src/HOL/Relation.ML

isatool fixgoal;

(*  Title:      Relation.ML
    ID:         $Id$
    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge
*)

open Relation;

(** Identity relation **)

Goalw [id_def] "(a,a) : id";  
by (Blast_tac 1);
qed "idI";

val major::prems = goalw thy [id_def]
    "[| p: id;  !!x.[| p = (x,x) |] ==> P  \
\    |] ==>  P";  
by (rtac (major RS CollectE) 1);
by (etac exE 1);
by (eresolve_tac prems 1);
qed "idE";

Goalw [id_def] "(a,b):id = (a=b)";
by (Blast_tac 1);
qed "pair_in_id_conv";
Addsimps [pair_in_id_conv];


(** Composition of two relations **)

Goalw [comp_def]
    "!!r s. [| (a,b):s; (b,c):r |] ==> (a,c) : r O s";
by (Blast_tac 1);
qed "compI";

(*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
val prems = goalw thy [comp_def]
    "[| xz : r O s;  \
\       !!x y z. [| xz = (x,z);  (x,y):s;  (y,z):r |] ==> P \
\    |] ==> P";
by (cut_facts_tac prems 1);
by (REPEAT (eresolve_tac [CollectE, splitE, exE, conjE] 1 
     ORELSE ares_tac prems 1));
qed "compE";

val prems = goal thy
    "[| (a,c) : r O s;  \
\       !!y. [| (a,y):s;  (y,c):r |] ==> P \
\    |] ==> P";
by (rtac compE 1);
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1));
qed "compEpair";

AddIs [compI, idI];
AddSEs [compE, idE];

Goal "R O id = R";
by (Fast_tac 1);
qed "R_O_id";

Goal "id O R = R";
by (Fast_tac 1);
qed "id_O_R";

Addsimps [R_O_id,id_O_R];

Goal "(R O S) O T = R O (S O T)";
by (Blast_tac 1);
qed "O_assoc";

Goal "!!r s. [| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
by (Blast_tac 1);
qed "comp_mono";

Goal
    "!!r s. [| s <= A Times B;  r <= B Times C |] ==> (r O s) <= A Times C";
by (Blast_tac 1);
qed "comp_subset_Sigma";

(** Natural deduction for trans(r) **)

val prems = goalw thy [trans_def]
    "(!! x y z. [| (x,y):r;  (y,z):r |] ==> (x,z):r) ==> trans(r)";
by (REPEAT (ares_tac (prems@[allI,impI]) 1));
qed "transI";

Goalw [trans_def]
    "!!r. [| trans(r);  (a,b):r;  (b,c):r |] ==> (a,c):r";
by (Blast_tac 1);
qed "transD";

(** Natural deduction for r^-1 **)

Goalw [converse_def] "!!a b r. ((a,b): r^-1) = ((b,a):r)";
by (Simp_tac 1);
qed "converse_iff";

AddIffs [converse_iff];

Goalw [converse_def] "!!a b r. (a,b):r ==> (b,a): r^-1";
by (Simp_tac 1);
qed "converseI";

Goalw [converse_def] "!!a b r. (a,b) : r^-1 ==> (b,a) : r";
by (Blast_tac 1);
qed "converseD";

(*More general than converseD, as it "splits" the member of the relation*)
qed_goalw "converseE" thy [converse_def]
    "[| yx : r^-1;  \
\       !!x y. [| yx=(y,x);  (x,y):r |] ==> P \
\    |] ==> P"
 (fn [major,minor]=>
  [ (rtac (major RS CollectE) 1),
    (REPEAT (eresolve_tac [splitE, bexE,exE, conjE, minor] 1)),
    (assume_tac 1) ]);

AddSEs [converseE];

Goalw [converse_def] "(r^-1)^-1 = r";
by (Blast_tac 1);
qed "converse_converse";
Addsimps [converse_converse];

Goal "(r O s)^-1 = s^-1 O r^-1";
by (Blast_tac 1);
qed "converse_comp";

Goal "id^-1 = id";
by (Blast_tac 1);
qed "converse_id";
Addsimps [converse_id];

(** Domain **)

qed_goalw "Domain_iff" thy [Domain_def]
    "a: Domain(r) = (EX y. (a,y): r)"
 (fn _=> [ (Blast_tac 1) ]);

qed_goal "DomainI" thy "!!a b r. (a,b): r ==> a: Domain(r)"
 (fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]);

qed_goal "DomainE" thy
    "[| a : Domain(r);  !!y. (a,y): r ==> P |] ==> P"
 (fn prems=>
  [ (rtac (Domain_iff RS iffD1 RS exE) 1),
    (REPEAT (ares_tac prems 1)) ]);

AddIs  [DomainI];
AddSEs [DomainE];

Goal "Domain id = UNIV";
by (Blast_tac 1);
qed "Domain_id";
Addsimps [Domain_id];

(** Range **)

qed_goalw "RangeI" thy [Range_def] "!!a b r.(a,b): r ==> b : Range(r)"
 (fn _ => [ (etac (converseI RS DomainI) 1) ]);

qed_goalw "RangeE" thy [Range_def]
    "[| b : Range(r);  !!x. (x,b): r ==> P |] ==> P"
 (fn major::prems=>
  [ (rtac (major RS DomainE) 1),
    (resolve_tac prems 1),
    (etac converseD 1) ]);

AddIs  [RangeI];
AddSEs [RangeE];

Goal "Range id = UNIV";
by (Blast_tac 1);
qed "Range_id";
Addsimps [Range_id];

(*** Image of a set under a relation ***)

qed_goalw "Image_iff" thy [Image_def]
    "b : r^^A = (? x:A. (x,b):r)"
 (fn _ => [ Blast_tac 1 ]);

qed_goalw "Image_singleton" thy [Image_def]
    "r^^{a} = {b. (a,b):r}"
 (fn _ => [ Blast_tac 1 ]);

qed_goal "Image_singleton_iff" thy
    "(b : r^^{a}) = ((a,b):r)"
 (fn _ => [ rtac (Image_iff RS trans) 1,
            Blast_tac 1 ]);

AddIffs [Image_singleton_iff];

qed_goalw "ImageI" thy [Image_def]
    "!!a b r. [| (a,b): r;  a:A |] ==> b : r^^A"
 (fn _ => [ (Blast_tac 1)]);

qed_goalw "ImageE" thy [Image_def]
    "[| b: r^^A;  !!x.[| (x,b): r;  x:A |] ==> P |] ==> P"
 (fn major::prems=>
  [ (rtac (major RS CollectE) 1),
    (Clarify_tac 1),
    (rtac (hd prems) 1),
    (REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ]);

AddIs  [ImageI];
AddSEs [ImageE];


qed_goal "Image_empty" thy
    "R^^{} = {}"
 (fn _ => [ Blast_tac 1 ]);

Addsimps [Image_empty];

Goal "id ^^ A = A";
by (Blast_tac 1);
qed "Image_id";

Addsimps [Image_id];

qed_goal "Image_Int_subset" thy
    "R ^^ (A Int B) <= R ^^ A Int R ^^ B"
 (fn _ => [ Blast_tac 1 ]);

qed_goal "Image_Un" thy "R ^^ (A Un B) = R ^^ A Un R ^^ B"
 (fn _ => [ Blast_tac 1 ]);

qed_goal "Image_subset" thy "!!A B r. r <= A Times B ==> r^^C <= B"
 (fn _ =>
  [ (rtac subsetI 1),
    (REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ]);

(*NOT suitable for rewriting*)
Goal "r^^B = (UN y: B. r^^{y})";
by (Blast_tac 1);
qed "Image_eq_UN";


section "Univalent";

qed_goalw "UnivalentI" Relation.thy [Univalent_def] 
   "!!r. !x y. (x,y):r --> (!z. (x,z):r --> y=z) ==> Univalent r" (K [atac 1]);

qed_goalw "UnivalentD" Relation.thy [Univalent_def] 
	"!!r. [| Univalent r; (x,y):r; (x,z):r|] ==> y=z" (K [Auto_tac]);