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 {* 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 = lookup_quot_consts defs *}

ML {*

fun lift_tac_fset lthy t =

  lift_tac lthy t rel_eqv rel_refl rty quot trans2 rsp_thms reps_same absrep defs

*}

lemma "PLUS a b = PLUS b a"

by (tactic {* lift_tac_fset @{context} @{thm plus_sym_pre} 1 *})

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

lemma plus_assoc: "PLUS (PLUS x xa) xb = PLUS x (PLUS xa xb)"

by (tactic {* lift_tac_fset @{context} @{thm plus_assoc_pre} 1 *})

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

apply simp

done

lemma "foldl PLUS x [] = x"

apply (tactic {* lift_tac_fset @{context} @{thm ho_tst} 1 *})

sorry

(*

  FIXME: All below is your construction code; mostly commented out as it

  does not work.

*)

ML {*

    @{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 "PLUS (PLUS i j) k = PLUS i (PLUS j k)"

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

(*

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 {* 

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

*}

*)