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theory LamEx
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imports Nominal QuotMain
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begin
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atom_decl name
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nominal_datatype rlam =
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rVar "name"
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| rApp "rlam" "rlam"
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| rLam "name" "rlam"
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inductive
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alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
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where
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a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
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| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
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229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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| a3: "\<lbrakk>t \<approx> ([(a,b)]\<bullet>s); a\<sharp>[b].s\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"
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quotient lam = rlam / alpha
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sorry
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print_quotients
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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quotient_def (for lam)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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Var :: "name \<Rightarrow> lam"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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where
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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"Var \<equiv> rVar"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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quotient_def (for lam)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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App :: "lam \<Rightarrow> lam \<Rightarrow> lam"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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where
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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"App \<equiv> rApp"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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quotient_def (for lam)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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Lam :: "name \<Rightarrow> lam \<Rightarrow> lam"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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where
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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"Lam \<equiv> rLam"
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thm Var_def
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thm App_def
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thm Lam_def
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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(* definition of overloaded permutation function *)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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(* for the lifted type lam *)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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overloading
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
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perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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begin
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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quotient_def (for lam)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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perm_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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where
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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"perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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end
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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thm perm_lam_def
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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(* lemmas that need to lift *)
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lemma pi_var_com:
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fixes pi::"'x prm"
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shows "(pi\<bullet>rVar a) \<approx> rVar (pi\<bullet>a)"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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sorry
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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lemma pi_app_com:
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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fixes pi::"'x prm"
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shows "(pi\<bullet>rApp t1 t2) \<approx> rApp (pi\<bullet>t1) (pi\<bullet>t2)"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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sorry
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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lemma pi_lam_com:
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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fixes pi::"'x prm"
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shows "(pi\<bullet>rLam a t) \<approx> rLam (pi\<bullet>a) (pi\<bullet>t)"
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229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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sorry
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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lemma real_alpha:
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assumes "t = ([(a,b)]\<bullet>s)" "a\<sharp>s"
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shows "Lam a t = Lam b s"
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sorry
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(* Construction Site code *)
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lemma perm_rsp: "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
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fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(auto)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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(* this is propably true if some type conditions are imposed ;o) *)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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sorry
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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lemma fresh_rsp: "(op = ===> alpha ===> op =) fresh fresh"
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fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(auto)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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(* this is probably only true if some type conditions are imposed *)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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sorry
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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lemma rVar_rsp: "(op = ===> alpha) rVar rVar"
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apply(auto)
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apply(rule a1)
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apply(simp)
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done
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lemma rApp_rsp: "(alpha ===> alpha ===> alpha) rApp rApp"
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apply(auto)
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apply(rule a2)
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apply (assumption)
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apply (assumption)
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done
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lemma rLam_rsp: "(op = ===> alpha ===> alpha) rLam rLam"
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(auto)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(rule a3)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(rule_tac t="[(x,x)]\<bullet>y" and s="y" in subst)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(rule sym)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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apply(rule trans)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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apply(rule pt_name3)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(rule at_ds1[OF at_name_inst])
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13f985a93dbc
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Christian Urban <urbanc@in.tum.de>
diff
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apply(simp add: pt_name1)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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apply(assumption)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
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changeset
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apply(simp add: abs_fresh)
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13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
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done
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ML {* val defs = @{thms Var_def App_def Lam_def perm_lam_def} *}
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ML {* val consts = [@{const_name "rVar"}, @{const_name "rApp"}, @{const_name "rLam"}, @{const_name "perm"}]; *}
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ML {* val rty = @{typ "rlam"} *}
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ML {* val qty = @{typ "lam"} *}
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ML {* val rel = @{term "alpha"} *}
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ML {* val rel_eqv = (#equiv_thm o hd) (quotdata_lookup @{context}) *}
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ML {* val rel_refl = @{thm EQUIV_REFL} OF [rel_eqv] *}
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ML {* val quot = @{thm QUOTIENT_lam} *}
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ML {* val rsp_thms = @{thms perm_rsp fresh_rsp rVar_rsp rApp_rsp rLam_rsp} @ @{thms ho_all_prs ho_ex_prs} *}
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ML {* val trans2 = @{thm QUOT_TYPE_I_lam.R_trans2} *}
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ML {* val reps_same = @{thm QUOT_TYPE_I_lam.REPS_same} *}
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ML {* add_lower_defs @{context} @{thms perm_lam_def} *}
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ML {* val rr = @{thm eq_reflection} OF [hd (rev (add_lower_defs @{context} @{thms perm_lam_def}))] *}
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ML {* val rrr = repeat_eqsubst_thm @{context} @{thms prod_fun_id map_id} rr *}
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ML {*
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fun lift_thm_lam lthy t =
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lift_thm lthy consts rty qty rel rel_eqv rel_refl quot rsp_thms trans2 reps_same defs t
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*}
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ML {* lift_thm_lam @{context} @{thm pi_var_com} *}
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ML {* lift_thm_lam @{context} @{thm pi_app_com} *}
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ML {* lift_thm_lam @{context} @{thm pi_lam_com} *}
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fun
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option_map::"('a \<Rightarrow> 'b) \<Rightarrow> ('a noption) \<Rightarrow> ('b noption)"
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where
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"option_map f (nSome x) = nSome (f x)"
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| "option_map f nNone = nNone"
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fun
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option_rel
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where
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"option_rel r (nSome x) (nSome y) = r x y"
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| "option_rel r _ _ = False"
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declare [[map noption = (option_map, option_rel)]]
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lemma OPT_QUOTIENT:
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assumes q: "QUOTIENT R Abs Rep"
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shows "QUOTIENT (option_rel R) (option_map Abs) (option_map Rep)"
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apply (unfold QUOTIENT_def)
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apply (auto)
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using q
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apply (unfold QUOTIENT_def)
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apply (case_tac "a :: 'b noption")
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apply (simp)
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apply (simp)
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apply (case_tac "a :: 'b noption")
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apply (simp only: option_map.simps)
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apply (subst option_rel.simps)
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(* Simp starts hanging so don't know how to continue *)
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sorry
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(* Christian: Does it make sense? *)
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lemma abs_fun_rsp: "(op = ===> alpha ===> op = ===> op =) abs_fun abs_fun"
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sorry
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(* Should not be needed *)
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lemma eq_rsp2: "((op = ===> op =) ===> (op = ===> op =) ===> op =) op = op ="
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apply auto
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apply (rule ext)
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apply auto
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apply (rule ext)
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apply auto
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done
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(* Should not be needed *)
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lemma perm_rsp_eq: "(op = ===> (op = ===> op =) ===> op = ===> op =) op \<bullet> op \<bullet>"
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apply auto
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thm arg_cong2
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apply (rule_tac f="perm x" in arg_cong2)
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apply (auto)
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apply (rule ext)
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apply (auto)
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done
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(* Should not be needed *)
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lemma fresh_rsp_eq: "(op = ===> (op = ===> op =) ===> op =) fresh fresh"
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apply (simp add: FUN_REL.simps)
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apply (metis ext)
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done
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(* It is just a test, it doesn't seem true... *)
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lemma quotient_cheat: "QUOTIENT op = (option_map ABS_lam) (option_map REP_lam)"
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sorry
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ML {* val rsp_thms = @{thms abs_fun_rsp OPT_QUOTIENT eq_rsp2 quotient_cheat perm_rsp_eq fresh_rsp_eq} @ rsp_thms *}
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ML {*
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fun lift_thm_lam lthy t =
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lift_thm lthy consts rty qty rel rel_eqv rel_refl quot rsp_thms trans2 reps_same defs t
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*}
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thm a3
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ML {* Toplevel.program (fn () => lift_thm_lam @{context} @{thm a3}) *}
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ML {* val t_u = MetaSimplifier.rewrite_rule @{thms fresh_def supp_def} @{thm a3} *}
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ML {* Toplevel.program (fn () => lift_thm_lam @{context} t_u) *}
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ML t_u
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ML {* val t_a = atomize_thm t_u *}
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ML {* val t_r = regularize t_a rty rel rel_eqv @{context} *}
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ML {* val t = (fst o Thm.dest_comb o snd o Thm.dest_comb) (cprop_of t_u) *}
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ML {* val t = (snd o Thm.dest_comb o snd o Thm.dest_comb) t *}
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ML {* val t = (snd o Thm.dest_comb o snd o Thm.dest_comb) t *}
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ML {* val t = (snd o (Thm.dest_abs NONE) o snd o Thm.dest_comb) t *}
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ML {* val t = (snd o Thm.dest_comb o snd o Thm.dest_comb) t *}
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ML {* val t = (snd o (Thm.dest_abs NONE) o snd o Thm.dest_comb) t *}
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ML {* val t = (fst o Thm.dest_comb o snd o Thm.dest_comb) t *}
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ML {* term_of t *}
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term "[b].(s::rlam)"
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thm abs_fun_def
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ML {* val t_t = repabs @{context} t_r consts rty qty quot rel_refl trans2 rsp_thms *}
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