theory FSet

imports QuotMain

begin

inductive

  list_eq (infix "\<approx>" 50)

where

  "a#b#xs \<approx> b#a#xs"

| "[] \<approx> []"

| "xs \<approx> ys \<Longrightarrow> ys \<approx> xs"

| "a#a#xs \<approx> a#xs"

| "xs \<approx> ys \<Longrightarrow> a#xs \<approx> a#ys"

| "\<lbrakk>xs1 \<approx> xs2; xs2 \<approx> xs3\<rbrakk> \<Longrightarrow> xs1 \<approx> xs3"

lemma list_eq_refl:

  shows "xs \<approx> xs"

  apply (induct xs)

   apply (auto intro: list_eq.intros)

  done

lemma equiv_list_eq:

  shows "EQUIV list_eq"

  unfolding EQUIV_REFL_SYM_TRANS REFL_def SYM_def TRANS_def

  apply(auto intro: list_eq.intros list_eq_refl)

  done

quotient fset = "'a list" / "list_eq"

  apply(rule equiv_list_eq)

  done

print_theorems

typ "'a fset"

thm "Rep_fset"

thm "ABS_fset_def"

ML {* @{term "Abs_fset"} *}

local_setup {*

  make_const_def @{binding EMPTY} @{term "[]"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term Nil

term EMPTY

thm EMPTY_def

local_setup {*

  make_const_def @{binding INSERT} @{term "op #"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term Cons

term INSERT

thm INSERT_def

local_setup {*

  make_const_def @{binding UNION} @{term "op @"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term append

term UNION

thm UNION_def

thm QUOTIENT_fset

thm QUOT_TYPE_I_fset.thm11

fun

  membship :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infix "memb" 100)

where

  m1: "(x memb []) = False"

| m2: "(x memb (y#xs)) = ((x=y) \<or> (x memb xs))"

fun

  card1 :: "'a list \<Rightarrow> nat"

where

  card1_nil: "(card1 []) = 0"

| card1_cons: "(card1 (x # xs)) = (if (x memb xs) then (card1 xs) else (Suc (card1 xs)))"

local_setup {*

  make_const_def @{binding card} @{term "card1"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term card1

term card

thm card_def

(* text {*

 Maybe make_const_def should require a theorem that says that the particular lifted function

 respects the relation. With it such a definition would be impossible:

 make_const_def @{binding CARD} @{term "length"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}*)

lemma card1_0:

  fixes a :: "'a list"

  shows "(card1 a = 0) = (a = [])"

  apply (induct a)

   apply (simp)

  apply (simp_all)

   apply meson

  apply (simp_all)

  done

lemma not_mem_card1:

  fixes x :: "'a"

  fixes xs :: "'a list"

  shows "~(x memb xs) \<Longrightarrow> card1 (x # xs) = Suc (card1 xs)"

  by simp

lemma mem_cons:

  fixes x :: "'a"

  fixes xs :: "'a list"

  assumes a : "x memb xs"

  shows "x # xs \<approx> xs"

  using a

  apply (induct xs)

  apply (auto intro: list_eq.intros)

  done

lemma card1_suc:

  fixes xs :: "'a list"

  fixes n :: "nat"

  assumes c: "card1 xs = Suc n"

  shows "\<exists>a ys. ~(a memb ys) \<and> xs \<approx> (a # ys)"

  using c

apply(induct xs)

apply (metis Suc_neq_Zero card1_0)

apply (metis QUOT_TYPE_I_fset.R_trans card1_cons list_eq_refl mem_cons)

done

primrec

  fold1

where

  "fold1 f (g :: 'a \<Rightarrow> 'b) (z :: 'b) [] = z"

| "fold1 f g z (a # A) =

     (if ((!u v. (f u v = f v u))

      \<and> (!u v w. ((f u (f v w) = f (f u v) w))))

     then (

       if (a memb A) then (fold1 f g z A) else (f (g a) (fold1 f g z A))

     ) else z)"

(* fold1_def is not usable, but: *)

thm fold1.simps

lemma fs1_strong_cases:

  fixes X :: "'a list"

  shows "(X = []) \<or> (\<exists>a. \<exists> Y. (~(a memb Y) \<and> (X \<approx> a # Y)))"

  apply (induct X)

  apply (simp)

  apply (metis QUOT_TYPE_I_fset.thm11 list_eq_refl mem_cons m1)

  done

local_setup {*

  make_const_def @{binding IN} @{term "membship"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term membship

term IN

thm IN_def

ML {*

  val consts = [@{const_name "Nil"}, @{const_name "Cons"},

                @{const_name "membship"}, @{const_name "card1"},

                @{const_name "append"}, @{const_name "fold1"}];

*}

ML {* val fset_defs = @{thms EMPTY_def IN_def UNION_def card_def INSERT_def} *}

ML {* val fset_defs_sym = map (fn t => symmetric (unabs_def @{context} t)) fset_defs *}

thm fun_map.simps

text {* Respectfullness *}

lemma mem_respects:

  fixes z

  assumes a: "list_eq x y"

  shows "(z memb x) = (z memb y)"

  using a by induct auto

lemma card1_rsp:

  fixes a b :: "'a list"

  assumes e: "a \<approx> b"

  shows "card1 a = card1 b"

  using e apply induct

  apply (simp_all add:mem_respects)

  done

lemma cons_preserves:

  fixes z

  assumes a: "xs \<approx> ys"

  shows "(z # xs) \<approx> (z # ys)"

  using a by (rule list_eq.intros(5))

lemma append_respects_fst:

  assumes a : "list_eq l1 l2"

  shows "list_eq (l1 @ s) (l2 @ s)"

  using a

  apply(induct)

  apply(auto intro: list_eq.intros)

  apply(simp add: list_eq_refl)

done

thm list.induct

lemma list_induct_hol4:

  fixes P :: "'a list \<Rightarrow> bool"

  assumes "((P []) \<and> (\<forall>t. (P t) \<longrightarrow> (\<forall>h. (P (h # t)))))"

  shows "(\<forall>l. (P l))"

  sorry

ML {* atomize_thm @{thm list_induct_hol4} *}

prove list_induct_r: {*

   build_regularize_goal (atomize_thm @{thm list_induct_hol4}) @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}

  apply (simp only: equiv_res_forall[OF equiv_list_eq])

  thm RIGHT_RES_FORALL_REGULAR

  apply (rule RIGHT_RES_FORALL_REGULAR)

  prefer 2

  apply (assumption)

  apply (metis)

  done

ML {*

fun instantiate_tac thm = Subgoal.FOCUS (fn {concl, ...} =>

let

  val pat = Drule.strip_imp_concl (cprop_of thm)

  val insts = Thm.match (pat, concl)

in

  rtac (Drule.instantiate insts thm) 1

end

handle _ => no_tac

)

*}

ML {*

fun res_forall_rsp_tac ctxt =

  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

  THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI}

  THEN' instantiate_tac @{thm RES_FORALL_RSP} ctxt THEN'

  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

*}

ML {*

fun quotient_tac quot_thm =

  REPEAT_ALL_NEW (FIRST' [

    rtac @{thm FUN_QUOTIENT},

    rtac quot_thm,

    rtac @{thm IDENTITY_QUOTIENT}

  ])

*}

ML {*

fun r_mk_comb_tac ctxt quot_thm reflex_thm trans_thm =

  (FIRST' [

    rtac @{thm FUN_QUOTIENT},

    rtac quot_thm,

    rtac @{thm IDENTITY_QUOTIENT},

    rtac @{thm ext},

    rtac trans_thm,

    res_forall_rsp_tac ctxt,

    (instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [quotient_tac quot_thm])),

    (instantiate_tac @{thm APPLY_RSP} ctxt THEN' (RANGE [quotient_tac quot_thm, quotient_tac quot_thm])),

    rtac reflex_thm,

    atac,

    (

     (simp_tac ((Simplifier.context @{context} HOL_ss) addsimps @{thms FUN_REL.simps}))

     THEN_ALL_NEW (fn _ => no_tac)

    )

    ])

*}

ML {*

fun r_mk_comb_tac_fset ctxt =

  r_mk_comb_tac ctxt @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2}

*}

ML {* val thm = @{thm list_induct_r} OF [atomize_thm @{thm list_induct_hol4}] *}

ML {* val trm_r = build_repabs_goal @{context} thm consts @{typ "'a list"} @{typ "'a fset"} *}

ML {* val trm = build_repabs_term @{context} thm consts @{typ "'a list"} @{typ "'a fset"} *}

prove list_induct_tr: trm_r

apply (atomize(full))

(* APPLY_RSP_TAC *)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

(* LAMBDA_RES_TAC *)

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* REFL_TAC *)

apply (simp)

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* REFL_TAC *)

apply (simp)

(* APPLY_RSP_TAC *)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* REFL_TAC *)

apply (simp)

(* APPLY_RSP *)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

(* MK_COMB_TAC *)

apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})

(* REFL_TAC *)

apply (simp)

(* W(C (curry op THEN) (G... *)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

(* CONS respects *)

apply (simp add: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (rule cons_preserves)

apply (assumption)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

done

prove list_induct_t: trm

apply (simp only: list_induct_tr[symmetric])

apply (tactic {* rtac thm 1 *})

done

ML {* val nthm = MetaSimplifier.rewrite_rule fset_defs_sym (snd (no_vars (Context.Theory @{theory}, @{thm list_induct_t}))) *}

thm list.recs(2)

thm m2

ML {* atomize_thm @{thm m2} *}

prove m2_r_p: {*

   build_regularize_goal (atomize_thm @{thm m2}) @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}

  apply (simp add: equiv_res_forall[OF equiv_list_eq])

done

ML {* val m2_r = @{thm m2_r_p} OF [atomize_thm @{thm m2}] *}

ML {* val m2_t_g = build_repabs_goal @{context} m2_r consts @{typ "'a list"} @{typ "'a fset"} *}

ML {* val m2_t = build_repabs_term @{context} m2_r consts @{typ "'a list"} @{typ "'a fset"} *}

prove m2_t_p: m2_t_g

apply (atomize(full))

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

apply (simp add: FUN_REL.simps)

prefer 2

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

prefer 2

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (simp add:mem_respects)

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (simp add:cons_preserves)

apply (simp only: FUN_REL.simps)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (rule allI)

apply (rule allI)

apply (rule impI)

apply (simp add:mem_respects)

apply (auto)

done

prove m2_t: m2_t

apply (simp only: m2_t_p[symmetric])

apply (tactic {* rtac m2_r 1 *})

done

lemma id_apply2 [simp]: "id x \<equiv> x"

  by (simp add: id_def)

thm LAMBDA_PRS

ML {*

 val t = prop_of @{thm LAMBDA_PRS}

 val tt = Drule.instantiate' [SOME @{ctyp "'a list"}, SOME @{ctyp "'a fset"}] [] @{thm LAMBDA_PRS}

 val ttt = @{thm LAMBDA_PRS} OF [@{thm QUOTIENT_fset}, @{thm IDENTITY_QUOTIENT}]

 val tttt = @{thm "HOL.sym"} OF [ttt]

*}

ML {*

 val ttttt = MetaSimplifier.rewrite_rule [@{thm id_apply2}] tttt

 val zzz = @{thm m2_t}

*}

ML {*

fun eqsubst_thm ctxt thms thm =

  let

    val goalstate = Goal.init (Thm.cprop_of thm)

    val a' = case (SINGLE (EqSubst.eqsubst_tac ctxt [0] thms 1) goalstate) of

      NONE => error "eqsubst_thm"

    | SOME th => cprem_of th 1

    val tac = (EqSubst.eqsubst_tac ctxt [0] thms 1) THEN simp_tac HOL_ss 1

    val cgoal = cterm_of (ProofContext.theory_of ctxt) (Logic.mk_equals (term_of (Thm.cprop_of thm), term_of a'))

    val rt = Toplevel.program (fn () => Goal.prove_internal [] cgoal (fn _ => tac));

  in

    Simplifier.rewrite_rule [rt] thm

  end

*}

ML {* val m2_t' = eqsubst_thm @{context} [ttttt] @{thm m2_t} *}

ML {* val rwr = @{thm FORALL_PRS[OF QUOTIENT_fset]} *}

ML {* val rwrs = @{thm "HOL.sym"} OF [ObjectLogic.rulify rwr] *}

ML {* rwrs; m2_t' *}

ML {* eqsubst_thm @{context} [rwrs] m2_t' *}

thm INSERT_def

ML {* val card1_suc_f = Thm.freezeT (atomize_thm @{thm card1_suc}) *}

prove card1_suc_r: {*

 Logic.mk_implies

   ((prop_of card1_suc_f),

   (regularise (prop_of card1_suc_f) @{typ "'a List.list"} @{term "op \<approx>"} @{context})) *}

  apply (simp add: equiv_res_forall[OF equiv_list_eq] equiv_res_exists[OF equiv_list_eq])

  done

ML {* @{thm card1_suc_r} OF [card1_suc_f] *}

ML {* val li = Thm.freezeT (atomize_thm @{thm fold1.simps(2)}) *}

prove fold1_def_2_r: {*

 Logic.mk_implies

   ((prop_of li),

   (regularise (prop_of li) @{typ "'a List.list"} @{term "op \<approx>"} @{context})) *}

  apply (simp add: equiv_res_forall[OF equiv_list_eq])

  done

ML {* @{thm fold1_def_2_r} OF [li] *}

lemma yy:

  shows "(False = x memb []) = (False = IN (x::nat) EMPTY)"

unfolding IN_def EMPTY_def

apply(rule_tac f="(op =) False" in arg_cong)

apply(rule mem_respects)

apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)

apply(rule list_eq.intros)

done

lemma

  shows "IN (x::nat) EMPTY = False"

using m1

apply -

apply(rule yy[THEN iffD1, symmetric])

apply(simp)

done

lemma

  shows "((x=y) \<or> (IN x xs) = (IN (x::nat) (INSERT y xs))) =

         ((x=y) \<or> x memb REP_fset xs = x memb (y # REP_fset xs))"

unfolding IN_def INSERT_def

apply(rule_tac f="(op \<or>) (x=y)" in arg_cong)

apply(rule_tac f="(op =) (x memb REP_fset xs)" in arg_cong)

apply(rule mem_respects)

apply(rule list_eq.intros(3))

apply(unfold REP_fset_def ABS_fset_def)

apply(simp only: QUOT_TYPE.REP_ABS_rsp[OF QUOT_TYPE_fset])

apply(rule list_eq_refl)

done

lemma yyy:

  shows "

    (

     (UNION EMPTY s = s) &

     ((UNION (INSERT e s1) s2) = (INSERT e (UNION s1 s2)))

    ) = (

     ((ABS_fset ([] @ REP_fset s)) = s) &

     ((ABS_fset ((e # (REP_fset s1)) @ REP_fset s2)) = ABS_fset (e # (REP_fset s1 @ REP_fset s2)))

    )"

  unfolding UNION_def EMPTY_def INSERT_def

  apply(rule_tac f="(op &)" in arg_cong2)

  apply(rule_tac f="(op =)" in arg_cong2)

  apply(simp only: QUOT_TYPE_I_fset.thm11[symmetric])

  apply(rule append_respects_fst)

  apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)

  apply(rule list_eq_refl)

  apply(simp)

  apply(rule_tac f="(op =)" in arg_cong2)

  apply(simp only: QUOT_TYPE_I_fset.thm11[symmetric])

  apply(rule append_respects_fst)

  apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)

  apply(rule list_eq_refl)

  apply(simp only: QUOT_TYPE_I_fset.thm11[symmetric])

  apply(rule list_eq.intros(5))

  apply(simp only: QUOT_TYPE_I_fset.REP_ABS_rsp)

  apply(rule list_eq_refl)

done