author | haftmann |
Sat, 05 Dec 2009 20:02:21 +0100 | |
changeset 34007 | aea892559fc5 |
parent 33027 | 9cf389429f6d |
child 36566 | f91342f218a9 |
permissions | -rw-r--r-- |
33027 | 1 |
(* Title: HOL/Metis_Examples/Abstraction.thy |
23449 | 2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Testing the metis method. |
23449 | 5 |
*) |
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theory Abstraction |
8 |
imports Main FuncSet |
|
23449 | 9 |
begin |
10 |
||
11 |
(*For Christoph Benzmueller*) |
|
12 |
lemma "x<1 & ((op=) = (op=)) ==> ((op=) = (op=)) & (x<(2::nat))"; |
|
13 |
by (metis One_nat_def less_Suc0 not_less0 not_less_eq numeral_2_eq_2) |
|
14 |
||
15 |
(*this is a theorem, but we can't prove it unless ext is applied explicitly |
|
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lemma "(op=) = (%x y. y=x)" |
|
17 |
*) |
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18 |
||
19 |
consts |
|
20 |
monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool" |
|
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pset :: "'a set => 'a set" |
|
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order :: "'a set => ('a * 'a) set" |
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23 |
||
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24 |
declare [[ atp_problem_prefix = "Abstraction__Collect_triv" ]] |
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lemma (*Collect_triv:*) "a \<in> {x. P x} ==> P a" |
26 |
proof (neg_clausify) |
|
27 |
assume 0: "(a\<Colon>'a\<Colon>type) \<in> Collect (P\<Colon>'a\<Colon>type \<Rightarrow> bool)" |
|
28 |
assume 1: "\<not> (P\<Colon>'a\<Colon>type \<Rightarrow> bool) (a\<Colon>'a\<Colon>type)" |
|
29 |
have 2: "(P\<Colon>'a\<Colon>type \<Rightarrow> bool) (a\<Colon>'a\<Colon>type)" |
|
30 |
by (metis CollectD 0) |
|
31 |
show "False" |
|
32 |
by (metis 2 1) |
|
33 |
qed |
|
34 |
||
35 |
lemma Collect_triv: "a \<in> {x. P x} ==> P a" |
|
23756 | 36 |
by (metis mem_Collect_eq) |
23449 | 37 |
|
38 |
||
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39 |
declare [[ atp_problem_prefix = "Abstraction__Collect_mp" ]] |
23449 | 40 |
lemma "a \<in> {x. P x --> Q x} ==> a \<in> {x. P x} ==> a \<in> {x. Q x}" |
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by (metis CollectI Collect_imp_eq ComplD UnE mem_Collect_eq); |
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--{*34 secs*} |
43 |
||
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declare [[ atp_problem_prefix = "Abstraction__Sigma_triv" ]] |
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lemma "(a,b) \<in> Sigma A B ==> a \<in> A & b \<in> B a" |
46 |
proof (neg_clausify) |
|
47 |
assume 0: "(a\<Colon>'a\<Colon>type, b\<Colon>'b\<Colon>type) \<in> Sigma (A\<Colon>'a\<Colon>type set) (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set)" |
|
48 |
assume 1: "(a\<Colon>'a\<Colon>type) \<notin> (A\<Colon>'a\<Colon>type set) \<or> (b\<Colon>'b\<Colon>type) \<notin> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) a" |
|
49 |
have 2: "(a\<Colon>'a\<Colon>type) \<in> (A\<Colon>'a\<Colon>type set)" |
|
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by (metis SigmaD1 0) |
|
51 |
have 3: "(b\<Colon>'b\<Colon>type) \<in> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) (a\<Colon>'a\<Colon>type)" |
|
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by (metis SigmaD2 0) |
|
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have 4: "(b\<Colon>'b\<Colon>type) \<notin> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) (a\<Colon>'a\<Colon>type)" |
|
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by (metis 1 2) |
|
55 |
show "False" |
|
56 |
by (metis 3 4) |
|
57 |
qed |
|
58 |
||
59 |
lemma Sigma_triv: "(a,b) \<in> Sigma A B ==> a \<in> A & b \<in> B a" |
|
60 |
by (metis SigmaD1 SigmaD2) |
|
61 |
||
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|
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declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect" ]] |
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lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b" |
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(*???metis says this is satisfiable! |
65 |
by (metis CollectD SigmaD1 SigmaD2) |
|
66 |
*) |
|
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by (meson CollectD SigmaD1 SigmaD2) |
68 |
||
69 |
||
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(*single-step*) |
71 |
lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b" |
|
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by (metis SigmaD1 SigmaD2 insert_def singleton_conv2 Un_empty_right vimage_Collect_eq vimage_def vimage_singleton_eq) |
23449 | 73 |
|
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|
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lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b" |
76 |
proof (neg_clausify) |
|
24827 | 77 |
assume 0: "(a\<Colon>'a\<Colon>type, b\<Colon>'b\<Colon>type) |
78 |
\<in> Sigma (A\<Colon>'a\<Colon>type set) |
|
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(COMBB Collect (COMBC (COMBB COMBB op =) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type)))" |
|
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assume 1: "(a\<Colon>'a\<Colon>type) \<notin> (A\<Colon>'a\<Colon>type set) \<or> a \<noteq> (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type)" |
|
81 |
have 2: "(a\<Colon>'a\<Colon>type) \<in> (A\<Colon>'a\<Colon>type set)" |
|
82 |
by (metis 0 SigmaD1) |
|
83 |
have 3: "(b\<Colon>'b\<Colon>type) |
|
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\<in> COMBB Collect (COMBC (COMBB COMBB op =) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type)) (a\<Colon>'a\<Colon>type)" |
|
85 |
by (metis 0 SigmaD2) |
|
86 |
have 4: "(b\<Colon>'b\<Colon>type) \<in> Collect (COMBB (op = (a\<Colon>'a\<Colon>type)) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type))" |
|
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by (metis 3) |
|
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have 5: "(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type) \<noteq> (a\<Colon>'a\<Colon>type)" |
|
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by (metis 1 2) |
|
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have 6: "(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type) = (a\<Colon>'a\<Colon>type)" |
|
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by (metis 4 vimage_singleton_eq insert_def singleton_conv2 Un_empty_right vimage_Collect_eq vimage_def) |
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show "False" |
24827 | 93 |
by (metis 5 6) |
94 |
qed |
|
95 |
||
96 |
(*Alternative structured proof, untyped*) |
|
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lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b" |
|
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proof (neg_clausify) |
|
99 |
assume 0: "(a, b) \<in> Sigma A (COMBB Collect (COMBC (COMBB COMBB op =) f))" |
|
100 |
have 1: "b \<in> Collect (COMBB (op = a) f)" |
|
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by (metis 0 SigmaD2) |
|
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have 2: "f b = a" |
|
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by (metis 1 vimage_Collect_eq singleton_conv2 insert_def Un_empty_right vimage_singleton_eq vimage_def) |
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assume 3: "a \<notin> A \<or> a \<noteq> f b" |
105 |
have 4: "a \<in> A" |
|
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by (metis 0 SigmaD1) |
|
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have 5: "f b \<noteq> a" |
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by (metis 4 3) |
|
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show "False" |
|
110 |
by (metis 5 2) |
|
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qed |
|
23449 | 112 |
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113 |
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declare [[ atp_problem_prefix = "Abstraction__CLF_eq_in_pp" ]] |
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lemma "(cl,f) \<in> CLF ==> CLF = (SIGMA cl: CL.{f. f \<in> pset cl}) ==> f \<in> pset cl" |
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by (metis Collect_mem_eq SigmaD2) |
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|
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118 |
lemma "(cl,f) \<in> CLF ==> CLF = (SIGMA cl: CL.{f. f \<in> pset cl}) ==> f \<in> pset cl" |
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|
119 |
proof (neg_clausify) |
24827 | 120 |
assume 0: "(cl, f) \<in> CLF" |
121 |
assume 1: "CLF = Sigma CL (COMBB Collect (COMBB (COMBC op \<in>) pset))" |
|
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assume 2: "f \<notin> pset cl" |
|
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have 3: "\<And>X1 X2. X2 \<in> COMBB Collect (COMBB (COMBC op \<in>) pset) X1 \<or> (X1, X2) \<notin> CLF" |
|
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by (metis SigmaD2 1) |
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have 4: "\<And>X1 X2. X2 \<in> pset X1 \<or> (X1, X2) \<notin> CLF" |
|
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by (metis 3 Collect_mem_eq) |
|
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have 5: "(cl, f) \<notin> CLF" |
|
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by (metis 2 4) |
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show "False" |
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by (metis 5 0) |
131 |
qed |
|
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|
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declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect_Pi" ]] |
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lemma |
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"(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl}) ==> |
|
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f \<in> pset cl \<rightarrow> pset cl" |
|
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proof (neg_clausify) |
138 |
assume 0: "f \<notin> Pi (pset cl) (COMBK (pset cl))" |
|
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assume 1: "(cl, f) |
|
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\<in> Sigma CL |
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(COMBB Collect |
|
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(COMBB (COMBC op \<in>) (COMBS (COMBB Pi pset) (COMBB COMBK pset))))" |
|
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show "False" |
|
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(* by (metis 0 Collect_mem_eq SigmaD2 1) ??doesn't terminate*) |
|
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by (insert 0 1, simp add: COMBB_def COMBS_def COMBC_def) |
|
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qed |
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148 |
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declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect_Int" ]] |
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lemma |
151 |
"(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==> |
|
152 |
f \<in> pset cl \<inter> cl" |
|
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proof (neg_clausify) |
154 |
assume 0: "(cl, f) |
|
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\<in> Sigma CL |
|
156 |
(COMBB Collect (COMBB (COMBC op \<in>) (COMBS (COMBB op \<inter> pset) COMBI)))" |
|
157 |
assume 1: "f \<notin> pset cl \<inter> cl" |
|
158 |
have 2: "f \<in> COMBB Collect (COMBB (COMBC op \<in>) (COMBS (COMBB op \<inter> pset) COMBI)) cl" |
|
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by (insert 0, simp add: COMBB_def) |
|
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(* by (metis SigmaD2 0) ??doesn't terminate*) |
|
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have 3: "f \<in> COMBS (COMBB op \<inter> pset) COMBI cl" |
|
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by (metis 2 Collect_mem_eq) |
|
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have 4: "f \<notin> cl \<inter> pset cl" |
|
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by (metis 1 Int_commute) |
|
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have 5: "f \<in> cl \<inter> pset cl" |
|
166 |
by (metis 3 Int_commute) |
|
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show "False" |
|
168 |
by (metis 5 4) |
|
169 |
qed |
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170 |
||
23449 | 171 |
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172 |
declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect_Pi_mono" ]] |
23449 | 173 |
lemma |
174 |
"(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)}) ==> |
|
175 |
(f \<in> pset cl \<rightarrow> pset cl) & (monotone f (pset cl) (order cl))" |
|
176 |
by auto |
|
177 |
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|
178 |
declare [[ atp_problem_prefix = "Abstraction__CLF_subset_Collect_Int" ]] |
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lemma "(cl,f) \<in> CLF ==> |
180 |
CLF \<subseteq> (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==> |
|
181 |
f \<in> pset cl \<inter> cl" |
|
24827 | 182 |
by auto |
27368 | 183 |
|
24827 | 184 |
(*??no longer terminates, with combinators |
23449 | 185 |
by (metis Collect_mem_eq Int_def SigmaD2 UnCI Un_absorb1) |
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--{*@{text Int_def} is redundant*} |
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*) |
23449 | 188 |
|
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|
189 |
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_Collect_Int" ]] |
23449 | 190 |
lemma "(cl,f) \<in> CLF ==> |
191 |
CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==> |
|
192 |
f \<in> pset cl \<inter> cl" |
|
24827 | 193 |
by auto |
194 |
(*??no longer terminates, with combinators |
|
23449 | 195 |
by (metis Collect_mem_eq Int_commute SigmaD2) |
24827 | 196 |
*) |
23449 | 197 |
|
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|
198 |
declare [[ atp_problem_prefix = "Abstraction__CLF_subset_Collect_Pi" ]] |
23449 | 199 |
lemma |
200 |
"(cl,f) \<in> CLF ==> |
|
201 |
CLF \<subseteq> (SIGMA cl': CL. {f. f \<in> pset cl' \<rightarrow> pset cl'}) ==> |
|
202 |
f \<in> pset cl \<rightarrow> pset cl" |
|
31754 | 203 |
by fast |
24827 | 204 |
(*??no longer terminates, with combinators |
23449 | 205 |
by (metis Collect_mem_eq SigmaD2 subsetD) |
24827 | 206 |
*) |
23449 | 207 |
|
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|
208 |
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_Collect_Pi" ]] |
23449 | 209 |
lemma |
210 |
"(cl,f) \<in> CLF ==> |
|
211 |
CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl}) ==> |
|
212 |
f \<in> pset cl \<rightarrow> pset cl" |
|
24827 | 213 |
by auto |
214 |
(*??no longer terminates, with combinators |
|
23449 | 215 |
by (metis Collect_mem_eq SigmaD2 contra_subsetD equalityE) |
24827 | 216 |
*) |
23449 | 217 |
|
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|
218 |
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_Collect_Pi_mono" ]] |
23449 | 219 |
lemma |
220 |
"(cl,f) \<in> CLF ==> |
|
221 |
CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)}) ==> |
|
222 |
(f \<in> pset cl \<rightarrow> pset cl) & (monotone f (pset cl) (order cl))" |
|
223 |
by auto |
|
224 |
||
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|
225 |
declare [[ atp_problem_prefix = "Abstraction__map_eq_zipA" ]] |
23449 | 226 |
lemma "map (%x. (f x, g x)) xs = zip (map f xs) (map g xs)" |
227 |
apply (induct xs) |
|
228 |
(*sledgehammer*) |
|
229 |
apply auto |
|
230 |
done |
|
231 |
||
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|
232 |
declare [[ atp_problem_prefix = "Abstraction__map_eq_zipB" ]] |
23449 | 233 |
lemma "map (%w. (w -> w, w \<times> w)) xs = |
234 |
zip (map (%w. w -> w) xs) (map (%w. w \<times> w) xs)" |
|
235 |
apply (induct xs) |
|
236 |
(*sledgehammer*) |
|
237 |
apply auto |
|
238 |
done |
|
239 |
||
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|
240 |
declare [[ atp_problem_prefix = "Abstraction__image_evenA" ]] |
23449 | 241 |
lemma "(%x. Suc(f x)) ` {x. even x} <= A ==> (\<forall>x. even x --> Suc(f x) \<in> A)"; |
242 |
(*sledgehammer*) |
|
243 |
by auto |
|
244 |
||
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|
245 |
declare [[ atp_problem_prefix = "Abstraction__image_evenB" ]] |
23449 | 246 |
lemma "(%x. f (f x)) ` ((%x. Suc(f x)) ` {x. even x}) <= A |
247 |
==> (\<forall>x. even x --> f (f (Suc(f x))) \<in> A)"; |
|
248 |
(*sledgehammer*) |
|
249 |
by auto |
|
250 |
||
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|
251 |
declare [[ atp_problem_prefix = "Abstraction__image_curry" ]] |
23449 | 252 |
lemma "f \<in> (%u v. b \<times> u \<times> v) ` A ==> \<forall>u v. P (b \<times> u \<times> v) ==> P(f y)" |
253 |
(*sledgehammer*) |
|
254 |
by auto |
|
255 |
||
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|
256 |
declare [[ atp_problem_prefix = "Abstraction__image_TimesA" ]] |
23449 | 257 |
lemma image_TimesA: "(%(x,y). (f x, g y)) ` (A \<times> B) = (f`A) \<times> (g`B)" |
258 |
(*sledgehammer*) |
|
259 |
apply (rule equalityI) |
|
260 |
(***Even the two inclusions are far too difficult |
|
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|
261 |
using [[ atp_problem_prefix = "Abstraction__image_TimesA_simpler"]] |
23449 | 262 |
***) |
263 |
apply (rule subsetI) |
|
264 |
apply (erule imageE) |
|
265 |
(*V manages from here with help: Abstraction__image_TimesA_simpler_1_b.p*) |
|
266 |
apply (erule ssubst) |
|
267 |
apply (erule SigmaE) |
|
268 |
(*V manages from here: Abstraction__image_TimesA_simpler_1_a.p*) |
|
269 |
apply (erule ssubst) |
|
270 |
apply (subst split_conv) |
|
271 |
apply (rule SigmaI) |
|
272 |
apply (erule imageI) + |
|
273 |
txt{*subgoal 2*} |
|
274 |
apply (clarify ); |
|
275 |
apply (simp add: ); |
|
276 |
apply (rule rev_image_eqI) |
|
277 |
apply (blast intro: elim:); |
|
278 |
apply (simp add: ); |
|
279 |
done |
|
280 |
||
281 |
(*Given the difficulty of the previous problem, these two are probably |
|
282 |
impossible*) |
|
283 |
||
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declare [[ atp_problem_prefix = "Abstraction__image_TimesB" ]] |
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lemma image_TimesB: |
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"(%(x,y,z). (f x, g y, h z)) ` (A \<times> B \<times> C) = (f`A) \<times> (g`B) \<times> (h`C)" |
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(*sledgehammer*) |
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by force |
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a226f29d4bdc
re-organized signature of AtpWrapper structure: records instead of unnamed parameters and return values,
boehmes
parents:
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declare [[ atp_problem_prefix = "Abstraction__image_TimesC" ]] |
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lemma image_TimesC: |
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"(%(x,y). (x \<rightarrow> x, y \<times> y)) ` (A \<times> B) = |
|
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((%x. x \<rightarrow> x) ` A) \<times> ((%y. y \<times> y) ` B)" |
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(*sledgehammer*) |
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by auto |
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||
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end |