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Quelle Meson_Test.thySprache: Isabelle |
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section ‹Meson test cases› theory Meson_Test imports Main begin text ‹ WARNING: there are many potential conflicts between variables used below and constants declared in HOL! › hide_const (open) implies union inter subset quotient sum or text ‹ Test data for the MESON proof procedure (Excludes the equality problems 51, 52, 56, 58) › subsection ‹Interactive examples› lemma problem_25: "(∃x. P x) & (∀x. L x --> ~ (M x & R x)) & (∀x. P x --> (M x & L x)) & apply (rule ccontr) ML_prf ‹ val ctxt = 🍋; val prem25 = Thm.assume 🍋‹¬ ?thesis›; val nnf25 = Meson.make_nnf Meson.simp_options_all_true ctxt prem25; val xsko25 = Meson.skolemize Meson.simp_options_all_true ctxt nnf25; › apply (tactic ‹cut_tac xsko25 1 THEN REPEAT (eresolve_tac 🍋 [exE] 1)›) ML_val ‹ val ctxt = 🍋; val [_, sko25] = #prems (#1 (Subgoal.focus ctxt 1 NONE (#goal @{Isar.goal})) val clauses25 = Meson.make_clauses ctxt [sko25]; (*7 clauses*) val horns25 = Meson.make_horns clauses25; (*16 Horn clauses*) val go25 :: _ = Meson.gocls clauses25; val ctxt' = fold Thm.declare_hyps (maps Thm.chyps_of (go25 :: horns25)) ctxt Goal.prove ctxt' [] [] 🍋‹False› (fn _ => resolve_tac ctxt' [go25] 1 THEN Meson.depth_prolog_tac ctxt' horns25); › oops lemma problem_26: "((∃x. p x) = (∃x. q x)) & (∀x. ∀y. p x & q y --> (r x = s y)) --> ((∀x. p x --> r x) = (∀x. q x --> s x))" apply (rule ccontr) ML_prf ‹ val ctxt = 🍋; val prem26 = Thm.assume 🍋‹¬ ?thesis› val nnf26 = Meson.make_nnf Meson.simp_options_all_true ctxt prem26; val xsko26 = Meson.skolemize Meson.simp_options_all_true ctxt nnf26; › apply (tactic ‹cut_tac xsko26 1 THEN REPEAT (eresolve_tac 🍋 [exE] 1)›) ML_val ‹ val ctxt = 🍋; val [_, sko26] = #prems (#1 (Subgoal.focus ctxt 1 NONE (#goal @{Isar.goal})) val clauses26 = Meson.make_clauses ctxt [sko26]; val _ = 🍋 (length clauses26 = 9); val horns26 = Meson.make_horns clauses26; val _ = 🍋 (length horns26 = 24); val go26 :: _ = Meson.gocls clauses26; val ctxt' = fold Thm.declare_hyps (maps Thm.chyps_of (go26 :: horns26)) ctxt Goal.prove ctxt' [] [] 🍋‹False› (fn _ => resolve_tac ctxt' [go26] 1 THEN Meson.depth_prolog_tac ctxt' horns26); (*7 ms*) (*Proof is of length 107!!*) › oops lemma problem_43: 🍋 ‹NOW PROVED AUTOMATICALLY!!› (*16 Horn clauses*) "(∀x. ∀y. q x y = (∀z. p z x = (p z y::bool))) --> (∀x. (∀y. q x y = (q y x::boo apply (rule ccontr) ML_prf ‹ val ctxt = 🍋; val prem43 = Thm.assume 🍋‹¬ ?thesis›; val nnf43 = Meson.make_nnf Meson.simp_options_all_true ctxt prem43; val xsko43 = Meson.skolemize Meson.simp_options_all_true ctxt nnf43; › apply (tactic ‹cut_tac xsko43 1 THEN REPEAT (eresolve_tac 🍋 [exE] 1)›) ML_val ‹ val ctxt = 🍋; val [_, sko43] = #prems (#1 (Subgoal.focus ctxt 1 NONE (#goal @{Isar.goal})) val clauses43 = Meson.make_clauses ctxt [sko43]; val _ = 🍋 (length clauses43 = 6); val horns43 = Meson.make_horns clauses43; val _ = 🍋 (length horns43 = 16); val go43 :: _ = Meson.gocls clauses43; val ctxt' = fold Thm.declare_hyps (maps Thm.chyps_of (go43 :: horns43)) ctxt Goal.prove ctxt' [] [] 🍋‹False› ( resolve_tac ctxt' [go43] 1 THEN Meson.best_prolog_tac ctxt' Meson.size_of_subgoals horns43); (*7ms*) › oops (* #1 (q x xa ==> ~ q x xa) ==> q xa x #2 (q xa x ==> ~ q xa x) ==> q x xa #3 (~ q x xa ==> q x xa) ==> ~ q xa x #4 (~ q xa x ==> q xa x) ==> ~ q x xa #5 [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?U ==> p ?W ?U |] ==> p ?W ?V #6 [| ~ p ?W ?U ==> p ?W ?U; p ?W ?V ==> ~ p ?W ?V |] ==> ~ q ?U ?V #7 [| p ?W ?V ==> ~ p ?W ?V; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?U #8 [| ~ q ?U ?V ==> q ?U ?V; ~ p ?W ?V ==> p ?W ?V |] ==> p ?W ?U #9 [| ~ p ?W ?V ==> p ?W ?V; p ?W ?U ==> ~ p ?W ?U |] ==> ~ q ?U ?V #10 [| p ?W ?U ==> ~ p ?W ?U; ~ q ?U ?V ==> q ?U ?V |] ==> ~ p ?W ?V #11 [| p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U; p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V |] ==> q ?U ?V #12 [| p (xb ?U ?V) ?V ==> ~ p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==> p (xb ?U ?V) ?U #13 [| q ?U ?V ==> ~ q ?U ?V; p (xb ?U ?V) ?U ==> ~ p (xb ?U ?V) ?U |] ==> p (xb ?U ?V) ?V #14 [| ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U; ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V |] ==> q ?U ?V #15 [| ~ p (xb ?U ?V) ?V ==> p (xb ?U ?V) ?V; q ?U ?V ==> ~ q ?U ?V |] ==> ~ p (xb ?U ?V) ?U #16 [| q ?U ?V ==> ~ q ?U ?V; ~ p (xb ?U ?V) ?U ==> p (xb ?U ?V) ?U |] ==> ~ p (xb ?U ?V) ?V And here is the proof! (Unkn is the start state after use of goal clause) [Unkn, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1), Res ([Thm "#1"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2, Asm 1, Res ([Thm "#13"], false, 1), Asm 1, Res ([Thm "#10"], false, 1), Res ([Thm "#16"], false, 1), Asm 2, Asm 1, Res ([Thm "#1"], false, 1), Asm 1, Res ([Thm "#14"], false, 1), Res ([Thm "#5"], false, 1), Res ([Thm "#2"], false, 1), Asm 1, Res ([Thm "#13"], false, 1), Asm 2, Asm 1, Res ([Thm "#8"], false, 1), Res ([Thm "#2"], false, 1), Asm 1, Res ([Thm "#12"], false, 1), Asm 2, Asm 1] : lderiv list *) text ‹ MORE and MUCH HARDER test data for the MESON proof procedure (courtesy John Harrison). › (* ========================================================================= *) (* 100 problems selected from the TPTP library *) (* ========================================================================= *) (* * Original timings for John Harrison's MESON_TAC. * Timings below on a 600MHz Pentium III (perch) * Some timings below refer to griffon, which is a dual 2.5GHz Power Mac G5. * * A few variable names have been changed to avoid clashing with constants. * * Changed numeric constants e.g. 0, 1, 2... to num0, num1, num2... * * Here's a list giving typical CPU times, as well as common names and * literature references. * * BOO003-1 34.6 B2 part 1 [McCharen, et al., 1976]; Lemma proved [Overbe * BOO004-1 36.7 B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbe * BOO005-1 47.4 B3 part 1 [McCharen, et al., 1976]; B5 [McCharen, et al. * BOO006-1 48.4 B3 part 2 [McCharen, et al., 1976]; B6 [McCharen, et al. * BOO011-1 19.0 B7 [McCharen, et al., 1976]; prob7.ver1 [ANL] * CAT001-3 45.2 C1 [McCharen, et al., 1976]; p1.ver3.in [ANL] * CAT003-3 10.5 C3 [McCharen, et al., 1976]; p3.ver3.in [ANL] * CAT005-1 480.1 C5 [McCharen, et al., 1976]; p5.ver1.in [ANL] * CAT007-1 11.9 C7 [McCharen, et al., 1976]; p7.ver1.in [ANL] * CAT018-1 81.3 p18.ver1.in [ANL] * COL001-2 16.0 C1 [Wos & McCune, 1988] * COL023-1 5.1 [McCune & Wos, 1988] * COL032-1 15.8 [McCune & Wos, 1988] * COL052-2 13.2 bird4.ver2.in [ANL] * COL075-2 116.9 [Jech, 1994] * COM001-1 1.7 shortburst [Wilson & Minker, 1976] * COM002-1 4.4 burstall [Wilson & Minker, 1976] * COM002-2 7.4 * COM003-2 22.1 [Brushi, 1991] * COM004-1 45.1 * GEO003-1 71.7 T3 [McCharen, et al., 1976]; t3.ver1.in [ANL] * GEO017-2 78.8 D4.1 [Quaife, 1989] * GEO027-3 181.5 D10.1 [Quaife, 1989] * GEO058-2 104.0 R4 [Quaife, 1989] * GEO079-1 2.4 GEOMETRY THEOREM [Slagle, 1967] * GRP001-1 47.8 CADE-11 Competition 1 [Overbeek, 1990]; G1 [McCharen, et * GRP008-1 50.4 Problem 4 [Wos, 1965]; wos4 [Wilson & Minker, 1976] * GRP013-1 40.2 Problem 11 [Wos, 1965]; wos11 [Wilson & Minker, 1976] * GRP037-3 43.8 Problem 17 [Wos, 1965]; wos17 [Wilson & Minker, 1976] * GRP031-2 3.2 ls23 [Lawrence & Starkey, 1974]; ls23 [Wilson & Minker, 1976] * GRP034-4 2.5 ls26 [Lawrence & Starkey, 1974]; ls26 [Wilson & Minker, 1976] * GRP047-2 11.7 [Veroff, 1992] * GRP130-1 170.6 Bennett QG8 [TPTP]; QG8 [Slaney, 1993] * GRP156-1 48.7 ax_mono1c [Schulz, 1995] * GRP168-1 159.1 p01a [Schulz, 1995] * HEN003-3 39.9 HP3 [McCharen, et al., 1976] * HEN007-2 125.7 H7 [McCharen, et al., 1976] * HEN008-4 62.0 H8 [McCharen, et al., 1976] * HEN009-5 136.3 H9 [McCharen, et al., 1976]; hp9.ver3.in [ANL] * HEN012-3 48.5 new.ver2.in [ANL] * LCL010-1 370.9 EC-73 [McCune & Wos, 1992]; ec_yq.in [OTTER] * LCL077-2 51.6 morgan.two.ver1.in [ANL] * LCL082-1 14.6 IC-1.1 [Wos, et al., 1990]; IC-65 [McCune & Wos, 1992]; ls2 [SETHEO]; * LCL111-1 585.6 CADE-11 Competition 6 [Overbeek, 1990]; mv25.in [OTTER]; * LCL143-1 10.9 Lattice structure theorem 2 [Bonacina, 1991] * LCL182-1 271.6 Problem 2.16 [Whitehead & Russell, 1927] * LCL200-1 12.0 Problem 2.46 [Whitehead & Russell, 1927] * LCL215-1 214.4 Problem 2.62 [Whitehead & Russell, 1927]; Problem 2.63 [Whitehead & Russell, 1927] * LCL230-2 0.2 Pelletier 5 [Pelletier, 1986] * LDA003-1 68.5 Problem 3 [Jech, 1993] * MSC002-1 9.2 DBABHP [Michie, et al., 1972]; DBABHP [Wilson & Minker, 1976] * MSC003-1 3.2 HASPARTS-T1 [Wilson & Minker, 1976] * MSC004-1 9.3 HASPARTS-T2 [Wilson & Minker, 1976] * MSC005-1 1.8 Problem 5.1 [Plaisted, 1982] * MSC006-1 39.0 nonob.lop [SETHEO] * NUM001-1 14.0 Chang-Lee-10a [Chang, 1970]; ls28 [Lawrence & Starkey, 1974]; ls28 [Wilso * NUM021-1 52.3 ls65 [Lawrence & Starkey, 1974]; ls65 [Wilson & Minker, 1976] * NUM024-1 64.6 ls75 [Lawrence & Starkey, 1974]; ls75 [Wilson & Minker, 1976] * NUM180-1 621.2 LIM2.1 [Quaife] * NUM228-1 575.9 TRECDEF4 cor. [Quaife] * PLA002-1 37.4 Problem 5.7 [Plaisted, 1982] * PLA006-1 7.2 [Segre & Elkan, 1994] * PLA017-1 484.8 [Segre & Elkan, 1994] * PLA022-1 19.1 [Segre & Elkan, 1994] * PLA022-2 19.7 [Segre & Elkan, 1994] * PRV001-1 10.3 PV1 [McCharen, et al., 1976] * PRV003-1 3.9 E2 [McCharen, et al., 1976]; v2.lop [SETHEO] * PRV005-1 4.3 E4 [McCharen, et al., 1976]; v4.lop [SETHEO] * PRV006-1 6.0 E5 [McCharen, et al., 1976]; v5.lop [SETHEO] * PRV009-1 2.2 Hoares FIND [Bledsoe, 1977]; Problem 5.5 [Plaisted, 1982 * PUZ012-1 3.5 Boxes-of-fruit [Wos, 1988]; Boxes-of-fruit [Wos, et al., * PUZ020-1 56.6 knightknave.in [ANL] * PUZ025-1 58.4 Problem 35 [Smullyan, 1978]; tandl35.ver1.in [ANL] * PUZ029-1 5.1 pigs.ver1.in [ANL] * RNG001-3 82.4 EX6-T? [Wilson & Minker, 1976]; ex6.lop [SETHEO]; Example 6a [Fleisig, e * RNG001-5 399.8 Problem 21 [Wos, 1965]; wos21 [Wilson & Minker, 1976] * RNG011-5 8.4 CADE-11 Competition Eq-10 [Overbeek, 1990]; PROBLEM 10 [ * RNG023-6 9.1 [Stevens, 1987] * RNG028-2 9.3 PROOF III [Anantharaman & Hsiang, 1990] * RNG038-2 16.2 Problem 27 [Wos, 1965]; wos27 [Wilson & Minker, 1976] * RNG040-2 180.5 Problem 29 [Wos, 1965]; wos29 [Wilson & Minker, 1976] * RNG041-1 35.8 Problem 30 [Wos, 1965]; wos30 [Wilson & Minker, 1976] * ROB010-1 205.0 Lemma 3.3 [Winker, 1990]; RA2 [Lusk & Wos, 1992] * ROB013-1 23.6 Lemma 3.5 [Winker, 1990] * ROB016-1 15.2 Corollary 3.7 [Winker, 1990] * ROB021-1 230.4 [McCune, 1992] * SET005-1 192.2 ls108 [Lawrence & Starkey, 1974]; ls108 [Wilson & Minker, 1976] * SET009-1 10.5 ls116 [Lawrence & Starkey, 1974]; ls116 [Wilson & Minker, 1976] * SET025-4 694.7 Lemma 10 [Boyer, et al, 1986] * SET046-5 2.3 p42.in [ANL]; Pelletier 42 [Pelletier, 1986] * SET047-5 3.7 p43.in [ANL]; Pelletier 43 [Pelletier, 1986] * SYN034-1 2.8 QW [Michie, et al., 1972]; QW [Wilson & Minker, 1976] * SYN071-1 1.9 Pelletier 48 [Pelletier, 1986] * SYN349-1 61.7 Ch17N5 [Tammet, 1994] * SYN352-1 5.5 Ch18N4 [Tammet, 1994] * TOP001-2 61.1 Lemma 1a [Wick & McCune, 1989] * TOP002-2 0.4 Lemma 1b [Wick & McCune, 1989] * TOP004-1 181.6 Lemma 1d [Wick & McCune, 1989] * TOP004-2 9.0 Lemma 1d [Wick & McCune, 1989] * TOP005-2 139.8 Lemma 1e [Wick & McCune, 1989] *) abbreviation "EQU001_0_ax equal ≡ (∀X. equal(X::'a,X)) & (∀Y X. equal(X::'a,Y) --> equal(Y::'a,X)) & (∀Y X Z. equal(X::'a,Y) & equal(Y::'a,Z) --> equal(X::'a,Z))" abbreviation "BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply product add sum ≡ (∀X Y. sum(X::'a,Y,add(X::'a,Y))) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & (∀Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) & (∀X. sum(additive_identity::'a,X,X)) & (∀X. sum(X::'a,additive_identity,X)) & (∀X. product(multiplicative_identity::'a,X,X)) & (∀X. product(X::'a,multiplicative_identity,X)) & (∀Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) & (∀Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) & (∀Y Z X V3 V1 V2 V4. sum(X::'a,Y,V1) & sum(X::'a,Z,V2) & product(Y::'a,Z,V3) & sum(X::'a,V3,V4) --> product(V1::'a,V2,V4)) & (∀Y Z V1 V2 X V3 V4. sum(X::'a,Y,V1) & sum(X::'a,Z,V2) & product(Y::'a,Z,V3) & product(V1::'a,V2,V4) --> sum(X::'a,V3,V4)) & (∀Y Z V3 X V1 V2 V4. sum(Y::'a,X,V1) & sum(Z::'a,X,V2) & product(Y::'a,Z,V3) & sum(V3::'a,X,V4) --> product(V1::'a,V2,V4)) & (∀Y Z V1 V2 V3 X V4. sum(Y::'a,X,V1) & sum(Z::'a,X,V2) & product(Y::'a,Z,V3) & product(V1::'a,V2,V4) --> sum(V3::'a,X,V4)) & (∀X. sum(INVERSE(X),X,multiplicative_identity)) & (∀X. sum(X::'a,INVERSE(X),multiplicative_identity)) & (∀X. product(INVERSE(X),X,additive_identity)) & (∀X. product(X::'a,INVERSE(X),additive_identity)) & (∀X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & (∀X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))" abbreviation "BOO002_0_eq INVERSE multiply add product sum equal ≡ (∀X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & (∀X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & (∀X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & (∀X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & (∀X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & (∀X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & (∀X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y)))" (*51194 inferences so far. Searching to depth 13. 232.9 secs*) lemma BOO003_1: "EQU001_0_ax equal & BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply pro BOO002_0_eq INVERSE multiply add product sum equal & (~product(x::'a,x,x)) --> False" by meson (*51194 inferences so far. Searching to depth 13. 204.6 secs Strange! The previous problem also has 51194 inferences at depth 13. They must be very similar!*) lemma BOO004_1: "EQU001_0_ax equal & BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply pro BOO002_0_eq INVERSE multiply add product sum equal & (~sum(x::'a,x,x)) --> False" by meson (*74799 inferences so far. Searching to depth 13. 290.0 secs*) lemma BOO005_1: "EQU001_0_ax equal & BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply pro BOO002_0_eq INVERSE multiply add product sum equal & (~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False" by meson (*74799 inferences so far. Searching to depth 13. 314.6 secs*) lemma BOO006_1: "EQU001_0_ax equal & BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply pro BOO002_0_eq INVERSE multiply add product sum equal & (~product(x::'a,additive_identity,additive_identity)) --> False" by meson (*5 inferences so far. Searching to depth 5. 1.3 secs*) lemma BOO011_1: "EQU001_0_ax equal & BOO002_0_ax equal INVERSE multiplicative_identity additive_identity multiply pro BOO002_0_eq INVERSE multiply add product sum equal & (~equal(INVERSE(additive_identity),multiplicative_identity)) --> False" by meson abbreviation "CAT003_0_ax f1 compos codomain domain equal there_exists equivalent (∀Y X. equivalent(X::'a,Y) --> there_exists(X)) & (∀X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) & (∀X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) & (∀X. there_exists(domain(X)) --> there_exists(X)) & (∀X. there_exists(codomain(X)) --> there_exists(X)) & (∀Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) & (∀X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) & (∀X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) & (∀X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) & (∀X. equal(compos(X::'a,domain(X)),X)) & (∀X. equal(compos(codomain(X),X),X)) & (∀X Y. equivalent(X::'a,Y) --> there_exists(Y)) & (∀X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) & (∀Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) & (∀X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) & (∀X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &an> (∀X Y. equal(X::'a,f1(X::'a,Y)) & equal(Y::'a,f1(X::'a,Y)) --> equal(X::'a,Y))" abbreviation "CAT003_0_eq f1 compos codomain domain equivalent there_exists equal (∀X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) & (∀X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) & (∀X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) & (∀X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & (∀X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & (∀X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) & (∀X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & (∀A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E)))" (*4007 inferences so far. Searching to depth 9. 13 secs*) lemma CAT001_3: "EQU001_0_ax equal & CAT003_0_ax f1 compos codomain domain equal there_exists equivalent & CAT003_0_eq f1 compos codomain domain equivalent there_exists equal & (there_exists(compos(a::'a,b))) & (∀Y X Z. equal(compos(compos(a::'a,b),X),Y) & equal(compos(compos(a::'a,b),Z),Y) --> equal(X::'a,Z)) & (there_exists(compos(b::'a,h))) & (equal(compos(b::'a,h),compos(b::'a,g))) & (~equal(h::'a,g)) --> False" by meson (*245 inferences so far. Searching to depth 7. 1.0 secs*) lemma CAT003_3: "EQU001_0_ax equal & CAT003_0_ax f1 compos codomain domain equal there_exists equivalent & CAT003_0_eq f1 compos codomain domain equivalent there_exists equal & (there_exists(compos(a::'a,b))) & (∀Y X Z. equal(compos(X::'a,compos(a::'a,b)),Y) & equal(compos(Z::'a,compos(a::'a,b)),Y) --> equal(X::'a,Z)) & (there_exists(h)) & (equal(compos(h::'a,a),compos(g::'a,a))) & (~equal(g::'a,h)) --> False" by meson abbreviation "CAT001_0_ax equal codomain domain identity_map compos product defin (∀X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) & (∀Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) & (∀X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) & (∀Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &an> (∀Xy Y Z X Yz Xyz. product(X::'a,Y,Xy) & product(Xy::'a,Z,Xyz) & product(Y::'a,Z,Yz) --> product(X::'a,Yz (∀Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) & (∀Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &an> (∀Yz X Y Xy Z Xyz. product(Y::'a,Z,Yz) & product(X::'a,Yz,Xyz) & product(X::'a,Y,Xy) --> product(Xy::'a,Z (∀Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) & (∀X. identity_map(domain(X))) & (∀X. identity_map(codomain(X))) & (∀X. defined(X::'a,domain(X))) & (∀X. defined(codomain(X),X)) & (∀X. product(X::'a,domain(X),X)) & (∀X. product(codomain(X),X,X)) & (∀X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) & (∀Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) & (∀X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W))" abbreviation "CAT001_0_eq compos defined identity_map codomain domain product equ (∀X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) & (∀X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) & (∀X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) & (∀X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) & (∀X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) & (∀X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) & (∀X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) & (∀X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) & (∀X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) & (∀X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z)))" (*54288 inferences so far. Searching to depth 14. 118.0 secs*) lemma CAT005_1: "EQU001_0_ax equal & CAT001_0_ax equal codomain domain identity_map compos product defined & CAT001_0_eq compos defined identity_map codomain domain product equal & (defined(a::'a,d)) & (identity_map(d)) & (~equal(domain(a),d)) --> False" by meson (*1728 inferences so far. Searching to depth 10. 5.8 secs*) lemma CAT007_1: "EQU001_0_ax equal & CAT001_0_ax equal codomain domain identity_map compos product defined & CAT001_0_eq compos defined identity_map codomain domain product equal & (equal(domain(a),codomain(b))) & (~defined(a::'a,b)) --> False" by meson (*82895 inferences so far. Searching to depth 13. 355 secs*) lemma CAT018_1: "EQU001_0_ax equal & CAT001_0_ax equal codomain domain identity_map compos product defined & CAT001_0_eq compos defined identity_map codomain domain product equal & (defined(a::'a,b)) & (defined(b::'a,c)) & (~defined(a::'a,compos(b::'a,c))) --> False" by meson (*1118 inferences so far. Searching to depth 8. 2.3 secs*) lemma COL001_2: "EQU001_0_ax equal & (∀X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a (∀Y X. equal(apply(apply(k::'a,X),Y),X)) & (∀X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &pan> (∀X. equal(apply(i::'a,X),X)) & (∀A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & (∀X. equal(apply(apply(apply(s::'a,apply(b::'a,X)),i),apply(apply(s::'a,apply(b: (∀Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False" by meson (*500 inferences so far. Searching to depth 8. 0.9 secs*) lemma COL023_1: "EQU001_0_ax equal & (∀X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &pan> (∀X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z) (∀A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & (∀Y. ~equal(Y::'a,apply(combinator::'a,Y))) --> False" by meson (*3018 inferences so far. Searching to depth 10. 4.3 secs*) lemma COL032_1: "EQU001_0_ax equal & (∀X. equal(apply(m::'a,X),apply(X::'a,X))) & (∀Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) &pan> (∀A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & (∀G H. equal(G::'a,H) --> equal(f(G),f(H))) & (∀Y. ~equal(apply(Y::'a,f(Y)),apply(f(Y),apply(Y::'a,f(Y))))) --> False" by meson (*381878 inferences so far. Searching to depth 13. 670.4 secs*) lemma COL052_2: "EQU001_0_ax equal & (∀X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) &span> (∀X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,comm (∀Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &n> (∀A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) & (∀C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) & (∀Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) & (∀A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) & (∀G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) &> (∀J L K'. equal(J::'a,K') --> equal(response(L::'a,J),response(L::'a,K'))) &> (agreeable(c)) & (~agreeable(a)) & (equal(c::'a,compos(a::'a,b))) --> False" by meson (*13201 inferences so far. Searching to depth 11. 31.9 secs*) lemma COL075_2: "EQU001_0_ax equal & (∀Y X. equal(apply(apply(k::'a,X),Y),X)) & (∀X Y Z. equal(apply(apply(apply(abstraction::'a,X),Y),Z),apply(apply(X::'a,appl (∀D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) & (∀G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) & (∀A B. equal(A::'a,B) --> equal(b(A),b(B))) & (∀C D. equal(C::'a,D) --> equal(c(C),c(D))) & (∀Y. ~equal(apply(apply(Y::'a,b(Y)),c(Y)),apply(b(Y),b(Y)))) --> False" by meson (*33 inferences so far. Searching to depth 7. 0.1 secs*) lemma COM001_1: "(∀Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal (∀Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediat (∀Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) & (∀Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goa (labels(loop::'a,p3)) & (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) & (has(p4::'a,goto(out))) & (follows(p5::'a,p4)) & (follows(p8::'a,p3)) & (has(p8::'a,goto(loop))) & (~succeeds(p3::'a,p3)) --> False" by meson (*533 inferences so far. Searching to depth 13. 0.3 secs*) lemma COM002_1: "(∀Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal (∀Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediat (∀Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) & (∀Start_state Condition Goal_state. has(Start_state::'a,ifthen(Condition::'a,Goa (has(p1::'a,assign(register_j::'a,num0))) & (follows(p2::'a,p1)) & (has(p2::'a,assign(register_k::'a,num1))) & (labels(loop::'a,p3)) & (follows(p3::'a,p2)) & (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) & (has(p4::'a,goto(out))) & (follows(p5::'a,p4)) & (follows(p6::'a,p3)) & (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) & (follows(p7::'a,p6)) & (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) & (follows(p8::'a,p7)) & (has(p8::'a,goto(loop))) & (~succeeds(p3::'a,p3)) --> False" by meson (*4821 inferences so far. Searching to depth 14. 1.3 secs*) lemma COM002_2: "(∀Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) & (∀Goal_state Intermediate_state Start_state. fails(Goal_state::'a,Start_state) - (∀Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Labe (∀Start_state Condition Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,ifthen(Condition::'a,Goal_state)))) & (has(p1::'a,assign(register_j::'a,num0))) & (follows(p2::'a,p1)) & (has(p2::'a,assign(register_k::'a,num1))) & (labels(loop::'a,p3)) & (follows(p3::'a,p2)) & (has(p3::'a,ifthen(equal(register_j::'a,n),p4))) & (has(p4::'a,goto(out))) & (follows(p5::'a,p4)) & (follows(p6::'a,p3)) & (has(p6::'a,assign(register_k::'a,mtimes(num2::'a,register_k)))) & (follows(p7::'a,p6)) & (has(p7::'a,assign(register_j::'a,mplus(register_j::'a,num1)))) & (follows(p8::'a,p7)) & (has(p8::'a,goto(loop))) & (fails(p3::'a,p3)) --> False" by meson (*98 inferences so far. Searching to depth 10. 1.1 secs*) lemma COM003_2: "(∀X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) & (∀X. program_decides(X) | program(f2(X))) & (∀X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) & (∀X. program_program_decides(X) --> program(X)) & (∀X. program_program_decides(X) --> program_decides(X)) & (∀X. program(X) & program_decides(X) --> program_program_decides(X)) & (∀X. algorithm_program_decides(X) --> algorithm(X)) & (∀X. algorithm_program_decides(X) --> program_decides(X)) & (∀X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) & (∀Y X. program_halts2(X::'a,Y) --> program(X)) & (∀X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) & (∀X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) & (∀W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) & (∀Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) & (∀Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) & (∀Y X. program_not_halts2(X::'a,Y) --> program(X)) & (∀X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) & (∀X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) & (∀W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) & (∀Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) & (∀Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) & (∀X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z (∀X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y (∀X Y Z W. program_halts2(Y::'a,Z) & halts3_outputs(X::'a,Y,Z,W) --> program_halts2_halts3_outputs(X::'a,Y,Z,W)) & (∀X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2 (∀X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X:: (∀X Y Z W. program_not_halts2(Y::'a,Z) & halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2_halts3_outputs(X::'a,Y,Z,W)) & (∀X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) &span> (∀X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) (∀X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) & (∀X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y:: (∀X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y (∀X Y W. program_not_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_not_halts2_halts2_outputs(X::'a,Y,W)) & (∀X. algorithm_program_decides(X) --> program_program_decides(c1)) & (∀W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z, (∀W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a, (∀W. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> pr (∀W Y. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> (∀W Y. program(W) & program_halts2_halts3_outputs(W::'a,f3(W),f3(W),good) & program_not_halts2_halts3_outputs(W::'a,f3(W),f3(W),bad) --> (∀V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program( (∀V Y. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) & program_halts2(Y::'a,Y) --> halts2(c3::'a,Y)) & (∀V Y. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> progra (algorithm_program_decides(c4)) --> False" by meson (*2100398 inferences so far. Searching to depth 12. 1256s (21 mins) on griffon*) lemma COM004_1: "EQU001_0_ax equal & (∀C D P Q X Y. failure_node(X::'a,or(C::'a,P)) & failure_node(Y::'a,or(D::'a,Q)) & contradictory(P::'a,Q) & siblings(X::'a,Y) --> failure_node(parent_of(X::'a,Y),or(C::'a,D))) & (∀X. contradictory(negate(X),X)) & (∀X. contradictory(X::'a,negate(X))) & (∀X. siblings(left_child_of(X),right_child_of(X))) & (∀D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) & (∀F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) & (∀H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) & (∀K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) & (∀N O' P. equal(N::'a,O') --> equal(parent_of(N::'a,P),parent_of(O'::'a,P))) &an> (∀Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &an> (∀T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) & (∀V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) & (∀Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) & (∀B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) & (∀E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) & (∀H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) & (∀K1 M1 L1. equal(K1::'a,L1) & siblings(M1::'a,K1) --> siblings(M1::'a,L1)) & (failure_node(n_left::'a,or(EMPTY::'a,atom))) & (failure_node(n_right::'a,or(EMPTY::'a,negate(atom)))) & (equal(n_left::'a,left_child_of(n))) & (equal(n_right::'a,right_child_of(n))) & (∀Z. ~failure_node(Z::'a,or(EMPTY::'a,EMPTY))) --> False" oops abbreviation "GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_poin lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal (∀X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) & (∀V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) & (∀Y X V Z. between(X::'a,Y,Z) & between(X::'a,Y,V) --> equal(X::'a,Y) | between(X::'a,Z,V) | between(X::'a,V,Z)) & (∀Y X. equidistant(X::'a,Y,Y,X)) & (∀Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) & (∀X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) & (∀W X Z V Y. between(X::'a,W,V) & between(Y::'a,V,Z) --> between(X::'a,outer_pasch(W::'a,X,Y,Z,V),Y)) & (∀W X Y Z V. between(X::'a,W,V) & between(Y::'a,V,Z) --> between(Z::'a,W,outer_pasch(W::'a,X,Y,Z,V))) & (∀W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(X::'a,Z,euclid1(W::'a,X,Y,Z,V))) & (∀W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(X::'a,Y,euclid2(W::'a,X,Y,Z,V))) & (∀W X Y Z V. between(X::'a,V,W) & between(Y::'a,V,Z) --> equal(X::'a,V) | between(euclid1(W::'a,X,Y,Z,V),W,euclid2(W::'a,X,Y,Z,V))) & (∀X1 Y1 X Y Z V Z1 V1. equidistant(X::'a,Y,X1,Y1) & equidistant(Y::'a,Z,Y1,Z1) & equidistant(X::'a,V,X1,V1) & equidistant(Y::'a,V,Y1,V1) & b (∀X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) & (∀X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_po (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_po (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_po (∀Z X Y W V. equidistant(X::'a,W,X,V) & equidistant(Y::'a,W,Y,V) & equidistant(Z::'a,W,Z,V) --> between(X::' (∀X Y Z X1 Z1 V. equidistant(V::'a,X,V,X1) & equidistant(V::'a,Z,V,Z1) & between(V::'a,X,Z) & between(X::'a,Y,Z) --> equidistant(V::'a,Y,Z,continuous(X::'a,Y,Z,X1,Z1,V))) & (∀X Y Z X1 V Z1. equidistant(V::'a,X,V,X1) & equidistant(V::'a,Z,V,Z1) & between(V::'a,X,Z) & between(X::'a,Y,Z) --> between(X1::'a,continuous(X::'a,Y,Z,X1,Z1,V),Z1))" abbreviation "GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidi between equal ≡ (∀X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) & (∀X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) & (∀X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) & (∀X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) & (∀X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) & (∀X Y V1 V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(X::'a,V1,V2,V3,V4),outer (∀X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,X,V2,V3,V4),outer (∀X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,X,V3,V4),outer (∀X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,X,V4),outer (∀X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,V4,X),outer (∀A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C (∀G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::' (∀M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a, (∀S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::' (∀Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A (∀E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),eucl (∀K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),eucl (∀Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),eucl (∀W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),eucl (∀C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),eucl (∀X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y:: (∀X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1: (∀X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1: (∀X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1: (∀X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5), (∀X V1 Y V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,X,V2,V3,V4,V5), (∀X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5), (∀X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5), (∀X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5), (∀X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X), (*179 inferences so far. Searching to depth 7. 3.9 secs*) lemma GEO003_1: "EQU001_0_ax equal & GEO001_0_ax continuous lower_dimension_point_3 lower_dimension_point_2 lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant between (~between(a::'a,b,b)) --> False" by meson abbreviation "GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension between equal equidistant ≡ (∀Y X. equidistant(X::'a,Y,Y,X)) & (∀X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) & (∀Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) & (∀X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) & (∀X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) & (∀X1 Y1 X Y Z V Z1 V1. equidistant(X::'a,Y,X1,Y1) & equidistant(Y::'a,Z,Y1,Z1) & equidistant(X::'a,V,X1,V1) & equidistant(Y::'a,V,Y1,V1) & b (∀X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) & (∀U V W X Y. between(U::'a,V,W) & between(Y::'a,X,W) --> between(V::'a,inner_pasch(U::'a,V,W,X,Y),Y)) & (∀V W X Y U. between(U::'a,V,W) & between(Y::'a,X,W) --> between(X::'a,inner_pasch(U::'a,V,W,X,Y),U)) & (~between(lower_dimension_point_1::'a,lower_dimension_point_2,lower_dimension_po (~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_po (~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_po (∀Z X Y W V. equidistant(X::'a,W,X,V) & equidistant(Y::'a,W,Y,V) & equidistant(Z::'a,W,Z,V) --> between(X::' (∀U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(U::'a,V,euclid1(U::'a,V,W,X,Y))) & (∀U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(U::'a,X,euclid2(U::'a,V,W,X,Y))) & (∀U V W X Y. between(U::'a,W,Y) & between(V::'a,W,X) --> equal(U::'a,W) | between(euclid1(U::'a,V,W,X,Y),Y,euclid2(U::'a,V,W,X,Y))) & (∀U V V1 W X X1. equidistant(U::'a,V,U,V1) & equidistant(U::'a,X,U,X1) & between(U::'a,V,X) & between(V::'a,W,X) --> between(V1::'a,continuous(U::'a,V,V1,W,X,X1),X1)) & (∀U V V1 W X X1. equidistant(U::'a,V,U,V1) & equidistant(U::'a,X,U,X1) & between(U::'a,V,X) & between(V::'a,W,X) --> equidistant(U::'a,W,U,continuous(U::'a,V,V1,W,X,X1))) & (∀X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) & (∀X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) & (∀X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) & (∀X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) & (∀X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) & (∀X Y V1 V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(X::'a,V1,V2,V3,V4),inner (∀X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,X,V2,V3,V4),inner (∀X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,X,V3,V4),inner (∀X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,X,V4),inner (∀X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,V4,X),inner (∀A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C (∀G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::' (∀M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a, (∀S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::' (∀Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A (∀E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),eucl (∀K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),eucl (∀Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),eucl (∀W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),eucl (∀C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),eucl (∀X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y:: (∀X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1: (∀X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1: (∀X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1: (∀X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5), (∀X V1 Y V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,X,V2,V3,V4,V5), (∀X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5), (∀X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5), (∀X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5), (∀X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X), (*4272 inferences so far. Searching to depth 10. 29.4 secs*) lemma GEO017_2: "EQU001_0_ax equal & GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension between equal equidistant & (equidistant(u::'a,v,w,x)) & (~equidistant(u::'a,v,x,w)) --> False" by meson (*382903 inferences so far. Searching to depth 9. 1022s (17 mins) on griffon*) lemma GEO027_3: "EQU001_0_ax equal & GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension between equal equidistant & (∀U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) & (∀X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &n> (∀A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B (∀U V. equidistant(U::'a,V,U,V)) & (∀W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) & (∀V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) & (∀U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) & (∀V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) & (∀W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) & (∀X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) & (∀X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) & (∀W X U V Y Z. equidistant(U::'a,V,W,X) & equidistant(W::'a,X,Y,Z) --> equidistant(U::'a,V,Y,Z)) & (∀U V W. equal(V::'a,extension(U::'a,V,W,W))) & (∀W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) & (∀U V. between(U::'a,V,reflection(U::'a,V))) & (∀U V. equidistant(V::'a,reflection(U::'a,V),U,V)) & (∀U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) & (∀U. equal(U::'a,reflection(U::'a,U))) & (∀U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) & (∀U V. equidistant(U::'a,U,V,V)) & (∀V V1 U W U1 W1. equidistant(U::'a,V,U1,V1) & equidistant(V::'a,W,V1,W1) & between(U::'a,V,W) & between(U1::'a,V1,W1) --> equidistant(U::'a,W,U1,W1)) & (∀U V W X. between(U::'a,V,W) & between(U::'a,V,X) & equidistant(V::'a,W,V,X) --> equal(U::'a,V) | equ (between(u::'a,v,w)) & (~equal(u::'a,v)) & (~equal(w::'a,extension(u::'a,v,v,w))) --> False" oops (*313884 inferences so far. Searching to depth 10. 887 secs: 15 mins.*) lemma GEO058_2: "EQU001_0_ax equal & GEO002_ax_eq continuous euclid2 euclid1 lower_dimension_point_3 lower_dimension_point_2 lower_dimension_point_1 inner_pasch extension between equal equidistant & (∀U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) & (∀X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &n> (∀A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B (equal(v::'a,reflection(u::'a,v))) & (~equal(u::'a,v)) --> False" oops (*0 inferences so far. Searching to depth 0. 0.2 secs*) lemma GEO079_1: "(∀U V W X Y Z. right_angle(U::'a,V,W) & right_angle(X::'a,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) & (∀U V W X Y Z. CONGRUENT(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) & (∀V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) & (∀U V X Y. parallel(U::'a,V,X,Y) --> eq(X::'a,V,U,V,X,Y)) & (trapezoid(a::'a,b,c,d)) & (~eq(a::'a,c,b,c,a,d)) --> False" by meson abbreviation "GRP003_0_ax equal multiply INVERSE identity product ≡ (∀X. product(identity::'a,X,X)) & (∀X. product(X::'a,identity,X)) & (∀X. product(INVERSE(X),X,identity)) & (∀X. product(X::'a,INVERSE(X),identity)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W))" abbreviation "GRP003_0_eq product multiply INVERSE equal ≡ (∀X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y))) & (∀X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & (∀X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & (∀X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))" (*2032008 inferences so far. Searching to depth 16. 658s (11 mins) on griffon*) lemma GRP001_1: "EQU001_0_ax equal & GRP003_0_ax equal multiply INVERSE identity product & GRP003_0_eq product multiply INVERSE equal & (∀X. product(X::'a,X,identity)) & (product(a::'a,b,c)) & (~product(b::'a,a,c)) --> False" oops (*2386 inferences so far. Searching to depth 11. 8.7 secs*) lemma GRP008_1: "EQU001_0_ax equal & GRP003_0_ax equal multiply INVERSE identity product & GRP003_0_eq product multiply INVERSE equal & (∀A B. equal(A::'a,B) --> equal(h(A),h(B))) & (∀C D. equal(C::'a,D) --> equal(j(C),j(D))) & (∀A B. equal(A::'a,B) & q(A) --> q(B)) & (∀B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) & (∀A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) & (∀A. product(j(A),A,h(A)) & product(A::'a,j(A),h(A)) --> q(A)) & (~q(identity)) --> False" by meson (*8625 inferences so far. Searching to depth 11. 20 secs*) lemma GRP013_1: "EQU001_0_ax equal & GRP003_0_ax equal multiply INVERSE identity product & GRP003_0_eq product multiply INVERSE equal & (∀A. product(A::'a,A,identity)) & (product(a::'a,b,c)) & (product(INVERSE(a),INVERSE(b),d)) & (∀A C B. product(INVERSE(A),INVERSE(B),C) --> product(A::'a,C,B)) & (~product(c::'a,d,identity)) --> False" by meson (*2448 inferences so far. Searching to depth 10. 7.2 secs*) lemma GRP037_3: "EQU001_0_ax equal & GRP003_0_ax equal multiply INVERSE identity product & GRP003_0_eq product multiply INVERSE equal & (∀A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,INVERSE(B),C) --> subgroup_member(C)) (∀A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) & (∀A. subgroup_member(A) --> product(Gidentity::'a,A,A)) & (∀A. subgroup_member(A) --> product(A::'a,Gidentity,A)) & (∀A. subgroup_member(A) --> product(A::'a,Ginverse(A),Gidentity)) & (∀A. subgroup_member(A) --> product(Ginverse(A),A,Gidentity)) & (∀A. subgroup_member(A) --> subgroup_member(Ginverse(A))) & (∀A B. equal(A::'a,B) --> equal(Ginverse(A),Ginverse(B))) & (∀A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) & (∀B C D A. product(A::'a,B,C) & product(D::'a,B,C) --> equal(D::'a,A)) & (subgroup_member(a)) & (subgroup_member(Gidentity)) & (~equal(INVERSE(a),Ginverse(a))) --> False" by meson (*163 inferences so far. Searching to depth 11. 0.3 secs*) lemma GRP031_2: "(∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &pan> (∀A. product(A::'a,INVERSE(A),identity)) & (∀A. product(A::'a,identity,A)) & (∀A. ~product(A::'a,a,identity)) --> False" by meson (*47 inferences so far. Searching to depth 11. 0.2 secs*) lemma GRP034_4: "(∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X. product(identity::'a,X,X)) & (∀X. product(X::'a,identity,X)) & (∀X. product(X::'a,INVERSE(X),identity)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &pan> (∀B A C. subgroup_member(A) & subgroup_member(B) & product(B::'a,INVERSE(A),C) --> subgroup_member(C)) (subgroup_member(a)) & (~subgroup_member(INVERSE(a))) --> False" by meson (*3287 inferences so far. Searching to depth 14. 3.5 secs*) lemma GRP047_2: "(∀X. product(identity::'a,X,X)) & (∀X. product(INVERSE(X),X,identity)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &pan> (∀X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) & (equal(a::'a,b)) & (~equal(multiply(c::'a,a),multiply(c::'a,b))) --> False" by meson (*25559 inferences so far. Searching to depth 19. 16.9 secs*) lemma GRP130_1_002: "(group_element(e_1)) & (group_element(e_2)) & (~equal(e_1::'a,e_2)) & (~equal(e_2::'a,e_1)) & (∀X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) & (∀X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) & (∀X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) & (∀Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) & (∀Z1 Z2 Y X. product(X::'a,Y,Z1) & product(X::'a,Z1,Z2) --> product(Z2::'a,Y,X)) --> False" by meson abbreviation "GRP004_0_ax INVERSE identity multiply equal ≡ (∀X. equal(multiply(identity::'a,X),X)) & (∀X. equal(multiply(INVERSE(X),X),identity)) & (∀X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) (∀A B. equal(A::'a,B) --> equal(INVERSE(A),INVERSE(B))) & (∀C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) & (∀F' H G. equal(F'::'a,G) --> equal(multiply(H::'a,F'),multiply(H::'a,G)))" abbreviation "GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal ≡ (∀Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &> (∀Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) & (∀X Y Z. equal(greatest_lower_bound(X::'a,greatest_lower_bound(Y::'a,Z)),greates (∀X Y Z. equal(least_upper_bound(X::'a,least_upper_bound(Y::'a,Z)),least_upper_b (∀X. equal(least_upper_bound(X::'a,X),X)) & (∀X. equal(greatest_lower_bound(X::'a,X),X)) & (∀Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) & (∀Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) & (∀Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(mult (∀Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_boun (∀Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply (∀Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(mu (∀A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_b (∀A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_b (∀A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B (∀A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C (∀A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) & (∀A C B. equal(A::'a,B) --> equal(multiply(C::'a,A),multiply(C::'a,B)))" (*3468 inferences so far. Searching to depth 10. 9.1 secs*) lemma GRP156_1: "EQU001_0_ax equal & GRP004_0_ax INVERSE identity multiply equal & GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal & (equal(least_upper_bound(a::'a,b),b)) & (~equal(greatest_lower_bound(multiply(a::'a,c),multiply(b::'a,c)),multiply(a::'a by meson (*4394 inferences so far. Searching to depth 10. 8.2 secs*) lemma GRP168_1: "EQU001_0_ax equal & GRP004_0_ax INVERSE identity multiply equal & GRP004_2_ax multiply least_upper_bound greatest_lower_bound equal & (equal(least_upper_bound(a::'a,b),b)) & (~equal(least_upper_bound(multiply(INVERSE(c),multiply(a::'a,c)),multiply(INVERS by meson abbreviation "HEN002_0_ax identity Zero Divide equal mless_equal ≡ (∀X Y. mless_equal(X::'a,Y) --> equal(Divide(X::'a,Y),Zero)) & (∀X Y. equal(Divide(X::'a,Y),Zero) --> mless_equal(X::'a,Y)) & (∀Y X. mless_equal(Divide(X::'a,Y),X)) & (∀X Y Z. mless_equal(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a (∀X. mless_equal(Zero::'a,X)) & (∀X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) & (∀X. mless_equal(X::'a,identity))" abbreviation "HEN002_0_eq mless_equal Divide equal ≡ (∀A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) & (∀G H I'. equal(G::'a,H) & mless_equal(G::'a,I') --> mless_equal(H::'a,I')) & (∀J L K'. equal(J::'a,K') & mless_equal(L::'a,J) --> mless_equal(L::'a,K'))" (*250258 inferences so far. Searching to depth 16. 406.2 secs*) lemma HEN003_3: "EQU001_0_ax equal & HEN002_0_ax identity Zero Divide equal mless_equal & HEN002_0_eq mless_equal Divide equal & (~equal(Divide(a::'a,a),Zero)) --> False" by meson (*202177 inferences so far. Searching to depth 14. 451 secs*) lemma HEN007_2: "EQU001_0_ax equal & (∀X Y. mless_equal(X::'a,Y) --> quotient(X::'a,Y,Zero)) & (∀X Y. quotient(X::'a,Y,Zero) --> mless_equal(X::'a,Y)) & (∀Y Z X. quotient(X::'a,Y,Z) --> mless_equal(Z::'a,X)) & (∀Y X V3 V2 V1 Z V4 V5. quotient(X::'a,Y,V1) & quotient(Y::'a,Z,V2) & quotient(X::'a,Z,V3) & quotient(V3::'a,V2,V4) & quotient(V1: (∀X. mless_equal(Zero::'a,X)) & (∀X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) & (∀X. mless_equal(X::'a,identity)) & (∀X Y. quotient(X::'a,Y,Divide(X::'a,Y))) & (∀X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) & (∀X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) & (∀X Z Y. equal(X::'a,Y) & mless_equal(Z::'a,X) --> mless_equal(Z::'a,Y)) & (∀X Y Z. equal(X::'a,Y) & mless_equal(X::'a,Z) --> mless_equal(Y::'a,Z)) & (∀X Y W. equal(X::'a,Y) --> equal(Divide(X::'a,W),Divide(Y::'a,W))) & (∀X W Y. equal(X::'a,Y) --> equal(Divide(W::'a,X),Divide(W::'a,Y))) & (∀X. quotient(X::'a,identity,Zero)) & (∀X. quotient(Zero::'a,X,Zero)) & (∀X. quotient(X::'a,X,Zero)) & (∀X. quotient(X::'a,Zero,X)) & (∀Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) & (∀W1 X Z W2 Y. quotient(X::'a,Y,W1) & mless_equal(W1::'a,Z) & quotient(X::'a,Z,W2) --> mless_equal(W2::'a (mless_equal(x::'a,y)) & (quotient(z::'a,y,zQy)) & (quotient(z::'a,x,zQx)) & (~mless_equal(zQy::'a,zQx)) --> False" by meson (*60026 inferences so far. Searching to depth 12. 42.2 secs*) lemma HEN008_4: "EQU001_0_ax equal & HEN002_0_ax identity Zero Divide equal mless_equal & HEN002_0_eq mless_equal Divide equal & (∀X. equal(Divide(X::'a,identity),Zero)) & (∀X. equal(Divide(Zero::'a,X),Zero)) & (∀X. equal(Divide(X::'a,X),Zero)) & (equal(Divide(a::'a,Zero),a)) & (∀Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) & (∀X Z Y. mless_equal(Divide(X::'a,Y),Z) --> mless_equal(Divide(X::'a,Z),Y)) &n> (∀Y Z X. mless_equal(X::'a,Y) --> mless_equal(Divide(Z::'a,Y),Divide(Z::'a,X))) &< (mless_equal(a::'a,b)) & (~mless_equal(Divide(a::'a,c),Divide(b::'a,c))) --> False" by meson (*3160 inferences so far. Searching to depth 11. 3.5 secs*) lemma HEN009_5: "EQU001_0_ax equal & (∀Y X. equal(Divide(Divide(X::'a,Y),X),Zero)) & (∀X Y Z. equal(Divide(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::' (∀X. equal(Divide(Zero::'a,X),Zero)) & (∀X Y. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,X),Zero) --> equal(X::'a,Y)) & (∀X. equal(Divide(X::'a,identity),Zero)) & (∀A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) & (∀Y X Z. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,Z),Zero) --> equal(Divide(X::'a,Z),Zero)) & (∀X Z Y. equal(Divide(Divide(X::'a,Y),Z),Zero) --> equal(Divide(Divide(X::'a,Z), (∀Y Z X. equal(Divide(X::'a,Y),Zero) --> equal(Divide(Divide(Z::'a,Y),Divide(Z:: (~equal(Divide(identity::'a,a),Divide(identity::'a,Divide(identity::'a,Divide(id (equal(Divide(identity::'a,a),b)) & (equal(Divide(identity::'a,b),c)) & (equal(Divide(identity::'a,c),d)) & (~equal(b::'a,d)) --> False" by meson (*970373 inferences so far. Searching to depth 17. 890.0 secs*) lemma HEN012_3: "EQU001_0_ax equal & HEN002_0_ax identity Zero Divide equal mless_equal & HEN002_0_eq mless_equal Divide equal & (~mless_equal(a::'a,a)) --> False" by meson (*1063 inferences so far. Searching to depth 20. 1.0 secs*) lemma LCL010_1: "(∀X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & (∀X Z Y. is_a_theorem(equivalent(equivalent(X::'a,Y),equivalent(equivalent(X::'a (~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equ by meson (*2549 inferences so far. Searching to depth 12. 1.4 secs*) lemma LCL077_2: "(∀X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & (∀Y X. is_a_theorem(implies(X,implies(Y,X)))) & (∀Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),impli (∀Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) & (∀X2 X1 X3. is_a_theorem(implies(X1,X2)) & is_a_theorem(implies(X2,X3)) --> is_a_theorem(implies(X1,X3))) & (~is_a_theorem(implies(not(not(a)),a))) --> False" by meson (*2036 inferences so far. Searching to depth 20. 1.5 secs*) lemma LCL082_1: "(∀X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & (∀Y Z U X. is_a_theorem(implies(implies(implies(X::'a,Y),Z),implies(implies(Z::' (~is_a_theorem(implies(a::'a,implies(b::'a,a)))) --> False" by meson (*1100 inferences so far. Searching to depth 13. 1.0 secs*) lemma LCL111_1: "(∀X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) & (∀Y X. is_a_theorem(implies(X,implies(Y,X)))) & (∀Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) (∀Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) &span> (∀Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) & (~is_a_theorem(implies(implies(a,b),implies(implies(c,a),implies(c,b))))) --> Fa by meson (*667 inferences so far. Searching to depth 9. 1.4 secs*) lemma LCL143_1: "(∀X. equal(X,X)) & (∀Y X. equal(X,Y) --> equal(Y,X)) & (∀Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) & (∀X. equal(implies(true,X),X)) & (∀Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) &span> (∀Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) & (∀Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) & (∀A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) & (∀D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) & (∀G H. equal(G,H) --> equal(not(G),not(H))) & (∀X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) & (∀X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) & (∀X Y. ordered(X,Y) --> equal(implies(X,Y),true)) & (∀X Y. equal(implies(X,Y),true) --> ordered(X,Y)) & (∀A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) & (∀D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) & (∀G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) & (∀J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) & (∀M N O'. equal(M,N) & ordered(M,O') --> ordered(N,O')) & (∀P R Q. equal(P,Q) & ordered(R,P) --> ordered(R,Q)) & (ordered(x,y)) & (~ordered(implies(z,x),implies(z,y))) --> False" by meson (*5245 inferences so far. Searching to depth 12. 4.6 secs*) lemma LCL182_1: "(∀A. axiom(or(not(or(A,A)),A))) & (∀B A. axiom(or(not(A),or(B,A)))) & (∀B A. axiom(or(not(or(A,B)),or(B,A)))) & (∀B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) & (∀A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) & (∀X. axiom(X) --> theorem(X)) & (∀X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) & (∀X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) & (~theorem(or(not(or(not(p),q)),or(not(not(q)),not(p))))) --> False" by meson (*410 inferences so far. Searching to depth 10. 0.3 secs*) lemma LCL200_1: "(∀A. axiom(or(not(or(A,A)),A))) & (∀B A. axiom(or(not(A),or(B,A)))) & (∀B A. axiom(or(not(or(A,B)),or(B,A)))) & (∀B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) & (∀A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) & (∀X. axiom(X) --> theorem(X)) & (∀X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) & (∀X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) & (~theorem(or(not(not(or(p,q))),not(q)))) --> False" by meson (*5849 inferences so far. Searching to depth 12. 5.6 secs*) lemma LCL215_1: "(∀A. axiom(or(not(or(A,A)),A))) & (∀B A. axiom(or(not(A),or(B,A)))) & (∀B A. axiom(or(not(or(A,B)),or(B,A)))) & (∀B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) & (∀A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) & (∀X. axiom(X) --> theorem(X)) & (∀X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) & (∀X Y Z. axiom(or(not(X),Y)) & theorem(or(not(Y),Z)) --> theorem(or(not(X),Z))) & (~theorem(or(not(or(not(p),q)),or(not(or(p,q)),q)))) --> False" by meson (*0 secs. Not sure that a search even starts!*) lemma LCL230_2: "(q --> p | r) & (~p) & (q) & (~r) --> False" by meson (*119585 inferences so far. Searching to depth 14. 262.4 secs*) lemma LDA003_1: "EQU001_0_ax equal & (∀Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) & (∀X Y. left(X::'a,f(X::'a,Y))) & (∀Y X Z. left(X::'a,Y) & left(Y::'a,Z) --> left(X::'a,Z)) & (equal(num2::'a,f(num1::'a,num1))) & (equal(num3::'a,f(num2::'a,num1))) & (equal(u::'a,f(num2::'a,num2))) & (∀A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) & (∀G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) & (∀J L K'. equal(J::'a,K') & left(L::'a,J) --> left(L::'a,K')) & (~left(num3::'a,u)) --> False" by meson (*2392 inferences so far. Searching to depth 12. 2.2 secs*) lemma MSC002_1: "(at(something::'a,here,now)) & (∀Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Sit (∀Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Anothe (∀Thing Situation. ~held(Thing::'a,let_go(Situation))) & (∀Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) & (∀Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,le (∀Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pi (∀Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pic (∀Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) & (∀Place Thing Another_place Situation. at(Thing::'a,Place,Situation) & grabbed(Thing::'a,Situation) --> put(Thing::'a,Another_place,go(Another_place::'a,Situation))) & (∀Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Th (∀One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) & (∀Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) & (∀Situation. ~answer(Situation)) --> False" by meson (*73 inferences so far. Searching to depth 10. 0.2 secs*) lemma MSC003_1: "(∀Number_of_small_parts Small_part Big_part Number_of_mid_parts Mid_part. has_p (∀Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_pa (in'(john::'a,boy)) & (∀X. in'(X::'a,boy) --> in'(X::'a,human)) & (∀X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) & (∀X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) & (∀X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) & (~has_parts(john::'a,mtimes(num2::'a,num1),hand)) --> False" by meson (*1486 inferences so far. Searching to depth 20. 1.2 secs*) lemma MSC004_1: "(∀Number_of_small_parts Small_part Big_part Number_of_mid_parts Mid_part. has_p (∀Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_pa (in'(john::'a,boy)) & (∀X. in'(X::'a,boy) --> in'(X::'a,human)) & (∀X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) & (∀X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) & (∀X. in'(X::'a,arm) --> has_parts(X::'a,num1,hand)) & (~has_parts(john::'a,mtimes(mtimes(num2::'a,num1),num5),fingers)) --> False" by meson (*100 inferences so far. Searching to depth 12. 0.1 secs*) lemma MSC005_1: "(value(truth::'a,truth)) & (value(falsity::'a,falsity)) & (∀X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) & (∀X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) & (∀X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) & (∀X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) & (∀Value. ~value(xor(xor(xor(xor(truth::'a,falsity),falsity),truth),falsity),Valu by meson (*19116 inferences so far. Searching to depth 16. 15.9 secs*) lemma MSC006_1: "(∀Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) & (∀Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) & (∀Y X. q(X::'a,Y) --> q(Y::'a,X)) & (∀X Y. p(X::'a,Y) | q(X::'a,Y)) & (~p(a::'a,b)) & (~q(c::'a,d)) --> False" by meson (*1713 inferences so far. Searching to depth 10. 2.8 secs*) lemma NUM001_1: "(∀A. equal(A::'a,A)) & (∀B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) & (∀B A. equal(add(A::'a,B),add(B::'a,A))) & (∀A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) & (∀B A. equal(subtract(add(A::'a,B),B),A)) & (∀A B. equal(A::'a,subtract(add(A::'a,B),B))) & (∀A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) & (∀A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) & (∀A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) & (∀A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) & (∀A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) & (∀A C D B. equal(A::'a,B) & equal(C::'a,subtract(D::'a,A)) --> equal(C::'a,subtract(D::'a,B))) & (~equal(add(add(a::'a,b),c),add(a::'a,add(b::'a,c)))) --> False" by meson abbreviation "NUM001_0_ax multiply successor num0 add equal ≡ (∀A. equal(add(A::'a,num0),A)) & (∀A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) & (∀A. equal(multiply(A::'a,num0),num0)) & (∀B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) & (∀A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) & (∀A B. equal(A::'a,B) --> equal(successor(A),successor(B)))" abbreviation "NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless (∀A C B. mless(A::'a,B) & mless(C::'a,A) --> mless(C::'a,B)) & (∀A B C. equal(add(successor(A),B),C) --> mless(B::'a,C)) & (∀A B. mless(A::'a,B) --> equal(add(successor(predecessor_of_1st_minus_2nd(B::'a abbreviation "NUM001_2_ax equal mless divides ≡ (∀A B. divides(A::'a,B) --> mless(A::'a,B) | equal(A::'a,B)) & (∀A B. mless(A::'a,B) --> divides(A::'a,B)) & (∀A B. equal(A::'a,B) --> divides(A::'a,B))" (*20717 inferences so far. Searching to depth 11. 13.7 secs*) lemma NUM021_1: "EQU001_0_ax equal & NUM001_0_ax multiply successor num0 add equal & NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless & NUM001_2_ax equal mless divides & (mless(b::'a,c)) & (~mless(b::'a,a)) & (divides(c::'a,a)) & (∀A. ~equal(successor(A),num0)) --> False" by meson (*26320 inferences so far. Searching to depth 10. 26.4 secs*) lemma NUM024_1: "EQU001_0_ax equal & NUM001_0_ax multiply successor num0 add equal & NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless & (∀B A. equal(add(A::'a,B),add(B::'a,A))) & (∀B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) & (mless(a::'a,a)) & (∀A. ~equal(successor(A),num0)) --> False" by meson abbreviation "SET004_0_ax not_homomorphism2 not_homomorphism1 homomorphism compatible operation cantor diagonalise subset_relation one_to_one choice apply regular function identity_relation single_valued_class compos powerClass sum_class omega inductive successor_relation successor image' rng domain range_of INVERSE flip rot domain_of null_class restrct difference union complement intersection element_relation second first cross_product ordered_pair singleton unordered_pair equal universal_class not_subclass_element member subclass ≡ (∀X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) & (∀X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) & (∀X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) & (∀X. subclass(X::'a,universal_class)) & (∀X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) & (∀Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) & (∀X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) & (∀X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a, (∀X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) &< (∀X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) &< (∀X Y. member(unordered_pair(X::'a,Y),universal_class)) & (∀X. equal(unordered_pair(X::'a,X),singleton(X))) & (∀X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ord (∀V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a (∀U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a (∀U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) & (∀X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> equal(ordered_pair(first(Z),se (subclass(element_relation::'a,cross_product(universal_class::'a,universal_class (∀X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) &> (∀X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_ (∀Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) & (∀X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) & (∀Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) & (∀Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) & (∀Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z: (∀X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y (∀X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersect (∀Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y)) (∀Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) &pan> (∀Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) & (∀Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),univer (∀X. subclass(rot(X),cross_product(cross_product(universal_class::'a,universal_c (∀V W U X. member(ordered_pair(ordered_pair(U::'a,V),W),rot(X)) --> member(order (∀U V W X. member(ordered_pair(ordered_pair(V::'a,W),U),X) & member(ordered_pair(ordered_pair(U::'a,V),W),cross_product(cross_product(universal_class::'a,universal_class),universal_class)) --> member(ordered_pair(ordered_pair(U::'a,V),W),rot(X))) & (∀X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_ (∀V U W X. member(ordered_pair(ordered_pair(U::'a,V),W),flip(X)) --> member(orde (∀U V W X. member(ordered_pair(ordered_pair(V::'a,U),W),X) & member(ordered_pair(ordered_pair(U::'a,V),W),cross_product(cross_product(universal_class::'a,universal_class),universal_class)) --> member(ordered_pair(ordered_pair(U::'a,V),W),flip(X))) & (∀Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),INVERSE(Y))) &pan> (∀Z. equal(domain_of(INVERSE(Z)),range_of(Z))) & (∀Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_cla (∀Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_cl (∀Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image'(Xr::'a,X))) &an> (∀X. equal(union(X::'a,singleton(X)),successor(X))) & (subclass(successor_relation::'a,cross_product(universal_class::'a,universal_cla (∀X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y (∀X Y. equal(successor(X),Y) & member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(X::'a,Y),successor_relation)) & (∀X. inductive(X) --> member(null_class::'a,X)) & (∀X. inductive(X) --> subclass(image'(successor_relation::'a,X),X)) & (∀X. member(null_class::'a,X) & subclass(image'(successor_relation::'a,X),X) --> inductive(X)) & (inductive(omega)) & (∀Y. inductive(Y) --> subclass(omega::'a,Y)) & (member(omega::'a,universal_class)) & (∀X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class( (∀X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &an> (∀X. equal(complement(image'(element_relation::'a,complement(X))),powerClass(X)) (∀U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &pan> (∀Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_ (∀Z Yr Xr Y. member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr)) --> member(Z::'a,im (∀Y Z Yr Xr. member(Z::'a,image'(Yr::'a,image'(Xr::'a,singleton(Y)))) & member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr))) & (∀X. single_valued_class(X) --> subclass(compos(X::'a,INVERSE(X)),identity_relat (∀X. subclass(compos(X::'a,INVERSE(X)),identity_relation) --> single_valued_clas (∀Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,univers (∀Xf. function(Xf) --> subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation)) &span> (∀Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation) --> function(Xf)) & (∀Xf X. function(Xf) & member(X::'a,universal_class) --> member(image'(Xf::'a,X),universal_class)) & (∀X. equal(X::'a,null_class) | member(regular(X),X)) & (∀X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) (∀Xf Y. equal(sum_class(image'(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) & (function(choice)) & (∀Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(ch (∀Xf. one_to_one(Xf) --> function(Xf)) & (∀Xf. one_to_one(Xf) --> function(INVERSE(Xf))) & (∀Xf. function(INVERSE(Xf)) & function(Xf) --> one_to_one(Xf)) & (equal(intersection(cross_product(universal_class::'a,universal_class),intersect (equal(intersection(INVERSE(subset_relation),subset_relation),identity_relation) (∀Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagon (∀X. equal(intersection(domain_of(X),diagonalise(compos(INVERSE(element_relation (∀Xf. operation(Xf) --> function(Xf)) & (∀Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(d (∀Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) & (∀Xf. function(Xf) & equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf)) & subclass(range_of(Xf),domain_of(domain_of(Xf))) --> operatio (∀Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) & (∀Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),dom (∀Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(dom (∀Xh Xh1 Xf1 Xf2. function(Xh) & equal(domain_of(domain_of(Xf1)),domain_of(Xh)) & subclass(range_of(Xh),domain_of(domain_of(Xf2))) (∀Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) & (∀Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) & (∀Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) &> (∀Xf2 Xh Xf1 X Y. homomorphism(Xh::'a,Xf1,Xf2) & member(ordered_pair(X::'a,Y),domain_of(Xf1)) --> equal(apply(Xf2::'a,ordered_pair(apply(Xh::'a,X),apply(Xh::'a,Y))),apply(Xh::'a,apply(Xf1::'a,ordered_pair(X::'a,Y))))) & (∀Xh Xf1 Xf2. operation(Xf1) & operation(Xf2) & compatible(Xh::'a,Xf1,Xf2) --> member(ordered_pair (∀Xh Xf1 Xf2. operation(Xf1) & operation(Xf2) & compatible(Xh::'a,Xf1,Xf2) & equal(apply(Xf2::'a,ordered_pair(apply(Xh::'a,not_homomorphism1(Xh::'a,Xf1,Xf2)),apply(Xh::'a,not_homomorphism2(Xh::'a,Xf1,Xf2)))),apply(Xh::'a,apply(Xf1::'a,ordered_pair(not_homomorphism1(Xh::'a,Xf1,Xf2),not_homomorphism2(Xh::'a,Xf1,Xf2))))) --> homomorphism(Xh::'a,Xf1,Xf2))" abbreviation "SET004_0_eq subclass single_valued_class operation one_to_one member inductive homomorphism function compatible unordered_pair union sum_class successor singleton second rot restrct regular range_of rng powerClass ordered_pair not_subclass_element not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip first domain_of domain difference diagonalise cross_product compos complement cantor apply equal ≡ (∀D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) & (∀G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) & (∀J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) & (∀L M. equal(L::'a,M) --> equal(complement(L),complement(M))) & (∀N O' P. equal(N::'a,O') --> equal(compos(N::'a,P),compos(O'::'a,P))) & (∀Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) & (∀T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a, (∀W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X)) (∀Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) & (∀B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D (∀E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F (∀H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1, (∀L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1, (∀P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1, (∀T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) & (∀V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) & (∀X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) & (∀Z1 A2 B2. equal(Z1::'a,A2) --> equal(image'(Z1::'a,B2),image'(A2::'a,B2))) &an> (∀C2 E2 D2. equal(C2::'a,D2) --> equal(image'(E2::'a,C2),image'(E2::'a,D2))) &an> (∀F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2:: (∀I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2:: (∀L2 M2. equal(L2::'a,M2) --> equal(INVERSE(L2),INVERSE(M2))) & (∀N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_ho (∀R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_ho (∀V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_ho (∀Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_ho (∀D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_ho (∀H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_ho (∀L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subcl (∀O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subcl (∀R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3:: (∀U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3:: (∀X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) & (∀Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) &< (∀D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) &< (∀H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) &< (∀L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) & (∀N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) & (∀P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R (∀T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U (∀X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A (∀B5 C5. equal(B5::'a,C5) --> equal(rot(B5),rot(C5))) & (∀D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) & (∀F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) & (∀H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) & (∀J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) & (∀L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) &> (∀O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) &> (∀R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair( (∀U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair( (∀X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) & (∀B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) & (∀F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) & (∀J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) & (∀L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) & (∀P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) & (∀T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) & (∀X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) & (∀Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) & (∀C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) & (∀F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) & (∀H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) & (∀J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) & (∀L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) & (∀O7 Q7 P7. equal(O7::'a,P7) & subclass(Q7::'a,O7) --> subclass(Q7::'a,P7))" abbreviation "SET004_1_ax range_of function maps apply application_function singleton_relation element_relation complement intersection single_valued3 singleton image' domain single_valued2 second single_valued1 identity_relation INVERSE not_subclass_element first domain_of domain_relation composition_function compos equal ordered_pair member universal_class cross_product compose_class subclass ≡ (∀X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class (∀X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y (∀Y Z X. member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universa (subclass(composition_function::'a,cross_product(universal_class::'a,cross_produ (∀X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) (∀X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_ (subclass(domain_relation::'a,cross_product(universal_class::'a,universal_class) (∀X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) &< (∀X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),d (∀X. equal(first(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation (∀X. equal(second(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relatio (∀X. equal(domain(X::'a,image'(INVERSE(X),singleton(single_valued1(X))),single_v (equal(intersection(complement(compos(element_relation::'a,complement(identity_r (subclass(application_function::'a,cross_product(universal_class::'a,cross_produ (∀Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) (∀X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) (∀Z X Y. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),cross_product(universa (∀X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) & (∀Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) & (∀X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) & (∀Xf Y. function(Xf) & subclass(range_of(Xf),Y) --> maps(Xf::'a,domain_of(Xf),Y))" abbreviation "SET004_1_eq maps single_valued3 single_valued2 single_valued1 compo (∀L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) & (∀N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) &n> (∀P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &n> (∀R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &n> (∀X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) & (∀B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) & (∀F3 H3 I3 G3. equal(F3::'a,G3) & maps(H3::'a,I3,F3) --> maps(H3::'a,I3,G3))" abbreviation "NUM004_0_ax integer_of omega ordinal_multiply add_relation ordinal_add recursion apply range_of union_of_range_map function recursion_equation_functions rest_relation rest_of limit_ordinals kind_1_ordinals successor_relation image' universal_class sum_class element_relation ordinal_numbers section not_well_ordering ordered_pair least member well_ordering singleton domain_of segment null_class intersection asymmetric compos transitive cross_product connected identity_relation complement restrct subclass irreflexive symmetrization_of INVERSE union equal ≡ (∀X. equal(union(X::'a,INVERSE(X)),symmetrization_of(X))) & (∀X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_ (∀X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexiv (∀Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_rel (∀X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrizatio (∀Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr: (∀Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a, (∀Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,INVERSE(Xr)), (∀Xr Y. equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class) --> asym (∀Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) &< (∀X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) & (∀Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) & (∀Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U), (∀Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) & (∀Y V U Xr. ~(well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) & member(ordered_pair(V::'a,least(Xr::'a,U)),Xr))) & (∀Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) & (∀Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_o (∀V Xr Y. member(V::'a,not_well_ordering(Xr::'a,Y)) & equal(segment(Xr::'a,not_well_ordering(Xr::'a,Y),V),null_class) & connected(Xr::'a,Y) --> wel (∀Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) & (∀Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) &pan> (∀Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) & (∀X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) &pan> (∀X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) & (∀X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> me (∀X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) --> member(X::'a,ordinal_numbers) | equal(X::'a,ordinal_numbers)) & (equal(union(singleton(null_class),image'(successor_relation::'a,ordinal_numbers (equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals) (∀X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) &pan> (∀V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X)) (∀X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,univ (∀U V X. member(U::'a,domain_of(X)) & equal(restrct(X::'a,U,universal_class),V) --> member(ordered_pair(U::'a,V),rest_of(X))) & (subclass(rest_relation::'a,cross_product(universal_class::'a,universal_class))) (∀X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) &n> (∀X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),res (∀X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) & (∀Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) & (∀Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ord (∀Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest (∀X Z. function(Z) & function(X) & member(domain_of(X),ordinal_numbers) & equal(compos(Z::'a,rest_of(X)),X) --> member(X::'a,recursion_equation_functions(Z))) & (subclass(union_of_range_map::'a,cross_product(universal_class::'a,universal_cla (∀X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(rang (∀X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_ (∀X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ord (∀X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_m (∀X. member(X::'a,omega) --> equal(integer_of(X),X)) & (∀X. member(X::'a,omega) | equal(integer_of(X),null_class))" abbreviation "NUM004_0_eq well_ordering transitive section irreflexive connected asymmetric symmetrization_of segment rest_of recursion_equation_functions recursion ordinal_multiply ordinal_add not_well_ordering least integer_of equal ≡ (∀D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) & (∀F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) & (∀I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) & (∀L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M (∀O' Q P. equal(O'::'a,P) --> equal(not_well_ordering(Q::'a,O'),not_well_orderin (∀R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T' (∀U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) &pan> (∀X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y:: (∀A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multi (∀F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1:: (∀J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1:: (∀N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1:: (∀R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_e (∀T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) & (∀V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X (∀Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A (∀D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G (∀H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2)) (∀J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) & (∀M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) & (∀P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) & (∀S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) & (∀V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) & (∀Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) & (∀B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) & (∀F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) & (∀J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) & (∀N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) & (∀Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) & (∀T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) & (∀W3 Y3 X3. equal(W3::'a,X3) & well_ordering(Y3::'a,W3) --> well_ordering(Y3::'a,X3))" (*1345 inferences so far. Searching to depth 7. 23.3 secs. BIG*) lemma NUM180_1: "EQU001_0_ax equal & SET004_0_ax not_homomorphism2 not_homomorphism1 homomorphism compatible operation cantor diagonalise subset_relation one_to_one choice apply regular function identity_relation single_valued_class compos powerClass sum_class omega inductive successor_relation successor image' rng domain range_of INVERSE flip rot domain_of null_class restrct difference union complement intersection element_relation second first cross_product ordered_pair singleton unordered_pair equal universal_class not_subclass_element member subclass & SET004_0_eq subclass single_valued_class operation one_to_one member inductive homomorphism function compatible unordered_pair union sum_class successor singleton second rot restrct regular range_of rng powerClass ordered_pair not_subclass_element not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip first domain_of domain difference diagonalise cross_product compos complement cantor apply equal & SET004_1_ax range_of function maps apply application_function singleton_relation element_relation complement intersection single_valued3 singleton image' domain single_valued2 second single_valued1 identity_relation INVERSE not_subclass_element first domain_of domain_relation composition_function compos equal ordered_pair member universal_class cross_product compose_class subclass & SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equa NUM004_0_ax integer_of omega ordinal_multiply add_relation ordinal_add recursion apply range_of union_of_range_map function recursion_equation_functions rest_relation rest_of limit_ordinals kind_1_ordinals successor_relation image' universal_class sum_class element_relation ordinal_numbers section not_well_ordering ordered_pair least member well_ordering singleton domain_of segment null_class intersection asymmetric compos transitive cross_product connected identity_relation complement restrct subclass irreflexive symmetrization_of INVERSE union equal & NUM004_0_eq well_ordering transitive section irreflexive connected asymmetric symmetrization_of segment rest_of recursion_equation_functions recursion ordinal_multiply ordinal_add not_well_ordering least integer_of equal & (~subclass(limit_ordinals::'a,ordinal_numbers)) --> False" by meson (*0 inferences so far. Searching to depth 0. 16.8 secs. BIG*) lemma NUM228_1: "EQU001_0_ax equal & SET004_0_ax not_homomorphism2 not_homomorphism1 homomorphism compatible operation cantor diagonalise subset_relation one_to_one choice apply regular function identity_relation single_valued_class compos powerClass sum_class omega inductive successor_relation successor image' rng domain range_of INVERSE flip rot domain_of null_class restrct difference union complement intersection element_relation second first cross_product ordered_pair singleton unordered_pair equal universal_class not_subclass_element member subclass & SET004_0_eq subclass single_valued_class operation one_to_one member inductive homomorphism function compatible unordered_pair union sum_class successor singleton second rot restrct regular range_of rng powerClass ordered_pair not_subclass_element not_homomorphism2 not_homomorphism1 INVERSE intersection image' flip first domain_of domain difference diagonalise cross_product compos complement cantor apply equal & SET004_1_ax range_of function maps apply application_function singleton_relation element_relation complement intersection single_valued3 singleton image' domain single_valued2 second single_valued1 identity_relation INVERSE not_subclass_element first domain_of domain_relation composition_function compos equal ordered_pair member universal_class cross_product compose_class subclass & SET004_1_eq maps single_valued3 single_valued2 single_valued1 compose_class equa NUM004_0_ax integer_of omega ordinal_multiply add_relation ordinal_add recursion apply range_of union_of_range_map function recursion_equation_functions rest_relation rest_of limit_ordinals kind_1_ordinals successor_relation image' universal_class sum_class element_relation ordinal_numbers section not_well_ordering ordered_pair least member well_ordering singleton domain_of segment null_class intersection asymmetric compos transitive cross_product connected identity_relation complement restrct subclass irreflexive symmetrization_of INVERSE union equal & NUM004_0_eq well_ordering transitive section irreflexive connected asymmetric symmetrization_of segment rest_of recursion_equation_functions recursion ordinal_multiply ordinal_add not_well_ordering least integer_of equal & (~function(z)) & (~equal(recursion_equation_functions(z),null_class)) --> False" by meson (*4868 inferences so far. Searching to depth 12. 4.3 secs*) lemma PLA002_1: "(∀Situation1 Situation2. warm(Situation1) | cold(Situation2)) & (∀Situation. at(a::'a,Situation) --> at(b::'a,walk(b::'a,Situation))) & (∀Situation. at(a::'a,Situation) --> at(b::'a,drive(b::'a,Situation))) & (∀Situation. at(b::'a,Situation) --> at(a::'a,walk(a::'a,Situation))) & (∀Situation. at(b::'a,Situation) --> at(a::'a,drive(a::'a,Situation))) & (∀Situation. cold(Situation) & at(b::'a,Situation) --> at(c::'a,skate(c::'a,Situation))) & (∀Situation. cold(Situation) & at(c::'a,Situation) --> at(b::'a,skate(b::'a,Situation))) & (∀Situation. warm(Situation) & at(b::'a,Situation) --> at(d::'a,climb(d::'a,Situation))) & (∀Situation. warm(Situation) & at(d::'a,Situation) --> at(b::'a,climb(b::'a,Situation))) & (∀Situation. at(c::'a,Situation) --> at(d::'a,go(d::'a,Situation))) & (∀Situation. at(d::'a,Situation) --> at(c::'a,go(c::'a,Situation))) & (∀Situation. at(c::'a,Situation) --> at(e::'a,go(e::'a,Situation))) & (∀Situation. at(e::'a,Situation) --> at(c::'a,go(c::'a,Situation))) & (∀Situation. at(d::'a,Situation) --> at(f::'a,go(f::'a,Situation))) & (∀Situation. at(f::'a,Situation) --> at(d::'a,go(d::'a,Situation))) & (at(f::'a,s0)) & (∀S'. ~at(a::'a,S')) --> False" by meson abbreviation "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY a (∀X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) & (∀State X. holds(EMPTY::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(p (∀Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(EMPTY::'a,State) --> holds(clear(Y) (∀Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) & (∀State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) & (∀X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(EMPTY::'a,do(putdown(X::'a,Y),State))) & (∀X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) & (∀X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) & (∀Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a (∀X State Z Y. holds(clear(Z),State) & differ(Z::'a,Y) --> holds(clear(Z),do(putdown(X::'a,Y),State)))" abbreviation "PLA001_1_ax EMPTY clear s0 on holds table d c b a differ ≡ (∀Y X. differ(Y::'a,X) --> differ(X::'a,Y)) & (differ(a::'a,b)) & (differ(a::'a,c)) & (differ(a::'a,d)) & (differ(a::'a,table)) & (differ(b::'a,c)) & (differ(b::'a,d)) & (differ(b::'a,table)) & (differ(c::'a,d)) & (differ(c::'a,table)) & (differ(d::'a,table)) & (holds(on(a::'a,table),s0)) & (holds(on(b::'a,table),s0)) & (holds(on(c::'a,d),s0)) & (holds(on(d::'a,table),s0)) & (holds(clear(a),s0)) & (holds(clear(b),s0)) & (holds(clear(c),s0)) & (holds(EMPTY::'a,s0)) & (∀State. holds(clear(table),State))" (*190 inferences so far. Searching to depth 10. 0.6 secs*) lemma PLA006_1: "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &pan> PLA001_1_ax EMPTY clear s0 on holds table d c b a differ & (∀State. ~holds(on(c::'a,table),State)) --> False" by meson (*190 inferences so far. Searching to depth 10. 0.5 secs*) lemma PLA017_1: "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &pan> PLA001_1_ax EMPTY clear s0 on holds table d c b a differ & (∀State. ~holds(on(a::'a,c),State)) --> False" by meson (*13732 inferences so far. Searching to depth 16. 9.8 secs*) lemma PLA022_1: "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &pan> PLA001_1_ax EMPTY clear s0 on holds table d c b a differ & (∀State. ~holds(and'(on(c::'a,d),on(a::'a,c)),State)) --> False" by meson (*217 inferences so far. Searching to depth 13. 0.7 secs*) lemma PLA022_2: "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds &pan> PLA001_1_ax EMPTY clear s0 on holds table d c b a differ & (∀State. ~holds(and'(on(a::'a,c),on(c::'a,d)),State)) --> False" by meson (*948 inferences so far. Searching to depth 18. 1.1 secs*) lemma PRV001_1: "(∀X Y Z. q1(X::'a,Y,Z) & mless_or_equal(X::'a,Y) --> q2(X::'a,Y,Z)) & (∀X Y Z. q1(X::'a,Y,Z) --> mless_or_equal(X::'a,Y) | q3(X::'a,Y,Z)) & (∀Z X Y. q2(X::'a,Y,Z) --> q4(X::'a,Y,Y)) & (∀Z Y X. q3(X::'a,Y,Z) --> q4(X::'a,Y,X)) & (∀X. mless_or_equal(X::'a,X)) & (∀X Y. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,X) --> equal(X::'a,Y)) & (∀Y X Z. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,Z) --> mless_or_equal(X::'a,Z)) & (∀Y X. mless_or_equal(X::'a,Y) | mless_or_equal(Y::'a,X)) & (∀X Y. equal(X::'a,Y) --> mless_or_equal(X::'a,Y)) & (∀X Y Z. equal(X::'a,Y) & mless_or_equal(X::'a,Z) --> mless_or_equal(Y::'a,Z)) & (∀X Z Y. equal(X::'a,Y) & mless_or_equal(Z::'a,X) --> mless_or_equal(Z::'a,Y)) & (q1(a::'a,b,c)) & (∀W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,a))) & (∀W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,b))) --> False" by meson (*PRV is now called SWV (software verification) *) abbreviation "SWV001_1_ax mless_THAN successor predecessor equal ≡ (∀X. equal(predecessor(successor(X)),X)) & (∀X. equal(successor(predecessor(X)),X)) & (∀X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) & (∀X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) & (∀X. mless_THAN(predecessor(X),X)) & (∀X. mless_THAN(X::'a,successor(X))) & (∀X Y Z. mless_THAN(X::'a,Y) & mless_THAN(Y::'a,Z) --> mless_THAN(X::'a,Z)) & (∀X Y. mless_THAN(X::'a,Y) | mless_THAN(Y::'a,X) | equal(X::'a,Y)) & (∀X. ~mless_THAN(X::'a,X)) & (∀Y X. ~(mless_THAN(X::'a,Y) & mless_THAN(Y::'a,X))) & (∀Y X Z. equal(X::'a,Y) & mless_THAN(X::'a,Z) --> mless_THAN(Y::'a,Z)) & (∀Y Z X. equal(X::'a,Y) & mless_THAN(Z::'a,X) --> mless_THAN(Z::'a,Y))" abbreviation "SWV001_0_eq a successor predecessor equal ≡ (∀X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) & (∀X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) & (∀X Y. equal(X::'a,Y) --> equal(a(X),a(Y)))" (*21 inferences so far. Searching to depth 5. 0.4 secs*) lemma PRV003_1: "EQU001_0_ax equal & SWV001_1_ax mless_THAN successor predecessor equal & SWV001_0_eq a successor predecessor equal & (~mless_THAN(n::'a,j)) & (mless_THAN(k::'a,j)) & (~mless_THAN(k::'a,i)) & (mless_THAN(i::'a,n)) & (mless_THAN(a(j),a(k))) & (∀X. mless_THAN(X::'a,j) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,i)) & (∀X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) & (∀X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) & (mless_THAN(j::'a,i)) --> False" by meson (*584 inferences so far. Searching to depth 7. 1.1 secs*) lemma PRV005_1: "EQU001_0_ax equal & SWV001_1_ax mless_THAN successor predecessor equal & SWV001_0_eq a successor predecessor equal & (~mless_THAN(n::'a,k)) & (~mless_THAN(k::'a,l)) & (~mless_THAN(k::'a,i)) & (mless_THAN(l::'a,n)) & (mless_THAN(One::'a,l)) & (mless_THAN(a(k),a(predecessor(l)))) & (∀X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,l)) & (∀X. mless_THAN(One::'a,l) & mless_THAN(a(X),a(predecessor(l))) --> mless_THAN(X::'a,l) | mless_THAN(n::'a,X)) & (∀X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,l) & mless_THAN(a(X),a(predecessor(X))))) --> False" by meson (*2343 inferences so far. Searching to depth 8. 3.5 secs*) lemma PRV006_1: "EQU001_0_ax equal & SWV001_1_ax mless_THAN successor predecessor equal & SWV001_0_eq a successor predecessor equal & (~mless_THAN(n::'a,m)) & (mless_THAN(i::'a,m)) & (mless_THAN(i::'a,n)) & (~mless_THAN(i::'a,One)) & (mless_THAN(a(i),a(m))) & (∀X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(m)) --> mless_THAN(X::'a,i)) & (∀X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) & (∀X. ~(mless_THAN(One::'a,X) & mless_THAN(X::'a,i) & mless_THAN(a(X),a(predecessor(X))))) --> False" by meson (*86 inferences so far. Searching to depth 14. 0.1 secs*) lemma PRV009_1: "(∀Y X. mless_or_equal(X::'a,Y) | mless(Y::'a,X)) & (mless(j::'a,i)) & (mless_or_equal(m::'a,p)) & (mless_or_equal(p::'a,q)) & (mless_or_equal(q::'a,n)) & (∀X Y. mless_or_equal(m::'a,X) & mless(X::'a,i) & mless(j::'a,Y) & mless_or_equal(Y::'a,n) --> mless_or_equal(a(X),a(Y))) & (∀X Y. mless_or_equal(m::'a,X) & mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,j) --> mless_or_equal(a(X), (∀X Y. mless_or_equal(i::'a,X) & mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,n) --> mless_or_equal(a(X), (~mless_or_equal(a(p),a(q))) --> False" by meson (*222 inferences so far. Searching to depth 8. 0.4 secs*) lemma PUZ012_1: "(∀X. equal_fruits(X::'a,X)) & (∀X. equal_boxes(X::'a,X)) & (∀X Y. ~(label(X::'a,Y) & contains(X::'a,Y))) & (∀X. contains(boxa::'a,X) | contains(boxb::'a,X) | contains(boxc::'a,X)) & (∀X. contains(X::'a,apples) | contains(X::'a,bananas) | contains(X::'a,oranges)) (∀X Y Z. contains(X::'a,Y) & contains(X::'a,Z) --> equal_fruits(Y::'a,Z)) & (∀Y X Z. contains(X::'a,Y) & contains(Z::'a,Y) --> equal_boxes(X::'a,Z)) & (~equal_boxes(boxa::'a,boxb)) & (~equal_boxes(boxb::'a,boxc)) & (~equal_boxes(boxa::'a,boxc)) & (~equal_fruits(apples::'a,bananas)) & (~equal_fruits(bananas::'a,oranges)) & (~equal_fruits(apples::'a,oranges)) & (label(boxa::'a,apples)) & (label(boxb::'a,oranges)) & (label(boxc::'a,bananas)) & (contains(boxb::'a,apples)) & (~(contains(boxa::'a,bananas) & contains(boxc::'a,oranges))) --> False" by meson (*35 inferences so far. Searching to depth 5. 3.2 secs*) lemma PUZ020_1: "EQU001_0_ax equal & (∀A B. equal(A::'a,B) --> equal(statement_by(A),statement_by(B))) & (∀X. person(X) --> knight(X) | knave(X)) & (∀X. ~(person(X) & knight(X) & knave(X))) & (∀X Y. says(X::'a,Y) & a_truth(Y) --> a_truth(Y)) & (∀X Y. ~(says(X::'a,Y) & equal(X::'a,Y))) & (∀Y X. says(X::'a,Y) --> equal(Y::'a,statement_by(X))) & (∀X Y. ~(person(X) & equal(X::'a,statement_by(Y)))) & (∀X. person(X) & a_truth(statement_by(X)) --> knight(X)) & (∀X. person(X) --> a_truth(statement_by(X)) | knave(X)) & (∀X Y. equal(X::'a,Y) & knight(X) --> knight(Y)) & (∀X Y. equal(X::'a,Y) & knave(X) --> knave(Y)) & (∀X Y. equal(X::'a,Y) & person(X) --> person(Y)) & (∀X Y Z. equal(X::'a,Y) & says(X::'a,Z) --> says(Y::'a,Z)) & (∀X Z Y. equal(X::'a,Y) & says(Z::'a,X) --> says(Z::'a,Y)) & (∀X Y. equal(X::'a,Y) & a_truth(X) --> a_truth(Y)) & (∀X Y. knight(X) & says(X::'a,Y) --> a_truth(Y)) & (∀X Y. ~(knave(X) & says(X::'a,Y) & a_truth(Y))) & (person(husband)) & (person(wife)) & (~equal(husband::'a,wife)) & (says(husband::'a,statement_by(husband))) & (a_truth(statement_by(husband)) & knight(husband) --> knight(wife)) & (knight(husband) --> a_truth(statement_by(husband))) & (a_truth(statement_by(husband)) | knight(wife)) & (knight(wife) --> a_truth(statement_by(husband))) & (~knight(husband)) --> False" by meson (*121806 inferences so far. Searching to depth 17. 63.0 secs*) lemma PUZ025_1: "(∀X. a_truth(truthteller(X)) | a_truth(liar(X))) & (∀X. ~(a_truth(truthteller(X)) & a_truth(liar(X)))) & (∀Truthteller Statement. a_truth(truthteller(Truthteller)) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(Statement)) & (∀Liar Statement. ~(a_truth(liar(Liar)) & a_truth(says(Liar::'a,Statement)) & a_truth(Statement))) & (∀Statement Truthteller. a_truth(Statement) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(truthteller(Truthteller))) & (∀Statement Liar. a_truth(says(Liar::'a,Statement)) --> a_truth(Statement) | a_t (∀Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) &span> (∀Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a (∀X Y. a_truth(equal_type(X::'a,Y)) & a_truth(truthteller(X)) --> a_truth(truthteller(Y))) & (∀X Y. a_truth(equal_type(X::'a,Y)) & a_truth(liar(X)) --> a_truth(liar(Y))) & (∀X Y. a_truth(truthteller(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(liar(Y (∀X Y. a_truth(liar(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(truthteller(Y (∀Y X. a_truth(equal_type(X::'a,Y)) --> a_truth(equal_type(Y::'a,X))) & (∀X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) & a_truth(Y) --> answer(yes)) & (∀X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) --> a_truth(Y) | answer(no)) & (∀X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) & a_truth(Y) --> answer(no)) & (∀X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) --> a_truth(Y) | answer(yes)) & (people(b::'a,c,a)) & (people(a::'a,b,a)) & (people(a::'a,c,b)) & (people(c::'a,b,a)) & (a_truth(says(a::'a,equal_type(b::'a,c)))) & (ask_1_if_2(c::'a,equal_type(a::'a,b))) & (∀Answer. ~answer(Answer)) --> False" by meson (*621 inferences so far. Searching to depth 18. 0.2 secs*) lemma PUZ029_1: "(∀X. dances_on_tightropes(X) | eats_pennybuns(X) | old(X)) & (∀X. pig(X) & liable_to_giddiness(X) --> treated_with_respect(X)) & (∀X. wise(X) & balloonist(X) --> has_umbrella(X)) & (∀X. ~(looks_ridiculous(X) & eats_pennybuns(X) & eats_lunch_in_public(X))) & (∀X. balloonist(X) & young(X) --> liable_to_giddiness(X)) & (∀X. fat(X) & looks_ridiculous(X) --> dances_on_tightropes(X) | eats_lunch_in_public(X)) & (∀X. ~(liable_to_giddiness(X) & wise(X) & dances_on_tightropes(X))) & (∀X. pig(X) & has_umbrella(X) --> looks_ridiculous(X)) & (∀X. treated_with_respect(X) --> dances_on_tightropes(X) | fat(X)) & (∀X. young(X) | old(X)) & (∀X. ~(young(X) & old(X))) & (wise(piggy)) & (young(piggy)) & (pig(piggy)) & (balloonist(piggy)) --> False" by meson abbreviation "RNG001_0_ax equal additive_inverse add multiply product additive_id (∀X. sum(additive_identity::'a,X,X)) & (∀X. sum(X::'a,additive_identity,X)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y. sum(X::'a,Y,add(X::'a,Y))) & (∀X. sum(additive_inverse(X),X,additive_identity)) & (∀X. sum(X::'a,additive_inverse(X),additive_identity)) & (∀Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) & (∀Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) & (∀Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &pan> (∀Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) & (∀Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) & (∀X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & (∀X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V))" abbreviation "RNG001_0_eq product multiply sum add additive_inverse equal ≡ (∀X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) & (∀X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & (∀X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) & (∀X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & (∀X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & (∀X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) & (∀X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))" (*93620 inferences so far. Searching to depth 24. 65.9 secs*) lemma RNG001_3: "(∀X. sum(additive_identity::'a,X,X)) & (∀X. sum(additive_inverse(X),X,additive_identity)) & (∀Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) & (∀Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) & (~product(a::'a,additive_identity,additive_identity)) --> False" by meson abbreviation "RNG_other_ax multiply add equal product additive_identity additive_ (∀X. sum(X::'a,additive_inverse(X),additive_identity)) & (∀Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) & (∀Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) & (∀Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &pan> (∀Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) & (∀Y Z V3 X V1 V2 V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & product(V3::'a,X,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 V3 X V4. product(Y::'a,X,V1) & product(Z::'a,X,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(V3::'a,X,V4)) & (∀X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & (∀X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & (∀X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) & (∀X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) & (∀X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) & (∀X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) & (∀X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) & (∀X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) & (∀X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y))" (****************SLOW 76385914 inferences so far. Searching to depth 18 No proof after 5 1/2 hours! (griffon) ****************) lemma RNG001_5: "EQU001_0_ax equal & (∀X. sum(additive_identity::'a,X,X)) & (∀X. sum(X::'a,additive_identity,X)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y. sum(X::'a,Y,add(X::'a,Y))) & (∀X. sum(additive_inverse(X),X,additive_identity)) & RNG_other_ax multiply add equal product additive_identity additive_inverse sum &span> (~product(a::'a,additive_identity,additive_identity)) --> False" oops (*0 inferences so far. Searching to depth 0. 0.5 secs*) lemma RNG011_5: "EQU001_0_ax equal & (∀A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & (∀G H. equal(G::'a,H) --> equal(additive_inverse(G),additive_inverse(H))) & (∀I' J K'. equal(I'::'a,J) --> equal(multiply(I'::'a,K'),multiply(J::'a,K'))) &pan> (∀L N M. equal(L::'a,M) --> equal(multiply(N::'a,L),multiply(N::'a,M))) & (∀A B C D. equal(A::'a,B) --> equal(associator(A::'a,C,D),associator(B::'a,C,D)) (∀E G F' H. equal(E::'a,F') --> equal(associator(G::'a,E,H),associator(G::'a,F', (∀I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a (∀M N O'. equal(M::'a,N) --> equal(commutator(M::'a,O'),commutator(N::'a,O'))) &span> (∀P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) &n> (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & (∀X. equal(add(X::'a,additive_identity),X)) & (∀X. equal(add(additive_identity::'a,X),X)) & (∀X. equal(add(X::'a,additive_inverse(X)),additive_identity)) & (∀X. equal(add(additive_inverse(X),X),additive_identity)) & (equal(additive_inverse(additive_identity),additive_identity)) & (∀X Y. equal(add(X::'a,add(additive_inverse(X),Y)),Y)) & (∀X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inv (∀X. equal(additive_inverse(additive_inverse(X)),X)) & (∀X. equal(multiply(X::'a,additive_identity),additive_identity)) & (∀X. equal(multiply(additive_identity::'a,X),additive_identity)) & (∀X Y. equal(multiply(additive_inverse(X),additive_inverse(Y)),multiply(X::'a,Y) (∀X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a (∀X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)) (∀Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a (∀X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)) (∀X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &span> (∀X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_ (∀X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply (∀X Y. equal(multiply(multiply(associator(X::'a,X,Y),X),associator(X::'a,X,Y)),a (~equal(multiply(multiply(associator(a::'a,a,b),a),associator(a::'a,a,b)),additi by meson (*202 inferences so far. Searching to depth 8. 0.6 secs*) lemma RNG023_6: "EQU001_0_ax equal & (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) & (∀X. equal(add(additive_identity::'a,X),X)) & (∀X. equal(add(X::'a,additive_identity),X)) & (∀X. equal(multiply(additive_identity::'a,X),additive_identity)) & (∀X. equal(multiply(X::'a,additive_identity),additive_identity)) & (∀X. equal(add(additive_inverse(X),X),additive_identity)) & (∀X. equal(add(X::'a,additive_inverse(X)),additive_identity)) & (∀Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a (∀X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)) (∀X. equal(additive_inverse(additive_inverse(X)),X)) & (∀X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &span> (∀X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &span> (∀X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_ (∀X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply (∀D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) & (∀G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) & (∀J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &an> (∀L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O (∀P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S (∀T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W, (∀X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) &n> (∀A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B (∀D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) (∀G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) (~equal(associator(x::'a,x,y),additive_identity)) --> False" by meson (*0 inferences so far. Searching to depth 0. 0.6 secs*) lemma RNG028_2: "EQU001_0_ax equal & (∀X. equal(add(additive_identity::'a,X),X)) & (∀X. equal(multiply(additive_identity::'a,X),additive_identity)) & (∀X. equal(multiply(X::'a,additive_identity),additive_identity)) & (∀X. equal(add(additive_inverse(X),X),additive_identity)) & (∀X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inv (∀X. equal(additive_inverse(additive_inverse(X)),X)) & (∀Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a (∀X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)) (∀X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &span> (∀X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &span> (∀X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)) (∀X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a (equal(additive_inverse(additive_identity),additive_identity)) & (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) & (∀Z X Y. equal(add(X::'a,Z),add(Y::'a,Z)) --> equal(X::'a,Y)) & (∀Z X Y. equal(add(Z::'a,X),add(Z::'a,Y)) --> equal(X::'a,Y)) & (∀D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) & (∀G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) & (∀J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &an> (∀D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) (∀G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) (∀X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_ (∀L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O (∀P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S (∀T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W, (∀X Y. ~equal(multiply(multiply(Y::'a,X),Y),multiply(Y::'a,multiply(X::'a,Y)))) &< (∀X Y Z. ~equal(associator(Y::'a,X,Z),additive_inverse(associator(X::'a,Y,Z)))) &< (∀X Y Z. ~equal(associator(Z::'a,Y,X),additive_inverse(associator(X::'a,Y,Z)))) &< (~equal(multiply(multiply(cx::'a,multiply(cy::'a,cx)),cz),multiply(cx::'a,multip by meson (*209 inferences so far. Searching to depth 9. 1.2 secs*) lemma RNG038_2: "(∀X. sum(X::'a,additive_identity,X)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y. sum(X::'a,Y,add(X::'a,Y))) & RNG_other_ax multiply add equal product additive_identity additive_inverse sum &span> (∀X. product(additive_identity::'a,X,additive_identity)) & (∀X. product(X::'a,additive_identity,additive_identity)) & (∀X Y. equal(X::'a,additive_identity) --> product(X::'a,h(X::'a,Y),Y)) & (product(a::'a,b,additive_identity)) & (~equal(a::'a,additive_identity)) & (~equal(b::'a,additive_identity)) --> False" by meson (*2660 inferences so far. Searching to depth 10. 7.0 secs*) lemma RNG040_2: "EQU001_0_ax equal & RNG001_0_eq product multiply sum add additive_inverse equal & (∀X. sum(additive_identity::'a,X,X)) & (∀X. sum(X::'a,additive_identity,X)) & (∀X Y. product(X::'a,Y,multiply(X::'a,Y))) & (∀X Y. sum(X::'a,Y,add(X::'a,Y))) & (∀X. sum(additive_inverse(X),X,additive_identity)) & (∀X. sum(X::'a,additive_inverse(X),additive_identity)) & (∀Y U Z X V W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(U::'a,Z,W) --> sum(X::'a,V,W)) & (∀Y X V U Z W. sum(X::'a,Y,U) & sum(Y::'a,Z,V) & sum(X::'a,V,W) --> sum(U::'a,Z,W)) & (∀Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) & (∀Y U Z X V W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(U::'a,Z,W) --> product(X::'a,V,W)) &pan> (∀Y X V U Z W. product(X::'a,Y,U) & product(Y::'a,Z,V) & product(X::'a,V,W) --> product(U::'a,Z,W)) &pan> (∀Y Z X V3 V1 V2 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & product(X::'a,V3,V4) --> sum(V1::'a,V2,V4)) & (∀Y Z V1 V2 X V3 V4. product(X::'a,Y,V1) & product(X::'a,Z,V2) & sum(Y::'a,Z,V3) & sum(V1::'a,V2,V4) --> product(X::'a,V3,V4)) & (∀X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) & (∀X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) & (∀A. product(A::'a,multiplicative_identity,A)) & (∀A. product(multiplicative_identity::'a,A,A)) & (∀A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity (∀A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &span> (∀B A C. product(A::'a,B,C) --> product(B::'a,A,C)) & (∀A B. equal(A::'a,B) --> equal(h(A),h(B))) & (sum(b::'a,c,d)) & (product(d::'a,a,additive_identity)) & (product(b::'a,a,l)) & (product(c::'a,a,n)) & (~sum(l::'a,n,additive_identity)) --> False" by meson (*8991 inferences so far. Searching to depth 9. 22.2 secs*) lemma RNG041_1: "EQU001_0_ax equal & RNG001_0_ax equal additive_inverse add multiply product additive_identity sum &pan> RNG001_0_eq product multiply sum add additive_inverse equal & (∀A B. equal(A::'a,B) --> equal(h(A),h(B))) & (∀A. product(additive_identity::'a,A,additive_identity)) & (∀A. product(A::'a,additive_identity,additive_identity)) & (∀A. product(A::'a,multiplicative_identity,A)) & (∀A. product(multiplicative_identity::'a,A,A)) & (∀A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity (∀A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &span> (product(a::'a,b,additive_identity)) & (~equal(a::'a,additive_identity)) & (~equal(b::'a,additive_identity)) --> False" by meson (*101319 inferences so far. Searching to depth 14. 76.0 secs*) lemma ROB010_1: "EQU001_0_ax equal & (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & (∀Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &< (∀A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & (∀G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & (equal(negate(add(a::'a,negate(b))),c)) & (~equal(negate(add(c::'a,negate(add(b::'a,a)))),a)) --> False" by meson (*6933 inferences so far. Searching to depth 12. 5.1 secs*) lemma ROB013_1: "EQU001_0_ax equal & (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & (∀Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &< (∀A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & (∀G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & (equal(negate(add(a::'a,b)),c)) & (~equal(negate(add(c::'a,negate(add(negate(b),a)))),a)) --> False" by meson (*6614 inferences so far. Searching to depth 11. 20.4 secs*) lemma ROB016_1: "EQU001_0_ax equal & (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & (∀Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &< (∀A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & (∀G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & (∀J K' L. equal(J::'a,K') --> equal(multiply(J::'a,L),multiply(K'::'a,L))) &> (∀M O' N. equal(M::'a,N) --> equal(multiply(O'::'a,M),multiply(O'::'a,N))) &> (∀P Q. equal(P::'a,Q) --> equal(successor(P),successor(Q))) & (∀R S'. equal(R::'a,S') & positive_integer(R) --> positive_integer(S')) & (∀X. equal(multiply(One::'a,X),X)) & (∀V X. positive_integer(X) --> equal(multiply(successor(V),X),add(X::'a,multiply (positive_integer(One)) & (∀X. positive_integer(X) --> positive_integer(successor(X))) & (equal(negate(add(d::'a,e)),negate(e))) & (positive_integer(k)) & (∀Vk X Y. equal(negate(add(negate(Y),negate(add(X::'a,negate(Y))))),X) & positive_integer(Vk) --> equal(negate(add(Y::'a,multiply(Vk::'a,add(X::'a,negate(add(X::'a,negate(Y))))))),negate(Y))) & (~equal(negate(add(e::'a,multiply(k::'a,add(d::'a,negate(add(d::'a,negate(e))))) by meson (*14077 inferences so far. Searching to depth 11. 32.8 secs*) lemma ROB021_1: "EQU001_0_ax equal & (∀Y X. equal(add(X::'a,Y),add(Y::'a,X))) & (∀X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) & (∀Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &< (∀A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) & (∀G H. equal(G::'a,H) --> equal(negate(G),negate(H))) & (∀X Y. equal(negate(X),negate(Y)) --> equal(X::'a,Y)) & (~equal(add(negate(add(a::'a,negate(b))),negate(add(negate(a),negate(b)))),b)) - by meson (*35532 inferences so far. Searching to depth 19. 54.3 secs*) lemma SET005_1: "(∀Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) & (∀Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Sub (∀Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) -- (∀Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superse (∀Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superse (∀Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) & (∀Set2 Intersection Element Set1. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set1)) & (∀Set1 Intersection Element Set2. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set2)) & (∀Set2 Set1 Element Intersection. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Set2) & memb (∀Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | i (∀Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | i (∀Set1 Set2 Intersection. member(h(Set1::'a,Set2,Intersection),Intersection) & member(h(Set1::'a,Set2,Intersection),Set2) & me (intersection(a::'a,b,aIb)) & (intersection(b::'a,c,bIc)) & (intersection(a::'a,bIc,aIbIc)) & (~intersection(aIb::'a,c,aIbIc)) --> False" by meson (*6450 inferences so far. Searching to depth 14. 4.2 secs*) lemma SET009_1: "(∀Subset Element Superset. member(Element::'a,Subset) & ssubset(Subset::'a,Superset) --> member(Element::'a,Superset)) & (∀Superset Subset. ssubset(Subset::'a,Superset) | member(member_of_1_not_of_2(Su (∀Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) -- (∀Subset Superset. equal_sets(Subset::'a,Superset) --> ssubset(Subset::'a,Supers (∀Subset Superset. equal_sets(Superset::'a,Subset) --> ssubset(Subset::'a,Supers (∀Set2 Set1. ssubset(Set1::'a,Set2) & ssubset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) & (∀Set2 Difference Element Set1. difference(Set1::'a,Set2,Difference) & member(Element::'a,Difference) --> member(Element::'a,Set1)) & (∀Element A_set Set1 Set2. ~(member(Element::'a,Set1) & member(Element::'a,Set2) & difference(A_set::'a,Set1 (∀Set1 Difference Element Set2. member(Element::'a,Set1) & difference(Set1::'a,Set2,Difference) --> member(Element::'a,Difference) | member(Element::'a,Set2)) & (∀Set1 Set2 Difference. difference(Set1::'a,Set2,Difference) | member(k(Set1::'a (∀Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Se (∀Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Difference) & member(k(Set1::'a,Set2,Difference),Set1) --> member(k(Set1::'a,Set2,Difference),Set2) | difference(Set1::'a,Set2,Difference)) & (ssubset(d::'a,a)) & (difference(b::'a,a,bDa)) & (difference(b::'a,d,bDd)) & (~ssubset(bDa::'a,bDd)) --> False" by meson (*34726 inferences so far. Searching to depth 6. 2420 secs: 40 mins! BIG*) lemma SET025_4: "EQU001_0_ax equal & (∀Y X. member(X::'a,Y) --> little_set(X)) & (∀X Y. little_set(f1(X::'a,Y)) | equal(X::'a,Y)) & (∀X Y. member(f1(X::'a,Y),X) | member(f1(X::'a,Y),Y) | equal(X::'a,Y)) & (∀X Y. member(f1(X::'a,Y),X) & member(f1(X::'a,Y),Y) --> equal(X::'a,Y)) & (∀X U Y. member(U::'a,non_ordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::' (∀Y U X. little_set(U) & equal(U::'a,X) --> member(U::'a,non_ordered_pair(X::'a,Y))) & (∀X U Y. little_set(U) & equal(U::'a,Y) --> member(U::'a,non_ordered_pair(X::'a,Y))) & (∀X Y. little_set(non_ordered_pair(X::'a,Y))) & (∀X. equal(singleton_set(X),non_ordered_pair(X::'a,X))) & (∀X Y. equal(ordered_pair(X::'a,Y),non_ordered_pair(singleton_set(X),non_ordered (∀X. ordered_pair_predicate(X) --> little_set(f2(X))) & (∀X. ordered_pair_predicate(X) --> little_set(f3(X))) & (∀X. ordered_pair_predicate(X) --> equal(X::'a,ordered_pair(f2(X),f3(X)))) &> (∀X Y Z. little_set(Y) & little_set(Z) & equal(X::'a,ordered_pair(Y::'a,Z)) --> ordered_pair_pred (∀Z X. member(Z::'a,first(X)) --> little_set(f4(Z::'a,X))) & (∀Z X. member(Z::'a,first(X)) --> little_set(f5(Z::'a,X))) & (∀Z X. member(Z::'a,first(X)) --> equal(X::'a,ordered_pair(f4(Z::'a,X),f5(Z::'a, (∀Z X. member(Z::'a,first(X)) --> member(Z::'a,f4(Z::'a,X))) & (∀X V Z U. little_set(U) & little_set(V) & equal(X::'a,ordered_pair(U::'a,V)) & member(Z::'a,U) --> member(Z::'a,first(X))) & (∀Z X. member(Z::'a,second(X)) --> little_set(f6(Z::'a,X))) & (∀Z X. member(Z::'a,second(X)) --> little_set(f7(Z::'a,X))) & (∀Z X. member(Z::'a,second(X)) --> equal(X::'a,ordered_pair(f6(Z::'a,X),f7(Z::'a (∀Z X. member(Z::'a,second(X)) --> member(Z::'a,f7(Z::'a,X))) & (∀X U Z V. little_set(U) & little_set(V) & equal(X::'a,ordered_pair(U::'a,V)) & member(Z::'a,V) --> member(Z::'a,second(X))) & (∀Z. member(Z::'a,estin) --> ordered_pair_predicate(Z)) & (∀Z. member(Z::'a,estin) --> member(first(Z),second(Z))) & (∀Z. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),second(Z)) --> member(Z::'a,estin)) & (∀Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) & (∀X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) & (∀X Z Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) & (∀Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) & (∀Z X. little_set(Z) --> member(Z::'a,complement(X)) | member(Z::'a,X)) & (∀X Y. equal(union(X::'a,Y),complement(intersection(complement(X),complement(Y)) (∀Z X. member(Z::'a,domain_of(X)) --> ordered_pair_predicate(f8(Z::'a,X))) &> (∀Z X. member(Z::'a,domain_of(X)) --> member(f8(Z::'a,X),X)) & (∀Z X. member(Z::'a,domain_of(X)) --> equal(Z::'a,first(f8(Z::'a,X)))) & (∀X Z Xp. little_set(Z) & ordered_pair_predicate(Xp) & member(Xp::'a,X) & equal(Z::'a,first(Xp)) --> member(Z::'a,domain_of(X))) & (∀X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> ordered_pair_predicate(Z)) &an> (∀Y Z X. member(Z::'a,cross_product(X::'a,Y)) --> member(first(Z),X)) & (∀X Z Y. member(Z::'a,cross_product(X::'a,Y)) --> member(second(Z),Y)) & (∀X Z Y. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),X) & member(second(Z),Y) --> member(Z::'a,cross_product(X::'a,Y))) & (∀X Z. member(Z::'a,inv1 X) --> ordered_pair_predicate(Z)) & (∀Z X. member(Z::'a,inv1 X) --> member(ordered_pair(second(Z),first(Z)),X)) &n> (∀Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::' (∀Z X. member(Z::'a,rot_right(X)) --> little_set(f9(Z::'a,X))) & (∀Z X. member(Z::'a,rot_right(X)) --> little_set(f10(Z::'a,X))) & (∀Z X. member(Z::'a,rot_right(X)) --> little_set(f11(Z::'a,X))) & (∀Z X. member(Z::'a,rot_right(X)) --> equal(Z::'a,ordered_pair(f9(Z::'a,X),order (∀Z X. member(Z::'a,rot_right(X)) --> member(ordered_pair(f10(Z::'a,X),ordered_p (∀Z V W U X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordere (∀Z X. member(Z::'a,flip_range_of(X)) --> little_set(f12(Z::'a,X))) & (∀Z X. member(Z::'a,flip_range_of(X)) --> little_set(f13(Z::'a,X))) & (∀Z X. member(Z::'a,flip_range_of(X)) --> little_set(f14(Z::'a,X))) & (∀Z X. member(Z::'a,flip_range_of(X)) --> equal(Z::'a,ordered_pair(f12(Z::'a,X), (∀Z X. member(Z::'a,flip_range_of(X)) --> member(ordered_pair(f12(Z::'a,X),order (∀Z U W V X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordere (∀X. equal(successor(X),union(X::'a,singleton_set(X)))) & (∀Z. ~member(Z::'a,empty_set)) & (∀Z. little_set(Z) --> member(Z::'a,universal_set)) & (little_set(infinity)) & (member(empty_set::'a,infinity)) & (∀X. member(X::'a,infinity) --> member(successor(X),infinity)) & (∀Z X. member(Z::'a,sigma(X)) --> member(f16(Z::'a,X),X)) & (∀Z X. member(Z::'a,sigma(X)) --> member(Z::'a,f16(Z::'a,X))) & (∀X Z Y. member(Y::'a,X) & member(Z::'a,Y) --> member(Z::'a,sigma(X))) & (∀U. little_set(U) --> little_set(sigma(U))) & (∀X U Y. ssubset(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) & (∀Y X. ssubset(X::'a,Y) | member(f17(X::'a,Y),X)) & (∀X Y. member(f17(X::'a,Y),Y) --> ssubset(X::'a,Y)) & (∀X Y. proper_subset(X::'a,Y) --> ssubset(X::'a,Y)) & (∀X Y. ~(proper_subset(X::'a,Y) & equal(X::'a,Y))) & (∀X Y. ssubset(X::'a,Y) --> proper_subset(X::'a,Y) | equal(X::'a,Y)) & (∀Z X. member(Z::'a,powerset(X)) --> ssubset(Z::'a,X)) & (∀Z X. little_set(Z) & ssubset(Z::'a,X) --> member(Z::'a,powerset(X))) & (∀U. little_set(U) --> little_set(powerset(U))) & (∀Z X. relation(Z) & member(X::'a,Z) --> ordered_pair_predicate(X)) & (∀Z. relation(Z) | member(f18(Z),Z)) & (∀Z. ordered_pair_predicate(f18(Z)) --> relation(Z)) & (∀U X V W. single_valued_set(X) & little_set(U) & little_set(V) & little_set(W) & member(ordered_pair(U::'a,V),X) & member(ordered_pair(U::'a,W),X) --> equal(V::'a,W)) & (∀X. single_valued_set(X) | little_set(f19(X))) & (∀X. single_valued_set(X) | little_set(f20(X))) & (∀X. single_valued_set(X) | little_set(f21(X))) & (∀X. single_valued_set(X) | member(ordered_pair(f19(X),f20(X)),X)) & (∀X. single_valued_set(X) | member(ordered_pair(f19(X),f21(X)),X)) & (∀X. equal(f20(X),f21(X)) --> single_valued_set(X)) & (∀Xf. function(Xf) --> relation(Xf)) & (∀Xf. function(Xf) --> single_valued_set(Xf)) & (∀Xf. relation(Xf) & single_valued_set(Xf) --> function(Xf)) & (∀Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> ordered_pair_predicate(f22(Z::'a,X, (∀Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> member(f22(Z::'a,X,Xf),Xf)) & (∀Z Xf X. member(Z::'a,image'(X::'a,Xf)) --> member(first(f22(Z::'a,X,Xf)),X)) &span> (∀X Xf Z. member(Z::'a,image'(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) &span> (∀Xf X Y Z. little_set(Z) & ordered_pair_predicate(Y) & member(Y::'a,Xf) & member(first(Y),X) & equal(second(Y),Z) --> member(Z::'a (∀X Xf. little_set(X) & function(Xf) --> little_set(image'(X::'a,Xf))) & (∀X U Y. ~(disjoint(X::'a,Y) & member(U::'a,X) & member(U::'a,Y))) & (∀Y X. disjoint(X::'a,Y) | member(f23(X::'a,Y),X)) & (∀X Y. disjoint(X::'a,Y) | member(f23(X::'a,Y),Y)) & (∀X. equal(X::'a,empty_set) | member(f24(X),X)) & (∀X. equal(X::'a,empty_set) | disjoint(f24(X),X)) & (function(f25)) & (∀X. little_set(X) --> equal(X::'a,empty_set) | member(f26(X),X)) & (∀X. little_set(X) --> equal(X::'a,empty_set) | member(ordered_pair(X::'a,f26(X) (∀Z X. member(Z::'a,range_of(X)) --> ordered_pair_predicate(f27(Z::'a,X))) &> (∀Z X. member(Z::'a,range_of(X)) --> member(f27(Z::'a,X),X)) & (∀Z X. member(Z::'a,range_of(X)) --> equal(Z::'a,second(f27(Z::'a,X)))) & (∀X Z Xp. little_set(Z) & ordered_pair_predicate(Xp) & member(Xp::'a,X) & equal(Z::'a,second(Xp)) --> member(Z::'a,range_of(X))) & (∀Z. member(Z::'a,identity_relation) --> ordered_pair_predicate(Z)) & (∀Z. member(Z::'a,identity_relation) --> equal(first(Z),second(Z))) & (∀Z. little_set(Z) & ordered_pair_predicate(Z) & equal(first(Z),second(Z)) --> member(Z::'a,identity_relation (∀X Y. equal(restrct(X::'a,Y),intersection(X::'a,cross_product(Y::'a,universal_s (∀Xf. one_to_one_function(Xf) --> function(Xf)) & (∀Xf. one_to_one_function(Xf) --> function(inv1 Xf)) & (∀Xf. function(Xf) & function(inv1 Xf) --> one_to_one_function(Xf)) & (∀Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> ordered_pair_predicate(f28(Z::'a,Xf, (∀Z Y Xf. member(Z::'a,apply(Xf::'a,Y)) --> member(f28(Z::'a,Xf,Y),Xf)) & (∀Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> equal(first(f28(Z::'a,Xf,Y)),Y)) &an> (∀Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> member(Z::'a,second(f28(Z::'a,Xf,Y)) (∀Xf Y Z W. ordered_pair_predicate(W) & member(W::'a,Xf) & equal(first(W),Y) & member(Z::'a,second(W)) --> member(Z::'a,apply(Xf::'a,Y))) & (∀Xf X Y. equal(apply_to_two_arguments(Xf::'a,X,Y),apply(Xf::'a,ordered_pair(X:: (∀X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) & (∀Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) & (∀X Xf Y. maps(Xf::'a,X,Y) --> ssubset(range_of(Xf),Y)) & (∀X Xf Y. function(Xf) & equal(domain_of(Xf),X) & ssubset(range_of(Xf),Y) --> maps(Xf::'a,X,Y)) & (∀Xf Xs. closed(Xs::'a,Xf) --> little_set(Xs)) & (∀Xs Xf. closed(Xs::'a,Xf) --> little_set(Xf)) & (∀Xf Xs. closed(Xs::'a,Xf) --> maps(Xf::'a,cross_product(Xs::'a,Xs),Xs)) & (∀Xf Xs. little_set(Xs) & little_set(Xf) & maps(Xf::'a,cross_product(Xs::'a,Xs),Xs) --> closed(Xs: (∀Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f29(Z::'a,Xf,Xg)) (∀Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f30(Z::'a,Xf,Xg)) (∀Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f31(Z::'a,Xf,Xg)) (∀Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> equal(Z::'a,ordered_pair(f29 (∀Z Xg Xf. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f29(Z::' (∀Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f31(Z::' (∀Z X Xf W Y Xg. little_set(Z) & little_set(X) & little_set(Y) & little_set(W) & equal(Z::'a,ordered_pair(X::'a,Y) (∀Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf (∀Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf (∀Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,X (∀Xs2 Xs1 Xf1 Xf2 X Xh Y. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) & member(X::'a,Xs1) & member(Y::'a,Xs1 (∀Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism (∀Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism (∀Xh Xs1 Xf1 Xs2 Xf2. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) & equal(apply(Xh::'a,apply_to_two_arguments(Xf1::'a,f32(Xh::'a,Xs1,Xf1,Xs2,Xf2),f33(Xh::'a,Xs1,Xf1,Xs2,Xf2))),apply_to_two_arguments(Xf2::'a,apply(Xh::'a,f32(Xh::'a,Xs1,Xf1,Xs2,Xf2)),apply(Xh::'a,f33(Xh::'a,Xs1,Xf1,Xs2,Xf2)))) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2)) & (∀A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) & (∀A2 B2. equal(A2::'a,B2) --> equal(f2(A2),f2(B2))) & (∀G4 H4. equal(G4::'a,H4) --> equal(f3(G4),f3(H4))) & (∀O7 P7 Q7. equal(O7::'a,P7) --> equal(f4(O7::'a,Q7),f4(P7::'a,Q7))) & (∀R7 T7 S7. equal(R7::'a,S7) --> equal(f4(T7::'a,R7),f4(T7::'a,S7))) & (∀U7 V7 W7. equal(U7::'a,V7) --> equal(f5(U7::'a,W7),f5(V7::'a,W7))) & (∀X7 Z7 Y7. equal(X7::'a,Y7) --> equal(f5(Z7::'a,X7),f5(Z7::'a,Y7))) & (∀A8 B8 C8. equal(A8::'a,B8) --> equal(f6(A8::'a,C8),f6(B8::'a,C8))) & (∀D8 F8 E8. equal(D8::'a,E8) --> equal(f6(F8::'a,D8),f6(F8::'a,E8))) & (∀G8 H8 I8. equal(G8::'a,H8) --> equal(f7(G8::'a,I8),f7(H8::'a,I8))) & (∀J8 L8 K8. equal(J8::'a,K8) --> equal(f7(L8::'a,J8),f7(L8::'a,K8))) & (∀M8 N8 O8. equal(M8::'a,N8) --> equal(f8(M8::'a,O8),f8(N8::'a,O8))) & (∀P8 R8 Q8. equal(P8::'a,Q8) --> equal(f8(R8::'a,P8),f8(R8::'a,Q8))) & (∀S8 T8 U8. equal(S8::'a,T8) --> equal(f9(S8::'a,U8),f9(T8::'a,U8))) & (∀V8 X8 W8. equal(V8::'a,W8) --> equal(f9(X8::'a,V8),f9(X8::'a,W8))) & (∀G H I'. equal(G::'a,H) --> equal(f10(G::'a,I'),f10(H::'a,I'))) & (∀J L K'. equal(J::'a,K') --> equal(f10(L::'a,J),f10(L::'a,K'))) & (∀M N O'. equal(M::'a,N) --> equal(f11(M::'a,O'),f11(N::'a,O'))) & (∀P R Q. equal(P::'a,Q) --> equal(f11(R::'a,P),f11(R::'a,Q))) & (∀S' T' U. equal(S'::'a,T') --> equal(f12(S'::'a,U),f12(T'::'a,U))) & (∀V X W. equal(V::'a,W) --> equal(f12(X::'a,V),f12(X::'a,W))) & (∀Y Z A1. equal(Y::'a,Z) --> equal(f13(Y::'a,A1),f13(Z::'a,A1))) & (∀B1 D1 C1. equal(B1::'a,C1) --> equal(f13(D1::'a,B1),f13(D1::'a,C1))) & (∀E1 F1 G1. equal(E1::'a,F1) --> equal(f14(E1::'a,G1),f14(F1::'a,G1))) & (∀H1 J1 I1. equal(H1::'a,I1) --> equal(f14(J1::'a,H1),f14(J1::'a,I1))) & (∀K1 L1 M1. equal(K1::'a,L1) --> equal(f16(K1::'a,M1),f16(L1::'a,M1))) & (∀N1 P1 O1. equal(N1::'a,O1) --> equal(f16(P1::'a,N1),f16(P1::'a,O1))) & (∀Q1 R1 S1. equal(Q1::'a,R1) --> equal(f17(Q1::'a,S1),f17(R1::'a,S1))) & (∀T1 V1 U1. equal(T1::'a,U1) --> equal(f17(V1::'a,T1),f17(V1::'a,U1))) & (∀W1 X1. equal(W1::'a,X1) --> equal(f18(W1),f18(X1))) & (∀Y1 Z1. equal(Y1::'a,Z1) --> equal(f19(Y1),f19(Z1))) & (∀C2 D2. equal(C2::'a,D2) --> equal(f20(C2),f20(D2))) & (∀E2 F2. equal(E2::'a,F2) --> equal(f21(E2),f21(F2))) & (∀G2 H2 I2 J2. equal(G2::'a,H2) --> equal(f22(G2::'a,I2,J2),f22(H2::'a,I2,J2))) &< (∀K2 M2 L2 N2. equal(K2::'a,L2) --> equal(f22(M2::'a,K2,N2),f22(M2::'a,L2,N2))) &< (∀O2 Q2 R2 P2. equal(O2::'a,P2) --> equal(f22(Q2::'a,R2,O2),f22(Q2::'a,R2,P2))) &< (∀S2 T2 U2. equal(S2::'a,T2) --> equal(f23(S2::'a,U2),f23(T2::'a,U2))) & (∀V2 X2 W2. equal(V2::'a,W2) --> equal(f23(X2::'a,V2),f23(X2::'a,W2))) & (∀Y2 Z2. equal(Y2::'a,Z2) --> equal(f24(Y2),f24(Z2))) & (∀A3 B3. equal(A3::'a,B3) --> equal(f26(A3),f26(B3))) & (∀C3 D3 E3. equal(C3::'a,D3) --> equal(f27(C3::'a,E3),f27(D3::'a,E3))) & (∀F3 H3 G3. equal(F3::'a,G3) --> equal(f27(H3::'a,F3),f27(H3::'a,G3))) & (∀I3 J3 K3 L3. equal(I3::'a,J3) --> equal(f28(I3::'a,K3,L3),f28(J3::'a,K3,L3))) &< (∀M3 O3 N3 P3. equal(M3::'a,N3) --> equal(f28(O3::'a,M3,P3),f28(O3::'a,N3,P3))) &< (∀Q3 S3 T3 R3. equal(Q3::'a,R3) --> equal(f28(S3::'a,T3,Q3),f28(S3::'a,T3,R3))) &< (∀U3 V3 W3 X3. equal(U3::'a,V3) --> equal(f29(U3::'a,W3,X3),f29(V3::'a,W3,X3))) &< (∀Y3 A4 Z3 B4. equal(Y3::'a,Z3) --> equal(f29(A4::'a,Y3,B4),f29(A4::'a,Z3,B4))) &< (∀C4 E4 F4 D4. equal(C4::'a,D4) --> equal(f29(E4::'a,F4,C4),f29(E4::'a,F4,D4))) &< (∀I4 J4 K4 L4. equal(I4::'a,J4) --> equal(f30(I4::'a,K4,L4),f30(J4::'a,K4,L4))) &< (∀M4 O4 N4 P4. equal(M4::'a,N4) --> equal(f30(O4::'a,M4,P4),f30(O4::'a,N4,P4))) &< (∀Q4 S4 T4 R4. equal(Q4::'a,R4) --> equal(f30(S4::'a,T4,Q4),f30(S4::'a,T4,R4))) &< (∀U4 V4 W4 X4. equal(U4::'a,V4) --> equal(f31(U4::'a,W4,X4),f31(V4::'a,W4,X4))) &< (∀Y4 A5 Z4 B5. equal(Y4::'a,Z4) --> equal(f31(A5::'a,Y4,B5),f31(A5::'a,Z4,B5))) &< (∀C5 E5 F5 D5. equal(C5::'a,D5) --> equal(f31(E5::'a,F5,C5),f31(E5::'a,F5,D5))) &< (∀G5 H5 I5 J5 K5 L5. equal(G5::'a,H5) --> equal(f32(G5::'a,I5,J5,K5,L5),f32(H5:: (∀M5 O5 N5 P5 Q5 R5. equal(M5::'a,N5) --> equal(f32(O5::'a,M5,P5,Q5,R5),f32(O5:: (∀S5 U5 V5 T5 W5 X5. equal(S5::'a,T5) --> equal(f32(U5::'a,V5,S5,W5,X5),f32(U5:: (∀Y5 A6 B6 C6 Z5 D6. equal(Y5::'a,Z5) --> equal(f32(A6::'a,B6,C6,Y5,D6),f32(A6:: (∀E6 G6 H6 I6 J6 F6. equal(E6::'a,F6) --> equal(f32(G6::'a,H6,I6,J6,E6),f32(G6:: (∀K6 L6 M6 N6 O6 P6. equal(K6::'a,L6) --> equal(f33(K6::'a,M6,N6,O6,P6),f33(L6:: (∀Q6 S6 R6 T6 U6 V6. equal(Q6::'a,R6) --> equal(f33(S6::'a,Q6,T6,U6,V6),f33(S6:: (∀W6 Y6 Z6 X6 A7 B7. equal(W6::'a,X6) --> equal(f33(Y6::'a,Z6,W6,A7,B7),f33(Y6:: (∀C7 E7 F7 G7 D7 H7. equal(C7::'a,D7) --> equal(f33(E7::'a,F7,G7,C7,H7),f33(E7:: (∀I7 K7 L7 M7 N7 J7. equal(I7::'a,J7) --> equal(f33(K7::'a,L7,M7,N7,I7),f33(K7:: (∀A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) & (∀D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) & (∀G H I' J. equal(G::'a,H) --> equal(apply_to_two_arguments(G::'a,I',J),apply_to (∀K' M L N. equal(K'::'a,L) --> equal(apply_to_two_arguments(M::'a,K',N),apply_t (∀O' Q R P. equal(O'::'a,P) --> equal(apply_to_two_arguments(Q::'a,R,O'),apply_t (∀S' T'. equal(S'::'a,T') --> equal(complement(S'),complement(T'))) & (∀U V W. equal(U::'a,V) --> equal(composition(U::'a,W),composition(V::'a,W))) &pan> (∀X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) &pan> (∀A1 B1. equal(A1::'a,B1) --> equal(inv1 A1,inv1 B1)) & (∀C1 D1 E1. equal(C1::'a,D1) --> equal(cross_product(C1::'a,E1),cross_product(D1 (∀F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1 (∀I1 J1. equal(I1::'a,J1) --> equal(domain_of(I1),domain_of(J1))) & (∀I10 J10. equal(I10::'a,J10) --> equal(first(I10),first(J10))) & (∀Q10 R10. equal(Q10::'a,R10) --> equal(flip_range_of(Q10),flip_range_of(R10))) &< (∀S10 T10 U10. equal(S10::'a,T10) --> equal(image'(S10::'a,U10),image'(T10::'a,U (∀V10 X10 W10. equal(V10::'a,W10) --> equal(image'(X10::'a,V10),image'(X10::'a,W (∀Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersecti (∀B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersecti (∀E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_or (∀H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_or (∀K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pa (∀N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pa (∀Q11 R11. equal(Q11::'a,R11) --> equal(powerset(Q11),powerset(R11))) & (∀S11 T11. equal(S11::'a,T11) --> equal(range_of(S11),range_of(T11))) & (∀U11 V11 W11. equal(U11::'a,V11) --> equal(restrct(U11::'a,W11),restrct(V11::'a (∀X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a (∀A12 B12. equal(A12::'a,B12) --> equal(rot_right(A12),rot_right(B12))) & (∀C12 D12. equal(C12::'a,D12) --> equal(second(C12),second(D12))) & (∀K12 L12. equal(K12::'a,L12) --> equal(sigma(K12),sigma(L12))) & (∀M12 N12. equal(M12::'a,N12) --> equal(singleton_set(M12),singleton_set(N12))) &< (∀O12 P12. equal(O12::'a,P12) --> equal(successor(O12),successor(P12))) & (∀Q12 R12 S12. equal(Q12::'a,R12) --> equal(union(Q12::'a,S12),union(R12::'a,S12 (∀T12 V12 U12. equal(T12::'a,U12) --> equal(union(V12::'a,T12),union(V12::'a,U12 (∀W12 X12 Y12. equal(W12::'a,X12) & closed(W12::'a,Y12) --> closed(X12::'a,Y12)) & (∀Z12 B13 A13. equal(Z12::'a,A13) & closed(B13::'a,Z12) --> closed(B13::'a,A13)) & (∀C13 D13 E13. equal(C13::'a,D13) & disjoint(C13::'a,E13) --> disjoint(D13::'a,E13)) & (∀F13 H13 G13. equal(F13::'a,G13) & disjoint(H13::'a,F13) --> disjoint(H13::'a,G13)) & (∀I13 J13. equal(I13::'a,J13) & function(I13) --> function(J13)) & (∀K13 L13 M13 N13 O13 P13. equal(K13::'a,L13) & homomorphism(K13::'a,M13,N13,O13,P13) --> homomorphism(L13::'a,M13,N13,O13,P13)) & (∀Q13 S13 R13 T13 U13 V13. equal(Q13::'a,R13) & homomorphism(S13::'a,Q13,T13,U13,V13) --> homomorphism(S13::'a,R13,T13,U13,V13)) & (∀W13 Y13 Z13 X13 A14 B14. equal(W13::'a,X13) & homomorphism(Y13::'a,Z13,W13,A14,B14) --> homomorphism(Y13::'a,Z13,X13,A14,B14)) & (∀C14 E14 F14 G14 D14 H14. equal(C14::'a,D14) & homomorphism(E14::'a,F14,G14,C14,H14) --> homomorphism(E14::'a,F14,G14,D14,H14)) & (∀I14 K14 L14 M14 N14 J14. equal(I14::'a,J14) & homomorphism(K14::'a,L14,M14,N14,I14) --> homomorphism(K14::'a,L14,M14,N14,J14)) & (∀O14 P14. equal(O14::'a,P14) & little_set(O14) --> little_set(P14)) & (∀Q14 R14 S14 T14. equal(Q14::'a,R14) & maps(Q14::'a,S14,T14) --> maps(R14::'a,S14,T14)) & (∀U14 W14 V14 X14. equal(U14::'a,V14) & maps(W14::'a,U14,X14) --> maps(W14::'a,V14,X14)) & (∀Y14 A15 B15 Z14. equal(Y14::'a,Z14) & maps(A15::'a,B15,Y14) --> maps(A15::'a,B15,Z14)) & (∀C15 D15 E15. equal(C15::'a,D15) & member(C15::'a,E15) --> member(D15::'a,E15)) & (∀F15 H15 G15. equal(F15::'a,G15) & member(H15::'a,F15) --> member(H15::'a,G15)) & (∀I15 J15. equal(I15::'a,J15) & one_to_one_function(I15) --> one_to_one_function(J15)) & (∀K15 L15. equal(K15::'a,L15) & ordered_pair_predicate(K15) --> ordered_pair_predicate(L15)) & (∀M15 N15 O15. equal(M15::'a,N15) & proper_subset(M15::'a,O15) --> proper_subset(N15::'a,O15)) & (∀P15 R15 Q15. equal(P15::'a,Q15) & proper_subset(R15::'a,P15) --> proper_subset(R15::'a,Q15)) & (∀S15 T15. equal(S15::'a,T15) & relation(S15) --> relation(T15)) & (∀U15 V15. equal(U15::'a,V15) & single_valued_set(U15) --> single_valued_set(V15)) & (∀W15 X15 Y15. equal(W15::'a,X15) & ssubset(W15::'a,Y15) --> ssubset(X15::'a,Y15)) & (∀Z15 B16 A16. equal(Z15::'a,A16) & ssubset(B16::'a,Z15) --> ssubset(B16::'a,A16)) & (~little_set(ordered_pair(a::'a,b))) --> False" oops (*13 inferences so far. Searching to depth 8. 0 secs*) lemma SET046_5: "(∀Y X. ~(element(X::'a,a) & element(X::'a,Y) & element(Y::'a,X))) & (∀X. element(X::'a,f(X)) | element(X::'a,a)) & (∀X. element(f(X),X) | element(X::'a,a)) --> False" by meson (*33 inferences so far. Searching to depth 9. 0.2 secs*) lemma SET047_5: "(∀X Z Y. set_equal(X::'a,Y) & element(Z::'a,X) --> element(Z::'a,Y)) & (∀Y Z X. set_equal(X::'a,Y) & element(Z::'a,Y) --> element(Z::'a,X)) & (∀X Y. element(f(X::'a,Y),X) | element(f(X::'a,Y),Y) | set_equal(X::'a,Y)) &> (∀X Y. element(f(X::'a,Y),Y) & element(f(X::'a,Y),X) --> set_equal(X::'a,Y)) & (set_equal(a::'a,b) | set_equal(b::'a,a)) & (~(set_equal(b::'a,a) & set_equal(a::'a,b))) --> False" by meson (*311 inferences so far. Searching to depth 12. 0.1 secs*) lemma SYN034_1: "(∀A. p(A::'a,a) | p(A::'a,f(A))) & (∀A. p(A::'a,a) | p(f(A),A)) & (∀A B. ~(p(A::'a,B) & p(B::'a,A) & p(B::'a,a))) --> False" by meson (*30 inferences so far. Searching to depth 6. 0.2 secs*) lemma SYN071_1: "EQU001_0_ax equal & (equal(a::'a,b) | equal(c::'a,d)) & (equal(a::'a,c) | equal(b::'a,d)) & (~equal(a::'a,d)) & (~equal(b::'a,c)) --> False" by meson (*1897410 inferences so far. Searching to depth 48 206s, nearly 4 mins on griffon.*) lemma SYN349_1: "(∀X Y. f(w(X),g(X::'a,Y)) --> f(X::'a,g(X::'a,Y))) & (∀X Y. f(X::'a,g(X::'a,Y)) --> f(w(X),g(X::'a,Y))) & (∀Y X. f(X::'a,g(X::'a,Y)) & f(Y::'a,g(X::'a,Y)) --> f(g(X::'a,Y),Y) | f(g(X::'a,Y),w(X))) & (∀Y X. f(g(X::'a,Y),Y) & f(Y::'a,g(X::'a,Y)) --> f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) & (∀Y X. f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),Y) | f(Y::'a,g(X::'a,Y)) | f(g(X::'a,Y (∀Y X. f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),Y) --> f(Y::'a,g(X::'a,Y)) | f(g(X::'a,Y),w(X))) & (∀Y X. f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)) --> f(g(X::'a,Y),Y) | f(Y::'a,g(X::'a,Y))) & (∀Y X. f(g(X::'a,Y),Y) & f(g(X::'a,Y),w(X)) --> f(X::'a,g(X::'a,Y)) | f(Y::'a,g(X::'a,Y))) & (∀Y X. f(Y::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)) --> f(X::'a,g(X::'a,Y)) | f(g(X::'a,Y),Y)) & (∀Y X. ~(f(X::'a,g(X::'a,Y)) & f(g(X::'a,Y),Y) & f(Y::'a,g(X::'a,Y)) & f(g(X::'a,Y),w(X)))) --> False" oops (*398 inferences so far. Searching to depth 12. 0.4 secs*) lemma SYN352_1: "(f(a::'a,b)) & (∀X Y. f(X::'a,Y) --> f(b::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) & (∀X Y. f(X::'a,Y) | f(z(X::'a,Y),z(X::'a,Y))) & (∀X Y. f(b::'a,z(X::'a,Y)) | f(X::'a,z(X::'a,Y)) | f(z(X::'a,Y),z(X::'a,Y))) &an> (∀X Y. f(b::'a,z(X::'a,Y)) & f(X::'a,z(X::'a,Y)) --> f(z(X::'a,Y),z(X::'a,Y))) & (∀X Y. ~(f(X::'a,Y) & f(X::'a,z(X::'a,Y)) & f(Y::'a,z(X::'a,Y)))) & (∀X Y. f(X::'a,Y) --> f(X::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) --> False" by meson (*5336 inferences so far. Searching to depth 15. 5.3 secs*) lemma TOP001_2: "(∀Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1( (∀U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1( (∀U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) & (∀Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) & (∀Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) & (∀U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) & (∀X. subset_sets(X::'a,X)) & (∀X U Y. subset_sets(X::'a,Y) & element_of_set(U::'a,X) --> element_of_set(U::'a,Y)) & (∀X Y. equal_sets(X::'a,Y) --> subset_sets(X::'a,Y)) & (∀Y X. subset_sets(X::'a,Y) | element_of_set(in_1st_set(X::'a,Y),X)) & (∀X Y. element_of_set(in_1st_set(X::'a,Y),Y) --> subset_sets(X::'a,Y)) & (basis(cx::'a,f)) & (~subset_sets(union_of_members(top_of_basis(f)),cx)) --> False" by meson (*0 inferences so far. Searching to depth 0. 0 secs*) lemma TOP002_2: "(∀Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf:: (∀X. ~element_of_set(X::'a,empty_set)) & (~element_of_collection(empty_set::'a,top_of_basis(f))) --> False" by meson (*0 inferences so far. Searching to depth 0. 6.5 secs. BIG*) lemma TOP004_1: "(∀Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1( (∀U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1( (∀U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) & (∀Vf U Va. element_of_set(U::'a,intersection_of_members(Vf)) & element_of_collection(Va::'a,Vf) --> element_of_set(U::'a,Va)) & (∀U Vf. element_of_set(U::'a,intersection_of_members(Vf)) | element_of_collectio (∀Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection (∀Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) &n> (∀X Vt. topological_space(X::'a,Vt) --> element_of_collection(empty_set::'a,Vt)) (∀X Vt. topological_space(X::'a,Vt) --> element_of_collection(X::'a,Vt)) & (∀X Y Z Vt. topological_space(X::'a,Vt) & element_of_collection(Y::'a,Vt) & element_of_collection(Z::'a,Vt) --> ele (∀X Vf Vt. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) --> element_of_collection(union_of_members(Vf),Vt)) & (∀X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) --> (∀X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | element_of_collection(f3(X::'a,Vt),Vt)) & (∀X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) --> (∀X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | element_of_collection(f4(X::'a,Vt),Vt)) & (∀X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(intersection_of_sets(f3(X::'a,Vt),f4(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt) | subset_collections(f5(X::'a,Vt),Vt)) & (∀X Vt. equal_sets(union_of_members(Vt),X) & element_of_collection(empty_set::'a,Vt) & element_of_collection(X::'a,Vt) & element_of_collection(intersection_of_sets(f3(X::'a,Vt),f4(X::'a,Vt)),Vt) & ele (∀U X Vt. open(U::'a,X,Vt) --> topological_space(X::'a,Vt)) & (∀X U Vt. open(U::'a,X,Vt) --> element_of_collection(U::'a,Vt)) & (∀X U Vt. topological_space(X::'a,Vt) & element_of_collection(U::'a,Vt) --> open(U::'a,X,Vt)) & (∀U X Vt. closed(U::'a,X,Vt) --> topological_space(X::'a,Vt)) & (∀U X Vt. closed(U::'a,X,Vt) --> open(relative_complement_sets(U::'a,X),X,Vt)) &span> (∀U X Vt. topological_space(X::'a,Vt) & open(relative_complement_sets(U::'a,X),X,Vt) --> closed(U::'a,X,Vt)) & (∀Vs X Vt. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vt)) & (∀Vt X Vs. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vs)) & (∀X Vs Vt. finer(Vt::'a,Vs,X) --> subset_collections(Vs::'a,Vt)) & (∀X Vs Vt. topological_space(X::'a,Vt) & topological_space(X::'a,Vs) & subset_collections(Vs::'a,Vt) --> finer( (∀Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) & (∀X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of (∀X Y Vb1 Vb2 Vf. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of (∀X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of (∀Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set( (∀X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_coll (∀X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_coll (∀X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set( (∀Uu9 X Vf. equal_sets(union_of_members(Vf),X) & element_of_set(f7(X::'a,Vf),Uu9) & element_of_collection(Uu9::'a,Vf (∀Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) & (∀U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) & (∀Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) & (∀Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::' (∀Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> ele (∀U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topol (∀U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subse (∀X Y U Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> eleme (∀X Vt Y U. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> equal (∀X Vt U Y Uu12. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_collection(Uu12::'a,Vt) (∀U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X:: (∀U Vt Y X. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) (∀Y X Vt U. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> element_of_set(U::'a, (∀X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a (∀Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U (∀U Y Uu13 X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(U::'a,Uu13) & subset_sets(Uu13::'a,Y) & open(U (∀U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::' (∀U Vt Y X. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &< (∀Y X Vt U V. element_of_set(U::'a,closure(Y::'a,X,Vt)) & subset_sets(Y::'a,V) & closed(V::'a,X,Vt) --> (∀Y X Vt U. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> element_of_set(U::'a,closure(Y::'a,X,Vt)) | subset_sets(Y::'a,f14(Y::'a,X,Vt,U))) & (∀Y U X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> element_of_set(U::'a,closure(Y::'a,X,Vt)) | closed(f14(Y::'a,X,Vt,U),X,Vt)) & (∀Y X Vt U. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(U::'a,f14(Y::'a,X,Vt,U)) (∀U Y X Vt. neighborhood(U::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) & (∀Y U X Vt. neighborhood(U::'a,Y,X,Vt) --> open(U::'a,X,Vt)) & (∀X Vt Y U. neighborhood(U::'a,Y,X,Vt) --> element_of_set(Y::'a,U)) & (∀X Vt Y U. topological_space(X::'a,Vt) & open(U::'a,X,Vt) & element_of_set(Y::'a,U) --> neighborhoo (∀Z Y X Vt. limit_point(Z::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) & (∀Z Vt Y X. limit_point(Z::'a,Y,X,Vt) --> subset_sets(Y::'a,X)) & (∀Z X Vt U Y. limit_point(Z::'a,Y,X,Vt) & neighborhood(U::'a,Z,X,Vt) --> element_of_set(f15(Z::'a,Y,X,Vt,U),intersection_of_sets(U::'a,Y))) & (∀Y X Vt U Z. ~(limit_point(Z::'a,Y,X,Vt) & neighborhood(U::'a,Z,X,Vt) & eq_p(f15(Z::'a,Y,X,Vt,U),Z))) & (∀Y Z X Vt. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) --> limit_point(Z::'a,Y,X,Vt) | neighborhood(f16(Z::'a,Y,X,Vt),Z,X,Vt)) & (∀X Vt Y Uu16 Z. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_set(Uu16::'a,intersecti (∀U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X:: (∀U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a, (∀U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a, (∀U Y X Vt. topological_space(X::'a,Vt) & element_of_set(U::'a,closure(Y::'a,X,Vt)) & element_of_set(U::'a,closure(relative_c (∀X Vt. hausdorff(X::'a,Vt) --> topological_space(X::'a,Vt)) & (∀X_2 X_1 X Vt. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a, (∀X_1 X_2 X Vt. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a, (∀X Vt X_1 X_2. hausdorff(X::'a,Vt) & element_of_set(X_1::'a,X) & element_of_set(X_2::'a,X) --> eq_p(X_1::'a, (∀Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19 (∀Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20 (∀X Vt. topological_space(X::'a,Vt) & eq_p(f19(X::'a,Vt),f20(X::'a,Vt)) --> hausdorff(X::'a,Vt)) & (∀X Vt Uu19 Uu20. topological_space(X::'a,Vt) & neighborhood(Uu19::'a,f19(X::'a,Vt),X,Vt) & neighborhood(Uu20::'a,f20(X::'a,V (∀Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) &pan> (∀Va2 X Vt Va1. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va1::'a,empty_set))) & (∀Va1 X Vt Va2. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va2::'a,empty_set))) & (∀Va2 X Va1 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va1::'a,V (∀Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,V (∀Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::' (∀X Vt Va1 Va2. separation(Va1::'a,Va2,X,Vt) --> disjoint_s(Va1::'a,Va2)) & (∀Vt X Va1 Va2. topological_space(X::'a,Vt) & element_of_collection(Va1::'a,Vt) & element_of_collection(Va2::'a,Vt) & equal_sets(union_of_sets(Va1::'a,Va2),X) & (∀X Vt. connected_space(X::'a,Vt) --> topological_space(X::'a,Vt)) & (∀Va1 Va2 X Vt. ~(connected_space(X::'a,Vt) & separation(Va1::'a,Va2,X,Vt))) & (∀X Vt. topological_space(X::'a,Vt) --> connected_space(X::'a,Vt) | separation(f (∀Va X Vt. connected_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) & (∀Vt Va X. connected_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) & (∀X Vt Va. connected_set(Va::'a,X,Vt) --> connected_space(Va::'a,subspace_topolo (∀X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & connected_space(Va::'a,subspace_topology (∀Vf X Vt. open_covering(Vf::'a,X,Vt) --> topological_space(X::'a,Vt)) & (∀X Vf Vt. open_covering(Vf::'a,X,Vt) --> subset_collections(Vf::'a,Vt)) & (∀Vt Vf X. open_covering(Vf::'a,X,Vt) --> equal_sets(union_of_members(Vf),X)) &pan> (∀Vt Vf X. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) & equal_sets(union_of_members(Vf),X) --> o (∀X Vt. compact_space(X::'a,Vt) --> topological_space(X::'a,Vt)) & (∀X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> finite'(f23(X::'a,Vt,Vf1))) & (∀X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> subset_collections(f23(X::'a,Vt,Vf1),Vf1)) & (∀Vf1 X Vt. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> open_covering(f23(X::'a,Vt,Vf1),X,Vt)) & (∀X Vt. topological_space(X::'a,Vt) --> compact_space(X::'a,Vt) | open_covering( (∀Uu24 X Vt. topological_space(X::'a,Vt) & finite'(Uu24) & subset_collections(Uu24::'a,f24(X::'a, (∀Va X Vt. compact_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) & (∀Vt Va X. compact_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) & (∀X Vt Va. compact_set(Va::'a,X,Vt) --> compact_space(Va::'a,subspace_topology(X (∀X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & compact_space(Va::'a,subspace_topology(X (basis(cx::'a,f)) & (∀U. element_of_collection(U::'a,top_of_basis(f))) & (∀V. element_of_collection(V::'a,top_of_basis(f))) & (∀U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) -- by meson (*0 inferences so far. Searching to depth 0. 0.8 secs*) lemma TOP004_2: "(∀U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) & (∀Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) & (∀X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of (∀X Y Vb1 Vb2 Vf. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of (∀X Vf Y Vb1 Vb2. basis(X::'a,Vf) & element_of_set(Y::'a,X) & element_of_collection(Vb1::'a,Vf) & element_of_collection(Vb2::'a,Vf) & element_of (∀Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) & (∀U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) & (∀Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) & (∀Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::' (∀Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> ele (∀Y X Z. subset_sets(X::'a,Y) & subset_sets(Y::'a,Z) --> subset_sets(X::'a,Z)) & (∀Y Z X. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set( (∀X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set( (∀X Z Y. element_of_set(Z::'a,X) & element_of_set(Z::'a,Y) --> element_of_set(Z::'a,intersection_of_sets(X::'a,Y))) & (∀X U Y V. subset_sets(X::'a,Y) & subset_sets(U::'a,V) --> subset_sets(intersection_of_sets(X::'a,U),intersection_of_sets(Y::'a,V))) & (∀X Z Y. equal_sets(X::'a,Y) & element_of_set(Z::'a,X) --> element_of_set(Z::'a,Y)) & (∀Y X. equal_sets(intersection_of_sets(X::'a,Y),intersection_of_sets(Y::'a,X))) &< (basis(cx::'a,f)) & (∀U. element_of_collection(U::'a,top_of_basis(f))) & (∀V. element_of_collection(V::'a,top_of_basis(f))) & (∀U V. ~element_of_collection(intersection_of_sets(U::'a,V),top_of_basis(f))) -- by meson (*53777 inferences so far. Searching to depth 20. 68.7 secs*) lemma TOP005_2: "(∀Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1( (∀U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1( (∀Vf U X. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_set(X::'a,f10(Vf::'a,U,X))) & (∀U X Vf. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> element_of_collection(f10(Vf::'a,U,X),Vf)) & (∀Vf X U. element_of_collection(U::'a,top_of_basis(Vf)) & element_of_set(X::'a,U) --> subset_sets(f10(Vf::'a,U,X),U)) & (∀Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::' (∀Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> ele (∀X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a (∀Y X Z. subset_sets(X::'a,Y) & element_of_collection(Y::'a,Z) --> subset_sets(X::'a,union_of_members(Z))) & (∀X U Y. subset_collections(X::'a,Y) & element_of_collection(U::'a,X) --> element_of_collection(U::'a,Y)) & (subset_collections(g::'a,top_of_basis(f))) & (~element_of_collection(union_of_members(g),top_of_basis(f))) --> False" by meson end
¤ Dauer der Verarbeitung: 0.91 Sekunden (vorverarbeitet am 2026-04-28) ¤*© Formatika GbR, Deutschland |
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2026-05-26