--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Examples/IntEx.thy	Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,276 @@
+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_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