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src/HOL/Relation.ML

author | paulson |

Fri, 22 Oct 1999 18:26:46 +0200 | |

changeset 7913 | 86be2946bb0b |

parent 7822 | 09aabe6d04b8 |

child 8004 | 6273f58ea2c1 |

permissions | -rw-r--r-- |

new theorems on Image

(* 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";