106 lemma mplus_fail2 [simp]: "m +++ fail = m" |
105 lemma mplus_fail2 [simp]: "m +++ fail = m" |
107 by (cases m, simp_all) |
106 by (cases m, simp_all) |
108 |
107 |
109 lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)" |
108 lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)" |
110 by (cases x, simp_all) |
109 by (cases x, simp_all) |
111 |
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112 subsubsection {* Fatbar combinator *} |
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113 |
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114 definition |
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115 fatbar :: "('a \<rightarrow> 'b match) \<rightarrow> ('a \<rightarrow> 'b match) \<rightarrow> ('a \<rightarrow> 'b match)" where |
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116 "fatbar = (\<Lambda> a b x. a\<cdot>x +++ b\<cdot>x)" |
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117 |
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118 abbreviation |
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119 fatbar_syn :: "['a \<rightarrow> 'b match, 'a \<rightarrow> 'b match] \<Rightarrow> 'a \<rightarrow> 'b match" (infixr "\<parallel>" 60) where |
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120 "m1 \<parallel> m2 == fatbar\<cdot>m1\<cdot>m2" |
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121 |
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122 lemma fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> (m \<parallel> ms)\<cdot>x = \<bottom>" |
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123 by (simp add: fatbar_def) |
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124 |
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125 lemma fatbar2: "m\<cdot>x = fail \<Longrightarrow> (m \<parallel> ms)\<cdot>x = ms\<cdot>x" |
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126 by (simp add: fatbar_def) |
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127 |
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128 lemma fatbar3: "m\<cdot>x = succeed\<cdot>y \<Longrightarrow> (m \<parallel> ms)\<cdot>x = succeed\<cdot>y" |
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129 by (simp add: fatbar_def) |
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130 |
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131 lemmas fatbar_simps = fatbar1 fatbar2 fatbar3 |
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132 |
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133 lemma run_fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = \<bottom>" |
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134 by (simp add: fatbar_def) |
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135 |
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136 lemma run_fatbar2: "m\<cdot>x = fail \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = run\<cdot>(ms\<cdot>x)" |
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137 by (simp add: fatbar_def) |
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138 |
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139 lemma run_fatbar3: "m\<cdot>x = succeed\<cdot>y \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = y" |
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140 by (simp add: fatbar_def) |
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141 |
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142 lemmas run_fatbar_simps [simp] = run_fatbar1 run_fatbar2 run_fatbar3 |
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143 |
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144 subsection {* Case branch combinator *} |
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145 |
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146 definition |
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147 branch :: "('a \<rightarrow> 'b match) \<Rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'c match)" where |
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148 "branch p \<equiv> \<Lambda> r x. match_case\<cdot>fail\<cdot>(\<Lambda> y. succeed\<cdot>(r\<cdot>y))\<cdot>(p\<cdot>x)" |
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149 |
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150 lemma branch_simps: |
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151 "p\<cdot>x = \<bottom> \<Longrightarrow> branch p\<cdot>r\<cdot>x = \<bottom>" |
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152 "p\<cdot>x = fail \<Longrightarrow> branch p\<cdot>r\<cdot>x = fail" |
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153 "p\<cdot>x = succeed\<cdot>y \<Longrightarrow> branch p\<cdot>r\<cdot>x = succeed\<cdot>(r\<cdot>y)" |
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154 by (simp_all add: branch_def) |
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155 |
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156 lemma branch_succeed [simp]: "branch succeed\<cdot>r\<cdot>x = succeed\<cdot>(r\<cdot>x)" |
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157 by (simp add: branch_def) |
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158 |
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159 subsubsection {* Cases operator *} |
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160 |
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161 definition |
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162 cases :: "'a match \<rightarrow> 'a::pcpo" where |
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163 "cases = match_case\<cdot>\<bottom>\<cdot>ID" |
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164 |
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165 text {* rewrite rules for cases *} |
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166 |
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167 lemma cases_strict [simp]: "cases\<cdot>\<bottom> = \<bottom>" |
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168 by (simp add: cases_def) |
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169 |
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170 lemma cases_fail [simp]: "cases\<cdot>fail = \<bottom>" |
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171 by (simp add: cases_def) |
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172 |
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173 lemma cases_succeed [simp]: "cases\<cdot>(succeed\<cdot>x) = x" |
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174 by (simp add: cases_def) |
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175 |
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176 subsection {* Case syntax *} |
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177 |
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178 nonterminals |
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179 Case_syn Cases_syn |
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180 |
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181 syntax |
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182 "_Case_syntax":: "['a, Cases_syn] => 'b" ("(Case _ of/ _)" 10) |
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183 "_Case1" :: "['a, 'b] => Case_syn" ("(2_ =>/ _)" 10) |
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184 "" :: "Case_syn => Cases_syn" ("_") |
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185 "_Case2" :: "[Case_syn, Cases_syn] => Cases_syn" ("_/ | _") |
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186 |
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187 syntax (xsymbols) |
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188 "_Case1" :: "['a, 'b] => Case_syn" ("(2_ \<Rightarrow>/ _)" 10) |
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189 |
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190 translations |
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191 "_Case_syntax x ms" == "CONST Fixrec.cases\<cdot>(ms\<cdot>x)" |
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192 "_Case2 m ms" == "m \<parallel> ms" |
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193 |
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194 text {* Parsing Case expressions *} |
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195 |
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196 syntax |
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197 "_pat" :: "'a" |
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198 "_variable" :: "'a" |
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199 "_noargs" :: "'a" |
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200 |
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201 translations |
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202 "_Case1 p r" => "CONST branch (_pat p)\<cdot>(_variable p r)" |
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203 "_variable (_args x y) r" => "CONST csplit\<cdot>(_variable x (_variable y r))" |
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204 "_variable _noargs r" => "CONST unit_when\<cdot>r" |
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205 |
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206 parse_translation {* |
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207 (* rewrite (_pat x) => (succeed) *) |
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208 (* rewrite (_variable x t) => (Abs_CFun (%x. t)) *) |
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209 [(@{syntax_const "_pat"}, fn _ => Syntax.const @{const_syntax Fixrec.succeed}), |
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210 mk_binder_tr (@{syntax_const "_variable"}, @{const_syntax Abs_CFun})]; |
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211 *} |
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212 |
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213 text {* Printing Case expressions *} |
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214 |
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215 syntax |
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216 "_match" :: "'a" |
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217 |
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218 print_translation {* |
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219 let |
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220 fun dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax unit_when},_) $ t) = |
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221 (Syntax.const @{syntax_const "_noargs"}, t) |
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222 | dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax csplit},_) $ t) = |
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223 let |
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224 val (v1, t1) = dest_LAM t; |
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225 val (v2, t2) = dest_LAM t1; |
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226 in (Syntax.const @{syntax_const "_args"} $ v1 $ v2, t2) end |
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227 | dest_LAM (Const (@{const_syntax Abs_CFun},_) $ t) = |
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228 let |
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229 val abs = |
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230 case t of Abs abs => abs |
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231 | _ => ("x", dummyT, incr_boundvars 1 t $ Bound 0); |
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232 val (x, t') = atomic_abs_tr' abs; |
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233 in (Syntax.const @{syntax_const "_variable"} $ x, t') end |
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234 | dest_LAM _ = raise Match; (* too few vars: abort translation *) |
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235 |
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236 fun Case1_tr' [Const(@{const_syntax branch},_) $ p, r] = |
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237 let val (v, t) = dest_LAM r in |
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238 Syntax.const @{syntax_const "_Case1"} $ |
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239 (Syntax.const @{syntax_const "_match"} $ p $ v) $ t |
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240 end; |
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241 |
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242 in [(@{const_syntax Rep_CFun}, Case1_tr')] end; |
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243 *} |
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244 |
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245 translations |
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246 "x" <= "_match (CONST Fixrec.succeed) (_variable x)" |
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247 |
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248 |
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249 subsection {* Pattern combinators for data constructors *} |
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250 |
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251 types ('a, 'b) pat = "'a \<rightarrow> 'b match" |
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252 |
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253 definition |
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254 cpair_pat :: "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a \<times> 'b, 'c \<times> 'd) pat" where |
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255 "cpair_pat p1 p2 = (\<Lambda>(x, y). |
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256 match_case\<cdot>fail\<cdot>(\<Lambda> a. match_case\<cdot>fail\<cdot>(\<Lambda> b. succeed\<cdot>(a, b))\<cdot>(p2\<cdot>y))\<cdot>(p1\<cdot>x))" |
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257 |
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258 definition |
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259 spair_pat :: |
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260 "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a::pcpo \<otimes> 'b::pcpo, 'c \<times> 'd) pat" where |
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261 "spair_pat p1 p2 = (\<Lambda>(:x, y:). cpair_pat p1 p2\<cdot>(x, y))" |
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262 |
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263 definition |
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264 sinl_pat :: "('a, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where |
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265 "sinl_pat p = sscase\<cdot>p\<cdot>(\<Lambda> x. fail)" |
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266 |
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267 definition |
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268 sinr_pat :: "('b, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where |
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269 "sinr_pat p = sscase\<cdot>(\<Lambda> x. fail)\<cdot>p" |
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270 |
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271 definition |
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272 up_pat :: "('a, 'b) pat \<Rightarrow> ('a u, 'b) pat" where |
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273 "up_pat p = fup\<cdot>p" |
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274 |
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275 definition |
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276 TT_pat :: "(tr, unit) pat" where |
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277 "TT_pat = (\<Lambda> b. If b then succeed\<cdot>() else fail fi)" |
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278 |
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279 definition |
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280 FF_pat :: "(tr, unit) pat" where |
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281 "FF_pat = (\<Lambda> b. If b then fail else succeed\<cdot>() fi)" |
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282 |
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283 definition |
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284 ONE_pat :: "(one, unit) pat" where |
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285 "ONE_pat = (\<Lambda> ONE. succeed\<cdot>())" |
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286 |
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287 text {* Parse translations (patterns) *} |
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288 translations |
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289 "_pat (XCONST Pair x y)" => "CONST cpair_pat (_pat x) (_pat y)" |
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290 "_pat (XCONST spair\<cdot>x\<cdot>y)" => "CONST spair_pat (_pat x) (_pat y)" |
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291 "_pat (XCONST sinl\<cdot>x)" => "CONST sinl_pat (_pat x)" |
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292 "_pat (XCONST sinr\<cdot>x)" => "CONST sinr_pat (_pat x)" |
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293 "_pat (XCONST up\<cdot>x)" => "CONST up_pat (_pat x)" |
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294 "_pat (XCONST TT)" => "CONST TT_pat" |
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295 "_pat (XCONST FF)" => "CONST FF_pat" |
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296 "_pat (XCONST ONE)" => "CONST ONE_pat" |
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297 |
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298 text {* CONST version is also needed for constructors with special syntax *} |
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299 translations |
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300 "_pat (CONST Pair x y)" => "CONST cpair_pat (_pat x) (_pat y)" |
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301 "_pat (CONST spair\<cdot>x\<cdot>y)" => "CONST spair_pat (_pat x) (_pat y)" |
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302 |
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303 text {* Parse translations (variables) *} |
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304 translations |
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305 "_variable (XCONST Pair x y) r" => "_variable (_args x y) r" |
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306 "_variable (XCONST spair\<cdot>x\<cdot>y) r" => "_variable (_args x y) r" |
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307 "_variable (XCONST sinl\<cdot>x) r" => "_variable x r" |
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308 "_variable (XCONST sinr\<cdot>x) r" => "_variable x r" |
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309 "_variable (XCONST up\<cdot>x) r" => "_variable x r" |
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310 "_variable (XCONST TT) r" => "_variable _noargs r" |
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311 "_variable (XCONST FF) r" => "_variable _noargs r" |
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312 "_variable (XCONST ONE) r" => "_variable _noargs r" |
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313 |
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314 translations |
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315 "_variable (CONST Pair x y) r" => "_variable (_args x y) r" |
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316 "_variable (CONST spair\<cdot>x\<cdot>y) r" => "_variable (_args x y) r" |
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317 |
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318 text {* Print translations *} |
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319 translations |
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320 "CONST Pair (_match p1 v1) (_match p2 v2)" |
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321 <= "_match (CONST cpair_pat p1 p2) (_args v1 v2)" |
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322 "CONST spair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)" |
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323 <= "_match (CONST spair_pat p1 p2) (_args v1 v2)" |
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324 "CONST sinl\<cdot>(_match p1 v1)" <= "_match (CONST sinl_pat p1) v1" |
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325 "CONST sinr\<cdot>(_match p1 v1)" <= "_match (CONST sinr_pat p1) v1" |
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326 "CONST up\<cdot>(_match p1 v1)" <= "_match (CONST up_pat p1) v1" |
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327 "CONST TT" <= "_match (CONST TT_pat) _noargs" |
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328 "CONST FF" <= "_match (CONST FF_pat) _noargs" |
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329 "CONST ONE" <= "_match (CONST ONE_pat) _noargs" |
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330 |
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331 lemma cpair_pat1: |
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332 "branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>(x, y) = \<bottom>" |
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333 apply (simp add: branch_def cpair_pat_def) |
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334 apply (cases "p\<cdot>x", simp_all) |
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335 done |
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336 |
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337 lemma cpair_pat2: |
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338 "branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>(x, y) = fail" |
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339 apply (simp add: branch_def cpair_pat_def) |
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340 apply (cases "p\<cdot>x", simp_all) |
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341 done |
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342 |
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343 lemma cpair_pat3: |
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344 "branch p\<cdot>r\<cdot>x = succeed\<cdot>s \<Longrightarrow> |
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345 branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>(x, y) = branch q\<cdot>s\<cdot>y" |
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346 apply (simp add: branch_def cpair_pat_def) |
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347 apply (cases "p\<cdot>x", simp_all) |
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348 apply (cases "q\<cdot>y", simp_all) |
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349 done |
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350 |
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351 lemmas cpair_pat [simp] = |
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352 cpair_pat1 cpair_pat2 cpair_pat3 |
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353 |
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354 lemma spair_pat [simp]: |
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355 "branch (spair_pat p1 p2)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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356 "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> |
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357 \<Longrightarrow> branch (spair_pat p1 p2)\<cdot>r\<cdot>(:x, y:) = |
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358 branch (cpair_pat p1 p2)\<cdot>r\<cdot>(x, y)" |
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359 by (simp_all add: branch_def spair_pat_def) |
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360 |
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361 lemma sinl_pat [simp]: |
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362 "branch (sinl_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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363 "x \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = branch p\<cdot>r\<cdot>x" |
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364 "y \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = fail" |
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365 by (simp_all add: branch_def sinl_pat_def) |
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366 |
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367 lemma sinr_pat [simp]: |
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368 "branch (sinr_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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369 "x \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = fail" |
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370 "y \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = branch p\<cdot>r\<cdot>y" |
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371 by (simp_all add: branch_def sinr_pat_def) |
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372 |
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373 lemma up_pat [simp]: |
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374 "branch (up_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>" |
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375 "branch (up_pat p)\<cdot>r\<cdot>(up\<cdot>x) = branch p\<cdot>r\<cdot>x" |
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376 by (simp_all add: branch_def up_pat_def) |
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377 |
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378 lemma TT_pat [simp]: |
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379 "branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>" |
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380 "branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = succeed\<cdot>r" |
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381 "branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = fail" |
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382 by (simp_all add: branch_def TT_pat_def) |
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383 |
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384 lemma FF_pat [simp]: |
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385 "branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>" |
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386 "branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = fail" |
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387 "branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = succeed\<cdot>r" |
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388 by (simp_all add: branch_def FF_pat_def) |
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389 |
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390 lemma ONE_pat [simp]: |
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391 "branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>" |
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392 "branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>ONE = succeed\<cdot>r" |
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393 by (simp_all add: branch_def ONE_pat_def) |
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394 |
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395 |
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396 subsection {* Wildcards, as-patterns, and lazy patterns *} |
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397 |
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398 definition |
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399 wild_pat :: "'a \<rightarrow> unit match" where |
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400 "wild_pat = (\<Lambda> x. succeed\<cdot>())" |
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401 |
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402 definition |
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403 as_pat :: "('a \<rightarrow> 'b match) \<Rightarrow> 'a \<rightarrow> ('a \<times> 'b) match" where |
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404 "as_pat p = (\<Lambda> x. match_case\<cdot>fail\<cdot>(\<Lambda> a. succeed\<cdot>(x, a))\<cdot>(p\<cdot>x))" |
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405 |
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406 definition |
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407 lazy_pat :: "('a \<rightarrow> 'b::pcpo match) \<Rightarrow> ('a \<rightarrow> 'b match)" where |
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408 "lazy_pat p = (\<Lambda> x. succeed\<cdot>(cases\<cdot>(p\<cdot>x)))" |
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409 |
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410 text {* Parse translations (patterns) *} |
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411 translations |
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412 "_pat _" => "CONST wild_pat" |
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413 |
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414 text {* Parse translations (variables) *} |
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415 translations |
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416 "_variable _ r" => "_variable _noargs r" |
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417 |
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418 text {* Print translations *} |
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419 translations |
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420 "_" <= "_match (CONST wild_pat) _noargs" |
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421 |
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422 lemma wild_pat [simp]: "branch wild_pat\<cdot>(unit_when\<cdot>r)\<cdot>x = succeed\<cdot>r" |
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423 by (simp add: branch_def wild_pat_def) |
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424 |
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425 lemma as_pat [simp]: |
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426 "branch (as_pat p)\<cdot>(csplit\<cdot>r)\<cdot>x = branch p\<cdot>(r\<cdot>x)\<cdot>x" |
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427 apply (simp add: branch_def as_pat_def) |
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428 apply (cases "p\<cdot>x", simp_all) |
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429 done |
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430 |
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431 lemma lazy_pat [simp]: |
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432 "branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = succeed\<cdot>(r\<cdot>\<bottom>)" |
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433 "branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = succeed\<cdot>(r\<cdot>\<bottom>)" |
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434 "branch p\<cdot>r\<cdot>x = succeed\<cdot>s \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = succeed\<cdot>s" |
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435 apply (simp_all add: branch_def lazy_pat_def) |
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436 apply (cases "p\<cdot>x", simp_all)+ |
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437 done |
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438 |
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439 |
110 |
440 subsection {* Match functions for built-in types *} |
111 subsection {* Match functions for built-in types *} |
441 |
112 |
442 default_sort pcpo |
113 default_sort pcpo |
443 |
114 |