author  oheimb 
Tue, 24 Mar 1998 15:46:08 +0100  
changeset 4749  35b47e36e615 
parent 4643  1b40fcac5a09 
child 4751  6fbd9838ccae 
permissions  rwrr 
3193  1 
(* Title: HOL/WF_Rel 
2 
ID: $Id$ 

3 
Author: Konrad Slind 

4 
Copyright 1996 TU Munich 

5 

3296  6 
Derived WF relations: inverse image, lexicographic product, measure, ... 
3193  7 
*) 
8 

9 
open WF_Rel; 

10 

11 

12 
(* 

3237
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

13 
* "Less than" on the natural numbers 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

14 
**) 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

15 

4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

16 
goalw thy [less_than_def] "wf less_than"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

17 
by (rtac (wf_pred_nat RS wf_trancl) 1); 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

18 
qed "wf_less_than"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

19 
AddIffs [wf_less_than]; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

20 

4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

21 
goalw thy [less_than_def] "trans less_than"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

22 
by (rtac trans_trancl 1); 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

23 
qed "trans_less_than"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

24 
AddIffs [trans_less_than]; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

25 

4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

26 
goalw thy [less_than_def, less_def] "((x,y): less_than) = (x<y)"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

27 
by (Simp_tac 1); 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

28 
qed "less_than_iff"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

29 
AddIffs [less_than_iff]; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

30 

4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

31 
(* 
3193  32 
* The inverse image into a wellfounded relation is wellfounded. 
33 
**) 

34 

35 
goal thy "!!r. wf(r) ==> wf(inv_image r (f::'a=>'b))"; 

4089  36 
by (full_simp_tac (simpset() addsimps [inv_image_def, wf_eq_minimal]) 1); 
3718  37 
by (Clarify_tac 1); 
3193  38 
by (subgoal_tac "? (w::'b). w : {w. ? (x::'a). x: Q & (f x = w)}" 1); 
4089  39 
by (blast_tac (claset() delrules [allE]) 2); 
3193  40 
by (etac allE 1); 
41 
by (mp_tac 1); 

42 
by (Blast_tac 1); 

43 
qed "wf_inv_image"; 

44 
AddSIs [wf_inv_image]; 

45 

3237
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

46 
goalw thy [trans_def,inv_image_def] 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

47 
"!!r. trans r ==> trans (inv_image r f)"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

48 
by (Simp_tac 1); 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

49 
by (Blast_tac 1); 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

50 
qed "trans_inv_image"; 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

51 

4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

52 

3193  53 
(* 
54 
* All measures are wellfounded. 

55 
**) 

56 

57 
goalw thy [measure_def] "wf (measure f)"; 

3237
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

58 
by (rtac (wf_less_than RS wf_inv_image) 1); 
3193  59 
qed "wf_measure"; 
60 
AddIffs [wf_measure]; 

61 

4643  62 
val measure_induct = standard 
63 
(asm_full_simplify (simpset() addsimps [measure_def,inv_image_def]) 

64 
(wf_measure RS wf_induct)); 

65 
store_thm("measure_induct",measure_induct); 

66 

3193  67 
(* 
68 
* Wellfoundedness of lexicographic combinations 

69 
**) 

70 

71 
val [wfa,wfb] = goalw thy [wf_def,lex_prod_def] 

72 
"[ wf(ra); wf(rb) ] ==> wf(ra**rb)"; 

3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

73 
by (EVERY1 [rtac allI,rtac impI]); 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

74 
by (simp_tac (HOL_basic_ss addsimps [split_paired_All]) 1); 
3193  75 
by (rtac (wfa RS spec RS mp) 1); 
76 
by (EVERY1 [rtac allI,rtac impI]); 

77 
by (rtac (wfb RS spec RS mp) 1); 

78 
by (Blast_tac 1); 

79 
qed "wf_lex_prod"; 

80 
AddSIs [wf_lex_prod]; 

81 

82 
(* 

83 
* Transitivity of WF combinators. 

84 
**) 

85 
goalw thy [trans_def, lex_prod_def] 

86 
"!!R1 R2. [ trans R1; trans R2 ] ==> trans (R1 ** R2)"; 

87 
by (Simp_tac 1); 

88 
by (Blast_tac 1); 

89 
qed "trans_lex_prod"; 

90 
AddSIs [trans_lex_prod]; 

91 

92 

93 
(* 

94 
* Wellfoundedness of proper subset on finite sets. 

95 
**) 

96 
goalw thy [finite_psubset_def] "wf(finite_psubset)"; 

97 
by (rtac (wf_measure RS wf_subset) 1); 

4089  98 
by (simp_tac (simpset() addsimps [measure_def, inv_image_def, less_than_def, 
3237
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

99 
symmetric less_def])1); 
4089  100 
by (fast_tac (claset() addSIs [psubset_card]) 1); 
3193  101 
qed "wf_finite_psubset"; 
102 

3237
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

103 
goalw thy [finite_psubset_def, trans_def] "trans finite_psubset"; 
4089  104 
by (simp_tac (simpset() addsimps [psubset_def]) 1); 
3237
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

105 
by (Blast_tac 1); 
4da86d44de33
Relation "less_than" internalizes "<" for easy use of TFL
paulson
parents:
3193
diff
changeset

106 
qed "trans_finite_psubset"; 
3193  107 

3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

108 
(* 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

109 
* Wellfoundedness of finite acyclic relations 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

110 
* Cannot go into WF because it needs Finite 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

111 
**) 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

112 

c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

113 
goal thy "!!r. finite r ==> acyclic r > wf r"; 
3457  114 
by (etac finite_induct 1); 
115 
by (Blast_tac 1); 

116 
by (split_all_tac 1); 

117 
by (Asm_full_simp_tac 1); 

3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

118 
qed_spec_mp "finite_acyclic_wf"; 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

119 

4749  120 
qed_goal "finite_acyclic_wf_converse" thy 
121 
"ÄX. Ëfinite r; acyclic rÌ êë wf (r^1)" (K [ 

122 
etac (finite_converse RS iffD2 RS finite_acyclic_wf) 1, 

123 
etac (acyclic_converse RS iffD2) 1]); 

124 

3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

125 
goal thy "!!r. finite r ==> wf r = acyclic r"; 
4089  126 
by (blast_tac (claset() addIs [finite_acyclic_wf,wf_acyclic]) 1); 
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

127 
qed "wf_iff_acyclic_if_finite"; 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

128 

c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

129 

c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

130 
(* 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

131 
* A relation is wellfounded iff it has no infinite descending chain 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

132 
**) 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

133 

c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

134 
goalw thy [wf_eq_minimal RS eq_reflection] 
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

135 
"wf r = (~(? f. !i. (f(Suc i),f i) : r))"; 
3457  136 
by (rtac iffI 1); 
137 
by (rtac notI 1); 

138 
by (etac exE 1); 

139 
by (eres_inst_tac [("x","{w. ? i. w=f i}")] allE 1); 

140 
by (Blast_tac 1); 

141 
by (etac swap 1); 

3446
a14e5451f613
Addition of not_imp (which pushes negation into implication) as a default
paulson
parents:
3436
diff
changeset

142 
by (Asm_full_simp_tac 1); 
3718  143 
by (Clarify_tac 1); 
3457  144 
by (subgoal_tac "!n. nat_rec x (%i y. @z. z:Q & (z,y):r) n : Q" 1); 
3436
d68dbb8f22d3
Tuned wf_iff_no_infinite_down_chain proof, based on Konrads ideas.
nipkow
parents:
3413
diff
changeset

145 
by (res_inst_tac[("x","nat_rec x (%i y. @z. z:Q & (z,y):r)")]exI 1); 
3457  146 
by (rtac allI 1); 
147 
by (Simp_tac 1); 

148 
by (rtac selectI2EX 1); 

149 
by (Blast_tac 1); 

150 
by (Blast_tac 1); 

151 
by (rtac allI 1); 

152 
by (induct_tac "n" 1); 

153 
by (Asm_simp_tac 1); 

154 
by (Simp_tac 1); 

155 
by (rtac selectI2EX 1); 

156 
by (Blast_tac 1); 

157 
by (Blast_tac 1); 

3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3296
diff
changeset

158 
qed "wf_iff_no_infinite_down_chain"; 