theory IntEx

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_equiv

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 plus_rsp[quotient_rsp]:

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

by (simp)

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

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 *}

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 test1: "my_plus a b = my_plus a b"

apply(rule refl)

done

lemma "PLUS a b = PLUS a b"

apply(tactic {* procedure_tac @{context} @{thm test1} 1 *})

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

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

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

done

thm lambda_prs

lemma test2: "my_plus a = my_plus a"

apply(rule refl)

done

lemma "PLUS a = PLUS a"

apply(tactic {* procedure_tac @{context} @{thm test2} 1 *})

apply(rule ballI)

apply(rule apply_rsp[OF Quotient_my_int plus_rsp])

apply(simp only: in_respects)

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

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

done

lemma test3: "my_plus = my_plus"

apply(rule refl)

done

lemma "PLUS = PLUS"

apply(tactic {* procedure_tac @{context} @{thm test3} 1 *})

apply(rule plus_rsp)

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

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

done

lemma "PLUS a b = PLUS b a"

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

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

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

apply(tactic {* clean_tac @{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} 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 "foldl PLUS x [] = x"

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

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

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

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

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

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} 1 *})

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

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

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

done

lemma ho_tst3: "foldl f (s::nat \<times> nat) ([]::(nat \<times> nat) list) = s"

by simp

lemma "foldl f (x::my_int) ([]::my_int list) = x"

apply(tactic {* procedure_tac @{context} @{thm ho_tst3} 1 *})

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

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

(* TODO: does not work when this is added *)

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

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

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

done

lemma lam_tst: "(\<lambda>x. (x, x)) y = (y, (y :: nat \<times> nat))"

by simp

lemma "(\<lambda>x. (x, x)) (y::my_int) = (y, y)"

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

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

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

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

apply(simp add: pair_prs)

done

lemma lam_tst2: "(\<lambda>(y :: nat \<times> nat). y) = (\<lambda>(x :: nat \<times> nat). x)"

by simp

lemma "(\<lambda>(y :: my_int). y) = (\<lambda>(x :: my_int). x)"

apply(tactic {* procedure_tac @{context} @{thm lam_tst2} 1 *})

defer

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

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

sorry

lemma lam_tst3: "(\<lambda>(y :: nat \<times> nat \<Rightarrow> nat \<times> nat). y) = (\<lambda>(x :: nat \<times> nat \<Rightarrow> nat \<times> nat). x)"

by auto

lemma "(\<lambda>(y :: my_int \<Rightarrow> my_int). y) = (\<lambda>(x :: my_int \<Rightarrow> my_int). x)"

apply(tactic {* procedure_tac @{context} @{thm lam_tst3} 1 *})

defer

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

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

sorry