(* Title: CTT/CTT.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section \<open>Constructive Type Theory\<close>
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
\<comment> \<open>Types\<close>
F :: "t"
T :: "t" \<comment> \<open>\<open>F\<close> is empty, \<open>T\<close> contains one element\<close>
contr :: "i\<Rightarrow>i"
tt :: "i"
\<comment> \<open>Natural numbers\<close>
N :: "t"
succ :: "i\<Rightarrow>i"
rec :: "[i, i, [i,i]\<Rightarrow>i] \<Rightarrow> i"
\<comment> \<open>Unions\<close>
inl :: "i\<Rightarrow>i"
inr :: "i\<Rightarrow>i"
"when" :: "[i, i\<Rightarrow>i, i\<Rightarrow>i]\<Rightarrow>i"
\<comment> \<open>General Sum and Binary Product\<close>
Sum :: "[t, i\<Rightarrow>t]\<Rightarrow>t"
fst :: "i\<Rightarrow>i"
snd :: "i\<Rightarrow>i"
split :: "[i, [i,i]\<Rightarrow>i] \<Rightarrow>i"
\<comment> \<open>General Product and Function Space\<close>
Prod :: "[t, i\<Rightarrow>t]\<Rightarrow>t"
\<comment> \<open>Types\<close>
Plus :: "[t,t]\<Rightarrow>t" (infixr "+" 40)
\<comment> \<open>Equality type\<close>
Eq :: "[t,i,i]\<Rightarrow>t"
eq :: "i"
\<comment> \<open>Judgements\<close>
Type :: "t \<Rightarrow> prop" ("(_ type)" [10] 5)
Eqtype :: "[t,t]\<Rightarrow>prop" ("(_ =/ _)" [10,10] 5)
Elem :: "[i, t]\<Rightarrow>prop" ("(_ /: _)" [10,10] 5)
Eqelem :: "[i,i,t]\<Rightarrow>prop" ("(_ =/ _ :/ _)" [10,10,10] 5)
Reduce :: "[i,i]\<Rightarrow>prop" ("Reduce[_,_]")
\<comment> \<open>Types\<close>
\<comment> \<open>Functions\<close>
lambda :: "(i \<Rightarrow> i) \<Rightarrow> i" (binder "\<^bold>\<lambda>" 10)
app :: "[i,i]\<Rightarrow>i" (infixl "`" 60)
\<comment> \<open>Natural numbers\<close>
Zero :: "i" ("0")
\<comment> \<open>Pairing\<close>
pair :: "[i,i]\<Rightarrow>i" ("(1<_,/_>)")
syntax
"_PROD" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Prod>_:_./ _)" 10)
"_SUM" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Sum>_:_./ _)" 10)
translations
"\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod(A, \<lambda>x. B)"
"\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum(A, \<lambda>x. B)"
abbreviation Arrow :: "[t,t]\<Rightarrow>t" (infixr "\<longrightarrow>" 30)
where "A \<longrightarrow> B \<equiv> \<Prod>_:A. B"
abbreviation Times :: "[t,t]\<Rightarrow>t" (infixr "\<times>" 50)
where "A \<times> B \<equiv> \<Sum>_:A. B"
text \<open>
Reduction: a weaker notion than equality; a hack for simplification.
\<open>Reduce[a,b]\<close> means either that \<open>a = b : A\<close> for some \<open>A\<close> or else
that \<open>a\<close> and \<open>b\<close> are textually identical.
Does not verify \<open>a:A\<close>! Sound because only \<open>trans_red\<close> uses a \<open>Reduce\<close>
premise. No new theorems can be proved about the standard judgements.
\<close>
axiomatization
where
refl_red: "\<And>a. Reduce[a,a]" and
red_if_equal: "\<And>a b A. a = b : A \<Longrightarrow> Reduce[a,b]" and
trans_red: "\<And>a b c A. \<lbrakk>a = b : A; Reduce[b,c]\<rbrakk> \<Longrightarrow> a = c : A" and
\<comment> \<open>Reflexivity\<close>
refl_type: "\<And>A. A type \<Longrightarrow> A = A" and
refl_elem: "\<And>a A. a : A \<Longrightarrow> a = a : A" and
\<comment> \<open>Symmetry\<close>
sym_type: "\<And>A B. A = B \<Longrightarrow> B = A" and
sym_elem: "\<And>a b A. a = b : A \<Longrightarrow> b = a : A" and
\<comment> \<open>Transitivity\<close>
trans_type: "\<And>A B C. \<lbrakk>A = B; B = C\<rbrakk> \<Longrightarrow> A = C" and
trans_elem: "\<And>a b c A. \<lbrakk>a = b : A; b = c : A\<rbrakk> \<Longrightarrow> a = c : A" and
equal_types: "\<And>a A B. \<lbrakk>a : A; A = B\<rbrakk> \<Longrightarrow> a : B" and
equal_typesL: "\<And>a b A B. \<lbrakk>a = b : A; A = B\<rbrakk> \<Longrightarrow> a = b : B" and
\<comment> \<open>Substitution\<close>
subst_type: "\<And>a A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> B(z) type\<rbrakk> \<Longrightarrow> B(a) type" and
subst_typeL: "\<And>a c A B D. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> B(z) = D(z)\<rbrakk> \<Longrightarrow> B(a) = D(c)" and
subst_elem: "\<And>a b A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> b(z):B(z)\<rbrakk> \<Longrightarrow> b(a):B(a)" and
subst_elemL:
"\<And>a b c d A B. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> b(z)=d(z) : B(z)\<rbrakk> \<Longrightarrow> b(a)=d(c) : B(a)" and
\<comment> \<open>The type \<open>N\<close> -- natural numbers\<close>
NF: "N type" and
NI0: "0 : N" and
NI_succ: "\<And>a. a : N \<Longrightarrow> succ(a) : N" and
NI_succL: "\<And>a b. a = b : N \<Longrightarrow> succ(a) = succ(b) : N" and
NE:
"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk>
\<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) : C(p)" and
NEL:
"\<And>p q a b c d C. \<lbrakk>p = q : N; a = c : C(0);
\<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v) = d(u,v): C(succ(u))\<rbrakk>
\<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) = rec(q,c,d) : C(p)" and
NC0:
"\<And>a b C. \<lbrakk>a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk>
\<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and
NC_succ:
"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> \<Longrightarrow>
rec(succ(p), a, \<lambda>u v. b(u,v)) = b(p, rec(p, a, \<lambda>u v. b(u,v))) : C(succ(p))" and
\<comment> \<open>The fourth Peano axiom. See page 91 of Martin-Löf's book.\<close>
zero_ne_succ: "\<And>a. \<lbrakk>a: N; 0 = succ(a) : N\<rbrakk> \<Longrightarrow> 0: F" and
\<comment> \<open>The Product of a family of types\<close>
ProdF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) type" and
ProdFL:
"\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) = \<Prod>x:C. D(x)" and
ProdI:
"\<And>b A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x):B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x. b(x) : \<Prod>x:A. B(x)" and
ProdIL: "\<And>b c A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x) = c(x) : B(x)\<rbrakk> \<Longrightarrow>
\<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>x. c(x) : \<Prod>x:A. B(x)" and
ProdE: "\<And>p a A B. \<lbrakk>p : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> p`a : B(a)" and
ProdEL: "\<And>p q a b A B. \<lbrakk>p = q: \<Prod>x:A. B(x); a = b : A\<rbrakk> \<Longrightarrow> p`a = q`b : B(a)" and
ProdC: "\<And>a b A B. \<lbrakk>a : A; \<And>x. x:A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x. b(x)) ` a = b(a) : B(a)" and
ProdC2: "\<And>p A B. p : \<Prod>x:A. B(x) \<Longrightarrow> (\<^bold>\<lambda>x. p`x) = p : \<Prod>x:A. B(x)" and
\<comment> \<open>The Sum of a family of types\<close>
SumF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) type" and
SumFL: "\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) = \<Sum>x:C. D(x)" and
SumI: "\<And>a b A B. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> <a,b> : \<Sum>x:A. B(x)" and
SumIL: "\<And>a b c d A B. \<lbrakk> a = c : A; b = d : B(a)\<rbrakk> \<Longrightarrow> <a,b> = <c,d> : \<Sum>x:A. B(x)" and
SumE: "\<And>p c A B C. \<lbrakk>p: \<Sum>x:A. B(x); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk>
\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and
SumEL: "\<And>p q c d A B C. \<lbrakk>p = q : \<Sum>x:A. B(x);
\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y)=d(x,y): C(<x,y>)\<rbrakk>
\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) = split(q, \<lambda>x y. d(x,y)) : C(p)" and
SumC: "\<And>a b c A B C. \<lbrakk>a: A; b: B(a); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk>
\<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and
fst_def: "\<And>a. fst(a) \<equiv> split(a, \<lambda>x y. x)" and
snd_def: "\<And>a. snd(a) \<equiv> split(a, \<lambda>x y. y)" and
\<comment> \<open>The sum of two types\<close>
PlusF: "\<And>A B. \<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> A+B type" and
PlusFL: "\<And>A B C D. \<lbrakk>A = C; B = D\<rbrakk> \<Longrightarrow> A+B = C+D" and
PlusI_inl: "\<And>a A B. \<lbrakk>a : A; B type\<rbrakk> \<Longrightarrow> inl(a) : A+B" and
PlusI_inlL: "\<And>a c A B. \<lbrakk>a = c : A; B type\<rbrakk> \<Longrightarrow> inl(a) = inl(c) : A+B" and
PlusI_inr: "\<And>b A B. \<lbrakk>A type; b : B\<rbrakk> \<Longrightarrow> inr(b) : A+B" and
PlusI_inrL: "\<And>b d A B. \<lbrakk>A type; b = d : B\<rbrakk> \<Longrightarrow> inr(b) = inr(d) : A+B" and
PlusE:
"\<And>p c d A B C. \<lbrakk>p: A+B;
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) : C(p)" and
PlusEL:
"\<And>p q c d e f A B C. \<lbrakk>p = q : A+B;
\<And>x. x: A \<Longrightarrow> c(x) = e(x) : C(inl(x));
\<And>y. y: B \<Longrightarrow> d(y) = f(y) : C(inr(y))\<rbrakk>
\<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) = when(q, \<lambda>x. e(x), \<lambda>y. f(y)) : C(p)" and
PlusC_inl:
"\<And>a c d A B C. \<lbrakk>a: A;
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk>
\<Longrightarrow> when(inl(a), \<lambda>x. c(x), \<lambda>y. d(y)) = c(a) : C(inl(a))" and
PlusC_inr:
"\<And>b c d A B C. \<lbrakk>b: B;
\<And>x. x:A \<Longrightarrow> c(x): C(inl(x));
\<And>y. y:B \<Longrightarrow> d(y): C(inr(y))\<rbrakk>
\<Longrightarrow> when(inr(b), \<lambda>x. c(x), \<lambda>y. d(y)) = d(b) : C(inr(b))" and
\<comment> \<open>The type \<open>Eq\<close>\<close>
EqF: "\<And>a b A. \<lbrakk>A type; a : A; b : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) type" and
EqFL: "\<And>a b c d A B. \<lbrakk>A = B; a = c : A; b = d : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) = Eq(B,c,d)" and
EqI: "\<And>a b A. a = b : A \<Longrightarrow> eq : Eq(A,a,b)" and
EqE: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> a = b : A" and
\<comment> \<open>By equality of types, can prove \<open>C(p)\<close> from \<open>C(eq)\<close>, an elimination rule\<close>
EqC: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> p = eq : Eq(A,a,b)" and
\<comment> \<open>The type \<open>F\<close>\<close>
FF: "F type" and
FE: "\<And>p C. \<lbrakk>p: F; C type\<rbrakk> \<Longrightarrow> contr(p) : C" and
FEL: "\<And>p q C. \<lbrakk>p = q : F; C type\<rbrakk> \<Longrightarrow> contr(p) = contr(q) : C" and
\<comment> \<open>The type T\<close>
\<comment> \<open>
Martin-Löf's book (page 68) discusses elimination and computation.
Elimination can be derived by computation and equality of types,
but with an extra premise \<open>C(x)\<close> type \<open>x:T\<close>.
Also computation can be derived from elimination.
\<close>
TF: "T type" and
TI: "tt : T" and
TE: "\<And>p c C. \<lbrakk>p : T; c : C(tt)\<rbrakk> \<Longrightarrow> c : C(p)" and
TEL: "\<And>p q c d C. \<lbrakk>p = q : T; c = d : C(tt)\<rbrakk> \<Longrightarrow> c = d : C(p)" and
TC: "\<And>p. p : T \<Longrightarrow> p = tt : T"
subsection "Tactics and derived rules for Constructive Type Theory"
text \<open>Formation rules.\<close>
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF
and formL_rls = ProdFL SumFL PlusFL EqFL
text \<open>
Introduction rules. OMITTED:
\<^item> \<open>EqI\<close>, because its premise is an \<open>eqelem\<close>, not an \<open>elem\<close>.
\<close>
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI
and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL
text \<open>
Elimination rules. OMITTED:
\<^item> \<open>EqE\<close>, because its conclusion is an \<open>eqelem\<close>, not an \<open>elem\<close>
\<^item> \<open>TE\<close>, because it does not involve a constructor.
\<close>
lemmas elim_rls = NE ProdE SumE PlusE FE
and elimL_rls = NEL ProdEL SumEL PlusEL FEL
text \<open>OMITTED: \<open>eqC\<close> are \<open>TC\<close> because they make rewriting loop: \<open>p = un = un = \<dots>\<close>\<close>
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr
text \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close>
lemmas element_rls = intr_rls elim_rls
text \<open>Definitions are (meta)equality axioms.\<close>
lemmas basic_defs = fst_def snd_def
text \<open>Compare with standard version: \<open>B\<close> is applied to UNSIMPLIFIED expression!\<close>
lemma SumIL2: "\<lbrakk>c = a : A; d = b : B(a)\<rbrakk> \<Longrightarrow> <c,d> = <a,b> : Sum(A,B)"
by (rule sym_elem) (rule SumIL; rule sym_elem)
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL
text \<open>
Exploit \<open>p:Prod(A,B)\<close> to create the assumption \<open>z:B(a)\<close>.
A more natural form of product elimination.
\<close>
lemma subst_prodE:
assumes "p: Prod(A,B)"
and "a: A"
and "\<And>z. z: B(a) \<Longrightarrow> c(z): C(z)"
shows "c(p`a): C(p`a)"
by (rule assms ProdE)+
subsection \<open>Tactics for type checking\<close>
ML \<open>
local
fun is_rigid_elem (Const(@{const_name Elem},_) $ a $ _) = not(is_Var (head_of a))
| is_rigid_elem (Const(@{const_name Eqelem},_) $ a $ _ $ _) = not(is_Var (head_of a))
| is_rigid_elem (Const(@{const_name 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!*)
fun test_assume_tac ctxt = SUBGOAL (fn (prem, i) =>
if is_rigid_elem (Logic.strip_assums_concl prem)
then assume_tac ctxt i else no_tac)
fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i
end
\<close>
text \<open>
For simplification: type formation and checking,
but no equalities between terms.
\<close>
lemmas routine_rls = form_rls formL_rls refl_type element_rls
ML \<open>
fun routine_tac rls ctxt prems =
ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls)));
(*Solve all subgoals "A type" using formation rules. *)
val form_net = Tactic.build_net @{thms form_rls};
fun form_tac ctxt =
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net));
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *)
fun typechk_tac ctxt thms =
let val tac =
filt_resolve_from_net_tac ctxt 3
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls}))
in REPEAT_FIRST (ASSUME ctxt 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 ctxt thms =
let val tac =
filt_resolve_from_net_tac ctxt 1
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls}))
in REPEAT_FIRST (ASSUME ctxt tac) end
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *)
fun equal_tac ctxt thms =
REPEAT_FIRST
(ASSUME ctxt
(filt_resolve_from_net_tac ctxt 3
(Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem}))))
\<close>
method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close>
method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close>
method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close>
method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close>
subsection \<open>Simplification\<close>
text \<open>To simplify the type in a goal.\<close>
lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B"
apply (rule equal_types)
apply (rule_tac [2] sym_type)
apply assumption+
done
text \<open>Simplify the parameter of a unary type operator.\<close>
lemma subst_eqtyparg:
assumes 1: "a=c : A"
and 2: "\<And>z. z:A \<Longrightarrow> 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
text \<open>Simplification rules for Constructive Type Theory.\<close>
lemmas reduction_rls = comp_rls [THEN trans_elem]
ML \<open>
(*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_net = Tactic.build_net @{thms EqI intr_rls}
fun eqintr_tac ctxt =
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net))
(** 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 (resolve_tac ctxt @{thms EqE} i) THEN
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i
fun SumE_tac ctxt sp i =
TRY (resolve_tac ctxt @{thms EqE} i) THEN
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i
fun PlusE_tac ctxt sp i =
TRY (resolve_tac ctxt @{thms EqE} i) THEN
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), 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 ctxt i =
resolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i
(*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *)
fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt 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 ctxt thms i =
form_tac ctxt ORELSE
resolve_tac ctxt thms i ORELSE
biresolve_tac ctxt safe0_brls i ORELSE mp_tac ctxt i ORELSE
DETERM (biresolve_tac ctxt safep_brls i)
fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i)
fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac ctxt unsafe_brls
(*Fails unless it solves the goal!*)
fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms)
\<close>
method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close>
method_setup NE = \<open>
Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s))
\<close>
method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close>
method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close>
ML_file "rew.ML"
method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close>
method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close>
subsection \<open>The elimination rules for fst/snd\<close>
lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A"
apply (unfold basic_defs)
apply (erule SumE)
apply assumption
done
text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close>
lemma SumE_snd:
assumes major: "p: Sum(A,B)"
and "A type"
and "\<And>x. x:A \<Longrightarrow> 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 (typechk assms)
done
end