1(* Title: CTT/CTT.thy 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3 Copyright 1993 University of Cambridge 4*) 5 6theory CTT 7imports Pure 8begin 9 10section \<open>Constructive Type Theory: axiomatic basis\<close> 11 12ML_file \<open>~~/src/Provers/typedsimp.ML\<close> 13setup Pure_Thy.old_appl_syntax_setup 14 15typedecl i 16typedecl t 17typedecl o 18 19consts 20 \<comment> \<open>Types\<close> 21 F :: "t" 22 T :: "t" \<comment> \<open>\<open>F\<close> is empty, \<open>T\<close> contains one element\<close> 23 contr :: "i\<Rightarrow>i" 24 tt :: "i" 25 \<comment> \<open>Natural numbers\<close> 26 N :: "t" 27 succ :: "i\<Rightarrow>i" 28 rec :: "[i, i, [i,i]\<Rightarrow>i] \<Rightarrow> i" 29 \<comment> \<open>Unions\<close> 30 inl :: "i\<Rightarrow>i" 31 inr :: "i\<Rightarrow>i" 32 "when" :: "[i, i\<Rightarrow>i, i\<Rightarrow>i]\<Rightarrow>i" 33 \<comment> \<open>General Sum and Binary Product\<close> 34 Sum :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 35 fst :: "i\<Rightarrow>i" 36 snd :: "i\<Rightarrow>i" 37 split :: "[i, [i,i]\<Rightarrow>i] \<Rightarrow>i" 38 \<comment> \<open>General Product and Function Space\<close> 39 Prod :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 40 \<comment> \<open>Types\<close> 41 Plus :: "[t,t]\<Rightarrow>t" (infixr "+" 40) 42 \<comment> \<open>Equality type\<close> 43 Eq :: "[t,i,i]\<Rightarrow>t" 44 eq :: "i" 45 \<comment> \<open>Judgements\<close> 46 Type :: "t \<Rightarrow> prop" ("(_ type)" [10] 5) 47 Eqtype :: "[t,t]\<Rightarrow>prop" ("(_ =/ _)" [10,10] 5) 48 Elem :: "[i, t]\<Rightarrow>prop" ("(_ /: _)" [10,10] 5) 49 Eqelem :: "[i,i,t]\<Rightarrow>prop" ("(_ =/ _ :/ _)" [10,10,10] 5) 50 Reduce :: "[i,i]\<Rightarrow>prop" ("Reduce[_,_]") 51 52 \<comment> \<open>Types\<close> 53 54 \<comment> \<open>Functions\<close> 55 lambda :: "(i \<Rightarrow> i) \<Rightarrow> i" (binder "\<^bold>\<lambda>" 10) 56 app :: "[i,i]\<Rightarrow>i" (infixl "`" 60) 57 \<comment> \<open>Natural numbers\<close> 58 Zero :: "i" ("0") 59 \<comment> \<open>Pairing\<close> 60 pair :: "[i,i]\<Rightarrow>i" ("(1<_,/_>)") 61 62syntax 63 "_PROD" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Prod>_:_./ _)" 10) 64 "_SUM" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Sum>_:_./ _)" 10) 65translations 66 "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod(A, \<lambda>x. B)" 67 "\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum(A, \<lambda>x. B)" 68 69abbreviation Arrow :: "[t,t]\<Rightarrow>t" (infixr "\<longrightarrow>" 30) 70 where "A \<longrightarrow> B \<equiv> \<Prod>_:A. B" 71 72abbreviation Times :: "[t,t]\<Rightarrow>t" (infixr "\<times>" 50) 73 where "A \<times> B \<equiv> \<Sum>_:A. B" 74 75text \<open> 76 Reduction: a weaker notion than equality; a hack for simplification. 77 \<open>Reduce[a,b]\<close> means either that \<open>a = b : A\<close> for some \<open>A\<close> or else 78 that \<open>a\<close> and \<open>b\<close> are textually identical. 79 80 Does not verify \<open>a:A\<close>! Sound because only \<open>trans_red\<close> uses a \<open>Reduce\<close> 81 premise. No new theorems can be proved about the standard judgements. 82\<close> 83axiomatization 84where 85 refl_red: "\<And>a. Reduce[a,a]" and 86 red_if_equal: "\<And>a b A. a = b : A \<Longrightarrow> Reduce[a,b]" and 87 trans_red: "\<And>a b c A. \<lbrakk>a = b : A; Reduce[b,c]\<rbrakk> \<Longrightarrow> a = c : A" and 88 89 \<comment> \<open>Reflexivity\<close> 90 91 refl_type: "\<And>A. A type \<Longrightarrow> A = A" and 92 refl_elem: "\<And>a A. a : A \<Longrightarrow> a = a : A" and 93 94 \<comment> \<open>Symmetry\<close> 95 96 sym_type: "\<And>A B. A = B \<Longrightarrow> B = A" and 97 sym_elem: "\<And>a b A. a = b : A \<Longrightarrow> b = a : A" and 98 99 \<comment> \<open>Transitivity\<close> 100 101 trans_type: "\<And>A B C. \<lbrakk>A = B; B = C\<rbrakk> \<Longrightarrow> A = C" and 102 trans_elem: "\<And>a b c A. \<lbrakk>a = b : A; b = c : A\<rbrakk> \<Longrightarrow> a = c : A" and 103 104 equal_types: "\<And>a A B. \<lbrakk>a : A; A = B\<rbrakk> \<Longrightarrow> a : B" and 105 equal_typesL: "\<And>a b A B. \<lbrakk>a = b : A; A = B\<rbrakk> \<Longrightarrow> a = b : B" and 106 107 \<comment> \<open>Substitution\<close> 108 109 subst_type: "\<And>a A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> B(z) type\<rbrakk> \<Longrightarrow> B(a) type" and 110 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 111 112 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 113 subst_elemL: 114 "\<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 115 116 117 \<comment> \<open>The type \<open>N\<close> -- natural numbers\<close> 118 119 NF: "N type" and 120 NI0: "0 : N" and 121 NI_succ: "\<And>a. a : N \<Longrightarrow> succ(a) : N" and 122 NI_succL: "\<And>a b. a = b : N \<Longrightarrow> succ(a) = succ(b) : N" and 123 124 NE: 125 "\<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> 126 \<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) : C(p)" and 127 128 NEL: 129 "\<And>p q a b c d C. \<lbrakk>p = q : N; a = c : C(0); 130 \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v) = d(u,v): C(succ(u))\<rbrakk> 131 \<Longrightarrow> rec(p, a, \<lambda>u v. b(u,v)) = rec(q,c,d) : C(p)" and 132 133 NC0: 134 "\<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> 135 \<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and 136 137 NC_succ: 138 "\<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> 139 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 140 141 \<comment> \<open>The fourth Peano axiom. See page 91 of Martin-L��f's book.\<close> 142 zero_ne_succ: "\<And>a. \<lbrakk>a: N; 0 = succ(a) : N\<rbrakk> \<Longrightarrow> 0: F" and 143 144 145 \<comment> \<open>The Product of a family of types\<close> 146 147 ProdF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) type" and 148 149 ProdFL: 150 "\<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 151 152 ProdI: 153 "\<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 154 155 ProdIL: "\<And>b c A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x) = c(x) : B(x)\<rbrakk> \<Longrightarrow> 156 \<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>x. c(x) : \<Prod>x:A. B(x)" and 157 158 ProdE: "\<And>p a A B. \<lbrakk>p : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> p`a : B(a)" and 159 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 160 161 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 162 163 ProdC2: "\<And>p A B. p : \<Prod>x:A. B(x) \<Longrightarrow> (\<^bold>\<lambda>x. p`x) = p : \<Prod>x:A. B(x)" and 164 165 166 \<comment> \<open>The Sum of a family of types\<close> 167 168 SumF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x) type" and 169 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 170 171 SumI: "\<And>a b A B. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> <a,b> : \<Sum>x:A. B(x)" and 172 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 173 174 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> 175 \<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and 176 177 SumEL: "\<And>p q c d A B C. \<lbrakk>p = q : \<Sum>x:A. B(x); 178 \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y)=d(x,y): C(<x,y>)\<rbrakk> 179 \<Longrightarrow> split(p, \<lambda>x y. c(x,y)) = split(q, \<lambda>x y. d(x,y)) : C(p)" and 180 181 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> 182 \<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and 183 184 fst_def: "\<And>a. fst(a) \<equiv> split(a, \<lambda>x y. x)" and 185 snd_def: "\<And>a. snd(a) \<equiv> split(a, \<lambda>x y. y)" and 186 187 188 \<comment> \<open>The sum of two types\<close> 189 190 PlusF: "\<And>A B. \<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> A+B type" and 191 PlusFL: "\<And>A B C D. \<lbrakk>A = C; B = D\<rbrakk> \<Longrightarrow> A+B = C+D" and 192 193 PlusI_inl: "\<And>a A B. \<lbrakk>a : A; B type\<rbrakk> \<Longrightarrow> inl(a) : A+B" and 194 PlusI_inlL: "\<And>a c A B. \<lbrakk>a = c : A; B type\<rbrakk> \<Longrightarrow> inl(a) = inl(c) : A+B" and 195 196 PlusI_inr: "\<And>b A B. \<lbrakk>A type; b : B\<rbrakk> \<Longrightarrow> inr(b) : A+B" and 197 PlusI_inrL: "\<And>b d A B. \<lbrakk>A type; b = d : B\<rbrakk> \<Longrightarrow> inr(b) = inr(d) : A+B" and 198 199 PlusE: 200 "\<And>p c d A B C. \<lbrakk>p: A+B; 201 \<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 202 \<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 203 204 PlusEL: 205 "\<And>p q c d e f A B C. \<lbrakk>p = q : A+B; 206 \<And>x. x: A \<Longrightarrow> c(x) = e(x) : C(inl(x)); 207 \<And>y. y: B \<Longrightarrow> d(y) = f(y) : C(inr(y))\<rbrakk> 208 \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) = when(q, \<lambda>x. e(x), \<lambda>y. f(y)) : C(p)" and 209 210 PlusC_inl: 211 "\<And>a c d A B C. \<lbrakk>a: A; 212 \<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 213 \<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> 214 \<Longrightarrow> when(inl(a), \<lambda>x. c(x), \<lambda>y. d(y)) = c(a) : C(inl(a))" and 215 216 PlusC_inr: 217 "\<And>b c d A B C. \<lbrakk>b: B; 218 \<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 219 \<And>y. y:B \<Longrightarrow> d(y): C(inr(y))\<rbrakk> 220 \<Longrightarrow> when(inr(b), \<lambda>x. c(x), \<lambda>y. d(y)) = d(b) : C(inr(b))" and 221 222 223 \<comment> \<open>The type \<open>Eq\<close>\<close> 224 225 EqF: "\<And>a b A. \<lbrakk>A type; a : A; b : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) type" and 226 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 227 EqI: "\<And>a b A. a = b : A \<Longrightarrow> eq : Eq(A,a,b)" and 228 EqE: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> a = b : A" and 229 230 \<comment> \<open>By equality of types, can prove \<open>C(p)\<close> from \<open>C(eq)\<close>, an elimination rule\<close> 231 EqC: "\<And>p a b A. p : Eq(A,a,b) \<Longrightarrow> p = eq : Eq(A,a,b)" and 232 233 234 \<comment> \<open>The type \<open>F\<close>\<close> 235 236 FF: "F type" and 237 FE: "\<And>p C. \<lbrakk>p: F; C type\<rbrakk> \<Longrightarrow> contr(p) : C" and 238 FEL: "\<And>p q C. \<lbrakk>p = q : F; C type\<rbrakk> \<Longrightarrow> contr(p) = contr(q) : C" and 239 240 241 \<comment> \<open>The type T\<close> 242 \<comment> \<open>Martin-L��f's book (page 68) discusses elimination and computation. 243 Elimination can be derived by computation and equality of types, 244 but with an extra premise \<open>C(x)\<close> type \<open>x:T\<close>. 245 Also computation can be derived from elimination.\<close> 246 247 TF: "T type" and 248 TI: "tt : T" and 249 TE: "\<And>p c C. \<lbrakk>p : T; c : C(tt)\<rbrakk> \<Longrightarrow> c : C(p)" and 250 TEL: "\<And>p q c d C. \<lbrakk>p = q : T; c = d : C(tt)\<rbrakk> \<Longrightarrow> c = d : C(p)" and 251 TC: "\<And>p. p : T \<Longrightarrow> p = tt : T" 252 253 254subsection "Tactics and derived rules for Constructive Type Theory" 255 256text \<open>Formation rules.\<close> 257lemmas form_rls = NF ProdF SumF PlusF EqF FF TF 258 and formL_rls = ProdFL SumFL PlusFL EqFL 259 260text \<open> 261 Introduction rules. OMITTED: 262 \<^item> \<open>EqI\<close>, because its premise is an \<open>eqelem\<close>, not an \<open>elem\<close>. 263\<close> 264lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI 265 and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL 266 267text \<open> 268 Elimination rules. OMITTED: 269 \<^item> \<open>EqE\<close>, because its conclusion is an \<open>eqelem\<close>, not an \<open>elem\<close> 270 \<^item> \<open>TE\<close>, because it does not involve a constructor. 271\<close> 272lemmas elim_rls = NE ProdE SumE PlusE FE 273 and elimL_rls = NEL ProdEL SumEL PlusEL FEL 274 275text \<open>OMITTED: \<open>eqC\<close> are \<open>TC\<close> because they make rewriting loop: \<open>p = un = un = \<dots>\<close>\<close> 276lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 277 278text \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close> 279lemmas element_rls = intr_rls elim_rls 280 281text \<open>Definitions are (meta)equality axioms.\<close> 282lemmas basic_defs = fst_def snd_def 283 284text \<open>Compare with standard version: \<open>B\<close> is applied to UNSIMPLIFIED expression!\<close> 285lemma SumIL2: "\<lbrakk>c = a : A; d = b : B(a)\<rbrakk> \<Longrightarrow> <c,d> = <a,b> : Sum(A,B)" 286 by (rule sym_elem) (rule SumIL; rule sym_elem) 287 288lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL 289 290text \<open> 291 Exploit \<open>p:Prod(A,B)\<close> to create the assumption \<open>z:B(a)\<close>. 292 A more natural form of product elimination. 293\<close> 294lemma subst_prodE: 295 assumes "p: Prod(A,B)" 296 and "a: A" 297 and "\<And>z. z: B(a) \<Longrightarrow> c(z): C(z)" 298 shows "c(p`a): C(p`a)" 299 by (rule assms ProdE)+ 300 301 302subsection \<open>Tactics for type checking\<close> 303 304ML \<open> 305local 306 307fun is_rigid_elem (Const(\<^const_name>\<open>Elem\<close>,_) $ a $ _) = not(is_Var (head_of a)) 308 | is_rigid_elem (Const(\<^const_name>\<open>Eqelem\<close>,_) $ a $ _ $ _) = not(is_Var (head_of a)) 309 | is_rigid_elem (Const(\<^const_name>\<open>Type\<close>,_) $ a) = not(is_Var (head_of a)) 310 | is_rigid_elem _ = false 311 312in 313 314(*Try solving a:A or a=b:A by assumption provided a is rigid!*) 315fun test_assume_tac ctxt = SUBGOAL (fn (prem, i) => 316 if is_rigid_elem (Logic.strip_assums_concl prem) 317 then assume_tac ctxt i else no_tac) 318 319fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i 320 321end 322\<close> 323 324text \<open> 325 For simplification: type formation and checking, 326 but no equalities between terms. 327\<close> 328lemmas routine_rls = form_rls formL_rls refl_type element_rls 329 330ML \<open> 331fun routine_tac rls ctxt prems = 332 ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls))); 333 334(*Solve all subgoals "A type" using formation rules. *) 335val form_net = Tactic.build_net @{thms form_rls}; 336fun form_tac ctxt = 337 REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net)); 338 339(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 340fun typechk_tac ctxt thms = 341 let val tac = 342 filt_resolve_from_net_tac ctxt 3 343 (Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls})) 344 in REPEAT_FIRST (ASSUME ctxt tac) end 345 346(*Solve a:A (a flexible, A rigid) by introduction rules. 347 Cannot use stringtrees (filt_resolve_tac) since 348 goals like ?a:SUM(A,B) have a trivial head-string *) 349fun intr_tac ctxt thms = 350 let val tac = 351 filt_resolve_from_net_tac ctxt 1 352 (Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls})) 353 in REPEAT_FIRST (ASSUME ctxt tac) end 354 355(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 356fun equal_tac ctxt thms = 357 REPEAT_FIRST 358 (ASSUME ctxt 359 (filt_resolve_from_net_tac ctxt 3 360 (Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem})))) 361\<close> 362 363method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close> 364method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close> 365method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close> 366method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close> 367 368 369subsection \<open>Simplification\<close> 370 371text \<open>To simplify the type in a goal.\<close> 372lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B" 373 apply (rule equal_types) 374 apply (rule_tac [2] sym_type) 375 apply assumption+ 376 done 377 378text \<open>Simplify the parameter of a unary type operator.\<close> 379lemma subst_eqtyparg: 380 assumes 1: "a=c : A" 381 and 2: "\<And>z. z:A \<Longrightarrow> B(z) type" 382 shows "B(a) = B(c)" 383 apply (rule subst_typeL) 384 apply (rule_tac [2] refl_type) 385 apply (rule 1) 386 apply (erule 2) 387 done 388 389text \<open>Simplification rules for Constructive Type Theory.\<close> 390lemmas reduction_rls = comp_rls [THEN trans_elem] 391 392ML \<open> 393(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 394 Uses other intro rules to avoid changing flexible goals.*) 395val eqintr_net = Tactic.build_net @{thms EqI intr_rls} 396fun eqintr_tac ctxt = 397 REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net)) 398 399(** Tactics that instantiate CTT-rules. 400 Vars in the given terms will be incremented! 401 The (rtac EqE i) lets them apply to equality judgements. **) 402 403fun NE_tac ctxt sp i = 404 TRY (resolve_tac ctxt @{thms EqE} i) THEN 405 Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i 406 407fun SumE_tac ctxt sp i = 408 TRY (resolve_tac ctxt @{thms EqE} i) THEN 409 Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i 410 411fun PlusE_tac ctxt sp i = 412 TRY (resolve_tac ctxt @{thms EqE} i) THEN 413 Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm PlusE} i 414 415(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 416 417(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) 418fun add_mp_tac ctxt i = 419 resolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i 420 421(*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *) 422fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i 423 424(*"safe" when regarded as predicate calculus rules*) 425val safe_brls = sort (make_ord lessb) 426 [ (true, @{thm FE}), (true,asm_rl), 427 (false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ] 428 429val unsafe_brls = 430 [ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}), 431 (true, @{thm subst_prodE}) ] 432 433(*0 subgoals vs 1 or more*) 434val (safe0_brls, safep_brls) = 435 List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 436 437fun safestep_tac ctxt thms i = 438 form_tac ctxt ORELSE 439 resolve_tac ctxt thms i ORELSE 440 biresolve_tac ctxt safe0_brls i ORELSE mp_tac ctxt i ORELSE 441 DETERM (biresolve_tac ctxt safep_brls i) 442 443fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i) 444 445fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac ctxt unsafe_brls 446 447(*Fails unless it solves the goal!*) 448fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms) 449\<close> 450 451method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close> 452method_setup NE = \<open> 453 Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s)) 454\<close> 455method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close> 456method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close> 457 458ML_file \<open>rew.ML\<close> 459method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close> 460method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close> 461 462 463subsection \<open>The elimination rules for fst/snd\<close> 464 465lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A" 466 apply (unfold basic_defs) 467 apply (erule SumE) 468 apply assumption 469 done 470 471text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close> 472lemma SumE_snd: 473 assumes major: "p: Sum(A,B)" 474 and "A type" 475 and "\<And>x. x:A \<Longrightarrow> B(x) type" 476 shows "snd(p) : B(fst(p))" 477 apply (unfold basic_defs) 478 apply (rule major [THEN SumE]) 479 apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 480 apply (typechk assms) 481 done 482 483 484section \<open>The two-element type (booleans and conditionals)\<close> 485 486definition Bool :: "t" 487 where "Bool \<equiv> T+T" 488 489definition true :: "i" 490 where "true \<equiv> inl(tt)" 491 492definition false :: "i" 493 where "false \<equiv> inr(tt)" 494 495definition cond :: "[i,i,i]\<Rightarrow>i" 496 where "cond(a,b,c) \<equiv> when(a, \<lambda>_. b, \<lambda>_. c)" 497 498lemmas bool_defs = Bool_def true_def false_def cond_def 499 500 501subsection \<open>Derivation of rules for the type \<open>Bool\<close>\<close> 502 503text \<open>Formation rule.\<close> 504lemma boolF: "Bool type" 505 unfolding bool_defs by typechk 506 507text \<open>Introduction rules for \<open>true\<close>, \<open>false\<close>.\<close> 508 509lemma boolI_true: "true : Bool" 510 unfolding bool_defs by typechk 511 512lemma boolI_false: "false : Bool" 513 unfolding bool_defs by typechk 514 515text \<open>Elimination rule: typing of \<open>cond\<close>.\<close> 516lemma boolE: "\<lbrakk>p:Bool; a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(p,a,b) : C(p)" 517 unfolding bool_defs 518 apply (typechk; erule TE) 519 apply typechk 520 done 521 522lemma boolEL: "\<lbrakk>p = q : Bool; a = c : C(true); b = d : C(false)\<rbrakk> 523 \<Longrightarrow> cond(p,a,b) = cond(q,c,d) : C(p)" 524 unfolding bool_defs 525 apply (rule PlusEL) 526 apply (erule asm_rl refl_elem [THEN TEL])+ 527 done 528 529text \<open>Computation rules for \<open>true\<close>, \<open>false\<close>.\<close> 530 531lemma boolC_true: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(true,a,b) = a : C(true)" 532 unfolding bool_defs 533 apply (rule comp_rls) 534 apply typechk 535 apply (erule_tac [!] TE) 536 apply typechk 537 done 538 539lemma boolC_false: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(false,a,b) = b : C(false)" 540 unfolding bool_defs 541 apply (rule comp_rls) 542 apply typechk 543 apply (erule_tac [!] TE) 544 apply typechk 545 done 546 547section \<open>Elementary arithmetic\<close> 548 549subsection \<open>Arithmetic operators and their definitions\<close> 550 551definition add :: "[i,i]\<Rightarrow>i" (infixr "#+" 65) 552 where "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))" 553 554definition diff :: "[i,i]\<Rightarrow>i" (infixr "-" 65) 555 where "a-b \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))" 556 557definition absdiff :: "[i,i]\<Rightarrow>i" (infixr "|-|" 65) 558 where "a|-|b \<equiv> (a-b) #+ (b-a)" 559 560definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 70) 561 where "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)" 562 563definition mod :: "[i,i]\<Rightarrow>i" (infixr "mod" 70) 564 where "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v) |-| b, 0, \<lambda>x y. succ(v)))" 565 566definition div :: "[i,i]\<Rightarrow>i" (infixr "div" 70) 567 where "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))" 568 569lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def 570 571 572subsection \<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close> 573 574subsubsection \<open>Addition\<close> 575 576text \<open>Typing of \<open>add\<close>: short and long versions.\<close> 577 578lemma add_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b : N" 579 unfolding arith_defs by typechk 580 581lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N" 582 unfolding arith_defs by equal 583 584 585text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close> 586 587lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N" 588 unfolding arith_defs by rew 589 590lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N" 591 unfolding arith_defs by rew 592 593 594subsubsection \<open>Multiplication\<close> 595 596text \<open>Typing of \<open>mult\<close>: short and long versions.\<close> 597 598lemma mult_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b : N" 599 unfolding arith_defs by (typechk add_typing) 600 601lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N" 602 unfolding arith_defs by (equal add_typingL) 603 604 605text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close> 606 607lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N" 608 unfolding arith_defs by rew 609 610lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N" 611 unfolding arith_defs by rew 612 613 614subsubsection \<open>Difference\<close> 615 616text \<open>Typing of difference.\<close> 617 618lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a - b : N" 619 unfolding arith_defs by typechk 620 621lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a - b = c - d : N" 622 unfolding arith_defs by equal 623 624 625text \<open>Computation for difference: 0 and successor cases.\<close> 626 627lemma diffC0: "a:N \<Longrightarrow> a - 0 = a : N" 628 unfolding arith_defs by rew 629 630text \<open>Note: \<open>rec(a, 0, \<lambda>z w.z)\<close> is \<open>pred(a).\<close>\<close> 631 632lemma diff_0_eq_0: "b:N \<Longrightarrow> 0 - b = 0 : N" 633 unfolding arith_defs 634 apply (NE b) 635 apply hyp_rew 636 done 637 638text \<open> 639 Essential to simplify FIRST!! (Else we get a critical pair) 640 \<open>succ(a) - succ(b)\<close> rewrites to \<open>pred(succ(a) - b)\<close>. 641\<close> 642lemma diff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) - succ(b) = a - b : N" 643 unfolding arith_defs 644 apply hyp_rew 645 apply (NE b) 646 apply hyp_rew 647 done 648 649 650subsection \<open>Simplification\<close> 651 652lemmas arith_typing_rls = add_typing mult_typing diff_typing 653 and arith_congr_rls = add_typingL mult_typingL diff_typingL 654 655lemmas congr_rls = arith_congr_rls intrL2_rls elimL_rls 656 657lemmas arithC_rls = 658 addC0 addC_succ 659 multC0 multC_succ 660 diffC0 diff_0_eq_0 diff_succ_succ 661 662ML \<open> 663 structure Arith_simp = TSimpFun( 664 val refl = @{thm refl_elem} 665 val sym = @{thm sym_elem} 666 val trans = @{thm trans_elem} 667 val refl_red = @{thm refl_red} 668 val trans_red = @{thm trans_red} 669 val red_if_equal = @{thm red_if_equal} 670 val default_rls = @{thms arithC_rls comp_rls} 671 val routine_tac = routine_tac @{thms arith_typing_rls routine_rls} 672 ) 673 674 fun arith_rew_tac ctxt prems = 675 make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems)) 676 677 fun hyp_arith_rew_tac ctxt prems = 678 make_rew_tac ctxt 679 (Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, @{thms congr_rls}, prems)) 680\<close> 681 682method_setup arith_rew = \<open> 683 Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths)) 684\<close> 685 686method_setup hyp_arith_rew = \<open> 687 Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths)) 688\<close> 689 690 691subsection \<open>Addition\<close> 692 693text \<open>Associative law for addition.\<close> 694lemma add_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #+ c = a #+ (b #+ c) : N" 695 apply (NE a) 696 apply hyp_arith_rew 697 done 698 699text \<open>Commutative law for addition. Can be proved using three inductions. 700 Must simplify after first induction! Orientation of rewrites is delicate.\<close> 701lemma add_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b = b #+ a : N" 702 apply (NE a) 703 apply hyp_arith_rew 704 apply (rule sym_elem) 705 prefer 2 706 apply (NE b) 707 prefer 4 708 apply (NE b) 709 apply hyp_arith_rew 710 done 711 712 713subsection \<open>Multiplication\<close> 714 715text \<open>Right annihilation in product.\<close> 716lemma mult_0_right: "a:N \<Longrightarrow> a #* 0 = 0 : N" 717 apply (NE a) 718 apply hyp_arith_rew 719 done 720 721text \<open>Right successor law for multiplication.\<close> 722lemma mult_succ_right: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* succ(b) = a #+ (a #* b) : N" 723 apply (NE a) 724 apply (hyp_arith_rew add_assoc [THEN sym_elem]) 725 apply (assumption | rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+ 726 done 727 728text \<open>Commutative law for multiplication.\<close> 729lemma mult_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b = b #* a : N" 730 apply (NE a) 731 apply (hyp_arith_rew mult_0_right mult_succ_right) 732 done 733 734text \<open>Addition distributes over multiplication.\<close> 735lemma add_mult_distrib: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #* c = (a #* c) #+ (b #* c) : N" 736 apply (NE a) 737 apply (hyp_arith_rew add_assoc [THEN sym_elem]) 738 done 739 740text \<open>Associative law for multiplication.\<close> 741lemma mult_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #* b) #* c = a #* (b #* c) : N" 742 apply (NE a) 743 apply (hyp_arith_rew add_mult_distrib) 744 done 745 746 747subsection \<open>Difference\<close> 748 749text \<open> 750 Difference on natural numbers, without negative numbers 751 \<^item> \<open>a - b = 0\<close> iff \<open>a \<le> b\<close> 752 \<^item> \<open>a - b = succ(c)\<close> iff \<open>a > b\<close> 753\<close> 754 755lemma diff_self_eq_0: "a:N \<Longrightarrow> a - a = 0 : N" 756 apply (NE a) 757 apply hyp_arith_rew 758 done 759 760 761lemma add_0_right: "\<lbrakk>c : N; 0 : N; c : N\<rbrakk> \<Longrightarrow> c #+ 0 = c : N" 762 by (rule addC0 [THEN [3] add_commute [THEN trans_elem]]) 763 764text \<open> 765 Addition is the inverse of subtraction: if \<open>b \<le> x\<close> then \<open>b #+ (x - b) = x\<close>. 766 An example of induction over a quantified formula (a product). 767 Uses rewriting with a quantified, implicative inductive hypothesis. 768\<close> 769schematic_goal add_diff_inverse_lemma: 770 "b:N \<Longrightarrow> ?a : \<Prod>x:N. Eq(N, b-x, 0) \<longrightarrow> Eq(N, b #+ (x-b), x)" 771 apply (NE b) 772 \<comment> \<open>strip one "universal quantifier" but not the "implication"\<close> 773 apply (rule_tac [3] intr_rls) 774 \<comment> \<open>case analysis on \<open>x\<close> in \<open>succ(u) \<le> x \<longrightarrow> succ(u) #+ (x - succ(u)) = x\<close>\<close> 775 prefer 4 776 apply (NE x) 777 apply assumption 778 \<comment> \<open>Prepare for simplification of types -- the antecedent \<open>succ(u) \<le> x\<close>\<close> 779 apply (rule_tac [2] replace_type) 780 apply (rule_tac [1] replace_type) 781 apply arith_rew 782 \<comment> \<open>Solves first 0 goal, simplifies others. Two sugbgoals remain. 783 Both follow by rewriting, (2) using quantified induction hyp.\<close> 784 apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close> 785 apply (hyp_arith_rew add_0_right) 786 apply assumption 787 done 788 789text \<open> 790 Version of above with premise \<open>b - a = 0\<close> i.e. \<open>a \<ge> b\<close>. 791 Using @{thm ProdE} does not work -- for \<open>?B(?a)\<close> is ambiguous. 792 Instead, @{thm add_diff_inverse_lemma} states the desired induction scheme; 793 the use of \<open>THEN\<close> below instantiates Vars in @{thm ProdE} automatically. 794\<close> 795lemma add_diff_inverse: "\<lbrakk>a:N; b:N; b - a = 0 : N\<rbrakk> \<Longrightarrow> b #+ (a-b) = a : N" 796 apply (rule EqE) 797 apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE]) 798 apply (assumption | rule EqI)+ 799 done 800 801 802subsection \<open>Absolute difference\<close> 803 804text \<open>Typing of absolute difference: short and long versions.\<close> 805 806lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b : N" 807 unfolding arith_defs by typechk 808 809lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a |-| b = c |-| d : N" 810 unfolding arith_defs by equal 811 812lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a |-| a = 0 : N" 813 unfolding absdiff_def by (arith_rew diff_self_eq_0) 814 815lemma absdiffC0: "a:N \<Longrightarrow> 0 |-| a = a : N" 816 unfolding absdiff_def by hyp_arith_rew 817 818lemma absdiff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) |-| succ(b) = a |-| b : N" 819 unfolding absdiff_def by hyp_arith_rew 820 821text \<open>Note how easy using commutative laws can be? ...not always...\<close> 822lemma absdiff_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a |-| b = b |-| a : N" 823 unfolding absdiff_def 824 apply (rule add_commute) 825 apply (typechk diff_typing) 826 done 827 828text \<open>If \<open>a + b = 0\<close> then \<open>a = 0\<close>. Surprisingly tedious.\<close> 829schematic_goal add_eq0_lemma: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> ?c : Eq(N,a#+b,0) \<longrightarrow> Eq(N,a,0)" 830 apply (NE a) 831 apply (rule_tac [3] replace_type) 832 apply arith_rew 833 apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close> 834 apply (rule_tac [2] zero_ne_succ [THEN FE]) 835 apply (erule_tac [3] EqE [THEN sym_elem]) 836 apply (typechk add_typing) 837 done 838 839text \<open> 840 Version of above with the premise \<open>a + b = 0\<close>. 841 Again, resolution instantiates variables in @{thm ProdE}. 842\<close> 843lemma add_eq0: "\<lbrakk>a:N; b:N; a #+ b = 0 : N\<rbrakk> \<Longrightarrow> a = 0 : N" 844 apply (rule EqE) 845 apply (rule add_eq0_lemma [THEN ProdE]) 846 apply (rule_tac [3] EqI) 847 apply typechk 848 done 849 850text \<open>Here is a lemma to infer \<open>a - b = 0\<close> and \<open>b - a = 0\<close> from \<open>a |-| b = 0\<close>, below.\<close> 851schematic_goal absdiff_eq0_lem: 852 "\<lbrakk>a:N; b:N; a |-| b = 0 : N\<rbrakk> \<Longrightarrow> ?a : Eq(N, a-b, 0) \<times> Eq(N, b-a, 0)" 853 apply (unfold absdiff_def) 854 apply intr 855 apply eqintr 856 apply (rule_tac [2] add_eq0) 857 apply (rule add_eq0) 858 apply (rule_tac [6] add_commute [THEN trans_elem]) 859 apply (typechk diff_typing) 860 done 861 862text \<open>If \<open>a |-| b = 0\<close> then \<open>a = b\<close> 863 proof: \<open>a - b = 0\<close> and \<open>b - a = 0\<close>, so \<open>b = a + (b - a) = a + 0 = a\<close>. 864\<close> 865lemma absdiff_eq0: "\<lbrakk>a |-| b = 0 : N; a:N; b:N\<rbrakk> \<Longrightarrow> a = b : N" 866 apply (rule EqE) 867 apply (rule absdiff_eq0_lem [THEN SumE]) 868 apply eqintr 869 apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem]) 870 apply (erule_tac [3] EqE) 871 apply (hyp_arith_rew add_0_right) 872 done 873 874 875subsection \<open>Remainder and Quotient\<close> 876 877text \<open>Typing of remainder: short and long versions.\<close> 878 879lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N" 880 unfolding mod_def by (typechk absdiff_typing) 881 882lemma mod_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a mod b = c mod d : N" 883 unfolding mod_def by (equal absdiff_typingL) 884 885 886text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close> 887 888lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N" 889 unfolding mod_def by (rew absdiff_typing) 890 891lemma modC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 892 succ(a) mod b = rec(succ(a mod b) |-| b, 0, \<lambda>x y. succ(a mod b)) : N" 893 unfolding mod_def by (rew absdiff_typing) 894 895 896text \<open>Typing of quotient: short and long versions.\<close> 897 898lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N" 899 unfolding div_def by (typechk absdiff_typing mod_typing) 900 901lemma div_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a div b = c div d : N" 902 unfolding div_def by (equal absdiff_typingL mod_typingL) 903 904lemmas div_typing_rls = mod_typing div_typing absdiff_typing 905 906 907text \<open>Computation for quotient: 0 and successor cases.\<close> 908 909lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N" 910 unfolding div_def by (rew mod_typing absdiff_typing) 911 912lemma divC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 913 succ(a) div b = rec(succ(a) mod b, succ(a div b), \<lambda>x y. a div b) : N" 914 unfolding div_def by (rew mod_typing) 915 916 917text \<open>Version of above with same condition as the \<open>mod\<close> one.\<close> 918lemma divC_succ2: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 919 succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), \<lambda>x y. a div b) : N" 920 apply (rule divC_succ [THEN trans_elem]) 921 apply (rew div_typing_rls modC_succ) 922 apply (NE "succ (a mod b) |-|b") 923 apply (rew mod_typing div_typing absdiff_typing) 924 done 925 926text \<open>For case analysis on whether a number is 0 or a successor.\<close> 927lemma iszero_decidable: "a:N \<Longrightarrow> rec(a, inl(eq), \<lambda>ka kb. inr(<ka, eq>)) : 928 Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))" 929 apply (NE a) 930 apply (rule_tac [3] PlusI_inr) 931 apply (rule_tac [2] PlusI_inl) 932 apply eqintr 933 apply equal 934 done 935 936text \<open>Main Result. Holds when \<open>b\<close> is 0 since \<open>a mod 0 = a\<close> and \<open>a div 0 = 0\<close>.\<close> 937lemma mod_div_equality: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b #+ (a div b) #* b = a : N" 938 apply (NE a) 939 apply (arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2) 940 apply (rule EqE) 941 \<comment> \<open>case analysis on \<open>succ(u mod b) |-| b\<close>\<close> 942 apply (rule_tac a1 = "succ (u mod b) |-| b" in iszero_decidable [THEN PlusE]) 943 apply (erule_tac [3] SumE) 944 apply (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2) 945 \<comment> \<open>Replace one occurrence of \<open>b\<close> by \<open>succ(u mod b)\<close>. Clumsy!\<close> 946 apply (rule add_typingL [THEN trans_elem]) 947 apply (erule EqE [THEN absdiff_eq0, THEN sym_elem]) 948 apply (rule_tac [3] refl_elem) 949 apply (hyp_arith_rew div_typing_rls) 950 done 951 952end 953