http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/ncfcont2.html#T25
E proof: http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t25_ncfcont2 261 axioms, 373 initial clauses, 1534 processed, 11.2 generated, 38s, 135 proof clause steps, 36/79 initial proof flas/clauses, lots of calculation
for r being real number st 0 <= r & r <= 2 * PI & sin r = 1 holds r = PI / 2
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/sin_cos6.html#T28
E proof: http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t28_sin_cos6 2175 proc, 30.6k generated, 27s, 185 steps, 71 clauses and 63 flas, - lots of calculation
S is_convergent_in_metrspace_to x iff S is_convergent_to x
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/frechet2.html#T28
E proof (using lgb): http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t28_frechet2.out
(base) mptp@air-03:/home/yan/1911-MIZAR40-ETF-IJCAR/00RESULTS/mizar40-all-T30/Enigma+mizar40-all-T10+mega-VHSLCAXPh2e15+lgb-t150-d30-l3600-e0.15+coop-mzr02$ less t28_frechet2.out
# Proof object clause steps : 199
# Proof object initial clauses used : 63
# Proof object initial formulas used : 28
# Parsed axioms : 308
# Initial clauses in saturation : 496
# Processed clauses : 4243
# ...remaining for further processing : 3165
# Generated clauses : 26575
# ...of the previous two non-trivial : 26184
# User time : 12.699 s
for k being natural_equivalence of F1,F3 st k = e1 `*` e & F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent holds k " = (e ") `*` (e1 ")
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/functor3.html#T42
E proof (lgb): http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t42_functor3.out
(base) mptp@air-03:/home/yan/1911-MIZAR40-ETF-IJCAR/00RESULTS/mizar40-all-T30/Enigma+mizar40-all-T10+mega-VHSLCAXPh2e15+lgb-t150-d30-l900-e0.15+coop-mzr02$ less t42_functor3.out
# Proof object clause steps : 263
# Proof object initial clauses used : 64
# Proof object initial formulas used : 32
# Proof object generating inferences : 101
# Proof object simplifying inferences : 744
# Parsed axioms : 182
# Initial clauses in saturation : 305
# Processed clauses : 2299
# ...remaining for further processing : 1230
# Generated clauses : 17490
# ...of the previous two non-trivial : 15995
# User time : 7.552 s
for a being Element of G for H being Subgroup of G holds ( a in H iff a * H = carr H )
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/group_2.html#T113 E proof: http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t113_group_2 217 fof axioms, 298 initial clauses, 1214 nontriv given, 8523 nontriv gener, 109 proof clause steps, 28/44 initial proof flas/clauses
Nice theorem about the derivation of ln(cos) proved by Enigma/lgb in loop2 (IJCAR'20 experiments):
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/fdiff_4.html#T42
The E proof is at http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t42_fdiff_4.out
It takes 190 clausal inferences and 37 fof axioms (52 clauses). The original problem size is 268 axioms (348 clauses). The search took 2k nontrivial given-clause loops and generated (only) 14k nontrivial clauses.
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/sin_cos9.html#T89
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) )
http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t89_sin_cos9
% Proof object clause steps : 143
% Proof object initial clauses used : 62
% Proof object initial formulas used : 45
% Parsed axioms : 643
% Initial clauses in saturation : 834
% Processed clauses : 4037
% ...remaining for further processing : 2341
% Generated clauses : 18764
% ...of the previous two non-trivial : 16507
% User time : 28.020 s
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/mesfunc5.html#T93
f is nonnegative & A c= B holds Integral (M,(f | A)) <= Integral (M,(f | B))
http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t93_mesfunc5
% Proof object clause steps : 126
% Proof object initial clauses used : 53
% Proof object initial formulas used : 29
% Parsed axioms : 441
% Initial clauses in saturation : 670
% Processed clauses : 3725
% ...remaining for further processing : 2236
% Generated clauses : 14044
% ...of the previous two non-trivial : 13075
(base) mptp@air-02:~/big1/bushy_np/en_gnn/convert-models/grid1500/l5-mzr02-premsel_enigma_01_2020_T10_loop01_epoch_4-query512-ctx768-w0-coop$
Enigma can now (sort of) prove that primes are infinite: for l being Nat ex p being Prime st ( p is prime & p > l )
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/newton.html#T72
The E proof is at http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t72_newton
Modulo the fact that we gave it 328 FOF axioms translated to 549 initial clauses, the proof stats seem easy:
% Proof object clause steps : 83
% Proof object initial clauses used : 42
% Proof object initial formulas used : 38
% Parsed axioms : 328
% Initial clauses in saturation : 549
% Processed clauses : 880
% ...remaining for further processing : 734
% Generated clauses : 3135
% ...of the previous two non-trivial : 2856
% User time : 6.772 s
The only issue is that no prover could do it before and that I got the proof with only one parameterization of Enigma out of 800 :-). (l8-mzr02-premsel_enigma_01_2020_T10_loop01_epoch_67-query128-ctx768-w0-solo).
The fact that it generated only 2.8k clauses during 734 nontrivial given clause loops (with 549 initial clauses) is pretty amazing (and probably the reason why this succeeded).
for T being non empty discrete TopSpace holds T is T_2
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/yellow13.html#T4
(base) mptp@air-02:~/big1/bushy_np/en_gnn/convert-models/grid28_60/mzr02-premsel_enigma_01_2020_T10_loop01_epoch_66-query768-ctx1536-w0-coop$ less t4_yellow13
% Parsed axioms : 194
% Initial clauses in saturation : 279
% Proof object clause steps : 103
% Proof object initial clauses used : 36
% Proof object initial formulas used : 23
% Processed clauses : 3866
% ...remaining for further processing : 1758
% Generated clauses : 36890
% ...of the previous two non-trivial : 32976
for GX being TopSpace holds ( ex x being Point of GX st for y being Point of GX holds x,y are_joined iff for x, y being Point of GX holds x,y are_joined )
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/connsp_1.html#T29
% Proof object clause steps : 99
% Proof object initial clauses used : 32
% Proof object initial formulas used : 17
% Parsed axioms : 74
% Initial clauses in saturation : 99
% Processed clauses : 1940
% ...remaining for further processing : 1069
% Generated clauses : 28829
% ...of the previous two non-trivial : 27799
P . ((A /\ B) /\ C) = ((P . A) * ((P .|. A) . B)) * ((P .|. (A /\ B)) . C)
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/prob_2.html#T30
E proof (lgb): http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t30_prob_2.out
(base) mptp@air-03:/home/yan/1911-MIZAR40-ETF-IJCAR/00RESULTS/mizar40-all-T30/Enigma+mizar40-all-T10+mega-VHSLCAXPh2e15+lgb-t150-d50-l900-e0.15+coop-mzr02$ less t30_prob_2.out
# Proof object clause steps : 179
# Proof object initial clauses used : 57
# Proof object initial formulas used : 40
# Parsed axioms : 233
# Initial clauses in saturation : 286
# Processed clauses : 2746
# ...remaining for further processing : 1856
# Generated clauses : 12511
# ...of the previous two non-trivial : 11325
# User time : 3.117 s
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/lopban_4.html#T33
E proof (lgb): http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t33_lopban_4.out
(base) mptp@air-03:/home/yan/1911-MIZAR40-ETF-IJCAR/00RESULTS/mizar40-all-T30/Enigma+mizar40-all-T10+mega-VHSLCAXPh2e15+lgb-t150-d30-l900-e0.15+coop-mzr02$ less t33_lopban_4.out
# Proof object clause steps : 267
# Proof object initial clauses used : 76
# Proof object initial formulas used : 33
# Parsed axioms : 306
# Initial clauses in saturation : 444
# Processed clauses : 4012
# ...remaining for further processing : 2859
# Generated clauses : 14692
# ...of the previous two non-trivial : 13478
# User time : 6.668 s
for U0 being non-empty MSAlgebra over S holds MSSubAlLattice U0 is bounded
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/msualg_2.html#T33
E proof (lgb): http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t33_msualg_2.out
(base) mptp@air-03:/home/yan/1911-MIZAR40-ETF-IJCAR/00RESULTS/mizar40-all-T30/Enigma+mizar40-all-T10+mega-VHSLCAXPh2e15+lgb-t150-d60-l700-e0.20+coop-mzr02$ less t33_msualg_2.out
# Proof object clause steps : 291
# Proof object initial clauses used : 81
# Proof object initial formulas used : 46
# Parsed axioms : 255
# Initial clauses in saturation : 475
# Processed clauses : 7051
# ...remaining for further processing : 2834
# Generated clauses : 32864
# ...of the previous two non-trivial : 28335
# User time : 8.732 s
for M being non empty set for J being non-Empty TopStruct-yielding ManySortedSet of M st ( for i being Element of M holds J . i is T_0 TopSpace ) holds product J is T_0
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/yellow14.html#T37
mptp@air-02:~/big1/bushy_np/en_gnn/convert-models/grid1500/l8-mzr02-premsel_enigma_01_2020_T10_loop01_epoch_51-query128-ctx1024-w0-coop$
% Proof object clause steps : 98
% Proof object initial clauses used : 44
% Proof object initial formulas used : 23
% Parsed axioms : 188
% Initial clauses in saturation : 319
% Processed clauses : 2185
% ...remaining for further processing : 1408
% Generated clauses : 9227
% ...of the previous two non-trivial : 7578
% User time : 20.956 s
for p being Rational for l,m,k holds not numerator p = m * l or not denominator p = k * l
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/rat_1.html#T29
E proof with lgb: http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t29_rat_1.out
(base) mptp@air-03:/home/yan/1911-MIZAR40-ETF-IJCAR/00RESULTS/mizar40-all-T30/Enigma+mizar40-all-T10+mega-VHSLCAXPh2e15+lgb-t150-d30-l1800-e0.15+coop-mzr02$ less t29_rat_1.out
# Proof object clause steps : 262
# Proof object initial clauses used : 69
# Proof object initial formulas used : 55
# Parsed axioms : 222
# Initial clauses in saturation : 277
# Processed clauses : 7815
# ...remaining for further processing : 3799
# Generated clauses : 163747
# ...of the previous two non-trivial : 151812
# User time : 27.157 s
1 -tuples_on D,D are_equipotent & card (1 -tuples_on D) = card D
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/card_4.html#T8
http://grid01.ciirc.cvut.cz/~mptp/enigma_prf/t8_card_4
% Proof object clause steps : 90
% Proof object initial clauses used : 35
% Proof object initial formulas used : 20
% Parsed axioms : 301
% Initial clauses in saturation : 508
% Processed clauses : 1262
% ...remaining for further processing : 932
% Generated clauses : 6827
% ...of the previous two non-trivial : 6260
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/limfunc2.html#T37
X,[:X,{x}:] are_equipotent & card X = card [:X,{x}:]
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/card_2.html#T6
/home/mptp/big1/bushy_np/en_gnn/convert-models/grid28_60/mzr02-premsel_enigma_01_2020_T10_loop01_epoch_66-query768-ctx1536-w0-coop/t64_compos_1 http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/compos_1.html#T64
% Generated clauses : 38354
% ...of the previous two non-trivial : 36601
% Processed clauses : 5785
% ...remaining for further processing : 3336
% Parsed axioms : 663
% Initial clauses in saturation : 1212
% Proof object initial clauses used : 78
% Proof object initial formulas used : 59
% Proof object clause steps : 158
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/int_5.html#T3
for fp being FinSequence of INT st len fp = 1 holds Poly-INT fp = INT --> (fp . 1)
(base) mptp@air-02:~/big1/bushy_np/en_gnn/convert-models/grid1500/l5-mzr02-premsel_enigma_01_2020_T10_loop01_epoch_45-query256-ctx1536-w0-coop$ less t3_int_5
% Proof object clause steps : 153
% Proof object initial clauses used : 76
% Proof object initial formulas used : 59
% Parsed axioms : 404
% Initial clauses in saturation : 656
% Processed clauses : 2547
% ...remaining for further processing : 1849
% Generated clauses : 11674
% ...of the previous two non-trivial : 10831
for R being Relation for X being set st R partially_orders X holds R |_ 2 X is Order of X
http://grid01.ciirc.cvut.cz/~mptp/7.13.01_4.181.1147/html/orders_1.html#T45
% Proof object clause steps : 132
% Proof object initial clauses used : 46
% Proof object initial formulas used : 26
% Parsed axioms : 218
% Initial clauses in saturation : 347
% Processed clauses : 3985
% ...remaining for further processing : 1852
% Generated clauses : 30744
% ...of the previous two non-trivial : 27676
% User time : 22.204 s