2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
3 Copyright 1993 University of Cambridge
6 header {* Constructive Type Theory *}
10 uses "~~/src/Provers/typedsimp.ML" ("rew.ML")
13 setup Pure_Thy.old_appl_syntax_setup
22 T :: "t" (*F is empty, T contains one element*)
28 rec :: "[i, i, [i,i]=>i] => i"
32 when :: "[i, i=>i, i=>i]=>i"
33 (*General Sum and Binary Product*)
37 split :: "[i, [i,i]=>i] =>i"
38 (*General Product and Function Space*)
39 Prod :: "[t, i=>t]=>t"
41 Plus :: "[t,t]=>t" (infixr "+" 40)
46 Type :: "t => prop" ("(_ type)" [10] 5)
47 Eqtype :: "[t,t]=>prop" ("(_ =/ _)" [10,10] 5)
48 Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5)
49 Eqelem :: "[i,i,t]=>prop" ("(_ =/ _ :/ _)" [10,10,10] 5)
50 Reduce :: "[i,i]=>prop" ("Reduce[_,_]")
54 lambda :: "(i => i) => i" (binder "lam " 10)
55 app :: "[i,i]=>i" (infixl "`" 60)
59 pair :: "[i,i]=>i" ("(1<_,/_>)")
62 "_PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10)
63 "_SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10)
65 "PROD x:A. B" == "CONST Prod(A, %x. B)"
66 "SUM x:A. B" == "CONST Sum(A, %x. B)"
69 Arrow :: "[t,t]=>t" (infixr "-->" 30) where
70 "A --> B == PROD _:A. B"
72 Times :: "[t,t]=>t" (infixr "*" 50) where
76 lambda (binder "\<lambda>\<lambda>" 10) and
77 Elem ("(_ /\<in> _)" [10,10] 5) and
78 Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
79 Arrow (infixr "\<longrightarrow>" 30) and
80 Times (infixr "\<times>" 50)
82 notation (HTML output)
83 lambda (binder "\<lambda>\<lambda>" 10) and
84 Elem ("(_ /\<in> _)" [10,10] 5) and
85 Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
86 Times (infixr "\<times>" 50)
89 "_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
90 "_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
93 "_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
94 "_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
98 (*Reduction: a weaker notion than equality; a hack for simplification.
99 Reduce[a,b] means either that a=b:A for some A or else that "a" and "b"
100 are textually identical.*)
102 (*does not verify a:A! Sound because only trans_red uses a Reduce premise
103 No new theorems can be proved about the standard judgements.*)
104 refl_red: "Reduce[a,a]"
105 red_if_equal: "a = b : A ==> Reduce[a,b]"
106 trans_red: "[| a = b : A; Reduce[b,c] |] ==> a = c : A"
110 refl_type: "A type ==> A = A"
111 refl_elem: "a : A ==> a = a : A"
115 sym_type: "A = B ==> B = A"
116 sym_elem: "a = b : A ==> b = a : A"
120 trans_type: "[| A = B; B = C |] ==> A = C"
121 trans_elem: "[| a = b : A; b = c : A |] ==> a = c : A"
123 equal_types: "[| a : A; A = B |] ==> a : B"
124 equal_typesL: "[| a = b : A; A = B |] ==> a = b : B"
128 subst_type: "[| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type"
129 subst_typeL: "[| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
131 subst_elem: "[| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
133 "[| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
136 (*The type N -- natural numbers*)
140 NI_succ: "a : N ==> succ(a) : N"
141 NI_succL: "a = b : N ==> succ(a) = succ(b) : N"
144 "[| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
145 ==> rec(p, a, %u v. b(u,v)) : C(p)"
148 "[| p = q : N; a = c : C(0);
149 !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |]
150 ==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"
153 "[| a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
154 ==> rec(0, a, %u v. b(u,v)) = a : C(0)"
158 !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>
159 rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"
161 (*The fourth Peano axiom. See page 91 of Martin-Lof's book*)
163 "[| a: N; 0 = succ(a) : N |] ==> 0: F"
166 (*The Product of a family of types*)
168 ProdF: "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"
171 "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
172 PROD x:A. B(x) = PROD x:C. D(x)"
175 "[| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"
178 "[| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
179 lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"
181 ProdE: "[| p : PROD x:A. B(x); a : A |] ==> p`a : B(a)"
182 ProdEL: "[| p=q: PROD x:A. B(x); a=b : A |] ==> p`a = q`b : B(a)"
185 "[| a : A; !!x. x:A ==> b(x) : B(x)|] ==>
186 (lam x. b(x)) ` a = b(a) : B(a)"
189 "p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"
192 (*The Sum of a family of types*)
194 SumF: "[| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"
196 "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"
198 SumI: "[| a : A; b : B(a) |] ==> <a,b> : SUM x:A. B(x)"
199 SumIL: "[| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"
202 "[| p: SUM x:A. B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
203 ==> split(p, %x y. c(x,y)) : C(p)"
206 "[| p=q : SUM x:A. B(x);
207 !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|]
208 ==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"
211 "[| a: A; b: B(a); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
212 ==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"
214 fst_def: "fst(a) == split(a, %x y. x)"
215 snd_def: "snd(a) == split(a, %x y. y)"
218 (*The sum of two types*)
220 PlusF: "[| A type; B type |] ==> A+B type"
221 PlusFL: "[| A = C; B = D |] ==> A+B = C+D"
223 PlusI_inl: "[| a : A; B type |] ==> inl(a) : A+B"
224 PlusI_inlL: "[| a = c : A; B type |] ==> inl(a) = inl(c) : A+B"
226 PlusI_inr: "[| A type; b : B |] ==> inr(b) : A+B"
227 PlusI_inrL: "[| A type; b = d : B |] ==> inr(b) = inr(d) : A+B"
230 "[| p: A+B; !!x. x:A ==> c(x): C(inl(x));
231 !!y. y:B ==> d(y): C(inr(y)) |]
232 ==> when(p, %x. c(x), %y. d(y)) : C(p)"
235 "[| p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x));
236 !!y. y: B ==> d(y) = f(y) : C(inr(y)) |]
237 ==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"
240 "[| a: A; !!x. x:A ==> c(x): C(inl(x));
241 !!y. y:B ==> d(y): C(inr(y)) |]
242 ==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"
245 "[| b: B; !!x. x:A ==> c(x): C(inl(x));
246 !!y. y:B ==> d(y): C(inr(y)) |]
247 ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
252 EqF: "[| A type; a : A; b : A |] ==> Eq(A,a,b) type"
253 EqFL: "[| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
254 EqI: "a = b : A ==> eq : Eq(A,a,b)"
255 EqE: "p : Eq(A,a,b) ==> a = b : A"
257 (*By equality of types, can prove C(p) from C(eq), an elimination rule*)
258 EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
263 FE: "[| p: F; C type |] ==> contr(p) : C"
264 FEL: "[| p = q : F; C type |] ==> contr(p) = contr(q) : C"
267 Martin-Lof's book (page 68) discusses elimination and computation.
268 Elimination can be derived by computation and equality of types,
269 but with an extra premise C(x) type x:T.
270 Also computation can be derived from elimination. *)
274 TE: "[| p : T; c : C(tt) |] ==> c : C(p)"
275 TEL: "[| p = q : T; c = d : C(tt) |] ==> c = d : C(p)"
276 TC: "p : T ==> p = tt : T"
279 subsection "Tactics and derived rules for Constructive Type Theory"
282 lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
283 and formL_rls = ProdFL SumFL PlusFL EqFL
286 OMITTED: EqI, because its premise is an eqelem, not an elem*)
287 lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI
288 and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL
291 OMITTED: EqE, because its conclusion is an eqelem, not an elem
292 TE, because it does not involve a constructor *)
293 lemmas elim_rls = NE ProdE SumE PlusE FE
294 and elimL_rls = NEL ProdEL SumEL PlusEL FEL
296 (*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *)
297 lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr
299 (*rules with conclusion a:A, an elem judgement*)
300 lemmas element_rls = intr_rls elim_rls
302 (*Definitions are (meta)equality axioms*)
303 lemmas basic_defs = fst_def snd_def
305 (*Compare with standard version: B is applied to UNSIMPLIFIED expression! *)
306 lemma SumIL2: "[| c=a : A; d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)"
307 apply (rule sym_elem)
309 apply (rule_tac [!] sym_elem)
313 lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL
315 (*Exploit p:Prod(A,B) to create the assumption z:B(a).
316 A more natural form of product elimination. *)
318 assumes "p: Prod(A,B)"
320 and "!!z. z: B(a) ==> c(z): C(z)"
321 shows "c(p`a): C(p`a)"
322 apply (rule assms ProdE)+
326 subsection {* Tactics for type checking *}
332 fun is_rigid_elem (Const("CTT.Elem",_) $ a $ _) = not(is_Var (head_of a))
333 | is_rigid_elem (Const("CTT.Eqelem",_) $ a $ _ $ _) = not(is_Var (head_of a))
334 | is_rigid_elem (Const("CTT.Type",_) $ a) = not(is_Var (head_of a))
335 | is_rigid_elem _ = false
339 (*Try solving a:A or a=b:A by assumption provided a is rigid!*)
340 val test_assume_tac = SUBGOAL(fn (prem,i) =>
341 if is_rigid_elem (Logic.strip_assums_concl prem)
342 then assume_tac i else no_tac)
344 fun ASSUME tf i = test_assume_tac i ORELSE tf i
350 (*For simplification: type formation and checking,
351 but no equalities between terms*)
352 lemmas routine_rls = form_rls formL_rls refl_type element_rls
356 val equal_rls = @{thms form_rls} @ @{thms element_rls} @ @{thms intrL_rls} @
357 @{thms elimL_rls} @ @{thms refl_elem}
360 fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4);
362 (*Solve all subgoals "A type" using formation rules. *)
363 val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac @{thms form_rls} 1));
365 (*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)
366 fun typechk_tac thms =
367 let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3
368 in REPEAT_FIRST (ASSUME tac) end
370 (*Solve a:A (a flexible, A rigid) by introduction rules.
371 Cannot use stringtrees (filt_resolve_tac) since
372 goals like ?a:SUM(A,B) have a trivial head-string *)
374 let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1
375 in REPEAT_FIRST (ASSUME tac) end
377 (*Equality proving: solve a=b:A (where a is rigid) by long rules. *)
379 REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3))
386 subsection {* Simplification *}
388 (*To simplify the type in a goal*)
389 lemma replace_type: "[| B = A; a : A |] ==> a : B"
390 apply (rule equal_types)
391 apply (rule_tac [2] sym_type)
395 (*Simplify the parameter of a unary type operator.*)
396 lemma subst_eqtyparg:
398 and 2: "!!z. z:A ==> B(z) type"
400 apply (rule subst_typeL)
401 apply (rule_tac [2] refl_type)
406 (*Simplification rules for Constructive Type Theory*)
407 lemmas reduction_rls = comp_rls [THEN trans_elem]
410 (*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification.
411 Uses other intro rules to avoid changing flexible goals.*)
412 val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1))
414 (** Tactics that instantiate CTT-rules.
415 Vars in the given terms will be incremented!
416 The (rtac EqE i) lets them apply to equality judgements. **)
418 fun NE_tac ctxt sp i =
419 TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i
421 fun SumE_tac ctxt sp i =
422 TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i
424 fun PlusE_tac ctxt sp i =
425 TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm PlusE} i
427 (** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **)
429 (*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *)
431 rtac @{thm subst_prodE} i THEN assume_tac i THEN assume_tac i
433 (*Finds P-->Q and P in the assumptions, replaces implication by Q *)
434 fun mp_tac i = etac @{thm subst_prodE} i THEN assume_tac i
436 (*"safe" when regarded as predicate calculus rules*)
437 val safe_brls = sort (make_ord lessb)
438 [ (true, @{thm FE}), (true,asm_rl),
439 (false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
442 [ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}),
443 (true, @{thm subst_prodE}) ]
445 (*0 subgoals vs 1 or more*)
446 val (safe0_brls, safep_brls) =
447 List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls
449 fun safestep_tac thms i =
451 resolve_tac thms i ORELSE
452 biresolve_tac safe0_brls i ORELSE mp_tac i ORELSE
453 DETERM (biresolve_tac safep_brls i)
455 fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i)
457 fun step_tac thms = safestep_tac thms ORELSE' biresolve_tac unsafe_brls
459 (*Fails unless it solves the goal!*)
460 fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms)
466 subsection {* The elimination rules for fst/snd *}
468 lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A"
469 apply (unfold basic_defs)
474 (*The first premise must be p:Sum(A,B) !!*)
476 assumes major: "p: Sum(A,B)"
478 and "!!x. x:A ==> B(x) type"
479 shows "snd(p) : B(fst(p))"
480 apply (unfold basic_defs)
481 apply (rule major [THEN SumE])
482 apply (rule SumC [THEN subst_eqtyparg, THEN replace_type])
483 apply (tactic {* typechk_tac @{thms assms} *})