(* Title: Relation.ML
ID: $Id$
Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
(** Identity relation **)
Goalw [Id_def] "(a,a) : Id";
by (Blast_tac 1);
qed "IdI";
val major::prems = Goalw [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];
Goalw [refl_def] "reflexive Id";
by Auto_tac;
qed "reflexive_Id";
(*A strange result, since Id is also symmetric.*)
Goalw [antisym_def] "antisym Id";
by Auto_tac;
qed "antisym_Id";
Goalw [trans_def] "trans Id";
by Auto_tac;
qed "trans_Id";
(** Diagonal relation: indentity restricted to some set **)
(*** Equality : the diagonal relation ***)
Goalw [diag_def] "[| a=b; a:A |] ==> (a,b) : diag(A)";
by (Blast_tac 1);
qed "diag_eqI";
val diagI = refl RS diag_eqI |> standard;
(*The general elimination rule*)
val major::prems = Goalw [diag_def]
"[| c : diag(A); \
\ !!x y. [| x:A; c = (x,x) |] ==> P \
\ |] ==> P";
by (rtac (major RS UN_E) 1);
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
qed "diagE";
AddSIs [diagI];
AddSEs [diagE];
Goal "((x,y) : diag A) = (x=y & x : A)";
by (Blast_tac 1);
qed "diag_iff";
Goal "diag(A) <= A Times A";
by (Blast_tac 1);
qed "diag_subset_Times";
(** Composition of two relations **)
Goalw [comp_def]
"[| (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 [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
"[| (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'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
by (Blast_tac 1);
qed "comp_mono";
Goal "[| 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 refl(r) **)
val prems = Goalw [refl_def]
"[| r <= A Times A; !! x. x:A ==> (x,x):r |] ==> refl A r";
by (REPEAT (ares_tac (prems@[ballI,conjI]) 1));
qed "reflI";
Goalw [refl_def] "[| refl A r; a:A |] ==> (a,a):r";
by (Blast_tac 1);
qed "reflD";
(** Natural deduction for antisym(r) **)
val prems = Goalw [antisym_def]
"(!! x y. [| (x,y):r; (y,x):r |] ==> x=y) ==> antisym(r)";
by (REPEAT (ares_tac (prems@[allI,impI]) 1));
qed "antisymI";
Goalw [antisym_def] "[| antisym(r); (a,b):r; (b,a):r |] ==> a=b";
by (Blast_tac 1);
qed "antisymD";
(** Natural deduction for trans(r) **)
val prems = Goalw [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] "[| 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^-1) = ((b,a):r)";
by (Simp_tac 1);
qed "converse_iff";
AddIffs [converse_iff];
Goalw [converse_def] "(a,b):r ==> (b,a): r^-1";
by (Simp_tac 1);
qed "converseI";
Goalw [converse_def] "(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*)
val [major,minor] = Goalw [converse_def]
"[| yx : r^-1; \
\ !!x y. [| yx=(y,x); (x,y):r |] ==> P \
\ |] ==> P";
by (rtac (major RS CollectE) 1);
by (REPEAT (eresolve_tac [splitE, bexE,exE, conjE, minor] 1));
by (assume_tac 1);
qed "converseE";
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];
Goal "(diag A) ^-1 = diag A";
by (Blast_tac 1);
qed "converse_diag";
Addsimps [converse_diag];
Goalw [refl_def] "refl A r ==> refl A (converse r)";
by (Blast_tac 1);
qed "refl_converse";
Goalw [antisym_def] "antisym (converse r) = antisym r";
by (Blast_tac 1);
qed "antisym_converse";
Goalw [trans_def] "trans (converse r) = trans r";
by (Blast_tac 1);
qed "trans_converse";
(** Domain **)
Goalw [Domain_def] "a: Domain(r) = (EX y. (a,y): r)";
by (Blast_tac 1);
qed "Domain_iff";
Goal "(a,b): r ==> a: Domain(r)";
by (etac (exI RS (Domain_iff RS iffD2)) 1) ;
qed "DomainI";
val prems= Goal "[| a : Domain(r); !!y. (a,y): r ==> P |] ==> P";
by (rtac (Domain_iff RS iffD1 RS exE) 1);
by (REPEAT (ares_tac prems 1)) ;
qed "DomainE";
AddIs [DomainI];
AddSEs [DomainE];
Goal "Domain Id = UNIV";
by (Blast_tac 1);
qed "Domain_Id";
Addsimps [Domain_Id];
Goal "Domain (diag A) = A";
by Auto_tac;
qed "Domain_diag";
Addsimps [Domain_diag];
Goal "Domain(A Un B) = Domain(A) Un Domain(B)";
by (Blast_tac 1);
qed "Domain_Un_eq";
Goal "Domain(A Int B) <= Domain(A) Int Domain(B)";
by (Blast_tac 1);
qed "Domain_Int_subset";
Goal "Domain(A) - Domain(B) <= Domain(A - B)";
by (Blast_tac 1);
qed "Domain_Diff_subset";
Goal "Domain (Union S) = (UN A:S. Domain A)";
by (Blast_tac 1);
qed "Domain_Union";
Goal "r <= s ==> Domain r <= Domain s";
by (Blast_tac 1);
qed "Domain_mono";
(** Range **)
Goalw [Domain_def, Range_def] "a: Range(r) = (EX y. (y,a): r)";
by (Blast_tac 1);
qed "Range_iff";
Goalw [Range_def] "(a,b): r ==> b : Range(r)";
by (etac (converseI RS DomainI) 1);
qed "RangeI";
val major::prems = Goalw [Range_def]
"[| b : Range(r); !!x. (x,b): r ==> P |] ==> P";
by (rtac (major RS DomainE) 1);
by (resolve_tac prems 1);
by (etac converseD 1) ;
qed "RangeE";
AddIs [RangeI];
AddSEs [RangeE];
Goal "Range Id = UNIV";
by (Blast_tac 1);
qed "Range_Id";
Addsimps [Range_Id];
Goal "Range (diag A) = A";
by Auto_tac;
qed "Range_diag";
Addsimps [Range_diag];
Goal "Range(A Un B) = Range(A) Un Range(B)";
by (Blast_tac 1);
qed "Range_Un_eq";
Goal "Range(A Int B) <= Range(A) Int Range(B)";
by (Blast_tac 1);
qed "Range_Int_subset";
Goal "Range(A) - Range(B) <= Range(A - B)";
by (Blast_tac 1);
qed "Range_Diff_subset";
Goal "Range (Union S) = (UN A:S. Range A)";
by (Blast_tac 1);
qed "Range_Union";
(*** Image of a set under a relation ***)
overload_1st_set "Relation.op ^^";
Goalw [Image_def] "b : r^^A = (? x:A. (x,b):r)";
by (Blast_tac 1);
qed "Image_iff";
Goalw [Image_def] "r^^{a} = {b. (a,b):r}";
by (Blast_tac 1);
qed "Image_singleton";
Goal "(b : r^^{a}) = ((a,b):r)";
by (rtac (Image_iff RS trans) 1);
by (Blast_tac 1);
qed "Image_singleton_iff";
AddIffs [Image_singleton_iff];
Goalw [Image_def] "[| (a,b): r; a:A |] ==> b : r^^A";
by (Blast_tac 1);
qed "ImageI";
val major::prems = Goalw [Image_def]
"[| b: r^^A; !!x.[| (x,b): r; x:A |] ==> P |] ==> P";
by (rtac (major RS CollectE) 1);
by (Clarify_tac 1);
by (rtac (hd prems) 1);
by (REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ;
qed "ImageE";
AddIs [ImageI];
AddSEs [ImageE];
Goal "R^^{} = {}";
by (Blast_tac 1);
qed "Image_empty";
Addsimps [Image_empty];
Goal "Id ^^ A = A";
by (Blast_tac 1);
qed "Image_Id";
Goal "diag A ^^ B = A Int B";
by (Blast_tac 1);
qed "Image_diag";
Addsimps [Image_Id, Image_diag];
Goal "R ^^ (A Int B) <= R ^^ A Int R ^^ B";
by (Blast_tac 1);
qed "Image_Int_subset";
Goal "R ^^ (A Un B) = R ^^ A Un R ^^ B";
by (Blast_tac 1);
qed "Image_Un";
Goal "r <= A Times B ==> r^^C <= B";
by (rtac subsetI 1);
by (REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ;
qed "Image_subset";
(*NOT suitable for rewriting*)
Goal "r^^B = (UN y: B. r^^{y})";
by (Blast_tac 1);
qed "Image_eq_UN";
Goal "[| r'<=r; A'<=A |] ==> (r' ^^ A') <= (r ^^ A)";
by (Blast_tac 1);
qed "Image_mono";
Goal "(r ^^ (UNION A B)) = (UN x:A.(r ^^ (B x)))";
by (Blast_tac 1);
qed "Image_UN";
(*Converse inclusion fails*)
Goal "(r ^^ (INTER A B)) <= (INT x:A.(r ^^ (B x)))";
by (Blast_tac 1);
qed "Image_INT_subset";
section "Univalent";
Goalw [Univalent_def]
"!x y. (x,y):r --> (!z. (x,z):r --> y=z) ==> Univalent r";
by (assume_tac 1);
qed "UnivalentI";
Goalw [Univalent_def]
"[| Univalent r; (x,y):r; (x,z):r|] ==> y=z";
by Auto_tac;
qed "UnivalentD";
(** Graphs of partial functions **)
Goal "Domain{(x,y). y = f x & P x} = {x. P x}";
by (Blast_tac 1);
qed "Domain_partial_func";
Goal "Range{(x,y). y = f x & P x} = f``{x. P x}";
by (Blast_tac 1);
qed "Range_partial_func";
(** Composition of function and relation **)
Goalw [fun_rel_comp_def] "A <= B ==> fun_rel_comp f A <= fun_rel_comp f B";
by (Fast_tac 1);
qed "fun_rel_comp_mono";
Goalw [fun_rel_comp_def] "! x. ?! y. (f x, y) : R ==> ?! g. g : fun_rel_comp f R";
by (res_inst_tac [("a","%x. @y. (f x, y) : R")] ex1I 1);
by (rtac CollectI 1);
by (rtac allI 1);
by (etac allE 1);
by (rtac (select_eq_Ex RS iffD2) 1);
by (etac ex1_implies_ex 1);
by (rtac ext 1);
by (etac CollectE 1);
by (REPEAT (etac allE 1));
by (rtac (select1_equality RS sym) 1);
by (atac 1);
by (atac 1);
qed "fun_rel_comp_unique";