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+ − 1 theory Terms
1129
+ − 2 imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs"
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+ − 3 begin
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+ − 4
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+ − 5 atom_decl name
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+ − 6
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+ − 7 text {* primrec seems to be genarally faster than fun *}
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+ − 8
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+ − 9 section {*** lets with binding patterns ***}
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+ − 10
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+ − 11 datatype rtrm1 =
+ − 12 rVr1 "name"
+ − 13 | rAp1 "rtrm1" "rtrm1"
+ − 14 | rLm1 "name" "rtrm1" --"name is bound in trm1"
+ − 15 | rLt1 "bp" "rtrm1" "rtrm1" --"all variables in bp are bound in the 2nd trm1"
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+ − 16 and bp =
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+ − 17 BUnit
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+ − 18 | BVr "name"
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+ − 19 | BPr "bp" "bp"
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+ − 20
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+ − 21 (* to be given by the user *)
1029
+ − 22
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+ − 23 primrec
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+ − 24 bv1
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+ − 25 where
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+ − 26 "bv1 (BUnit) = {}"
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+ − 27 | "bv1 (BVr x) = {atom x}"
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+ − 28 | "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"
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+ − 29
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+ − 30 (* needs to be calculated by the package *)
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+ − 31 primrec
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+ − 32 rfv_trm1 and rfv_bp
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+ − 33 where
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+ − 34 "rfv_trm1 (rVr1 x) = {atom x}"
+ − 35 | "rfv_trm1 (rAp1 t1 t2) = (rfv_trm1 t1) \<union> (rfv_trm1 t2)"
+ − 36 | "rfv_trm1 (rLm1 x t) = (rfv_trm1 t) - {atom x}"
+ − 37 | "rfv_trm1 (rLt1 bp t1 t2) = (rfv_trm1 t1) \<union> (rfv_trm1 t2 - bv1 bp)"
+ − 38 | "rfv_bp (BUnit) = {}"
+ − 39 | "rfv_bp (BVr x) = {atom x}"
+ − 40 | "rfv_bp (BPr b1 b2) = (rfv_bp b1) \<union> (rfv_bp b2)"
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+ − 41
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+ − 42 (* needs to be stated by the package *)
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+ − 43 instantiation
1028
+ − 44 rtrm1 and bp :: pt
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+ − 45 begin
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+ − 46
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+ − 47 primrec
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+ − 48 permute_rtrm1 and permute_bp
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+ − 49 where
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+ − 50 "permute_rtrm1 pi (rVr1 a) = rVr1 (pi \<bullet> a)"
+ − 51 | "permute_rtrm1 pi (rAp1 t1 t2) = rAp1 (permute_rtrm1 pi t1) (permute_rtrm1 pi t2)"
+ − 52 | "permute_rtrm1 pi (rLm1 a t) = rLm1 (pi \<bullet> a) (permute_rtrm1 pi t)"
+ − 53 | "permute_rtrm1 pi (rLt1 bp t1 t2) = rLt1 (permute_bp pi bp) (permute_rtrm1 pi t1) (permute_rtrm1 pi t2)"
957
+ − 54 | "permute_bp pi (BUnit) = BUnit"
+ − 55 | "permute_bp pi (BVr a) = BVr (pi \<bullet> a)"
+ − 56 | "permute_bp pi (BPr bp1 bp2) = BPr (permute_bp pi bp1) (permute_bp pi bp2)"
+ − 57
1028
+ − 58 lemma pt_rtrm1_bp_zero:
+ − 59 fixes t::rtrm1
957
+ − 60 and b::bp
+ − 61 shows "0 \<bullet> t = t"
+ − 62 and "0 \<bullet> b = b"
1028
+ − 63 apply(induct t and b rule: rtrm1_bp.inducts)
957
+ − 64 apply(simp_all)
+ − 65 done
+ − 66
1028
+ − 67 lemma pt_rtrm1_bp_plus:
+ − 68 fixes t::rtrm1
957
+ − 69 and b::bp
+ − 70 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 71 and "((p + q) \<bullet> b) = p \<bullet> (q \<bullet> b)"
1028
+ − 72 apply(induct t and b rule: rtrm1_bp.inducts)
957
+ − 73 apply(simp_all)
+ − 74 done
+ − 75
+ − 76 instance
+ − 77 apply default
1028
+ − 78 apply(simp_all add: pt_rtrm1_bp_zero pt_rtrm1_bp_plus)
957
+ − 79 done
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+ − 80
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+ − 81 end
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+ − 82
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+ − 83 inductive
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+ − 84 alpha1 :: "rtrm1 \<Rightarrow> rtrm1 \<Rightarrow> bool" ("_ \<approx>1 _" [100, 100] 100)
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+ − 85 where
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+ − 86 a1: "a = b \<Longrightarrow> (rVr1 a) \<approx>1 (rVr1 b)"
+ − 87 | a2: "\<lbrakk>t1 \<approx>1 t2; s1 \<approx>1 s2\<rbrakk> \<Longrightarrow> rAp1 t1 s1 \<approx>1 rAp1 t2 s2"
+ − 88 | a3: "(\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 rfv_trm1 pi ({atom ab}, s))) \<Longrightarrow> rLm1 aa t \<approx>1 rLm1 ab s"
+ − 89 | a4: "t1 \<approx>1 t2 \<Longrightarrow> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 rfv_trm1 pi ((bv1 b2), s2))) \<Longrightarrow> rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2"
+ − 90
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+ − 91 lemma alpha1_inj:
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+ − 92 "(rVr1 a \<approx>1 rVr1 b) = (a = b)"
+ − 93 "(rAp1 t1 s1 \<approx>1 rAp1 t2 s2) = (t1 \<approx>1 t2 \<and> s1 \<approx>1 s2)"
+ − 94 "(rLm1 aa t \<approx>1 rLm1 ab s) = (\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 rfv_trm1 pi ({atom ab}, s)))"
+ − 95 "(rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2) = (t1 \<approx>1 t2 \<and> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 rfv_trm1 pi ((bv1 b2), s2))))"
+ − 96 apply -
+ − 97 apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
+ − 98 apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
+ − 99 apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
+ − 100 apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
+ − 101 done
+ − 102
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+ − 103 (* Shouyld we derive it? But bv is given by the user? *)
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+ − 104 lemma bv1_eqvt[eqvt]:
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+ − 105 shows "(pi \<bullet> bv1 x) = bv1 (pi \<bullet> x)"
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+ − 106 apply (induct x)
1043
+ − 107 apply (simp_all add: empty_eqvt insert_eqvt atom_eqvt)
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+ − 108 done
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+ − 109
1031
+ − 110 lemma rfv_trm1_eqvt[eqvt]:
+ − 111 shows "(pi\<bullet>rfv_trm1 t) = rfv_trm1 (pi\<bullet>t)"
+ − 112 apply (induct t)
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+ − 113 apply (simp_all add: insert_eqvt atom_eqvt empty_eqvt union_eqvt Diff_eqvt bv1_eqvt)
1031
+ − 114 done
+ − 115
+ − 116
1028
+ − 117 lemma alpha1_eqvt:
+ − 118 shows "t \<approx>1 s \<Longrightarrow> (pi \<bullet> t) \<approx>1 (pi \<bullet> s)"
1031
+ − 119 apply (induct t s rule: alpha1.inducts)
+ − 120 apply (simp_all add:eqvts alpha1_inj)
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+ − 121 apply (erule exE)
+ − 122 apply (rule_tac x="pi \<bullet> pia" in exI)
+ − 123 apply (simp add: alpha_gen)
+ − 124 apply(erule conjE)+
+ − 125 apply(rule conjI)
+ − 126 apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
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+ − 127 apply(simp add: atom_eqvt Diff_eqvt insert_eqvt empty_eqvt rfv_trm1_eqvt)
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+ − 128 apply(rule conjI)
+ − 129 apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
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+ − 130 apply(simp add: atom_eqvt Diff_eqvt rfv_trm1_eqvt insert_eqvt empty_eqvt)
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+ − 131 apply(simp add: permute_eqvt[symmetric])
+ − 132 apply (erule exE)
+ − 133 apply (rule_tac x="pi \<bullet> pia" in exI)
+ − 134 apply (simp add: alpha_gen)
+ − 135 apply(erule conjE)+
+ − 136 apply(rule conjI)
+ − 137 apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
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+ − 138 apply(simp add: rfv_trm1_eqvt Diff_eqvt bv1_eqvt)
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+ − 139 apply(rule conjI)
+ − 140 apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
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+ − 141 apply(simp add: atom_eqvt rfv_trm1_eqvt Diff_eqvt bv1_eqvt)
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+ − 142 apply(simp add: permute_eqvt[symmetric])
+ − 143 done
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+ − 144
957
+ − 145 lemma alpha1_equivp: "equivp alpha1"
+ − 146 sorry
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+ − 147
1028
+ − 148 quotient_type trm1 = rtrm1 / alpha1
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+ − 149 by (rule alpha1_equivp)
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+ − 150
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+ − 151 quotient_definition
+ − 152 "Vr1 :: name \<Rightarrow> trm1"
+ − 153 as
+ − 154 "rVr1"
+ − 155
1029
+ − 156 quotient_definition
+ − 157 "Ap1 :: trm1 \<Rightarrow> trm1 \<Rightarrow> trm1"
+ − 158 as
+ − 159 "rAp1"
+ − 160
+ − 161 quotient_definition
+ − 162 "Lm1 :: name \<Rightarrow> trm1 \<Rightarrow> trm1"
+ − 163 as
+ − 164 "rLm1"
+ − 165
+ − 166 quotient_definition
+ − 167 "Lt1 :: bp \<Rightarrow> trm1 \<Rightarrow> trm1 \<Rightarrow> trm1"
+ − 168 as
+ − 169 "rLt1"
+ − 170
+ − 171 quotient_definition
+ − 172 "fv_trm1 :: trm1 \<Rightarrow> atom set"
+ − 173 as
+ − 174 "rfv_trm1"
+ − 175
+ − 176 lemma alpha_rfv1:
+ − 177 shows "t \<approx>1 s \<Longrightarrow> rfv_trm1 t = rfv_trm1 s"
+ − 178 apply(induct rule: alpha1.induct)
+ − 179 apply(simp_all add: alpha_gen.simps)
+ − 180 done
+ − 181
+ − 182 lemma [quot_respect]:
+ − 183 "(op = ===> alpha1) rVr1 rVr1"
+ − 184 "(alpha1 ===> alpha1 ===> alpha1) rAp1 rAp1"
+ − 185 "(op = ===> alpha1 ===> alpha1) rLm1 rLm1"
+ − 186 "(op = ===> alpha1 ===> alpha1 ===> alpha1) rLt1 rLt1"
+ − 187 apply (auto intro: alpha1.intros)
+ − 188 apply(rule a3) apply (rule_tac x="0" in exI)
+ − 189 apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv1 alpha_gen)
+ − 190 apply(rule a4) apply assumption apply (rule_tac x="0" in exI)
+ − 191 apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv1 alpha_gen)
+ − 192 done
+ − 193
+ − 194 lemma [quot_respect]:
+ − 195 "(op = ===> alpha1 ===> alpha1) permute permute"
+ − 196 apply auto
+ − 197 apply (rule alpha1_eqvt)
+ − 198 apply simp
+ − 199 done
+ − 200
+ − 201 lemma [quot_respect]:
+ − 202 "(alpha1 ===> op =) rfv_trm1 rfv_trm1"
+ − 203 apply (simp add: alpha_rfv1)
+ − 204 done
+ − 205
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+ − 206 lemmas trm1_bp_induct = rtrm1_bp.induct[quot_lifted]
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+ − 207 lemmas trm1_bp_inducts = rtrm1_bp.inducts[quot_lifted]
1029
+ − 208
+ − 209 instantiation trm1 and bp :: pt
+ − 210 begin
+ − 211
+ − 212 quotient_definition
+ − 213 "permute_trm1 :: perm \<Rightarrow> trm1 \<Rightarrow> trm1"
+ − 214 as
+ − 215 "permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"
+ − 216
1083
+ − 217 lemmas permute_trm1[simp] = permute_rtrm1_permute_bp.simps[quot_lifted]
+ − 218
1029
+ − 219 instance
+ − 220 apply default
+ − 221 apply(induct_tac [!] x rule: trm1_bp_inducts(1))
1083
+ − 222 apply(simp_all)
1029
+ − 223 done
+ − 224
+ − 225 end
+ − 226
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+ − 227 lemmas fv_trm1 = rfv_trm1_rfv_bp.simps[quot_lifted]
1029
+ − 228
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+ − 229 lemmas fv_trm1_eqvt = rfv_trm1_eqvt[quot_lifted]
1029
+ − 230
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+ − 231 lemmas alpha1_INJ = alpha1_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
1029
+ − 232
1031
+ − 233 lemma lm1_supp_pre:
+ − 234 shows "(supp (atom x, t)) supports (Lm1 x t) "
+ − 235 apply(simp add: supports_def)
+ − 236 apply(fold fresh_def)
+ − 237 apply(simp add: fresh_Pair swap_fresh_fresh)
+ − 238 apply(clarify)
+ − 239 apply(subst swap_at_base_simps(3))
+ − 240 apply(simp_all add: fresh_atom)
+ − 241 done
+ − 242
+ − 243 lemma lt1_supp_pre:
+ − 244 shows "(supp (x, t, s)) supports (Lt1 t x s) "
+ − 245 apply(simp add: supports_def)
+ − 246 apply(fold fresh_def)
+ − 247 apply(simp add: fresh_Pair swap_fresh_fresh)
+ − 248 done
+ − 249
+ − 250 lemma bp_supp: "finite (supp (bp :: bp))"
+ − 251 apply (induct bp)
+ − 252 apply(simp_all add: supp_def)
+ − 253 apply (fold supp_def)
+ − 254 apply (simp add: supp_at_base)
+ − 255 apply(simp add: Collect_imp_eq)
+ − 256 apply(simp add: Collect_neg_eq[symmetric])
+ − 257 apply (fold supp_def)
+ − 258 apply (simp)
+ − 259 done
+ − 260
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+ − 261 instance trm1 :: fs
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
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+ − 262 apply default
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
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+ − 263 apply(induct_tac x rule: trm1_bp_inducts(1))
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
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+ − 264 apply(simp_all)
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
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+ − 265 apply(simp add: supp_def alpha1_INJ eqvts)
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
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+ − 266 apply(simp add: supp_def[symmetric] supp_at_base)
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
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+ − 267 apply(simp only: supp_def alpha1_INJ eqvts permute_trm1)
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
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+ − 268 apply(simp add: Collect_imp_eq Collect_neg_eq)
1031
+ − 269 apply(rule supports_finite)
+ − 270 apply(rule lm1_supp_pre)
+ − 271 apply(simp add: supp_Pair supp_atom)
+ − 272 apply(rule supports_finite)
+ − 273 apply(rule lt1_supp_pre)
+ − 274 apply(simp add: supp_Pair supp_atom bp_supp)
+ − 275 done
1030
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Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
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+ − 276
1029
+ − 277 lemma supp_fv:
+ − 278 shows "supp t = fv_trm1 t"
+ − 279 apply(induct t rule: trm1_bp_inducts(1))
1030
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 280 apply(simp_all)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 281 apply(simp add: supp_def permute_trm1 alpha1_INJ fv_trm1)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 282 apply(simp only: supp_at_base[simplified supp_def])
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 283 apply(simp add: supp_def permute_trm1 alpha1_INJ fv_trm1)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 284 apply(simp add: Collect_imp_eq Collect_neg_eq)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 285 apply(subgoal_tac "supp (Lm1 name rtrm1) = supp (Abs {atom name} rtrm1)")
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 286 apply(simp add: supp_Abs fv_trm1)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 287 apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 288 apply(simp add: alpha1_INJ)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 289 apply(simp add: Abs_eq_iff)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 290 apply(simp add: alpha_gen.simps)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 291 apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric])
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 292 apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp(rtrm11) \<union> supp (Abs (bv1 bp) rtrm12)")
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 293 apply(simp add: supp_Abs fv_trm1)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 294 apply(simp (no_asm) add: supp_def)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 295 apply(simp add: alpha1_INJ)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 296 apply(simp add: Abs_eq_iff)
1033
+ − 297 apply(simp add: alpha_gen)
1030
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 298 apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 299 apply(simp add: Collect_imp_eq Collect_neg_eq)
07f97267a392
Finished the supp_fv proof; first proof that analyses the structure of 'Let' :)
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 300 done
1029
+ − 301
1032
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 302 lemma trm1_supp:
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 303 "supp (Vr1 x) = {atom x}"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 304 "supp (Ap1 t1 t2) = supp t1 \<union> supp t2"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 305 "supp (Lm1 x t) = (supp t) - {atom x}"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 306 "supp (Lt1 b t s) = supp t \<union> (supp s - bv1 b)"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 307 by (simp_all only: supp_fv fv_trm1)
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 308
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 309 lemma trm1_induct_strong:
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 310 assumes "\<And>name b. P b (Vr1 name)"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 311 and "\<And>rtrm11 rtrm12 b. \<lbrakk>\<And>c. P c rtrm11; \<And>c. P c rtrm12\<rbrakk> \<Longrightarrow> P b (Ap1 rtrm11 rtrm12)"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 312 and "\<And>name rtrm1 b. \<lbrakk>\<And>c. P c rtrm1; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lm1 name rtrm1)"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 313 and "\<And>bp rtrm11 rtrm12 b. \<lbrakk>\<And>c. P c rtrm11; \<And>c. P c rtrm12; bp1 bp \<sharp>* b\<rbrakk> \<Longrightarrow> P b (Lt1 bp rtrm11 rtrm12)"
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 314 shows "P a rtrma"
1035
+ − 315 sorry
1032
135bf399c036
The trm1_support lemma explicitly and stated a strong induction principle.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 316
950
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parents:
diff
changeset
+ − 317 section {*** lets with single assignments ***}
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parents:
diff
changeset
+ − 318
1091
+ − 319 datatype rtrm2 =
+ − 320 rVr2 "name"
+ − 321 | rAp2 "rtrm2" "rtrm2"
1093
+ − 322 | rLm2 "name" "rtrm2" --"bind (name) in (rtrm2)"
+ − 323 | rLt2 "rassign" "rtrm2" --"bind (bv2 rassign) in (rtrm2)"
1091
+ − 324 and rassign =
+ − 325 rAs "name" "rtrm2"
950
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parents:
diff
changeset
+ − 326
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 327 (* to be given by the user *)
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parents:
diff
changeset
+ − 328 primrec
1091
+ − 329 rbv2
950
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parents:
diff
changeset
+ − 330 where
1091
+ − 331 "rbv2 (rAs x t) = {atom x}"
950
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parents:
diff
changeset
+ − 332
98764f25f012
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parents:
diff
changeset
+ − 333 (* needs to be calculated by the package *)
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 334 primrec
1091
+ − 335 fv_rtrm2 and fv_rassign
950
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parents:
diff
changeset
+ − 336 where
1091
+ − 337 "fv_rtrm2 (rVr2 x) = {atom x}"
+ − 338 | "fv_rtrm2 (rAp2 t1 t2) = (fv_rtrm2 t1) \<union> (fv_rtrm2 t2)"
+ − 339 | "fv_rtrm2 (rLm2 x t) = (fv_rtrm2 t) - {atom x}"
+ − 340 | "fv_rtrm2 (rLt2 as t) = (fv_rtrm2 t - rbv2 as) \<union> (fv_rassign as)"
+ − 341 | "fv_rassign (rAs x t) = fv_rtrm2 t"
950
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parents:
diff
changeset
+ − 342
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 343 (* needs to be stated by the package *)
1091
+ − 344 instantiation
+ − 345 rtrm2 and rassign :: pt
950
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parents:
diff
changeset
+ − 346 begin
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parents:
diff
changeset
+ − 347
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parents:
diff
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+ − 348 primrec
1091
+ − 349 permute_rtrm2 and permute_rassign
950
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 350 where
1091
+ − 351 "permute_rtrm2 pi (rVr2 a) = rVr2 (pi \<bullet> a)"
+ − 352 | "permute_rtrm2 pi (rAp2 t1 t2) = rAp2 (permute_rtrm2 pi t1) (permute_rtrm2 pi t2)"
+ − 353 | "permute_rtrm2 pi (rLm2 a t) = rLm2 (pi \<bullet> a) (permute_rtrm2 pi t)"
+ − 354 | "permute_rtrm2 pi (rLt2 as t) = rLt2 (permute_rassign pi as) (permute_rtrm2 pi t)"
+ − 355 | "permute_rassign pi (rAs a t) = rAs (pi \<bullet> a) (permute_rtrm2 pi t)"
957
+ − 356
1091
+ − 357 lemma pt_rtrm2_rassign_zero:
+ − 358 fixes t::rtrm2
+ − 359 and b::rassign
957
+ − 360 shows "0 \<bullet> t = t"
+ − 361 and "0 \<bullet> b = b"
1091
+ − 362 apply(induct t and b rule: rtrm2_rassign.inducts)
957
+ − 363 apply(simp_all)
+ − 364 done
+ − 365
1091
+ − 366 lemma pt_rtrm2_rassign_plus:
+ − 367 fixes t::rtrm2
+ − 368 and b::rassign
957
+ − 369 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 370 and "((p + q) \<bullet> b) = p \<bullet> (q \<bullet> b)"
1091
+ − 371 apply(induct t and b rule: rtrm2_rassign.inducts)
957
+ − 372 apply(simp_all)
+ − 373 done
+ − 374
+ − 375 instance
+ − 376 apply default
1091
+ − 377 apply(simp_all add: pt_rtrm2_rassign_zero pt_rtrm2_rassign_plus)
957
+ − 378 done
+ − 379
950
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parents:
diff
changeset
+ − 380
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parents:
diff
changeset
+ − 381 end
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parents:
diff
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+ − 382
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parents:
diff
changeset
+ − 383 inductive
1091
+ − 384 alpha2 :: "rtrm2 \<Rightarrow> rtrm2 \<Rightarrow> bool" ("_ \<approx>2 _" [100, 100] 100)
+ − 385 and
+ − 386 alpha2a :: "rassign \<Rightarrow> rassign \<Rightarrow> bool" ("_ \<approx>2a _" [100, 100] 100)
950
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parents:
diff
changeset
+ − 387 where
1091
+ − 388 a1: "a = b \<Longrightarrow> (rVr2 a) \<approx>2 (rVr2 b)"
+ − 389 | a2: "\<lbrakk>t1 \<approx>2 t2; s1 \<approx>2 s2\<rbrakk> \<Longrightarrow> rAp2 t1 s1 \<approx>2 rAp2 t2 s2"
+ − 390 | a3: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha2 fv_rtrm2 pi ({atom b}, s))) \<Longrightarrow> rLm2 a t \<approx>2 rLm2 b s"
1092
+ − 391 | a4: "\<lbrakk>\<exists>pi. ((rbv2 bt, t) \<approx>gen alpha2 fv_rtrm2 pi ((rbv2 bs), s));
+ − 392 \<exists>pi. ((rbv2 bt, bt) \<approx>gen alpha2a fv_rassign pi (rbv2 bs, bs))\<rbrakk>
+ − 393 \<Longrightarrow> rLt2 bt t \<approx>2 rLt2 bs s"
+ − 394 | a5: "\<lbrakk>a = b; t \<approx>2 s\<rbrakk> \<Longrightarrow> rAs a t \<approx>2a rAs b s" (* This way rbv2 can be lifted *)
950
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parents:
diff
changeset
+ − 395
1091
+ − 396 lemma alpha2_equivp:
+ − 397 "equivp alpha2"
+ − 398 "equivp alpha2a"
957
+ − 399 sorry
950
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parents:
diff
changeset
+ − 400
1091
+ − 401 quotient_type
+ − 402 trm2 = rtrm2 / alpha2
+ − 403 and
+ − 404 assign = rassign / alpha2a
+ − 405 by (auto intro: alpha2_equivp)
+ − 406
950
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parents:
diff
changeset
+ − 407
1092
+ − 408
950
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parents:
diff
changeset
+ − 409 section {*** lets with many assignments ***}
98764f25f012
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parents:
diff
changeset
+ − 410
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 411 datatype trm3 =
98764f25f012
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parents:
diff
changeset
+ − 412 Vr3 "name"
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 413 | Ap3 "trm3" "trm3"
1093
+ − 414 | Lm3 "name" "trm3" --"bind (name) in (trm3)"
+ − 415 | Lt3 "assigns" "trm3" --"bind (bv3 assigns) in (trm3)"
950
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parents:
diff
changeset
+ − 416 and assigns =
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 417 ANil
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
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parents:
diff
changeset
+ − 418 | ACons "name" "trm3" "assigns"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 419
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 420 (* to be given by the user *)
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 421 primrec
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 422 bv3
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 423 where
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 424 "bv3 ANil = {}"
957
+ − 425 | "bv3 (ACons x t as) = {atom x} \<union> (bv3 as)"
950
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parents:
diff
changeset
+ − 426
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 427 primrec
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 428 fv_trm3 and fv_assigns
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 429 where
957
+ − 430 "fv_trm3 (Vr3 x) = {atom x}"
950
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parents:
diff
changeset
+ − 431 | "fv_trm3 (Ap3 t1 t2) = (fv_trm3 t1) \<union> (fv_trm3 t2)"
957
+ − 432 | "fv_trm3 (Lm3 x t) = (fv_trm3 t) - {atom x}"
950
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parents:
diff
changeset
+ − 433 | "fv_trm3 (Lt3 as t) = (fv_trm3 t - bv3 as) \<union> (fv_assigns as)"
98764f25f012
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parents:
diff
changeset
+ − 434 | "fv_assigns (ANil) = {}"
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parents:
diff
changeset
+ − 435 | "fv_assigns (ACons x t as) = (fv_trm3 t) \<union> (fv_assigns as)"
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parents:
diff
changeset
+ − 436
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 437 (* needs to be stated by the package *)
957
+ − 438 instantiation
+ − 439 trm3 and assigns :: pt
950
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parents:
diff
changeset
+ − 440 begin
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parents:
diff
changeset
+ − 441
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parents:
diff
changeset
+ − 442 primrec
957
+ − 443 permute_trm3 and permute_assigns
950
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parents:
diff
changeset
+ − 444 where
957
+ − 445 "permute_trm3 pi (Vr3 a) = Vr3 (pi \<bullet> a)"
+ − 446 | "permute_trm3 pi (Ap3 t1 t2) = Ap3 (permute_trm3 pi t1) (permute_trm3 pi t2)"
+ − 447 | "permute_trm3 pi (Lm3 a t) = Lm3 (pi \<bullet> a) (permute_trm3 pi t)"
+ − 448 | "permute_trm3 pi (Lt3 as t) = Lt3 (permute_assigns pi as) (permute_trm3 pi t)"
+ − 449 | "permute_assigns pi (ANil) = ANil"
+ − 450 | "permute_assigns pi (ACons a t as) = ACons (pi \<bullet> a) (permute_trm3 pi t) (permute_assigns pi as)"
+ − 451
+ − 452 lemma pt_trm3_assigns_zero:
+ − 453 fixes t::trm3
+ − 454 and b::assigns
+ − 455 shows "0 \<bullet> t = t"
+ − 456 and "0 \<bullet> b = b"
+ − 457 apply(induct t and b rule: trm3_assigns.inducts)
+ − 458 apply(simp_all)
+ − 459 done
+ − 460
+ − 461 lemma pt_trm3_assigns_plus:
+ − 462 fixes t::trm3
+ − 463 and b::assigns
+ − 464 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 465 and "((p + q) \<bullet> b) = p \<bullet> (q \<bullet> b)"
+ − 466 apply(induct t and b rule: trm3_assigns.inducts)
+ − 467 apply(simp_all)
+ − 468 done
+ − 469
+ − 470 instance
+ − 471 apply default
+ − 472 apply(simp_all add: pt_trm3_assigns_zero pt_trm3_assigns_plus)
+ − 473 done
+ − 474
950
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 475
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 476 end
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 477
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 478 inductive
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 479 alpha3 :: "trm3 \<Rightarrow> trm3 \<Rightarrow> bool" ("_ \<approx>3 _" [100, 100] 100)
1092
+ − 480 and
+ − 481 alpha3a :: "assigns \<Rightarrow> assigns \<Rightarrow> bool" ("_ \<approx>3a _" [100, 100] 100)
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 482 where
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added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 483 a1: "a = b \<Longrightarrow> (Vr3 a) \<approx>3 (Vr3 b)"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 484 | a2: "\<lbrakk>t1 \<approx>3 t2; s1 \<approx>3 s2\<rbrakk> \<Longrightarrow> Ap3 t1 s1 \<approx>3 Ap3 t2 s2"
1092
+ − 485 | a3: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha3 fv_rtrm3 pi ({atom b}, s))) \<Longrightarrow> Lm3 a t \<approx>3 Lm3 b s"
+ − 486 | a4: "\<lbrakk>\<exists>pi. ((bv3 bt, t) \<approx>gen alpha3 fv_trm3 pi ((bv3 bs), s));
+ − 487 \<exists>pi. ((bv3 bt, bt) \<approx>gen alpha3a fv_assign pi (bv3 bs, bs))\<rbrakk>
+ − 488 \<Longrightarrow> Lt3 bt t \<approx>3 Lt3 bs s"
+ − 489 | a5: "ANil \<approx>3a ANil"
+ − 490 | a6: "\<lbrakk>a = b; t \<approx>3 s; tt \<approx>3a st\<rbrakk> \<Longrightarrow> ACons a t tt \<approx>3a ACons b s st"
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 491
1092
+ − 492 lemma alpha3_equivp:
+ − 493 "equivp alpha3"
+ − 494 "equivp alpha3a"
957
+ − 495 sorry
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 496
1092
+ − 497 quotient_type
+ − 498 qtrm3 = trm3 / alpha3
+ − 499 and
+ − 500 qassigns = assigns / alpha3a
+ − 501 by (auto intro: alpha3_equivp)
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 502
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 503
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 504 section {*** lam with indirect list recursion ***}
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 505
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 506 datatype trm4 =
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 507 Vr4 "name"
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 508 | Ap4 "trm4" "trm4 list"
1093
+ − 509 | Lm4 "name" "trm4" --"bind (name) in (trm)"
950
98764f25f012
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 510
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 511 thm trm4.recs
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 512
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 513 primrec
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 514 fv_trm4 and fv_trm4_list
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 515 where
957
+ − 516 "fv_trm4 (Vr4 x) = {atom x}"
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 517 | "fv_trm4 (Ap4 t ts) = (fv_trm4 t) \<union> (fv_trm4_list ts)"
957
+ − 518 | "fv_trm4 (Lm4 x t) = (fv_trm4 t) - {atom x}"
950
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 519 | "fv_trm4_list ([]) = {}"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 520 | "fv_trm4_list (t#ts) = (fv_trm4 t) \<union> (fv_trm4_list ts)"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 521
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 522
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 523 (* needs to be stated by the package *)
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 524 (* there cannot be a clause for lists, as *)
1092
+ − 525 (* permutations are already defined in Nominal (also functions, options, and so on) *)
957
+ − 526 instantiation
+ − 527 trm4 :: pt
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 528 begin
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 529
957
+ − 530 (* does not work yet *)
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 531 primrec
963
+ − 532 permute_trm4 and permute_trm4_list
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 533 where
957
+ − 534 "permute_trm4 pi (Vr4 a) = Vr4 (pi \<bullet> a)"
963
+ − 535 | "permute_trm4 pi (Ap4 t ts) = Ap4 (permute_trm4 pi t) (permute_trm4_list pi ts)"
957
+ − 536 | "permute_trm4 pi (Lm4 a t) = Lm4 (pi \<bullet> a) (permute_trm4 pi t)"
963
+ − 537 | "permute_trm4_list pi ([]) = []"
+ − 538 | "permute_trm4_list pi (t#ts) = (permute_trm4 pi t) # (permute_trm4_list pi ts)"
+ − 539
+ − 540 lemma pt_trm4_list_zero:
+ − 541 fixes t::trm4
+ − 542 and ts::"trm4 list"
+ − 543 shows "0 \<bullet> t = t"
+ − 544 and "permute_trm4_list 0 ts = ts"
+ − 545 apply(induct t and ts rule: trm4.inducts)
+ − 546 apply(simp_all)
+ − 547 done
+ − 548
+ − 549 lemma pt_trm4_list_plus:
+ − 550 fixes t::trm4
+ − 551 and ts::"trm4 list"
+ − 552 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 553 and "(permute_trm4_list (p + q) ts) = permute_trm4_list p (permute_trm4_list q ts)"
+ − 554 apply(induct t and ts rule: trm4.inducts)
+ − 555 apply(simp_all)
+ − 556 done
+ − 557
+ − 558
+ − 559 instance
+ − 560 apply(default)
+ − 561 apply(simp_all add: pt_trm4_list_zero pt_trm4_list_plus)
+ − 562 done
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 563
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 564 end
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 565
963
+ − 566 (* "repairing" of the permute function *)
+ − 567 lemma repaired:
+ − 568 fixes ts::"trm4 list"
+ − 569 shows "permute_trm4_list p ts = p \<bullet> ts"
+ − 570 apply(induct ts)
+ − 571 apply(simp_all)
+ − 572 done
+ − 573
976
+ − 574 thm permute_trm4_permute_trm4_list.simps
963
+ − 575 thm permute_trm4_permute_trm4_list.simps[simplified repaired]
+ − 576
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 577 inductive
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 578 alpha4 :: "trm4 \<Rightarrow> trm4 \<Rightarrow> bool" ("_ \<approx>4 _" [100, 100] 100)
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 579 and alpha4list :: "trm4 list \<Rightarrow> trm4 list \<Rightarrow> bool" ("_ \<approx>4list _" [100, 100] 100)
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 580 where
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 581 a1: "a = b \<Longrightarrow> (Vr4 a) \<approx>4 (Vr4 b)"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 582 | a2: "\<lbrakk>t1 \<approx>4 t2; s1 \<approx>4list s2\<rbrakk> \<Longrightarrow> Ap4 t1 s1 \<approx>4 Ap4 t2 s2"
1092
+ − 583 | a3: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha4 fv_rtrm4 pi ({atom b}, s))) \<Longrightarrow> Lm4 a t \<approx>4 Lm4 b s"
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 584 | a5: "[] \<approx>4list []"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 585 | a6: "\<lbrakk>t \<approx>4 s; ts \<approx>4list ss\<rbrakk> \<Longrightarrow> (t#ts) \<approx>4list (s#ss)"
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 586
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 587 lemma alpha4_equivp: "equivp alpha4" sorry
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 588 lemma alpha4list_equivp: "equivp alpha4list" sorry
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 589
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 590 quotient_type
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 591 qtrm4 = trm4 / alpha4 and
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 592 qtrm4list = "trm4 list" / alpha4list
1042
+ − 593 by (simp_all add: alpha4_equivp alpha4list_equivp)
1036
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 594
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 595
1035
+ − 596 datatype rtrm5 =
+ − 597 rVr5 "name"
+ − 598 | rAp5 "rtrm5" "rtrm5"
1093
+ − 599 | rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"
1035
+ − 600 and rlts =
+ − 601 rLnil
+ − 602 | rLcons "name" "rtrm5" "rlts"
+ − 603
+ − 604 primrec
1050
+ − 605 rbv5
1035
+ − 606 where
1050
+ − 607 "rbv5 rLnil = {}"
+ − 608 | "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"
1035
+ − 609
+ − 610 primrec
+ − 611 rfv_trm5 and rfv_lts
+ − 612 where
+ − 613 "rfv_trm5 (rVr5 n) = {atom n}"
+ − 614 | "rfv_trm5 (rAp5 t s) = (rfv_trm5 t) \<union> (rfv_trm5 s)"
1050
+ − 615 | "rfv_trm5 (rLt5 lts t) = (rfv_trm5 t - rbv5 lts) \<union> (rfv_lts lts - rbv5 lts)"
1035
+ − 616 | "rfv_lts (rLnil) = {}"
+ − 617 | "rfv_lts (rLcons n t ltl) = (rfv_trm5 t) \<union> (rfv_lts ltl)"
+ − 618
+ − 619 instantiation
+ − 620 rtrm5 and rlts :: pt
+ − 621 begin
+ − 622
+ − 623 primrec
+ − 624 permute_rtrm5 and permute_rlts
+ − 625 where
+ − 626 "permute_rtrm5 pi (rVr5 a) = rVr5 (pi \<bullet> a)"
+ − 627 | "permute_rtrm5 pi (rAp5 t1 t2) = rAp5 (permute_rtrm5 pi t1) (permute_rtrm5 pi t2)"
1050
+ − 628 | "permute_rtrm5 pi (rLt5 ls t) = rLt5 (permute_rlts pi ls) (permute_rtrm5 pi t)"
1035
+ − 629 | "permute_rlts pi (rLnil) = rLnil"
+ − 630 | "permute_rlts pi (rLcons n t ls) = rLcons (pi \<bullet> n) (permute_rtrm5 pi t) (permute_rlts pi ls)"
+ − 631
+ − 632 lemma pt_rtrm5_zero:
+ − 633 fixes t::rtrm5
+ − 634 and l::rlts
+ − 635 shows "0 \<bullet> t = t"
+ − 636 and "0 \<bullet> l = l"
+ − 637 apply(induct t and l rule: rtrm5_rlts.inducts)
+ − 638 apply(simp_all)
+ − 639 done
+ − 640
+ − 641 lemma pt_rtrm5_plus:
+ − 642 fixes t::rtrm5
+ − 643 and l::rlts
+ − 644 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 645 and "((p + q) \<bullet> l) = p \<bullet> (q \<bullet> l)"
+ − 646 apply(induct t and l rule: rtrm5_rlts.inducts)
+ − 647 apply(simp_all)
+ − 648 done
+ − 649
+ − 650 instance
+ − 651 apply default
+ − 652 apply(simp_all add: pt_rtrm5_zero pt_rtrm5_plus)
+ − 653 done
+ − 654
950
98764f25f012
added Terms to Nominal - Instantiation of two types does not work (ask Florian)
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 655 end
1035
+ − 656
1036
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 657 inductive
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 658 alpha5 :: "rtrm5 \<Rightarrow> rtrm5 \<Rightarrow> bool" ("_ \<approx>5 _" [100, 100] 100)
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 659 and
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 660 alphalts :: "rlts \<Rightarrow> rlts \<Rightarrow> bool" ("_ \<approx>l _" [100, 100] 100)
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 661 where
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 662 a1: "a = b \<Longrightarrow> (rVr5 a) \<approx>5 (rVr5 b)"
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 663 | a2: "\<lbrakk>t1 \<approx>5 t2; s1 \<approx>5 s2\<rbrakk> \<Longrightarrow> rAp5 t1 s1 \<approx>5 rAp5 t2 s2"
1057
+ − 664 | a3: "\<lbrakk>\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 rfv_trm5 pi (rbv5 l2, t2));
+ − 665 \<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts rfv_lts pi (rbv5 l2, l2))\<rbrakk>
1036
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 666 \<Longrightarrow> rLt5 l1 t1 \<approx>5 rLt5 l2 t2"
aaac8274f08c
The alpha-equivalence relation for let-rec. Not sure if correct...
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 667 | a4: "rLnil \<approx>l rLnil"
1040
+ − 668 | a5: "ls1 \<approx>l ls2 \<Longrightarrow> t1 \<approx>5 t2 \<Longrightarrow> n1 = n2 \<Longrightarrow> rLcons n1 t1 ls1 \<approx>l rLcons n2 t2 ls2"
+ − 669
+ − 670 print_theorems
1035
+ − 671
1040
+ − 672 lemma alpha5_inj:
+ − 673 "((rVr5 a) \<approx>5 (rVr5 b)) = (a = b)"
+ − 674 "(rAp5 t1 s1 \<approx>5 rAp5 t2 s2) = (t1 \<approx>5 t2 \<and> s1 \<approx>5 s2)"
1057
+ − 675 "(rLt5 l1 t1 \<approx>5 rLt5 l2 t2) = ((\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 rfv_trm5 pi (rbv5 l2, t2))) \<and>
+ − 676 (\<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts rfv_lts pi (rbv5 l2, l2))))"
1040
+ − 677 "rLnil \<approx>l rLnil"
1057
+ − 678 "(rLcons n1 t1 ls1 \<approx>l rLcons n2 t2 ls2) = (n1 = n2 \<and> ls1 \<approx>l ls2 \<and> t1 \<approx>5 t2)"
1040
+ − 679 apply -
+ − 680 apply (simp_all add: alpha5_alphalts.intros)
+ − 681 apply rule
+ − 682 apply (erule alpha5.cases)
+ − 683 apply (simp_all add: alpha5_alphalts.intros)
+ − 684 apply rule
+ − 685 apply (erule alpha5.cases)
+ − 686 apply (simp_all add: alpha5_alphalts.intros)
+ − 687 apply rule
+ − 688 apply (erule alpha5.cases)
+ − 689 apply (simp_all add: alpha5_alphalts.intros)
+ − 690 apply rule
+ − 691 apply (erule alphalts.cases)
+ − 692 apply (simp_all add: alpha5_alphalts.intros)
+ − 693 done
+ − 694
+ − 695 lemma alpha5_equivps:
+ − 696 shows "equivp alpha5"
+ − 697 and "equivp alphalts"
+ − 698 sorry
+ − 699
+ − 700 quotient_type
+ − 701 trm5 = rtrm5 / alpha5
+ − 702 and
+ − 703 lts = rlts / alphalts
+ − 704 by (auto intro: alpha5_equivps)
+ − 705
1046
+ − 706 quotient_definition
+ − 707 "Vr5 :: name \<Rightarrow> trm5"
+ − 708 as
+ − 709 "rVr5"
+ − 710
+ − 711 quotient_definition
+ − 712 "Ap5 :: trm5 \<Rightarrow> trm5 \<Rightarrow> trm5"
+ − 713 as
+ − 714 "rAp5"
+ − 715
+ − 716 quotient_definition
+ − 717 "Lt5 :: lts \<Rightarrow> trm5 \<Rightarrow> trm5"
+ − 718 as
+ − 719 "rLt5"
+ − 720
+ − 721 quotient_definition
+ − 722 "Lnil :: lts"
+ − 723 as
+ − 724 "rLnil"
+ − 725
+ − 726 quotient_definition
+ − 727 "Lcons :: name \<Rightarrow> trm5 \<Rightarrow> lts \<Rightarrow> lts"
+ − 728 as
+ − 729 "rLcons"
+ − 730
+ − 731 quotient_definition
+ − 732 "fv_trm5 :: trm5 \<Rightarrow> atom set"
+ − 733 as
+ − 734 "rfv_trm5"
+ − 735
+ − 736 quotient_definition
1050
+ − 737 "fv_lts :: lts \<Rightarrow> atom set"
1046
+ − 738 as
+ − 739 "rfv_lts"
+ − 740
1050
+ − 741 quotient_definition
+ − 742 "bv5 :: lts \<Rightarrow> atom set"
+ − 743 as
+ − 744 "rbv5"
+ − 745
1092
+ − 746 lemma rbv5_eqvt:
+ − 747 "pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"
+ − 748 sorry
+ − 749
+ − 750 lemma rfv_trm5_eqvt:
+ − 751 "pi \<bullet> (rfv_trm5 x) = rfv_trm5 (pi \<bullet> x)"
+ − 752 sorry
+ − 753
+ − 754 lemma rfv_lts_eqvt:
+ − 755 "pi \<bullet> (rfv_lts x) = rfv_lts (pi \<bullet> x)"
+ − 756 sorry
+ − 757
+ − 758 lemma alpha5_eqvt:
+ − 759 "xa \<approx>5 y \<Longrightarrow> (x \<bullet> xa) \<approx>5 (x \<bullet> y)"
+ − 760 "xb \<approx>l ya \<Longrightarrow> (x \<bullet> xb) \<approx>l (x \<bullet> ya)"
+ − 761 apply(induct rule: alpha5_alphalts.inducts)
+ − 762 apply (simp_all add: alpha5_inj)
+ − 763 apply (erule exE)+
+ − 764 apply(unfold alpha_gen)
+ − 765 apply (erule conjE)+
+ − 766 apply (rule conjI)
+ − 767 apply (rule_tac x="x \<bullet> pi" in exI)
+ − 768 apply (rule conjI)
+ − 769 apply(rule_tac ?p1="- x" in permute_eq_iff[THEN iffD1])
+ − 770 apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_trm5_eqvt)
+ − 771 apply(rule conjI)
+ − 772 apply(rule_tac ?p1="- x" in fresh_star_permute_iff[THEN iffD1])
+ − 773 apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_trm5_eqvt)
+ − 774 apply (subst permute_eqvt[symmetric])
+ − 775 apply (simp)
+ − 776 apply (rule_tac x="x \<bullet> pia" in exI)
+ − 777 apply (rule conjI)
+ − 778 apply(rule_tac ?p1="- x" in permute_eq_iff[THEN iffD1])
+ − 779 apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_lts_eqvt)
+ − 780 apply(rule conjI)
+ − 781 apply(rule_tac ?p1="- x" in fresh_star_permute_iff[THEN iffD1])
+ − 782 apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_lts_eqvt)
+ − 783 apply (subst permute_eqvt[symmetric])
+ − 784 apply (simp)
+ − 785 done
+ − 786
1050
+ − 787 lemma alpha5_rfv:
+ − 788 "(t \<approx>5 s \<Longrightarrow> rfv_trm5 t = rfv_trm5 s)"
+ − 789 "(l \<approx>l m \<Longrightarrow> rfv_lts l = rfv_lts m)"
+ − 790 apply(induct rule: alpha5_alphalts.inducts)
+ − 791 apply(simp_all add: alpha_gen)
+ − 792 done
+ − 793
1053
+ − 794 lemma bv_list_rsp:
1055
+ − 795 shows "x \<approx>l y \<Longrightarrow> rbv5 x = rbv5 y"
1092
+ − 796 apply(induct rule: alpha5_alphalts.inducts(2))
+ − 797 apply(simp_all)
+ − 798 done
1053
+ − 799
1092
+ − 800 lemma [quot_respect]:
+ − 801 "(alphalts ===> op =) rfv_lts rfv_lts"
+ − 802 "(alpha5 ===> op =) rfv_trm5 rfv_trm5"
+ − 803 "(alphalts ===> op =) rbv5 rbv5"
+ − 804 "(op = ===> alpha5) rVr5 rVr5"
+ − 805 "(alpha5 ===> alpha5 ===> alpha5) rAp5 rAp5"
+ − 806 "(alphalts ===> alpha5 ===> alpha5) rLt5 rLt5"
+ − 807 "(alphalts ===> alpha5 ===> alpha5) rLt5 rLt5"
+ − 808 "(op = ===> alpha5 ===> alphalts ===> alphalts) rLcons rLcons"
+ − 809 "(op = ===> alpha5 ===> alpha5) permute permute"
+ − 810 "(op = ===> alphalts ===> alphalts) permute permute"
+ − 811 apply (simp_all add: alpha5_inj alpha5_rfv alpha5_eqvt bv_list_rsp)
+ − 812 apply (auto)
+ − 813 apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
+ − 814 apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
+ − 815 apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
+ − 816 apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
+ − 817 done
1057
+ − 818
1092
+ − 819 lemma
1053
+ − 820 shows "(alphalts ===> op =) rbv5 rbv5"
+ − 821 by (simp add: bv_list_rsp)
+ − 822
1073
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 823 lemmas trm5_lts_inducts = rtrm5_rlts.inducts[quot_lifted]
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 824
1050
+ − 825 instantiation trm5 and lts :: pt
+ − 826 begin
+ − 827
+ − 828 quotient_definition
+ − 829 "permute_trm5 :: perm \<Rightarrow> trm5 \<Rightarrow> trm5"
+ − 830 as
+ − 831 "permute :: perm \<Rightarrow> rtrm5 \<Rightarrow> rtrm5"
+ − 832
+ − 833 quotient_definition
+ − 834 "permute_lts :: perm \<Rightarrow> lts \<Rightarrow> lts"
+ − 835 as
+ − 836 "permute :: perm \<Rightarrow> rlts \<Rightarrow> rlts"
+ − 837
+ − 838 lemma trm5_lts_zero:
+ − 839 "0 \<bullet> (x\<Colon>trm5) = x"
+ − 840 "0 \<bullet> (y\<Colon>lts) = y"
1073
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 841 apply(induct x and y rule: trm5_lts_inducts)
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 842 apply(simp_all add: permute_rtrm5_permute_rlts.simps[quot_lifted])
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 843 done
1050
+ − 844
+ − 845 lemma trm5_lts_plus:
+ − 846 "(p + q) \<bullet> (x\<Colon>trm5) = p \<bullet> q \<bullet> x"
+ − 847 "(p + q) \<bullet> (y\<Colon>lts) = p \<bullet> q \<bullet> y"
1073
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 848 apply(induct x and y rule: trm5_lts_inducts)
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 849 apply(simp_all add: permute_rtrm5_permute_rlts.simps[quot_lifted])
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 850 done
1050
+ − 851
+ − 852 instance
+ − 853 apply default
+ − 854 apply (simp_all add: trm5_lts_zero trm5_lts_plus)
+ − 855 done
1040
+ − 856
1035
+ − 857 end
1050
+ − 858
1073
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 859 lemmas permute_trm5_lts = permute_rtrm5_permute_rlts.simps[quot_lifted]
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 860
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 861 lemmas alpha5_INJ = alpha5_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
1050
+ − 862
1073
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 863 lemmas bv5[simp] = rbv5.simps[quot_lifted]
1050
+ − 864
1073
53350d409473
Cleaned Terms using [lifted] and found a workaround for the instantiation problem.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 865 lemmas fv_trm5_lts[simp] = rfv_trm5_rfv_lts.simps[quot_lifted]
1050
+ − 866
+ − 867 lemma lets_ok:
+ − 868 "(Lt5 (Lcons x (Vr5 x) Lnil) (Vr5 x)) = (Lt5 (Lcons y (Vr5 y) Lnil) (Vr5 y))"
+ − 869 apply (subst alpha5_INJ)
1057
+ − 870 apply (rule conjI)
+ − 871 apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
+ − 872 apply (simp only: alpha_gen)
+ − 873 apply (simp add: permute_trm5_lts fresh_star_def)
1050
+ − 874 apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
+ − 875 apply (simp only: alpha_gen)
1052
+ − 876 apply (simp add: permute_trm5_lts fresh_star_def)
+ − 877 done
1050
+ − 878
1058
+ − 879 lemma lets_ok2:
+ − 880 "(Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) =
+ − 881 (Lt5 (Lcons y (Vr5 y) (Lcons x (Vr5 x) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
+ − 882 apply (subst alpha5_INJ)
+ − 883 apply (rule conjI)
+ − 884 apply (rule_tac x="0 :: perm" in exI)
+ − 885 apply (simp only: alpha_gen)
+ − 886 apply (simp add: permute_trm5_lts fresh_star_def)
+ − 887 apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
+ − 888 apply (simp only: alpha_gen)
+ − 889 apply (simp add: permute_trm5_lts fresh_star_def)
+ − 890 done
+ − 891
+ − 892
1057
+ − 893 lemma lets_not_ok1:
1056
+ − 894 "x \<noteq> y \<Longrightarrow> (Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
1057
+ − 895 (Lt5 (Lcons y (Vr5 x) (Lcons x (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
+ − 896 apply (subst alpha5_INJ(3))
+ − 897 apply(clarify)
1056
+ − 898 apply (simp add: alpha_gen)
+ − 899 apply (simp add: permute_trm5_lts fresh_star_def)
1057
+ − 900 apply (simp add: alpha5_INJ(5))
+ − 901 apply(clarify)
+ − 902 apply (simp add: alpha5_INJ(2))
+ − 903 apply (simp only: alpha5_INJ(1))
1056
+ − 904 done
+ − 905
1058
+ − 906 lemma distinct_helper:
+ − 907 shows "\<not>(rVr5 x \<approx>5 rAp5 y z)"
+ − 908 apply auto
+ − 909 apply (erule alpha5.cases)
+ − 910 apply (simp_all only: rtrm5.distinct)
+ − 911 done
1057
+ − 912
1058
+ − 913 lemma distinct_helper2:
+ − 914 shows "(Vr5 x) \<noteq> (Ap5 y z)"
+ − 915 by (lifting distinct_helper)
1057
+ − 916
1058
+ − 917 lemma lets_nok:
+ − 918 "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
+ − 919 (Lt5 (Lcons x (Ap5 (Vr5 z) (Vr5 z)) (Lcons y (Vr5 z) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
+ − 920 (Lt5 (Lcons y (Vr5 z) (Lcons x (Ap5 (Vr5 z) (Vr5 z)) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
+ − 921 apply (subst alpha5_INJ)
+ − 922 apply (simp only: alpha_gen permute_trm5_lts fresh_star_def)
+ − 923 apply (subst alpha5_INJ(5))
+ − 924 apply (subst alpha5_INJ(5))
+ − 925 apply (simp add: distinct_helper2)
+ − 926 done
1057
+ − 927
1103
+ − 928
1105
+ − 929 (* example with a bn function defined over the type itself *)
1103
+ − 930 datatype rtrm6 =
+ − 931 rVr6 "name"
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 932 | rLm6 "name" "rtrm6"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 933 | rLt6 "rtrm6" "rtrm6" --"bind (bv6 left) in (right)"
1103
+ − 934
+ − 935 primrec
+ − 936 rbv6
+ − 937 where
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 938 "rbv6 (rVr6 n) = {}"
1117
+ − 939 | "rbv6 (rLm6 n t) = {atom n} \<union> rbv6 t"
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 940 | "rbv6 (rLt6 l r) = rbv6 l \<union> rbv6 r"
1103
+ − 941
+ − 942 primrec
+ − 943 rfv_trm6
+ − 944 where
+ − 945 "rfv_trm6 (rVr6 n) = {atom n}"
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 946 | "rfv_trm6 (rLm6 n t) = (rfv_trm6 t) - {atom n}"
1119
+ − 947 | "rfv_trm6 (rLt6 l r) = (rfv_trm6 r - rbv6 l) \<union> rfv_trm6 l"
1103
+ − 948
+ − 949 instantiation
+ − 950 rtrm6 :: pt
+ − 951 begin
+ − 952
+ − 953 primrec
+ − 954 permute_rtrm6
+ − 955 where
+ − 956 "permute_rtrm6 pi (rVr6 a) = rVr6 (pi \<bullet> a)"
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 957 | "permute_rtrm6 pi (rLm6 n t) = rLm6 (pi \<bullet> n) (permute_rtrm6 pi t)"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 958 | "permute_rtrm6 pi (rLt6 l r) = rLt6 (permute_rtrm6 pi l) (permute_rtrm6 pi r)"
1103
+ − 959
+ − 960 lemma pt_rtrm6_zero:
+ − 961 fixes t::rtrm6
+ − 962 shows "0 \<bullet> t = t"
+ − 963 apply(induct t)
+ − 964 apply(simp_all)
+ − 965 done
+ − 966
+ − 967 lemma pt_rtrm6_plus:
+ − 968 fixes t::rtrm6
+ − 969 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 970 apply(induct t)
+ − 971 apply(simp_all)
+ − 972 done
+ − 973
+ − 974 instance
+ − 975 apply default
+ − 976 apply(simp_all add: pt_rtrm6_zero pt_rtrm6_plus)
+ − 977 done
+ − 978
+ − 979 end
+ − 980
+ − 981 inductive
+ − 982 alpha6 :: "rtrm6 \<Rightarrow> rtrm6 \<Rightarrow> bool" ("_ \<approx>6 _" [100, 100] 100)
+ − 983 where
+ − 984 a1: "a = b \<Longrightarrow> (rVr6 a) \<approx>6 (rVr6 b)"
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 985 | a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha6 rfv_trm6 pi ({atom b}, s))) \<Longrightarrow> rLm6 a t \<approx>6 rLm6 b s"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 986 | a3: "(\<exists>pi. (((rbv6 t1), s1) \<approx>gen alpha6 rfv_trm6 pi ((rbv6 t2), s2))) \<Longrightarrow> rLt6 t1 s1 \<approx>6 rLt6 t2 s2"
1103
+ − 987
+ − 988 lemma alpha6_equivps:
+ − 989 shows "equivp alpha6"
+ − 990 sorry
+ − 991
+ − 992 quotient_type
+ − 993 trm6 = rtrm6 / alpha6
+ − 994 by (auto intro: alpha6_equivps)
+ − 995
+ − 996 quotient_definition
+ − 997 "Vr6 :: name \<Rightarrow> trm6"
+ − 998 as
+ − 999 "rVr6"
+ − 1000
+ − 1001 quotient_definition
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1002 "Lm6 :: name \<Rightarrow> trm6 \<Rightarrow> trm6"
1103
+ − 1003 as
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1004 "rLm6"
1103
+ − 1005
+ − 1006 quotient_definition
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1007 "Lt6 :: trm6 \<Rightarrow> trm6 \<Rightarrow> trm6"
1103
+ − 1008 as
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1009 "rLt6"
1103
+ − 1010
+ − 1011 quotient_definition
+ − 1012 "fv_trm6 :: trm6 \<Rightarrow> atom set"
+ − 1013 as
+ − 1014 "rfv_trm6"
+ − 1015
+ − 1016 quotient_definition
+ − 1017 "bv6 :: trm6 \<Rightarrow> atom set"
+ − 1018 as
+ − 1019 "rbv6"
+ − 1020
1106
+ − 1021 lemma [quot_respect]:
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1022 "(op = ===> alpha6 ===> alpha6) permute permute"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1023 apply auto (* will work with eqvt *)
1106
+ − 1024 sorry
+ − 1025
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1026 (* Definitely not true , see lemma below *)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1027
1104
+ − 1028 lemma [quot_respect]:"(alpha6 ===> op =) rbv6 rbv6"
+ − 1029 apply simp apply clarify
+ − 1030 apply (erule alpha6.induct)
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1031 oops
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1032
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1033 lemma "(a :: name) \<noteq> b \<Longrightarrow> \<not> (alpha6 ===> op =) rbv6 rbv6"
1104
+ − 1034 apply simp
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1035 apply (rule_tac x="rLm6 (a::name) (rVr6 (a :: name))" in exI)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1036 apply (rule_tac x="rLm6 (b::name) (rVr6 (b :: name))" in exI)
1104
+ − 1037 apply simp
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1038 apply (rule a2)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1039 apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1040 apply (simp add: alpha_gen fresh_star_def)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1041 apply (rule a1)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1042 apply (rule refl)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1043 done
1103
+ − 1044
1104
+ − 1045 lemma [quot_respect]:"(alpha6 ===> op =) rfv_trm6 rfv_trm6"
1106
+ − 1046 apply simp apply clarify
+ − 1047 apply (induct_tac x y rule: alpha6.induct)
+ − 1048 apply simp_all
+ − 1049 apply (erule exE)
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1050 apply (simp_all add: alpha_gen)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1051 apply (erule conjE)+
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1052 apply (erule exE)
1106
+ − 1053 apply (erule conjE)+
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1054 apply (simp)
1106
+ − 1055 oops
1103
+ − 1056
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1057
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1058 lemma [quot_respect]: "(op = ===> alpha6) rVr6 rVr6"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1059 by (simp_all add: a1)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1060
1103
+ − 1061 lemma [quot_respect]:
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1062 "(op = ===> alpha6 ===> alpha6) rLm6 rLm6"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1063 "(alpha6 ===> alpha6 ===> alpha6) rLt6 rLt6"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1064 apply simp_all apply (clarify)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1065 apply (rule a2)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1066 apply (rule_tac x="0::perm" in exI)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1067 apply (simp add: alpha_gen)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1068 (* needs rfv6_rsp *) defer
1103
+ − 1069 apply clarify
+ − 1070 apply (rule a3)
+ − 1071 apply (rule_tac x="0::perm" in exI)
+ − 1072 apply (simp add: alpha_gen)
+ − 1073 (* needs rbv6_rsp *)
1106
+ − 1074 oops
1103
+ − 1075
+ − 1076 instantiation trm6 :: pt begin
+ − 1077
1104
+ − 1078 quotient_definition
+ − 1079 "permute_trm6 :: perm \<Rightarrow> trm6 \<Rightarrow> trm6"
+ − 1080 as
+ − 1081 "permute :: perm \<Rightarrow> rtrm6 \<Rightarrow> rtrm6"
+ − 1082
1103
+ − 1083 instance
1106
+ − 1084 apply default
1103
+ − 1085 sorry
+ − 1086 end
+ − 1087
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1088 lemma lifted_induct:
1106
+ − 1089 "\<lbrakk>x1 = x2; \<And>a b. a = b \<Longrightarrow> P (Vr6 a) (Vr6 b);
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1090 \<And>a t b s.
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1091 \<exists>pi. fv_trm6 t - {atom a} = fv_trm6 s - {atom b} \<and>
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1092 (fv_trm6 t - {atom a}) \<sharp>* pi \<and> pi \<bullet> t = s \<and> P (pi \<bullet> t) s \<Longrightarrow>
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1093 P (Lm6 a t) (Lm6 b s);
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1094 \<And>t1 s1 t2 s2.
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1095 \<exists>pi. fv_trm6 s1 - bv6 t1 = fv_trm6 s2 - bv6 t2 \<and>
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1096 (fv_trm6 s1 - bv6 t1) \<sharp>* pi \<and> pi \<bullet> s1 = s2 \<and> P (pi \<bullet> s1) s2 \<Longrightarrow>
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1097 P (Lt6 t1 s1) (Lt6 t2 s2)\<rbrakk>
1106
+ − 1098 \<Longrightarrow> P x1 x2"
+ − 1099 unfolding alpha_gen
+ − 1100 apply (lifting alpha6.induct[unfolded alpha_gen])
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1101 apply injection
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1102 (* notice unsolvable goals: (alpha6 ===> op =) rbv6 rbv6 *)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1103 oops
1106
+ − 1104
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1105 lemma lifted_inject_a3:
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1106 "\<exists>pi. fv_trm6 s1 - bv6 t1 = fv_trm6 s2 - bv6 t2 \<and>
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1107 (fv_trm6 s1 - bv6 t1) \<sharp>* pi \<and> pi \<bullet> s1 = s2 \<Longrightarrow> Lt6 t1 s1 = Lt6 t2 s2"
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1108 apply(lifting a3[unfolded alpha_gen])
1103
+ − 1109 apply injection
1111
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1110 (* notice unsolvable goals: (alpha6 ===> op =) rbv6 rbv6 *)
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1111 oops
ee276c9f12f0
A concrete example, with a proof that rbv is not regular and
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1112
1103
+ − 1113
+ − 1114
+ − 1115
1121
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1116 (* example with a respectful bn function defined over the type itself *)
1131
+ − 1117
1121
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1118 datatype rtrm7 =
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1119 rVr7 "name"
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1120 | rLm7 "name" "rtrm7"
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1121 | rLt7 "rtrm7" "rtrm7" --"bind (bv7 left) in (right)"
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1122
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1123 primrec
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1124 rbv7
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1125 where
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1126 "rbv7 (rVr7 n) = {atom n}"
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1127 | "rbv7 (rLm7 n t) = rbv7 t - {atom n}"
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1128 | "rbv7 (rLt7 l r) = rbv7 l \<union> rbv7 r"
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1129
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1130 primrec
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1131 rfv_trm7
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1132 where
1131
+ − 1133 "rfv_trm7 (rVr7 n) = {atom n}"
+ − 1134 | "rfv_trm7 (rLm7 n t) = (rfv_trm7 t) - {atom n}"
+ − 1135 | "rfv_trm7 (rLt7 l r) = (rfv_trm7 l) \<union> (rfv_trm7 r - rbv7 l)"
+ − 1136
+ − 1137 instantiation
+ − 1138 rtrm7 :: pt
+ − 1139 begin
+ − 1140
+ − 1141 primrec
+ − 1142 permute_rtrm7
+ − 1143 where
+ − 1144 "permute_rtrm7 pi (rVr7 a) = rVr7 (pi \<bullet> a)"
+ − 1145 | "permute_rtrm7 pi (rLm7 a t) = rLm7 (pi \<bullet> a) (permute_rtrm7 pi t)"
+ − 1146 | "permute_rtrm7 pi (rLt7 t1 t2) = rLt7 (permute_rtrm7 pi t1) (permute_rtrm7 pi t2)"
+ − 1147
+ − 1148 lemma pt_rtrm7_zero:
+ − 1149 fixes t::rtrm7
+ − 1150 shows "0 \<bullet> t = t"
+ − 1151 apply(induct t)
+ − 1152 apply(simp_all)
+ − 1153 done
+ − 1154
+ − 1155 lemma pt_rtrm7_plus:
+ − 1156 fixes t::rtrm7
+ − 1157 shows "((p + q) \<bullet> t) = p \<bullet> (q \<bullet> t)"
+ − 1158 apply(induct t)
+ − 1159 apply(simp_all)
+ − 1160 done
1121
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1161
1131
+ − 1162 instance
+ − 1163 apply default
+ − 1164 apply(simp_all add: pt_rtrm7_zero pt_rtrm7_plus)
+ − 1165 done
1121
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1166
1131
+ − 1167 end
+ − 1168
+ − 1169 inductive
+ − 1170 alpha7 :: "rtrm7 \<Rightarrow> rtrm7 \<Rightarrow> bool" ("_ \<approx>7 _" [100, 100] 100)
+ − 1171 where
+ − 1172 a1: "a = b \<Longrightarrow> (rVr7 a) \<approx>7 (rVr7 b)"
+ − 1173 | a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha7 rfv_trm7 pi ({atom b}, s))) \<Longrightarrow> rLm7 a t \<approx>7 rLm7 b s"
+ − 1174 | a3: "(\<exists>pi. (((rbv7 t1), s1) \<approx>gen alpha7 rfv_trm7 pi ((rbv7 t2), s2))) \<Longrightarrow> rLt7 t1 s1 \<approx>7 rLt7 t2 s2"
1121
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1175
1131
+ − 1176 lemma bvfv7: "rbv7 x = rfv_trm7 x"
+ − 1177 apply induct
+ − 1178 apply simp_all
+ − 1179 done
1121
8d3f92694e85
example with a respectful bn function defined over the type itself
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1180
1131
+ − 1181 lemma "(x::name) \<noteq> y \<Longrightarrow> \<not> (alpha7 ===> op =) rbv7 rbv7"
+ − 1182 apply simp
+ − 1183 apply (rule_tac x="rLt7 (rVr7 x) (rVr7 x)" in exI)
+ − 1184 apply (rule_tac x="rLt7 (rVr7 y) (rVr7 y)" in exI)
+ − 1185 apply simp
+ − 1186 apply (rule a3)
+ − 1187 apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
+ − 1188 apply (simp_all add: alpha_gen fresh_star_def)
+ − 1189 apply (rule a1)
+ − 1190 apply (rule refl)
+ − 1191 done
1103
+ − 1192
1133
+ − 1193
+ − 1194
+ − 1195
+ − 1196
+ − 1197 atom_decl name2
+ − 1198
1132
+ − 1199 datatype rfoo8 =
+ − 1200 Foo0 "name"
+ − 1201 | Foo1 "rbar8" "rfoo8" --"bind bv(bar) in foo"
+ − 1202 and rbar8 =
1133
+ − 1203 Bar0 "name2"
+ − 1204 | Bar1 "name" "name2" "rbar8" --"bind second name in b"
1132
+ − 1205
+ − 1206 primrec
+ − 1207 rbv8
+ − 1208 where
+ − 1209 "rbv8 (Bar0 x) = {}"
+ − 1210 | "rbv8 (Bar1 v x b) = {atom v}"
+ − 1211
+ − 1212 primrec
+ − 1213 rfv_foo8 and rfv_bar8
+ − 1214 where
+ − 1215 "rfv_foo8 (Foo0 x) = {atom x}"
+ − 1216 | "rfv_foo8 (Foo1 b t) = (rfv_foo8 t - rbv8 b) \<union> (rfv_bar8 b)"
+ − 1217 | "rfv_bar8 (Bar0 x) = {atom x}"
+ − 1218 | "rfv_bar8 (Bar1 v x t) = {atom v} \<union> (rfv_bar8 t - {atom x})"
+ − 1219
+ − 1220 instantiation
+ − 1221 rfoo8 and rbar8 :: pt
+ − 1222 begin
+ − 1223
+ − 1224 primrec
+ − 1225 permute_rfoo8 and permute_rbar8
+ − 1226 where
+ − 1227 "permute_rfoo8 pi (Foo0 a) = Foo0 (pi \<bullet> a)"
+ − 1228 | "permute_rfoo8 pi (Foo1 b t) = Foo1 (permute_rbar8 pi b) (permute_rfoo8 pi t)"
+ − 1229 | "permute_rbar8 pi (Bar0 a) = Bar0 (pi \<bullet> a)"
+ − 1230 | "permute_rbar8 pi (Bar1 v x t) = Bar1 (pi \<bullet> v) (pi \<bullet> x) (permute_rbar8 pi t)"
+ − 1231
+ − 1232 instance sorry
+ − 1233
+ − 1234 end
+ − 1235
+ − 1236 inductive
+ − 1237 alpha8f :: "rfoo8 \<Rightarrow> rfoo8 \<Rightarrow> bool" ("_ \<approx>f _" [100, 100] 100)
+ − 1238 and
+ − 1239 alpha8b :: "rbar8 \<Rightarrow> rbar8 \<Rightarrow> bool" ("_ \<approx>b _" [100, 100] 100)
+ − 1240 where
+ − 1241 a1: "a = b \<Longrightarrow> (Foo0 a) \<approx>f (Foo0 b)"
+ − 1242 | a2: "a = b \<Longrightarrow> (Bar0 a) \<approx>b (Bar0 b)"
+ − 1243 | a3: "(\<exists>pi. (((rbv8 b1), t1) \<approx>gen alpha8f rfv_foo8 pi ((rbv8 b2), t2))) \<Longrightarrow> Foo1 b1 t1 \<approx>f Foo1 b2 t2"
+ − 1244 | a4: "v1 = v2 \<Longrightarrow> (\<exists>pi. (({atom x1}, t1) \<approx>gen alpha8b rfv_bar8 pi ({atom x2}, t2))) \<Longrightarrow> Bar1 v1 x1 t1 \<approx>b Bar1 v2 x2 t2"
+ − 1245
+ − 1246 lemma "(alpha8b ===> op =) rbv8 rbv8"
+ − 1247 apply simp apply clarify
+ − 1248 apply (erule alpha8f_alpha8b.inducts(2))
+ − 1249 apply (simp_all)
+ − 1250 done
+ − 1251 lemma "(alpha8b ===> op =) rfv_bar8 rfv_bar8"
+ − 1252 apply simp apply clarify
+ − 1253 apply (erule alpha8f_alpha8b.inducts(2))
1133
+ − 1254 apply (simp_all add: alpha_gen)
1132
+ − 1255 done
+ − 1256
1133
+ − 1257 lemma "(c :: name2) \<noteq> b \<Longrightarrow> (atom (a :: name)) \<noteq> (atom b) \<Longrightarrow> \<not> (alpha8f ===> op =) rfv_foo8 rfv_foo8"
+ − 1258 apply (simp_all)
+ − 1259 apply (rule_tac x="Foo1 (Bar1 a c (Bar0 c)) (Foo0 a)" in exI)
+ − 1260 apply (rule_tac x="Foo1 (Bar1 a c (Bar0 b)) (Foo0 a)" in exI)
+ − 1261 apply (rule conjI)
+ − 1262 apply (rule a3)
+ − 1263 apply (rule_tac x="0 :: perm" in exI)
+ − 1264 apply (simp add: alpha_gen fresh_star_def)
+ − 1265 apply (rule a1)
+ − 1266 apply (rule refl)
+ − 1267 apply simp
+ − 1268 done
1132
+ − 1269
+ − 1270
+ − 1271
1103
+ − 1272
1048
+ − 1273 text {* type schemes *}
+ − 1274 datatype ty =
+ − 1275 Var "name"
+ − 1276 | Fun "ty" "ty"
+ − 1277
+ − 1278 instantiation
+ − 1279 ty :: pt
+ − 1280 begin
+ − 1281
+ − 1282 primrec
+ − 1283 permute_ty
+ − 1284 where
+ − 1285 "permute_ty pi (Var a) = Var (pi \<bullet> a)"
+ − 1286 | "permute_ty pi (Fun T1 T2) = Fun (permute_ty pi T1) (permute_ty pi T2)"
1046
+ − 1287
1048
+ − 1288 lemma pt_ty_zero:
+ − 1289 fixes T::ty
+ − 1290 shows "0 \<bullet> T = T"
+ − 1291 apply(induct T rule: ty.inducts)
+ − 1292 apply(simp_all)
+ − 1293 done
1046
+ − 1294
1048
+ − 1295 lemma pt_ty_plus:
+ − 1296 fixes T::ty
+ − 1297 shows "((p + q) \<bullet> T) = p \<bullet> (q \<bullet> T)"
+ − 1298 apply(induct T rule: ty.inducts)
+ − 1299 apply(simp_all)
+ − 1300 done
+ − 1301
+ − 1302 instance
+ − 1303 apply default
+ − 1304 apply(simp_all add: pt_ty_zero pt_ty_plus)
+ − 1305 done
1046
+ − 1306
1035
+ − 1307 end
1048
+ − 1308
+ − 1309 datatype tyS =
+ − 1310 All "name set" "ty"
1046
+ − 1311
1048
+ − 1312 instantiation
+ − 1313 tyS :: pt
+ − 1314 begin
+ − 1315
+ − 1316 primrec
+ − 1317 permute_tyS
+ − 1318 where
+ − 1319 "permute_tyS pi (All xs T) = All (pi \<bullet> xs) (pi \<bullet> T)"
1046
+ − 1320
1048
+ − 1321 lemma pt_tyS_zero:
+ − 1322 fixes T::tyS
+ − 1323 shows "0 \<bullet> T = T"
+ − 1324 apply(induct T rule: tyS.inducts)
+ − 1325 apply(simp_all)
+ − 1326 done
1046
+ − 1327
1048
+ − 1328 lemma pt_tyS_plus:
+ − 1329 fixes T::tyS
+ − 1330 shows "((p + q) \<bullet> T) = p \<bullet> (q \<bullet> T)"
+ − 1331 apply(induct T rule: tyS.inducts)
+ − 1332 apply(simp_all)
+ − 1333 done
+ − 1334
+ − 1335 instance
+ − 1336 apply default
+ − 1337 apply(simp_all add: pt_tyS_zero pt_tyS_plus)
+ − 1338 done
+ − 1339
+ − 1340 end
1046
+ − 1341
1040
+ − 1342
1048
+ − 1343 abbreviation
+ − 1344 "atoms xs \<equiv> {atom x| x. x \<in> xs}"
+ − 1345
+ − 1346 primrec
+ − 1347 rfv_ty
+ − 1348 where
+ − 1349 "rfv_ty (Var n) = {atom n}"
+ − 1350 | "rfv_ty (Fun T1 T2) = (rfv_ty T1) \<union> (rfv_ty T2)"
+ − 1351
+ − 1352 primrec
+ − 1353 rfv_tyS
+ − 1354 where
+ − 1355 "rfv_tyS (All xs T) = (rfv_ty T - atoms xs)"
+ − 1356
+ − 1357 inductive
+ − 1358 alpha_tyS :: "tyS \<Rightarrow> tyS \<Rightarrow> bool" ("_ \<approx>tyS _" [100, 100] 100)
+ − 1359 where
+ − 1360 a1: "\<exists>pi. ((atoms xs1, T1) \<approx>gen (op =) rfv_ty pi (atoms xs2, T2))
+ − 1361 \<Longrightarrow> All xs1 T1 \<approx>tyS All xs2 T2"
+ − 1362
+ − 1363 lemma
+ − 1364 shows "All {a, b} (Fun (Var a) (Var b)) \<approx>tyS All {b, a} (Fun (Var a) (Var b))"
+ − 1365 apply(rule a1)
+ − 1366 apply(simp add: alpha_gen)
+ − 1367 apply(rule_tac x="0::perm" in exI)
+ − 1368 apply(simp add: fresh_star_def)
+ − 1369 done
+ − 1370
+ − 1371 lemma
+ − 1372 shows "All {a, b} (Fun (Var a) (Var b)) \<approx>tyS All {a, b} (Fun (Var b) (Var a))"
+ − 1373 apply(rule a1)
+ − 1374 apply(simp add: alpha_gen)
+ − 1375 apply(rule_tac x="(atom a \<rightleftharpoons> atom b)" in exI)
+ − 1376 apply(simp add: fresh_star_def)
+ − 1377 done
+ − 1378
+ − 1379 lemma
+ − 1380 shows "All {a, b, c} (Fun (Var a) (Var b)) \<approx>tyS All {a, b} (Fun (Var a) (Var b))"
+ − 1381 apply(rule a1)
+ − 1382 apply(simp add: alpha_gen)
+ − 1383 apply(rule_tac x="0::perm" in exI)
+ − 1384 apply(simp add: fresh_star_def)
+ − 1385 done
+ − 1386
+ − 1387 lemma
+ − 1388 assumes a: "a \<noteq> b"
+ − 1389 shows "\<not>(All {a, b} (Fun (Var a) (Var b)) \<approx>tyS All {c} (Fun (Var c) (Var c)))"
+ − 1390 using a
+ − 1391 apply(clarify)
+ − 1392 apply(erule alpha_tyS.cases)
+ − 1393 apply(simp add: alpha_gen)
+ − 1394 apply(erule conjE)+
+ − 1395 apply(erule exE)
+ − 1396 apply(erule conjE)+
+ − 1397 apply(clarify)
+ − 1398 apply(simp)
+ − 1399 apply(simp add: fresh_star_def)
+ − 1400 apply(auto)
+ − 1401 done
1051
+ − 1402
1048
+ − 1403
1035
+ − 1404 end