[| a = c ∈ A; B type |] ==> inl(a) = inl(c) ∈ A + B
[| A type; b = d ∈ B |] ==> inr(b) = inr(d) ∈ A + B
lemma elim_rls:
[| p ∈ N; a ∈ C(0); !!u v. [| u ∈ N; v ∈ C(u) |] ==> b(u, v) ∈ C(succ(u)) |]
==> rec(p, a, b) ∈ C(p)
[| p ∈ Prod(A, B); a ∈ A |] ==> p ` a ∈ B(a)
[| p ∈ Sum(A, B); !!x y. [| x ∈ A; y ∈ B(x) |] ==> c(x, y) ∈ C(<x,y>) |]
==> split(p, c) ∈ C(p)
[| p ∈ A + B; !!x. x ∈ A ==> c(x) ∈ C(inl(x));
!!y. y ∈ B ==> d(y) ∈ C(inr(y)) |]
==> when(p, c, d) ∈ C(p)
[| p ∈ F; C type |] ==> contr(p) ∈ C
and elimL_rls:
[| p = q ∈ N; a = c ∈ C(0);
!!u v. [| u ∈ N; v ∈ C(u) |] ==> b(u, v) = d(u, v) ∈ C(succ(u)) |]
==> rec(p, a, b) = rec(q, c, d) ∈ C(p)
[| p = q ∈ Prod(A, B); a = b ∈ A |] ==> p ` a = q ` b ∈ B(a)
[| p = q ∈ Sum(A, B);
!!x y. [| x ∈ A; y ∈ B(x) |] ==> c(x, y) = d(x, y) ∈ C(<x,y>) |]
==> split(p, c) = split(q, d) ∈ C(p)
[| p = q ∈ A + B; !!x. x ∈ A ==> c(x) = e(x) ∈ C(inl(x));
!!y. y ∈ B ==> d(y) = f(y) ∈ C(inr(y)) |]
==> when(p, c, d) = when(q, e, f) ∈ C(p)
[| p = q ∈ F; C type |] ==> contr(p) = contr(q) ∈ C
lemma comp_rls:
[| a ∈ C(0); !!u v. [| u ∈ N; v ∈ C(u) |] ==> b(u, v) ∈ C(succ(u)) |]
==> rec(0, a, b) = a ∈ C(0)
[| p ∈ N; a ∈ C(0); !!u v. [| u ∈ N; v ∈ C(u) |] ==> b(u, v) ∈ C(succ(u)) |]
==> rec(succ(p), a, b) = b(p, rec(p, a, b)) ∈ C(succ(p))
[| a ∈ A; !!x. x ∈ A ==> b(x) ∈ B(x) |] ==> (λλx. b(x)) ` a = b(a) ∈ B(a)
[| a ∈ A; b ∈ B(a); !!x y. [| x ∈ A; y ∈ B(x) |] ==> c(x, y) ∈ C(<x,y>) |]
==> split(<a,b>, c) = c(a, b) ∈ C(<a,b>)
[| a ∈ A; !!x. x ∈ A ==> c(x) ∈ C(inl(x)); !!y. y ∈ B ==> d(y) ∈ C(inr(y)) |]
==> when(inl(a), c, d) = c(a) ∈ C(inl(a))
[| b ∈ B; !!x. x ∈ A ==> c(x) ∈ C(inl(x)); !!y. y ∈ B ==> d(y) ∈ C(inr(y)) |]
==> when(inr(b), c, d) = d(b) ∈ C(inr(b))
lemma element_rls:
0 ∈ N
a ∈ N ==> succ(a) ∈ N
[| A type; !!x. x ∈ A ==> b(x) ∈ B(x) |] ==> λλx. b(x) ∈ Prod(A, B)
[| a ∈ A; b ∈ B(a) |] ==> <a,b> ∈ Sum(A, B)
[| a ∈ A; B type |] ==> inl(a) ∈ A + B
[| A type; b ∈ B |] ==> inr(b) ∈ A + B
tt ∈ T
[| p ∈ N; a ∈ C(0); !!u v. [| u ∈ N; v ∈ C(u) |] ==> b(u, v) ∈ C(succ(u)) |]
==> rec(p, a, b) ∈ C(p)
[| p ∈ Prod(A, B); a ∈ A |] ==> p ` a ∈ B(a)
[| p ∈ Sum(A, B); !!x y. [| x ∈ A; y ∈ B(x) |] ==> c(x, y) ∈ C(<x,y>) |]
==> split(p, c) ∈ C(p)
[| p ∈ A + B; !!x. x ∈ A ==> c(x) ∈ C(inl(x));
!!y. y ∈ B ==> d(y) ∈ C(inr(y)) |]
==> when(p, c, d) ∈ C(p)
[| p ∈ F; C type |] ==> contr(p) ∈ C
lemma basic_defs:
fst(a) == split(a, λx y. x)
snd(a) == split(a, λx y. y)
lemma SumIL2:
[| c = a ∈ A; d = b ∈ B(a) |] ==> <c,d> = <a,b> ∈ Sum(A, B)
lemma intrL2_rls:
a = b ∈ N ==> succ(a) = succ(b) ∈ N
[| A type; !!x. x ∈ A ==> b(x) = c(x) ∈ B(x) |]
==> λλx. b(x) = λλx. c(x) ∈ Prod(A, B)
[| c = a ∈ A; d = b ∈ B(a) |] ==> <c,d> = <a,b> ∈ Sum(A, B)
[| a = c ∈ A; B type |] ==> inl(a) = inl(c) ∈ A + B
[| A type; b = d ∈ B |] ==> inr(b) = inr(d) ∈ A + B
lemma subst_prodE:
[| p ∈ Prod(A, B); a ∈ A; !!z. z ∈ B(a) ==> c(z) ∈ C(z) |]
==> c(p ` a) ∈ C(p ` a)
lemma routine_rls:
N type
[| A type; !!x. x ∈ A ==> B(x) type |] ==> Prod(A, B) type
[| A type; !!x. x ∈ A ==> B(x) type |] ==> Sum(A, B) type
[| A type; B type |] ==> A + B type
[| A type; a ∈ A; b ∈ A |] ==> Eq(A, a, b) type
F type
T type
[| A = C; !!x. x ∈ A ==> B(x) = D(x) |] ==> Prod(A, B) = Prod(C, D)
[| A = C; !!x. x ∈ A ==> B(x) = D(x) |] ==> Sum(A, B) = Sum(C, D)
[| A = C; B = D |] ==> A + B = C + D
[| A = B; a = c ∈ A; b = d ∈ A |] ==> Eq(A, a, b) = Eq(B, c, d)
A type ==> A = A
0 ∈ N
a ∈ N ==> succ(a) ∈ N
[| A type; !!x. x ∈ A ==> b(x) ∈ B(x) |] ==> λλx. b(x) ∈ Prod(A, B)
[| a ∈ A; b ∈ B(a) |] ==> <a,b> ∈ Sum(A, B)
[| a ∈ A; B type |] ==> inl(a) ∈ A + B
[| A type; b ∈ B |] ==> inr(b) ∈ A + B
tt ∈ T
[| p ∈ N; a ∈ C(0); !!u v. [| u ∈ N; v ∈ C(u) |] ==> b(u, v) ∈ C(succ(u)) |]
==> rec(p, a, b) ∈ C(p)
[| p ∈ Prod(A, B); a ∈ A |] ==> p ` a ∈ B(a)
[| p ∈ Sum(A, B); !!x y. [| x ∈ A; y ∈ B(x) |] ==> c(x, y) ∈ C(<x,y>) |]
==> split(p, c) ∈ C(p)
[| p ∈ A + B; !!x. x ∈ A ==> c(x) ∈ C(inl(x));
!!y. y ∈ B ==> d(y) ∈ C(inr(y)) |]
==> when(p, c, d) ∈ C(p)
[| p ∈ F; C type |] ==> contr(p) ∈ C
lemma replace_type:
[| B = A; a ∈ A |] ==> a ∈ B
lemma subst_eqtyparg:
[| a = c ∈ A; !!z. z ∈ A ==> B(z) type |] ==> B(a) = B(c)
lemma reduction_rls:
[| b ∈ C1(0); !!u v. [| u ∈ N; v ∈ C1(u) |] ==> b1(u, v) ∈ C1(succ(u));
b = c ∈ C1(0) |]
==> rec(0, b, b1) = c ∈ C1(0)
[| p1 ∈ N; a1 ∈ C1(0); !!u v. [| u ∈ N; v ∈ C1(u) |] ==> b1(u, v) ∈ C1(succ(u));
b1(p1, rec(p1, a1, b1)) = c ∈ C1(succ(p1)) |]
==> rec(succ(p1), a1, b1) = c ∈ C1(succ(p1))
[| a1 ∈ A1; !!x. x ∈ A1 ==> b1(x) ∈ B1(x); b1(a1) = c ∈ B1(a1) |]
==> (λλx. b1(x)) ` a1 = c ∈ B1(a1)
[| a1 ∈ A1; b1 ∈ B1(a1);
!!x y. [| x ∈ A1; y ∈ B1(x) |] ==> c1(x, y) ∈ C1(<x,y>);
c1(a1, b1) = c ∈ C1(<a1,b1>) |]
==> split(<a1,b1>, c1) = c ∈ C1(<a1,b1>)
[| a1 ∈ A1; !!x. x ∈ A1 ==> c1(x) ∈ C1(inl(x));
!!y. y ∈ B1 ==> d1(y) ∈ C1(inr(y)); c1(a1) = c ∈ C1(inl(a1)) |]
==> when(inl(a1), c1, d1) = c ∈ C1(inl(a1))
[| b1 ∈ B1; !!x. x ∈ A1 ==> c1(x) ∈ C1(inl(x));
!!y. y ∈ B1 ==> d1(y) ∈ C1(inr(y)); d1(b1) = c ∈ C1(inr(b1)) |]
==> when(inr(b1), c1, d1) = c ∈ C1(inr(b1))
lemma SumE_fst:
p ∈ Sum(A, B) ==> fst(p) ∈ A
lemma SumE_snd:
[| p ∈ Sum(A, B); A type; !!x. x ∈ A ==> B(x) type |] ==> snd(p) ∈ B(fst(p))