(* Title: CTT/CTT.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge*)section \<open>Constructive Type Theory\<close>theory CTTimports PurebeginML_file "~~/src/Provers/typedsimp.ML"setup Pure_Thy.old_appl_syntax_setuptypedecl itypedecl ttypedecl oconsts \<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>axiomatizationwhere 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 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 EqFLtext \<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_inrLtext \<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 FELtext \<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_inrtext \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close>lemmas element_rls = intr_rls elim_rlstext \<open>Definitions are (meta)equality axioms.\<close>lemmas basic_defs = fst_def snd_deftext \<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)" apply (rule sym_elem) apply (rule SumIL) apply (rule_tac [!] sym_elem) apply assumption+ donelemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrLtext \<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>localfun 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 _ = falsein(*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 iend\<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_rlsML \<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+ donetext \<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) donetext \<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} ifun 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} ifun 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_brlsfun 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 donetext \<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) doneend