(* Title: CTT/CTT.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
header {* Constructive Type Theory *}
theory CTT
imports Pure
begin
ML_file "~~/src/Provers/typedsimp.ML"
setup Pure_Thy.old_appl_syntax_setup
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*)
Zero :: "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" == "CONST Prod(A, %x. B)"
"SUM x:A. B" == "CONST 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 "\<lambda>\<lambda>" 10) and
Elem ("(_ /\<in> _)" [10,10] 5) and
Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
Arrow (infixr "\<longrightarrow>" 30) and
Times (infixr "\<times>" 50)
notation (HTML output)
lambda (binder "\<lambda>\<lambda>" 10) and
Elem ("(_ /\<in> _)" [10,10] 5) and
Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and
Times (infixr "\<times>" 50)
syntax (xsymbols)
"_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
"_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
syntax (HTML output)
"_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
"_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
(*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.*)
axiomatization where
refl_red: "\<And>a. Reduce[a,a]" and
red_if_equal: "\<And>a b A. a = b : A ==> Reduce[a,b]" and
trans_red: "\<And>a b c A. [| a = b : A; Reduce[b,c] |] ==> a = c : A" and
(*Reflexivity*)
refl_type: "\<And>A. A type ==> A = A" and
refl_elem: "\<And>a A. a : A ==> a = a : A" and
(*Symmetry*)
sym_type: "\<And>A B. A = B ==> B = A" and
sym_elem: "\<And>a b A. a = b : A ==> b = a : A" and
(*Transitivity*)
trans_type: "\<And>A B C. [| A = B; B = C |] ==> A = C" and
trans_elem: "\<And>a b c A. [| a = b : A; b = c : A |] ==> a = c : A" and
equal_types: "\<And>a A B. [| a : A; A = B |] ==> a : B" and
equal_typesL: "\<And>a b A B. [| a = b : A; A = B |] ==> a = b : B" and
(*Substitution*)
subst_type: "\<And>a A B. [| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type" and
subst_typeL: "\<And>a c A B D. [| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)" and
subst_elem: "\<And>a b A B. [| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)" and
subst_elemL:
"\<And>a b c d A B. [| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)" and
(*The type N -- natural numbers*)
NF: "N type" and
NI0: "0 : N" and
NI_succ: "\<And>a. a : N ==> succ(a) : N" and
NI_succL: "\<And>a b. a = b : N ==> succ(a) = succ(b) : N" and
NE:
"\<And>p a b C. [| 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)" and
NEL:
"\<And>p q a b c d C. [| 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)" and
NC0:
"\<And>a b C. [| 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)" and
NC_succ:
"\<And>p a b C. [| 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))" and
(*The fourth Peano axiom. See page 91 of Martin-Lof's book*)
zero_ne_succ:
"\<And>a. [| a: N; 0 = succ(a) : N |] ==> 0: F" and
(*The Product of a family of types*)
ProdF: "\<And>A B. [| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type" and
ProdFL:
"\<And>A B C D. [| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
PROD x:A. B(x) = PROD x:C. D(x)" and
ProdI:
"\<And>b A B. [| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)" and
ProdIL:
"\<And>b c A B. [| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
lam x. b(x) = lam x. c(x) : PROD x:A. B(x)" and
ProdE: "\<And>p a A B. [| p : PROD x:A. B(x); a : A |] ==> p`a : B(a)" and
ProdEL: "\<And>p q a b A B. [| p=q: PROD x:A. B(x); a=b : A |] ==> p`a = q`b : B(a)" and
ProdC:
"\<And>a b A B. [| a : A; !!x. x:A ==> b(x) : B(x)|] ==>
(lam x. b(x)) ` a = b(a) : B(a)" and
ProdC2:
"\<And>p A B. p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)" and
(*The Sum of a family of types*)
SumF: "\<And>A B. [| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type" and
SumFL:
"\<And>A B C D. [| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)" and
SumI: "\<And>a b A B. [| a : A; b : B(a) |] ==> <a,b> : SUM x:A. B(x)" and
SumIL: "\<And>a b c d A B. [| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)" and
SumE:
"\<And>p c A B C. [| 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)" and
SumEL:
"\<And>p q c d A B C. [| 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)" and
SumC:
"\<And>a b c A B C. [| 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>)" and
fst_def: "\<And>a. fst(a) == split(a, %x y. x)" and
snd_def: "\<And>a. snd(a) == split(a, %x y. y)" and
(*The sum of two types*)
PlusF: "\<And>A B. [| A type; B type |] ==> A+B type" and
PlusFL: "\<And>A B C D. [| A = C; B = D |] ==> A+B = C+D" and
PlusI_inl: "\<And>a A B. [| a : A; B type |] ==> inl(a) : A+B" and
PlusI_inlL: "\<And>a c A B. [| a = c : A; B type |] ==> inl(a) = inl(c) : A+B" and
PlusI_inr: "\<And>b A B. [| A type; b : B |] ==> inr(b) : A+B" and
PlusI_inrL: "\<And>b d A B. [| A type; b = d : B |] ==> inr(b) = inr(d) : A+B" and
PlusE:
"\<And>p c d A B C. [| 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)" and
PlusEL:
"\<And>p q c d e f A B C. [| 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)" and
PlusC_inl:
"\<And>a c d A C. [| 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))" and
PlusC_inr:
"\<And>b c d A B C. [| 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))" and
(*The type Eq*)
EqF: "\<And>a b A. [| A type; a : A; b : A |] ==> Eq(A,a,b) type" and
EqFL: "\<And>a b c d A B. [| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)" and
EqI: "\<And>a b A. a = b : A ==> eq : Eq(A,a,b)" and
EqE: "\<And>p a b A. p : Eq(A,a,b) ==> a = b : A" and
(*By equality of types, can prove C(p) from C(eq), an elimination rule*)
EqC: "\<And>p a b A. p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)" and
(*The type F*)
FF: "F type" and
FE: "\<And>p C. [| p: F; C type |] ==> contr(p) : C" and
FEL: "\<And>p q C. [| p = q : F; C type |] ==> contr(p) = contr(q) : C" and
(*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" and
TI: "tt : T" and
TE: "\<And>p c C. [| p : T; c : C(tt) |] ==> c : C(p)" and
TEL: "\<And>p q c d C. [| p = q : T; c = d : C(tt) |] ==> c = d : C(p)" and
TC: "\<And>p. 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 assms 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 equal_rls = @{thms form_rls} @ @{thms 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 @{thms 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 @ @{thms form_rls} @ @{thms 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 @ @{thms form_rls} @ @{thms 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 {*
(*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 (@{thm EqI} :: @{thms 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 ctxt sp i =
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i
fun SumE_tac ctxt sp i =
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i
fun PlusE_tac ctxt sp i =
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm 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 @{thm 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 @{thm subst_prodE} i THEN assume_tac i
(*"safe" when regarded as predicate calculus rules*)
val safe_brls = sort (make_ord lessb)
[ (true, @{thm FE}), (true,asm_rl),
(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ]
val unsafe_brls =
[ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}),
(true, @{thm 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)
*}
ML_file "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 assms} *})
done
end