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 equivp_def)

  apply(auto simp add: mem_def expand_fun_eq)

  done

thm quotient_thm

thm my_int_equivp

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[quotient_rsp]:

  shows "(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} *}

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 {* fun lift_tac_intex lthy t = lift_tac lthy t [rel_eqv] *}

ML {* fun inj_repabs_tac_intex lthy = inj_repabs_tac lthy [rel_refl] [trans2] *}

ML {* fun all_inj_repabs_tac_intex lthy = all_inj_repabs_tac lthy [rel_refl] [trans2] *}

lemma "PLUS a b = PLUS b a"

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

apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})

prefer 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

done

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)"

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

apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})

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

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

done

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

apply simp

done

lemma foldr_prs:

  assumes a: "Quotient R1 abs1 rep1"

  and     b: "Quotient R2 abs2 rep2"

  shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"

apply (induct l)

apply (simp add: Quotient_ABS_REP[OF b])

apply (simp add: Quotient_ABS_REP[OF a] Quotient_ABS_REP[OF b])

done

lemma foldl_prs:

  assumes a: "Quotient R1 abs1 rep1"

  and     b: "Quotient R2 abs2 rep2"

  shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"

apply (induct l arbitrary:e)

apply (simp add: Quotient_ABS_REP[OF a])

apply (simp add: Quotient_ABS_REP[OF a] Quotient_ABS_REP[OF b])

done

lemma map_prs:

  assumes a: "Quotient R1 abs1 rep1"

  and     b: "Quotient R2 abs2 rep2"

  shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"

apply (induct l)

apply (simp)

apply (simp add: Quotient_ABS_REP[OF a] Quotient_ABS_REP[OF b])

done

(* Removed unneeded assumption: "QUOTIENT R abs1 rep1" *)

lemma nil_prs:

  shows "map abs1 [] = []"

by simp

lemma cons_prs:

  assumes a: "Quotient R1 abs1 rep1"

  shows "(map abs1) ((rep1 h) # (map rep1 t)) = h # t"

  apply (induct t)

by (simp_all add: Quotient_ABS_REP[OF a])

lemma cons_rsp[quotient_rsp]:

  "(op \<approx> ===> list_rel op \<approx> ===> list_rel op \<approx>) op # op #"

by simp

(* I believe it's true. *)

lemma foldl_rsp[quotient_rsp]:

  "((op \<approx> ===> op \<approx> ===> op \<approx>) ===> op \<approx> ===> list_rel op \<approx> ===> op \<approx>) 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 (induct_tac xb yb rule: list_induct2) (* To finish I need to give it: arbitrary:xa ya *)

sorry

lemma nil_listrel_rsp[quotient_rsp]:

  "(list_rel R) [] []"

by simp

lemma "foldl PLUS x [] = x"

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

apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})

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

apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int])

apply(simp only: nil_prs)

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

done

lemma ho_tst2: "foldl my_plus x (h # t) \<approx> my_plus h (foldl my_plus x t)"

sorry

lemma "foldl PLUS x (h # t) = PLUS h (foldl PLUS x t)"

apply(tactic {* procedure_tac @{context} @{thm ho_tst2} 1 *})

apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})

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

apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int])

apply(simp only: cons_prs[OF Quotient_my_int])

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

done