src/HOL/Trancl.ML
author berghofe
Tue, 30 May 2000 18:02:49 +0200
changeset 9001 93af64f54bf2
parent 8703 816d8f6513be
child 9022 a389be05c06f
permissions -rw-r--r--
the is now defined using primrec, avoiding explicit use of arbitrary.
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(*  Title:      HOL/Trancl
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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Theorems about the transitive closure of a relation
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*)
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(** The relation rtrancl **)
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section "^*";
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Goal "mono(%s. Id Un (r O s))";
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by (rtac monoI 1);
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by (REPEAT (ares_tac [monoI, subset_refl, comp_mono, Un_mono] 1));
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qed "rtrancl_fun_mono";
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val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski);
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(*Reflexivity of rtrancl*)
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Goal "(a,a) : r^*";
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by (stac rtrancl_unfold 1);
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by (Blast_tac 1);
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qed "rtrancl_refl";
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Addsimps [rtrancl_refl];
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AddSIs   [rtrancl_refl];
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(*Closure under composition with r*)
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Goal "[| (a,b) : r^*;  (b,c) : r |] ==> (a,c) : r^*";
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by (stac rtrancl_unfold 1);
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by (Blast_tac 1);
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qed "rtrancl_into_rtrancl";
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(*rtrancl of r contains r*)
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Goal "!!p. p : r ==> p : r^*";
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by (split_all_tac 1);
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by (etac (rtrancl_refl RS rtrancl_into_rtrancl) 1);
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qed "r_into_rtrancl";
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(*monotonicity of rtrancl*)
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Goalw [rtrancl_def] "r <= s ==> r^* <= s^*";
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by (REPEAT(ares_tac [lfp_mono,Un_mono,comp_mono,subset_refl] 1));
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qed "rtrancl_mono";
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(** standard induction rule **)
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val major::prems = Goal 
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  "[| (a,b) : r^*; \
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\     !!x. P(x,x); \
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\     !!x y z.[| P(x,y); (x,y): r^*; (y,z): r |]  ==>  P(x,z) |] \
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\  ==>  P(a,b)";
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by (rtac ([rtrancl_def, rtrancl_fun_mono, major] MRS def_induct) 1);
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by (blast_tac (claset() addIs prems) 1);
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qed "rtrancl_full_induct";
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(*nice induction rule*)
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val major::prems = Goal
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    "[| (a::'a,b) : r^*;    \
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\       P(a); \
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\       !!y z.[| (a,y) : r^*;  (y,z) : r;  P(y) |] ==> P(z) |]  \
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\     ==> P(b)";
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(*by induction on this formula*)
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by (subgoal_tac "! y. (a::'a,b) = (a,y) --> P(y)" 1);
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(*now solve first subgoal: this formula is sufficient*)
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by (Blast_tac 1);
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(*now do the induction*)
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by (resolve_tac [major RS rtrancl_full_induct] 1);
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by (blast_tac (claset() addIs prems) 1);
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by (blast_tac (claset() addIs prems) 1);
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qed "rtrancl_induct";
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bind_thm ("rtrancl_induct2", split_rule
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  (read_instantiate [("a","(ax,ay)"), ("b","(bx,by)")] rtrancl_induct));
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(*transitivity of transitive closure!! -- by induction.*)
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Goalw [trans_def] "trans(r^*)";
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by Safe_tac;
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by (eres_inst_tac [("b","z")] rtrancl_induct 1);
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by (ALLGOALS(blast_tac (claset() addIs [rtrancl_into_rtrancl])));
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qed "trans_rtrancl";
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bind_thm ("rtrancl_trans", trans_rtrancl RS transD);
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(*elimination of rtrancl -- by induction on a special formula*)
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val major::prems = Goal
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    "[| (a::'a,b) : r^*;  (a = b) ==> P;        \
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\       !!y.[| (a,y) : r^*; (y,b) : r |] ==> P  \
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\    |] ==> P";
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by (subgoal_tac "(a::'a) = b  | (? y. (a,y) : r^* & (y,b) : r)" 1);
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by (rtac (major RS rtrancl_induct) 2);
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by (blast_tac (claset() addIs prems) 2);
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by (blast_tac (claset() addIs prems) 2);
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by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
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qed "rtranclE";
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bind_thm ("rtrancl_into_rtrancl2", r_into_rtrancl RS rtrancl_trans);
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(*** More r^* equations and inclusions ***)
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Goal "(r^*)^* = r^*";
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by Auto_tac;
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by (etac r_into_rtrancl 2);
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by (etac rtrancl_induct 1);
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by (rtac rtrancl_refl 1);
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by (blast_tac (claset() addIs [rtrancl_trans]) 1);
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qed "rtrancl_idemp";
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Addsimps [rtrancl_idemp];
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Goal "R^* O R^* = R^*";
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by (rtac set_ext 1);
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by (split_all_tac 1);
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by (blast_tac (claset() addIs [rtrancl_trans]) 1);
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qed "rtrancl_idemp_self_comp";
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Addsimps [rtrancl_idemp_self_comp];
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Goal "r <= s^* ==> r^* <= s^*";
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by (dtac rtrancl_mono 1);
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by (Asm_full_simp_tac 1);
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qed "rtrancl_subset_rtrancl";
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Goal "[| R <= S; S <= R^* |] ==> S^* = R^*";
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by (dtac rtrancl_mono 1);
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by (dtac rtrancl_mono 1);
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by (Asm_full_simp_tac 1);
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by (Blast_tac 1);
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qed "rtrancl_subset";
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Goal "(R^* Un S^*)^* = (R Un S)^*";
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by (blast_tac (claset() addSIs [rtrancl_subset]
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                        addIs [r_into_rtrancl, rtrancl_mono RS subsetD]) 1);
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qed "rtrancl_Un_rtrancl";
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Goal "(R^=)^* = R^*";
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by (blast_tac (claset() addSIs [rtrancl_subset] addIs [r_into_rtrancl]) 1);
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qed "rtrancl_reflcl";
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Addsimps [rtrancl_reflcl];
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Goal "(r - Id)^* = r^*";
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by (rtac sym 1);
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by (rtac rtrancl_subset 1);
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 by (Blast_tac 1);
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by (Clarify_tac 1);
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by (rename_tac "a b" 1);
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by (case_tac "a=b" 1);
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 by (Blast_tac 1);
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by (blast_tac (claset() addSIs [r_into_rtrancl]) 1);
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qed "rtrancl_r_diff_Id";
187cada50e19 A few lemmas and some Adds.
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parents: 8114
diff changeset
   152
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   153
Goal "(x,y) : (r^-1)^* ==> (x,y) : (r^*)^-1";
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   154
by (rtac converseI 1);
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   155
by (etac rtrancl_induct 1);
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   156
by (rtac rtrancl_refl 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   157
by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   158
qed "rtrancl_converseD";
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   159
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   160
Goal "(x,y) : (r^*)^-1 ==> (x,y) : (r^-1)^*";
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   161
by (dtac converseD 1);
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   162
by (etac rtrancl_induct 1);
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   163
by (rtac rtrancl_refl 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   164
by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1);
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   165
qed "rtrancl_converseI";
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   166
5069
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wenzelm
parents: 4838
diff changeset
   167
Goal "(r^-1)^* = (r^*)^-1";
4838
196100237656 cleanup for split_all_tac as wrapper in claset()
oheimb
parents: 4830
diff changeset
   168
by (safe_tac (claset() addSDs [rtrancl_converseD] addSIs [rtrancl_converseI]));
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   169
qed "rtrancl_converse";
1642
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paulson
parents: 1552
diff changeset
   170
5316
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paulson
parents: 5281
diff changeset
   171
val major::prems = Goal
1706
4e0d5c7f57fa Added backwards rtrancl_induct and special versions for pairs.
nipkow
parents: 1642
diff changeset
   172
    "[| (a,b) : r^*; P(b); \
4e0d5c7f57fa Added backwards rtrancl_induct and special versions for pairs.
nipkow
parents: 1642
diff changeset
   173
\       !!y z.[| (y,z) : r;  (z,b) : r^*;  P(z) |] ==> P(y) |]  \
4e0d5c7f57fa Added backwards rtrancl_induct and special versions for pairs.
nipkow
parents: 1642
diff changeset
   174
\     ==> P(a)";
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   175
by (rtac ((major RS converseI RS rtrancl_converseI) RS rtrancl_induct) 1);
2031
03a843f0f447 Ran expandshort
paulson
parents: 1985
diff changeset
   176
by (resolve_tac prems 1);
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   177
by (blast_tac (claset() addIs prems addSDs[rtrancl_converseD])1);
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   178
qed "converse_rtrancl_induct";
1706
4e0d5c7f57fa Added backwards rtrancl_induct and special versions for pairs.
nipkow
parents: 1642
diff changeset
   179
5347
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
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   180
bind_thm ("converse_rtrancl_induct2", split_rule
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
diff changeset
   181
  (read_instantiate [("a","(ax,ay)"),("b","(bx,by)")]converse_rtrancl_induct));
1496
c443b2adaf52 Added a few thms and the new theory RelPow.
nipkow
parents: 1465
diff changeset
   182
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5281
diff changeset
   183
val major::prems = Goal
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   184
 "[| (x,z):r^*; \
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   185
\    x=z ==> P; \
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   186
\    !!y. [| (x,y):r; (y,z):r^* |] ==> P \
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   187
\ |] ==> P";
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   188
by (subgoal_tac "x = z  | (? y. (x,y) : r & (y,z) : r^*)" 1);
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   189
by (rtac (major RS converse_rtrancl_induct) 2);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   190
by (blast_tac (claset() addIs prems) 2);
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   191
by (blast_tac (claset() addIs prems) 2);
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   192
by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
5347
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
diff changeset
   193
qed "converse_rtranclE";
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
diff changeset
   194
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
diff changeset
   195
bind_thm ("converse_rtranclE2", split_rule
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
diff changeset
   196
  (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] converse_rtranclE));
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   197
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   198
Goal "r O r^* = r^* O r";
5347
d014d7e57337 Added converse_rtranclE(2)
nipkow
parents: 5316
diff changeset
   199
by (blast_tac (claset() addEs [rtranclE, converse_rtranclE] 
3723
034f0f5ca43f Tidied proof of r_comp_rtrancl_eq
paulson
parents: 3457
diff changeset
   200
	               addIs [rtrancl_into_rtrancl, rtrancl_into_rtrancl2]) 1);
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   201
qed "r_comp_rtrancl_eq";
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   202
923
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parents:
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   203
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
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   204
(**** The relation trancl ****)
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clasohm
parents:
diff changeset
   205
5771
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   206
section "^+";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   207
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   208
Goalw [trancl_def] "[| p:r^+; r <= s |] ==> p:s^+";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   209
by (blast_tac (claset() addIs [rtrancl_mono RS subsetD]) 1);
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   210
qed "trancl_mono";
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   211
923
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clasohm
parents:
diff changeset
   212
(** Conversions between trancl and rtrancl **)
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parents:
diff changeset
   213
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   214
Goalw [trancl_def]
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   215
    "!!p. p : r^+ ==> p : r^*";
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   216
by (split_all_tac 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   217
by (etac compEpair 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   218
by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   219
qed "trancl_into_rtrancl";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   220
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   221
(*r^+ contains r*)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   222
Goalw [trancl_def]
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   223
   "!!p. p : r ==> p : r^+";
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   224
by (split_all_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   225
by (REPEAT (ares_tac [prem,compI,rtrancl_refl] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   226
qed "r_into_trancl";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   227
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   228
(*intro rule by definition: from rtrancl and r*)
5255
paulson
parents: 5148
diff changeset
   229
Goalw [trancl_def] "[| (a,b) : r^*;  (b,c) : r |]   ==>  (a,c) : r^+";
paulson
parents: 5148
diff changeset
   230
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   231
qed "rtrancl_into_trancl1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   232
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   233
(*intro rule from r and rtrancl*)
5255
paulson
parents: 5148
diff changeset
   234
Goal "[| (a,b) : r;  (b,c) : r^* |]   ==>  (a,c) : r^+";
paulson
parents: 5148
diff changeset
   235
by (etac rtranclE 1);
paulson
parents: 5148
diff changeset
   236
by (blast_tac (claset() addIs [r_into_trancl]) 1);
1122
20b708827030 renamed trans_rtrancl to rtrancl_trans and modified it by expanding trans.
nipkow
parents: 1121
diff changeset
   237
by (rtac (rtrancl_trans RS rtrancl_into_trancl1) 1);
5255
paulson
parents: 5148
diff changeset
   238
by (REPEAT (ares_tac [r_into_rtrancl] 1));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   239
qed "rtrancl_into_trancl2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   240
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   241
(*Nice induction rule for trancl*)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5281
diff changeset
   242
val major::prems = Goal
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   243
  "[| (a,b) : r^+;                                      \
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   244
\     !!y.  [| (a,y) : r |] ==> P(y);                   \
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   245
\     !!y z.[| (a,y) : r^+;  (y,z) : r;  P(y) |] ==> P(z)       \
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   246
\  |] ==> P(b)";
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   247
by (rtac (rewrite_rule [trancl_def] major  RS  compEpair) 1);
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   248
(*by induction on this formula*)
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   249
by (subgoal_tac "ALL z. (y,z) : r --> P(z)" 1);
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   250
(*now solve first subgoal: this formula is sufficient*)
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2031
diff changeset
   251
by (Blast_tac 1);
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   252
by (etac rtrancl_induct 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   253
by (ALLGOALS (blast_tac (claset() addIs (rtrancl_into_trancl1::prems))));
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   254
qed "trancl_induct";
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   255
6856
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   256
(*Another induction rule for trancl, incorporating transitivity.*)
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   257
val major::prems = goal thy
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   258
 "[| (x,y) : r^+; \
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   259
\    !!x y. (x,y) : r ==> P x y; \
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   260
\    !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z \
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   261
\ |] ==> P x y";
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   262
by (blast_tac (claset() addIs ([r_into_trancl,major RS trancl_induct]@prems))1);
6856
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   263
qed "trancl_trans_induct";
0364007b4bb3 New thm trancl_trans_induct
nipkow
parents: 6162
diff changeset
   264
923
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clasohm
parents:
diff changeset
   265
(*elimination of r^+ -- NOT an induction rule*)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5281
diff changeset
   266
val major::prems = Goal
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   267
    "[| (a::'a,b) : r^+;  \
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   268
\       (a,b) : r ==> P; \
1465
5d7a7e439cec expanded tabs
clasohm
parents: 1301
diff changeset
   269
\       !!y.[| (a,y) : r^+;  (y,b) : r |] ==> P  \
923
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clasohm
parents:
diff changeset
   270
\    |] ==> P";
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   271
by (subgoal_tac "(a::'a,b) : r | (? y. (a,y) : r^+  &  (y,b) : r)" 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   272
by (REPEAT (eresolve_tac ([asm_rl,disjE,exE,conjE]@prems) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   273
by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   274
by (etac rtranclE 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2031
diff changeset
   275
by (Blast_tac 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   276
by (blast_tac (claset() addSIs [rtrancl_into_trancl1]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   277
qed "tranclE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   278
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   279
(*Transitivity of r^+.
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   280
  Proved by unfolding since it uses transitivity of rtrancl. *)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   281
Goalw [trancl_def] "trans(r^+)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   282
by (rtac transI 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   283
by (REPEAT (etac compEpair 1));
1122
20b708827030 renamed trans_rtrancl to rtrancl_trans and modified it by expanding trans.
nipkow
parents: 1121
diff changeset
   284
by (rtac (rtrancl_into_rtrancl RS (rtrancl_trans RS compI)) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   285
by (REPEAT (assume_tac 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   286
qed "trans_trancl";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   287
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   288
bind_thm ("trancl_trans", trans_trancl RS transD);
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   289
5255
paulson
parents: 5148
diff changeset
   290
Goalw [trancl_def] "[| (x,y):r^*; (y,z):r^+ |] ==> (x,z):r^+";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   291
by (blast_tac (claset() addIs [rtrancl_trans,r_into_rtrancl]) 1);
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   292
qed "rtrancl_trancl_trancl";
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   293
5255
paulson
parents: 5148
diff changeset
   294
(* "[| (a,b) : r;  (b,c) : r^+ |]   ==>  (a,c) : r^+" *)
paulson
parents: 5148
diff changeset
   295
bind_thm ("trancl_into_trancl2", [trans_trancl, r_into_trancl] MRS transD);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   296
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   297
(* primitive recursion for trancl over finite relations: *)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   298
Goal "(insert (y,x) r)^+ = r^+ Un {(a,b). (a,y):r^* & (x,b):r^*}";
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3439
diff changeset
   299
by (rtac equalityI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3439
diff changeset
   300
 by (rtac subsetI 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3439
diff changeset
   301
 by (split_all_tac 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3439
diff changeset
   302
 by (etac trancl_induct 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   303
  by (blast_tac (claset() addIs [r_into_trancl]) 1);
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   304
 by (blast_tac (claset() addIs
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   305
     [rtrancl_into_trancl1,trancl_into_rtrancl,r_into_trancl,trancl_trans]) 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3439
diff changeset
   306
by (rtac subsetI 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   307
by (blast_tac (claset() addIs
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   308
     [rtrancl_into_trancl2, rtrancl_trancl_trancl,
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   309
      impOfSubs rtrancl_mono, trancl_mono]) 1);
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   310
qed "trancl_insert";
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   311
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   312
Goalw [trancl_def] "(r^-1)^+ = (r^+)^-1";
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   313
by (simp_tac (simpset() addsimps [rtrancl_converse,converse_comp]) 1);
5451
08ca6e067ee6 tided the unused rule irrefl_tranclI
paulson
parents: 5347
diff changeset
   314
by (simp_tac (simpset() addsimps [rtrancl_converse RS sym,
08ca6e067ee6 tided the unused rule irrefl_tranclI
paulson
parents: 5347
diff changeset
   315
				  r_comp_rtrancl_eq]) 1);
4746
a5dcd7e4a37d inverse -> converse
paulson
parents: 4153
diff changeset
   316
qed "trancl_converse";
3413
c1f63cc3a768 Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents: 2935
diff changeset
   317
5771
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   318
Goal "(x,y) : (r^+)^-1 ==> (x,y) : (r^-1)^+";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   319
by (asm_full_simp_tac (simpset() addsimps [trancl_converse]) 1);
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   320
qed "trancl_converseI";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   321
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   322
Goal "(x,y) : (r^-1)^+ ==> (x,y) : (r^+)^-1";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   323
by (asm_full_simp_tac (simpset() addsimps [trancl_converse]) 1);
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   324
qed "trancl_converseD";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   325
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   326
val major::prems = Goal
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   327
    "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y); \
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   328
\       !!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y) |]  \
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   329
\     ==> P(a)";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   330
by (rtac ((major RS converseI RS trancl_converseI) RS trancl_induct) 1);
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   331
 by (resolve_tac prems 1);
6162
484adda70b65 expandshort
paulson
parents: 5771
diff changeset
   332
 by (etac converseD 1);
5771
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   333
by (blast_tac (claset() addIs prems addSDs [trancl_converseD])1);
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   334
qed "converse_trancl_induct";
7c2c8cf20221 added nat_diff_split and a few lemmas in Trancl.
nipkow
parents: 5608
diff changeset
   335
5451
08ca6e067ee6 tided the unused rule irrefl_tranclI
paulson
parents: 5347
diff changeset
   336
(*Unused*)
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   337
Goal "r^-1 Int r^+ = {} ==> (x, x) ~: r^+";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   338
by (subgoal_tac "!y. (x, y) : r^+ --> x~=y" 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   339
by (Fast_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   340
by (strip_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   341
by (etac trancl_induct 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   342
by (auto_tac (claset() addIs [r_into_trancl], simpset()));
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   343
qed "irrefl_tranclI";
1130
0df0df1685a8 Added trancl_cs
nipkow
parents: 1128
diff changeset
   344
8703
816d8f6513be Times -> <*>
nipkow
parents: 8320
diff changeset
   345
Goal "[| (a,b) : r^*;  r <= A <*> A |] ==> a=b | a:A";
5255
paulson
parents: 5148
diff changeset
   346
by (etac rtrancl_induct 1);
paulson
parents: 5148
diff changeset
   347
by Auto_tac;
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1552
diff changeset
   348
val lemma = result();
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   349
8703
816d8f6513be Times -> <*>
nipkow
parents: 8320
diff changeset
   350
Goalw [trancl_def] "r <= A <*> A ==> r^+ <= A <*> A";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3723
diff changeset
   351
by (blast_tac (claset() addSDs [lemma]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   352
qed "trancl_subset_Sigma";
1130
0df0df1685a8 Added trancl_cs
nipkow
parents: 1128
diff changeset
   353
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   354
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   355
Goal "(r^+)^= = r^*";
4838
196100237656 cleanup for split_all_tac as wrapper in claset()
oheimb
parents: 4830
diff changeset
   356
by Safe_tac;
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   357
by  (etac trancl_into_rtrancl 1);
8265
187cada50e19 A few lemmas and some Adds.
nipkow
parents: 8114
diff changeset
   358
by (blast_tac (claset() addEs [rtranclE] addDs [rtrancl_into_trancl1]) 1);
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   359
qed "reflcl_trancl";
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   360
Addsimps[reflcl_trancl];
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   361
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4838
diff changeset
   362
Goal "(r^=)^+ = r^*";
4838
196100237656 cleanup for split_all_tac as wrapper in claset()
oheimb
parents: 4830
diff changeset
   363
by Safe_tac;
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   364
by  (dtac trancl_into_rtrancl 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   365
by  (Asm_full_simp_tac 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   366
by (etac rtranclE 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   367
by  Safe_tac;
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   368
by  (rtac r_into_trancl 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   369
by  (Simp_tac 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   370
by (rtac rtrancl_into_trancl1 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   371
by (etac (rtrancl_reflcl RS equalityD2 RS subsetD) 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   372
by (Fast_tac 1);
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   373
qed "trancl_reflcl";
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   374
Addsimps[trancl_reflcl];
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   375
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   376
Goal "{}^+ = {}";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   377
by (auto_tac (claset() addEs [trancl_induct], simpset()));
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   378
qed "trancl_empty";
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   379
Addsimps[trancl_empty];
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   380
7007
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   381
Goal "{}^* = Id";
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   382
by (rtac (reflcl_trancl RS subst) 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   383
by (Simp_tac 1);
b46ccfee8e59 qed_goal -> Goal
paulson
parents: 6856
diff changeset
   384
qed "rtrancl_empty";
4764
9b3293646b5d generalized appearance of trancl_into_rtrancl and r_into_trancl
oheimb
parents: 4746
diff changeset
   385
Addsimps[rtrancl_empty];