theory Multi_Recs2

imports "../Nominal2"

begin

(* 

  multiple recursive binders - multiple letrecs with multiple 

  clauses for each functions

  example 8 from Peter Sewell's bestiary (originally due

  to James Cheney) 

*)

atom_decl name

nominal_datatype fun_recs: exp =

  Var name

| Unit 

| Pair exp exp

| LetRec l::lrbs e::exp bind (set) "b_lrbs l" in l e

and fnclause =

  K x::name p::pat f::exp bind (set) "b_pat p" in f

and fnclauses =

  S fnclause

| ORs fnclause fnclauses

and lrb =

  Clause fnclauses

and lrbs =

  Single lrb

| More lrb lrbs

and pat =

  PVar name

| PUnit

| PPair pat pat

binder

  b_lrbs :: "lrbs \<Rightarrow> atom set" and

  b_pat :: "pat \<Rightarrow> atom set" and

  b_fnclauses :: "fnclauses \<Rightarrow> atom set" and

  b_fnclause :: "fnclause \<Rightarrow> atom set" and

  b_lrb :: "lrb \<Rightarrow> atom set"

where

  "b_lrbs (Single l) = b_lrb l"

| "b_lrbs (More l ls) = b_lrb l \<union> b_lrbs ls"

| "b_pat (PVar x) = {atom x}"

| "b_pat (PUnit) = {}"

| "b_pat (PPair p1 p2) = b_pat p1 \<union> b_pat p2"

| "b_fnclauses (S fc) = (b_fnclause fc)"

| "b_fnclauses (ORs fc fcs) = (b_fnclause fc) \<union> (b_fnclauses fcs)"

| "b_lrb (Clause fcs) = (b_fnclauses fcs)"

| "b_fnclause (K x pat exp) = {atom x}"

thm fun_recs.permute_bn

thm fun_recs.perm_bn_alpha

thm fun_recs.perm_bn_simps

thm fun_recs.bn_finite

thm fun_recs.inducts

thm fun_recs.distinct

thm fun_recs.induct

thm fun_recs.inducts

thm fun_recs.exhaust

thm fun_recs.fv_defs

thm fun_recs.bn_defs

thm fun_recs.perm_simps

thm fun_recs.eq_iff

thm fun_recs.fv_bn_eqvt

thm fun_recs.size_eqvt

thm fun_recs.supports

thm fun_recs.fsupp

thm fun_recs.supp

thm fun_recs.distinct

thm fun_recs.induct

thm fun_recs.inducts

thm fun_recs.exhaust

thm fun_recs.fv_defs

thm fun_recs.bn_defs

thm fun_recs.perm_simps

thm fun_recs.eq_iff

thm fun_recs.fv_bn_eqvt

thm fun_recs.size_eqvt

thm fun_recs.supports

thm fun_recs.fsupp

thm fun_recs.supp

lemma

  fixes c::"'a::fs"

  assumes "\<And>name c. P1 c (Var name)"

  and "\<And>c. P1 c Unit"

  and "\<And>exp1 exp2 c. \<lbrakk>\<And>c. P1 c exp1; \<And>c. P1 c exp2\<rbrakk> \<Longrightarrow> P1 c (Multi_Recs2.Pair exp1 exp2)"

  and "\<And>lrbs exp c. \<lbrakk>b_lrbs lrbs \<sharp>* c; \<And>c. P5 c lrbs; \<And>c. P1 c exp\<rbrakk> \<Longrightarrow> P1 c (LetRec lrbs exp)"

  and "\<And>name pat exp c. \<lbrakk>b_pat pat \<sharp>* c; \<And>c. P6 c pat; \<And>c. P1 c exp\<rbrakk> \<Longrightarrow> P2 c (K name pat exp)"

  and "\<And>fnclause c. (\<And>c. P2 c fnclause) \<Longrightarrow> P3 c (S fnclause)"

  and "\<And>fnclause fnclauses c. \<lbrakk>\<And>c. P2 c fnclause; \<And>c. P3 c fnclauses\<rbrakk> \<Longrightarrow> 

    P3 c (ORs fnclause fnclauses)"

  and "\<And>fnclauses c. (\<And>c. P3 c fnclauses) \<Longrightarrow> P4 c (Clause fnclauses)"

  and "\<And>lrb c. (\<And>c. P4 c lrb) \<Longrightarrow> P5 c (Single lrb)"

  and "\<And>lrb lrbs c. \<lbrakk>\<And>c. P4 c lrb; \<And>c. P5 c lrbs\<rbrakk> \<Longrightarrow> P5 c (More lrb lrbs)"

  and "\<And>name c. P6 c (PVar name)"

  and "\<And>c. P6 c PUnit"

  and "\<And>pat1 pat2 c. \<lbrakk>\<And>c. P6 c pat1; \<And>c. P6 c pat2\<rbrakk> \<Longrightarrow> P6 c (PPair pat1 pat2)"

  shows "P1 c exp" and "P2 c fnclause" and "P3 c fnclauses" and "P4 c lrb" and "P5 c lrbs" and "P6 c pat"

  apply(raw_tactic {* Induction_Schema.induction_schema_tac @{context} @{thms assms} *})

  apply(rule_tac y="exp" and c="c" in fun_recs.strong_exhaust(1))

  apply(simp_all)[4]

  apply(rule_tac ya="fnclause" and c="c" in fun_recs.strong_exhaust(2))

  apply(simp_all)[1]

  apply(rule_tac yb="fnclauses" in fun_recs.strong_exhaust(3))

  apply(simp_all)[2]

  apply(rule_tac yc="lrb" in fun_recs.strong_exhaust(4))

  apply(simp_all)[1]

  apply(rule_tac yd="lrbs" in fun_recs.strong_exhaust(5))

  apply(simp_all)[2]

  apply(rule_tac ye="pat" in fun_recs.strong_exhaust(6))

  apply(simp_all)[3]

  apply(tactic {* prove_termination_ind @{context} 1 *})

  apply(simp_all add: fun_recs.size)

  done

end