theory IntEx

imports QuotMain

begin

fun

  intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infix "\<approx>" 50)

where

  "intrel (x, y) (u, v) = (x + v = u + y)"

quotient my_int = "nat \<times> nat" / intrel

  apply(unfold EQUIV_def)

  apply(auto simp add: mem_def expand_fun_eq)

  done

thm my_int_equiv

print_theorems

print_quotients

quotient_def 

  ZERO::"my_int"

where

  "ZERO \<equiv> (0::nat, 0::nat)"

ML {* print_qconstinfo @{context} *}

term ZERO

thm ZERO_def

ML {* prop_of @{thm ZERO_def} *}

ML {* separate *}

quotient_def 

  ONE::"my_int"

where

  "ONE \<equiv> (1::nat, 0::nat)"

ML {* print_qconstinfo @{context} *}

term ONE

thm ONE_def

fun

  my_plus :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"

where

  "my_plus (x, y) (u, v) = (x + u, y + v)"

quotient_def 

  PLUS::"my_int \<Rightarrow> my_int \<Rightarrow> my_int"

where

  "PLUS \<equiv> my_plus"

term my_plus

term PLUS

thm PLUS_def

fun

  my_neg :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"

where

  "my_neg (x, y) = (y, x)"

quotient_def 

  NEG::"my_int \<Rightarrow> my_int"

where

  "NEG \<equiv> my_neg"

term NEG

thm NEG_def

definition

  MINUS :: "my_int \<Rightarrow> my_int \<Rightarrow> my_int"

where

  "MINUS z w = PLUS z (NEG w)"

fun

  my_mult :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"

where

  "my_mult (x, y) (u, v) = (x*u + y*v, x*v + y*u)"

quotient_def 

  MULT::"my_int \<Rightarrow> my_int \<Rightarrow> my_int"

where

  "MULT \<equiv> my_mult"

term MULT

thm MULT_def

(* NOT SURE WETHER THIS DEFINITION IS CORRECT *)

fun

  my_le :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"

where

  "my_le (x, y) (u, v) = (x+v \<le> u+y)"

quotient_def 

  LE :: "my_int \<Rightarrow> my_int \<Rightarrow> bool"

where

  "LE \<equiv> my_le"

term LE

thm LE_def

definition

  LESS :: "my_int \<Rightarrow> my_int \<Rightarrow> bool"

where

  "LESS z w = (LE z w \<and> z \<noteq> w)"

term LESS

thm LESS_def

definition

  ABS :: "my_int \<Rightarrow> my_int"

where

  "ABS i = (if (LESS i ZERO) then (NEG i) else i)"

definition

  SIGN :: "my_int \<Rightarrow> my_int"

where

 "SIGN i = (if i = ZERO then ZERO else if (LESS ZERO i) then ONE else (NEG ONE))"

ML {* print_qconstinfo @{context} *}

lemma plus_sym_pre:

  shows "my_plus a b \<approx> my_plus b a"

  apply(cases a)

  apply(cases b)

  apply(auto)

  done

lemma ho_plus_rsp:

  "(intrel ===> intrel ===> intrel) my_plus my_plus"

  by (simp)

ML {* val qty = @{typ "my_int"} *}

ML {* val ty_name = "my_int" *}

ML {* val rsp_thms = @{thms ho_plus_rsp} @ @{thms ho_all_prs ho_ex_prs} *}

ML {* val defs = @{thms PLUS_def} *}

ML {*

fun lift_thm_my_int lthy t =

  lift_thm lthy qty ty_name rsp_thms defs t

*}

ML {*

fun lift_thm_g_my_int lthy t g =

  lift_thm_goal lthy qty ty_name rsp_thms defs t g

*}

print_quotients

ML {* quotdata_lookup @{context} "IntEx.my_int" *}

ML {* quotdata_lookup @{context} "my_int" *}

ML {*

  val test = lift_thm_my_int @{context} @{thm plus_sym_pre}

*}

ML {*

  lift_thm_g_my_int @{context} @{thm plus_sym_pre} @{term "PLUS a b = PLUS b a"}

*}

lemma plus_assoc_pre:

  shows "my_plus (my_plus i j) k \<approx> my_plus i (my_plus j k)"

  apply (cases i)

  apply (cases j)

  apply (cases k)

  apply (simp)

  done

ML {* val test2 = lift_thm_my_int @{context} @{thm plus_assoc_pre} *}

ML {* lift_thm_g_my_int @{context} @{thm plus_assoc_pre} 

         @{term "PLUS (PLUS x xa) xb = PLUS x (PLUS xa xb)"} *}

ML {*

  mk_REGULARIZE_goal @{context} 

    @{term "\<forall>i j k. my_plus (my_plus i j) k \<approx> my_plus i (my_plus j k)"}

    @{term "\<forall>i j k. PLUS (PLUS i j) k = PLUS i (PLUS j k)"}

  |> Syntax.string_of_term @{context}

  |> writeln

*}

lemma procedure: 

  assumes a: "A"

  and     b: "A \<Longrightarrow> B"

  and     c: "B = C"

  and     d: "C = D"

  shows   "D"

  using a b c d

  by simp

ML {* 

fun procedure_inst ctxt rtrm qtrm =

let

  val thy = ProofContext.theory_of ctxt

  val rtrm' = HOLogic.dest_Trueprop rtrm

  val qtrm' = HOLogic.dest_Trueprop qtrm

  val reg_goal = Syntax.check_term ctxt (REGULARIZE_trm ctxt rtrm' qtrm')

  val inj_goal = Syntax.check_term ctxt (inj_REPABS ctxt (reg_goal, qtrm'))

in

  Drule.instantiate' [] 

    [SOME (cterm_of thy rtrm'), 

     SOME (cterm_of thy reg_goal),

     SOME (cterm_of thy inj_goal)] 

  @{thm procedure}

end

fun procedure_tac rthm =

  Subgoal.FOCUS (fn {context, concl, ...} =>

    let

      val rthm' = atomize_thm rthm

      val rule = procedure_inst context (prop_of rthm') (term_of concl)

    in

      EVERY1 [rtac rule, rtac rthm']

    end)

*}

lemma "\<forall>i j k. PLUS (PLUS i j) k = PLUS i (PLUS j k)"

apply(tactic {* procedure_tac @{thm plus_assoc_pre} @{context} 1 *})

(* phase 2 *)

ML_prf {*

 val lower = add_lower_defs @{context} @{thm PLUS_def}

*}

(*

does not work.

ML {*

fun r_mk_comb_tac ctxt rty quot_thm reflex_thm trans_thm rsp_thms =

  (REPEAT1 o FIRST1) 

    [(K (print_tac "start")) THEN' (K no_tac), 

     DT ctxt "1" (rtac trans_thm),

     DT ctxt "2" (LAMBDA_RES_TAC ctxt),

     DT ctxt "3" (ball_rsp_tac ctxt),

     DT ctxt "4" (bex_rsp_tac ctxt),

     DT ctxt "5" (FIRST' (map rtac rsp_thms)),

     DT ctxt "6" (instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [quotient_tac quot_thm])),

     DT ctxt "7" (rtac refl),

     DT ctxt "8" (APPLY_RSP_TAC rty ctxt THEN' (RANGE [quotient_tac quot_thm, quotient_tac quot_thm])),

     DT ctxt "9" (Cong_Tac.cong_tac @{thm cong}),

     DT ctxt "A" (rtac @{thm ext}),

     DT ctxt "B" (rtac reflex_thm),

     DT ctxt "C" (atac),

     DT ctxt "D" (SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))),

     DT ctxt "E" (WEAK_LAMBDA_RES_TAC ctxt),

     DT ctxt "F" (CHANGED' (asm_full_simp_tac (HOL_ss addsimps @{thms FUN_REL_EQ})))]

*}

*)

ML {*

val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} @{typ "my_int"}

val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "my_int"

val consts = lookup_quot_consts defs

*}

ML {* cprem_of *}

ML {* 

mk_inj_REPABS_goal @{context} (reg_atrm, aqtrm) 

  |> Syntax.check_term @{context}

*}

ML {* val my_goal = cterm_of @{theory} (mk_inj_REPABS_goal @{context} (reg_atrm, aqtrm)) *}

ML {* val yr_goal = cterm_of @{theory} (build_repabs_goal @{context} (@{thm testtest} OF [arthm]) consts rty qty) *}

prove {* mk_inj_REPABS_goal @{context} (reg_atrm, aqtrm) *}

apply(tactic {* ObjectLogic.full_atomize_tac 1 *})

apply(tactic {* REPEAT_ALL_NEW (r_mk_comb_tac @{context} rty quot rel_refl trans2 rsp_thms) 1 *})

done

ML {*

inj_REPABS @{context} (reg_atrm, aqtrm)  

|> Syntax.string_of_term @{context}

|> writeln

*}

lemma ho_tst: "foldl my_plus x [] = x"

apply simp

done

text {* Below is the construction site code used if things do not work *}

ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}

ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "my_int" *}

(* ML {* val consts = [@{const_name my_plus}] *}*)

ML {* val consts = lookup_quot_consts defs *}

ML {* val t_a = atomize_thm @{thm ho_tst} *}

(*

prove t_r: {* build_regularize_goal t_a rty rel @{context} *}

ML_prf {*   fun tac ctxt =

      (ObjectLogic.full_atomize_tac) THEN'

     REPEAT_ALL_NEW (FIRST' [

      rtac rel_refl,

      atac,

      rtac @{thm universal_twice},

      (rtac @{thm impI} THEN' atac),

      (*rtac @{thm equality_twice},*)

      EqSubst.eqsubst_tac ctxt [0]

        [(@{thm equiv_res_forall} OF [rel_eqv]),

         (@{thm equiv_res_exists} OF [rel_eqv])],

      (rtac @{thm impI} THEN' (asm_full_simp_tac (Simplifier.context ctxt HOL_ss)) THEN' rtac rel_refl),

      (rtac @{thm RIGHT_RES_FORALL_REGULAR})

     ]);*}

apply (atomize(full))

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

apply (auto)

done

ML {* val t_r = @{thm t_r} OF [t_a] *}*)

ML {* val t_r = regularize t_a rty rel rel_eqv rel_refl @{context} *}

ML {*

  val rt = build_repabs_term @{context} t_r consts rty qty

  val rg = Logic.mk_equals ((Thm.prop_of t_r), rt);

*}

lemma foldl_rsp:

  "((IntEx.intrel ===> IntEx.intrel ===> IntEx.intrel) ===>

           IntEx.intrel ===> op = ===> IntEx.intrel)

           foldl foldl"

  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 (rule allI)

  apply (rule allI)

  apply (rule impI)

  apply (simp only:)

  apply (rule list.induct)

  apply (simp)

  apply (simp only: foldl.simps)

  sorry

ML {* val rsp_thms = @{thm foldl_rsp} :: rsp_thms *}

prove t_t: rg

apply(atomize(full))

ML_prf {* fun r_mk_comb_tac_int lthy = r_mk_comb_tac lthy rty quot rel_refl trans2 rsp_thms *}

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

done

ML {* val t_t = @{thm Pure.equal_elim_rule1} OF [@{thm t_t},t_r] *}

ML {* val abs = findabs rty (prop_of t_a) *}

ML {* val aps = findaps rty (prop_of t_a); *}

ML {* val simp_lam_prs_thms = map (make_simp_prs_thm @{context} quot @{thm LAMBDA_PRS}) abs *}

(*ML {* val t_t = repabs @{context} @{thm t_r} consts rty qty quot rel_refl trans2 rsp_thms *}*)

ML findallex

ML {* val (alls, exs) = findallex @{context} rty qty (prop_of t_a) *}

ML {* val allthms = map (make_allex_prs_thm @{context} quot @{thm FORALL_PRS}) alls *}