Theory Meson_Test

theory Meson_Test
imports Main

header {* 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

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)) & ((∀x. P x --> Q x) | (∃x. P x & R x)) --> (∃x. Q x & P x)"
apply (rule ccontr)
ML_prf {*
val prem25 = Thm.assume @{cprop "¬ ?thesis"};
val nnf25 = Meson.make_nnf @{context} prem25;
val xsko25 = Meson.skolemize @{context} nnf25;
*}

apply (tactic {* cut_tac xsko25 1 THEN REPEAT (etac exE 1) *})
ML_val {*
val [_, sko25] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
val clauses25 = Meson.make_clauses @{context} [sko25]; (*7 clauses*)
val horns25 = Meson.make_horns clauses25; (*16 Horn clauses*)
val go25 :: _ = Meson.gocls clauses25;

Goal.prove @{context} [] [] @{prop False} (fn _ =>
rtac go25 1 THEN
Meson.depth_prolog_tac 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 prem26 = Thm.assume @{cprop "¬ ?thesis"}
val nnf26 = Meson.make_nnf @{context} prem26;
val xsko26 = Meson.skolemize @{context} nnf26;
*}

apply (tactic {* cut_tac xsko26 1 THEN REPEAT (etac exE 1) *})
ML_val {*
val [_, sko26] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
val clauses26 = Meson.make_clauses @{context} [sko26]; (*9 clauses*)
val horns26 = Meson.make_horns clauses26; (*24 Horn clauses*)
val go26 :: _ = Meson.gocls clauses26;

Goal.prove @{context} [] [] @{prop False} (fn _ =>
rtac go26 1 THEN
Meson.depth_prolog_tac 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::bool)))"
apply (rule ccontr)
ML_prf {*
val prem43 = Thm.assume @{cprop "¬ ?thesis"};
val nnf43 = Meson.make_nnf @{context} prem43;
val xsko43 = Meson.skolemize @{context} nnf43;
*}

apply (tactic {* cut_tac xsko43 1 THEN REPEAT (etac exE 1) *})
ML_val {*
val [_, sko43] = #prems (#1 (Subgoal.focus @{context} 1 (#goal @{Isar.goal})));
val clauses43 = Meson.make_clauses @{context} [sko43]; (*6*)
val horns43 = Meson.make_horns clauses43; (*16*)
val go43 :: _ = Meson.gocls clauses43;

Goal.prove @{context} [] [] @{prop False} (fn _ =>
rtac go43 1 THEN
Meson.best_prolog_tac 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 [Overbeek, et al., 1976]; prob2_part1.ver1.in [ANL]
* BOO004-1 36.7 B2 part 2 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob2_part2.ver1 [ANL]
* BOO005-1 47.4 B3 part 1 [McCharen, et al., 1976]; B5 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob3_part1.ver1.in [ANL]
* BOO006-1 48.4 B3 part 2 [McCharen, et al., 1976]; B6 [McCharen, et al., 1976]; Lemma proved [Overbeek, et al., 1976]; prob3_part2.ver1 [ANL]
* 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 al., 1976]; THEOREM 1 [Lusk & McCune, 1993]; wos10 [Wilson & Minker, 1976]; xsquared.ver1.in [ANL]; [Robinson, 1963]
* 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]; S1 [Pfenning, 1988]
* LCL111-1 585.6 CADE-11 Competition 6 [Overbeek, 1990]; mv25.in [OTTER]; MV-57 [McCune & Wos, 1992]; mv.in part 2 [OTTER]; ovb6 [SETHEO]; THEOREM 6 [Lusk & McCune, 1993]
* 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 [Wilson & Minker, 1976]
* 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., 1992]; boxes.ver1.in [ANL]
* 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, et al., 1974]; FEX6T1 [SPRFN]; FEX6T2 [SPRFN]
* 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 [Zhang, 1993]; THEOREM EQ-10 [Lusk & McCune, 1993]
* 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 product add sum &
BOO002_0_eq INVERSE multiply add product sum equal &
(~product(x::'a,x,x)) --> False"

oops

(*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 product add sum &
BOO002_0_eq INVERSE multiply add product sum equal &
(~sum(x::'a,x,x)) --> False"

oops

(*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 product add sum &
BOO002_0_eq INVERSE multiply add product sum equal &
(~sum(x::'a,multiplicative_identity,multiplicative_identity)) --> False"

oops

(*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 product add sum &
BOO002_0_eq INVERSE multiply add product sum equal &
(~product(x::'a,additive_identity,additive_identity)) --> False"

oops

(*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 product add sum &
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)) &
(∀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 defined ≡
(∀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)) &
(∀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,Xyz)) &
(∀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)) &
(∀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,Xyz)) &
(∀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 equal ≡
(∀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"

oops

(*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"

oops

(*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,Z)))) &
(∀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)))) &
(∀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::'a,X)),i)),apply(x::'a,apply(apply(apply(s::'a,apply(b::'a,X)),i),apply(apply(s::'a,apply(b::'a,X)),i))))) &
(∀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)))) &
(∀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)))) &
(∀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)))) &
(∀X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) &
(∀Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &
(∀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"

oops

(*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,apply(k::'a,Z)),apply(Y::'a,Z)))) &
(∀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"

oops

(*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_state::'a,Start_state)) &
(∀Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
(∀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,Goal_state)) --> succeeds(Goal_state::'a,Start_state)) &
(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_state::'a,Start_state)) &
(∀Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
(∀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,Goal_state)) --> succeeds(Goal_state::'a,Start_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))) &
(~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) --> fails(Goal_state::'a,Intermediate_state) | fails(Intermediate_state::'a,Start_state)) &
(∀Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) &
(∀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,Z,W)) &
(∀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(Y::'a,Z)) &
(∀X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
(∀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)) &
(∀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::'a,Y)) &
(∀X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
(∀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,good)) &
(∀W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) &
(∀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) --> program(c2)) &
(∀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) --> program_halts2_halts2_outputs(c2::'a,Y,good)) &
(∀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) --> program_not_halts2_halts2_outputs(c2::'a,Y,bad)) &
(∀V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program(c3)) &
(∀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) --> program_not_halts2_halts2_outputs(c3::'a,Y,bad)) &
(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))) &
(∀Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &
(∀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_point_2
lower_dimension_point_1 extension euclid2 euclid1 outer_pasch equidistant equal between ≡
(∀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) & between(X::'a,Y,Z) & between(X1::'a,Y1,Z1) --> equal(X::'a,Y) | equidistant(Z::'a,V,Z1,V1)) &
(∀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_point_3)) &
(~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
(~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
(∀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::'a,Y,Z) | between(Y::'a,Z,X) | between(Z::'a,X,Y) | equal(W::'a,V)) &
(∀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 equidistant
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_pasch(Y::'a,V1,V2,V3,V4))) &
(∀X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,X,V2,V3,V4),outer_pasch(V1::'a,Y,V2,V3,V4))) &
(∀X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,X,V3,V4),outer_pasch(V1::'a,V2,Y,V3,V4))) &
(∀X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,X,V4),outer_pasch(V1::'a,V2,V3,Y,V4))) &
(∀X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(outer_pasch(V1::'a,V2,V3,V4,X),outer_pasch(V1::'a,V2,V3,V4,Y))) &
(∀A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C,D,E,F'))) &
(∀G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::'a,H,J,K',L))) &
(∀M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a,P,N,Q,R))) &
(∀S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::'a,V,W,T',X))) &
(∀Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A1::'a,B1,C1,D1,Z))) &
(∀E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),euclid2(F1::'a,G1,H1,I1,J1))) &
(∀K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),euclid2(M1::'a,L1,N1,O1,P1))) &
(∀Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),euclid2(S1::'a,T1,R1,U1,V1))) &
(∀W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),euclid2(Y1::'a,Z1,A2,X1,B2))) &
(∀C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),euclid2(E2::'a,F2,G2,H2,D2))) &
(∀X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
(∀X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
(∀X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
(∀X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
(∀X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5),continuous(Y::'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),continuous(V1::'a,Y,V2,V3,V4,V5))) &
(∀X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5),continuous(V1::'a,V2,Y,V3,V4,V5))) &
(∀X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5),continuous(V1::'a,V2,V3,Y,V4,V5))) &
(∀X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5),continuous(V1::'a,V2,V3,V4,Y,V5))) &
(∀X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X),continuous(V1::'a,V2,V3,V4,V5,Y)))"



(*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 between &
GEO001_0_eq continuous extension euclid2 euclid1 outer_pasch equidistant between equal &
(~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) & between(X::'a,Y,Z) & between(X1::'a,Y1,Z1) --> equal(X::'a,Y) | equidistant(Z::'a,V,Z1,V1)) &
(∀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_point_3)) &
(~between(lower_dimension_point_2::'a,lower_dimension_point_3,lower_dimension_point_1)) &
(~between(lower_dimension_point_3::'a,lower_dimension_point_1,lower_dimension_point_2)) &
(∀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::'a,Y,Z) | between(Y::'a,Z,X) | between(Z::'a,X,Y) | equal(W::'a,V)) &
(∀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_pasch(Y::'a,V1,V2,V3,V4))) &
(∀X V1 Y V2 V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,X,V2,V3,V4),inner_pasch(V1::'a,Y,V2,V3,V4))) &
(∀X V1 V2 Y V3 V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,X,V3,V4),inner_pasch(V1::'a,V2,Y,V3,V4))) &
(∀X V1 V2 V3 Y V4. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,X,V4),inner_pasch(V1::'a,V2,V3,Y,V4))) &
(∀X V1 V2 V3 V4 Y. equal(X::'a,Y) --> equal(inner_pasch(V1::'a,V2,V3,V4,X),inner_pasch(V1::'a,V2,V3,V4,Y))) &
(∀A B C D E F'. equal(A::'a,B) --> equal(euclid1(A::'a,C,D,E,F'),euclid1(B::'a,C,D,E,F'))) &
(∀G I' H J K' L. equal(G::'a,H) --> equal(euclid1(I'::'a,G,J,K',L),euclid1(I'::'a,H,J,K',L))) &
(∀M O' P N Q R. equal(M::'a,N) --> equal(euclid1(O'::'a,P,M,Q,R),euclid1(O'::'a,P,N,Q,R))) &
(∀S' U V W T' X. equal(S'::'a,T') --> equal(euclid1(U::'a,V,W,S',X),euclid1(U::'a,V,W,T',X))) &
(∀Y A1 B1 C1 D1 Z. equal(Y::'a,Z) --> equal(euclid1(A1::'a,B1,C1,D1,Y),euclid1(A1::'a,B1,C1,D1,Z))) &
(∀E1 F1 G1 H1 I1 J1. equal(E1::'a,F1) --> equal(euclid2(E1::'a,G1,H1,I1,J1),euclid2(F1::'a,G1,H1,I1,J1))) &
(∀K1 M1 L1 N1 O1 P1. equal(K1::'a,L1) --> equal(euclid2(M1::'a,K1,N1,O1,P1),euclid2(M1::'a,L1,N1,O1,P1))) &
(∀Q1 S1 T1 R1 U1 V1. equal(Q1::'a,R1) --> equal(euclid2(S1::'a,T1,Q1,U1,V1),euclid2(S1::'a,T1,R1,U1,V1))) &
(∀W1 Y1 Z1 A2 X1 B2. equal(W1::'a,X1) --> equal(euclid2(Y1::'a,Z1,A2,W1,B2),euclid2(Y1::'a,Z1,A2,X1,B2))) &
(∀C2 E2 F2 G2 H2 D2. equal(C2::'a,D2) --> equal(euclid2(E2::'a,F2,G2,H2,C2),euclid2(E2::'a,F2,G2,H2,D2))) &
(∀X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
(∀X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
(∀X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
(∀X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
(∀X Y V1 V2 V3 V4 V5. equal(X::'a,Y) --> equal(continuous(X::'a,V1,V2,V3,V4,V5),continuous(Y::'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),continuous(V1::'a,Y,V2,V3,V4,V5))) &
(∀X V1 V2 Y V3 V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,X,V3,V4,V5),continuous(V1::'a,V2,Y,V3,V4,V5))) &
(∀X V1 V2 V3 Y V4 V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,X,V4,V5),continuous(V1::'a,V2,V3,Y,V4,V5))) &
(∀X V1 V2 V3 V4 Y V5. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,X,V5),continuous(V1::'a,V2,V3,V4,Y,V5))) &
(∀X V1 V2 V3 V4 V5 Y. equal(X::'a,Y) --> equal(continuous(V1::'a,V2,V3,V4,V5,X),continuous(V1::'a,V2,V3,V4,V5,Y)))"


(*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"

oops

(*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))) &
(∀A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
(∀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) | equal(W::'a,X)) &
(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))) &
(∀A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
(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)) &
(∀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"

oops

(*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)) &
(∀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)) &
(∀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)) &
(∀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)) &
(∀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)) &
(∀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)) &
(∀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"

oops

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)),greatest_lower_bound(greatest_lower_bound(X::'a,Y),Z))) &
(∀X Y Z. equal(least_upper_bound(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(least_upper_bound(X::'a,Y),Z))) &
(∀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(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(∀Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(∀Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
(∀Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
(∀A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &
(∀A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &
(∀A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &
(∀A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &
(∀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,c))) --> False"

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(INVERSE(c),multiply(b::'a,c))),multiply(INVERSE(c),multiply(b::'a,c)))) --> False"

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,Y),Z))) &
(∀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"

oops

(*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::'a,Z,V5) --> mless_equal(V4::'a,V5)) &
(∀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,Y)) &
(mless_equal(x::'a,y)) &
(quotient(z::'a,y,zQy)) &
(quotient(z::'a,x,zQx)) &
(~mless_equal(zQy::'a,zQx)) --> False"

oops

(*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)) &
(∀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"

oops

(*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::'a,Y),Z)),Zero)) &
(∀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),Zero)) &
(∀Y Z X. equal(Divide(X::'a,Y),Zero) --> equal(Divide(Divide(Z::'a,Y),Divide(Z::'a,X)),Zero)) &
(~equal(Divide(identity::'a,a),Divide(identity::'a,Divide(identity::'a,Divide(identity::'a,a))))) &
(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"

oops


(*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,Z),equivalent(Z::'a,Y))))) &
(~is_a_theorem(equivalent(equivalent(a::'a,b),equivalent(equivalent(c::'a,b),equivalent(a::'a,c))))) --> False"

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),implies(X,Z))))) &
(∀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::'a,X),implies(U::'a,X))))) &
(~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)))) &
(∀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))))) --> False"

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)) &
(∀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"

oops


(*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(Situation))) &
(∀Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) &
(∀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,let_go(Situation))) &
(∀Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) &
(∀Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) &
(∀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(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) &
(∀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_parts(Big_part::'a,Number_of_mid_parts,Mid_part) --> in'(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Mid_part) | has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
(∀Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) & has_parts(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Number_of_small_parts,Small_part) --> has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
(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_parts(Big_part::'a,Number_of_mid_parts,Mid_part) --> in'(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Mid_part) | has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
(∀Big_part Mid_part Number_of_mid_parts Number_of_small_parts Small_part. has_parts(Big_part::'a,Number_of_mid_parts,Mid_part) & has_parts(object_in(Big_part::'a,Mid_part,Small_part,Number_of_mid_parts,Number_of_small_parts),Number_of_small_parts,Small_part) --> has_parts(Big_part::'a,mtimes(Number_of_mid_parts::'a,Number_of_small_parts),Small_part)) &
(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),Value)) --> False"

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,A)),A),B))"


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"

oops

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,Y)) &
(∀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))),ordered_pair(X::'a,Y))) &
(∀V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) &
(∀U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) &
(∀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),second(Z)),Z)) &
(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_class)) & member(X::'a,Y) --> member(ordered_pair(X::'a,Y),element_relation)) &
(∀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::'a,X)) &
(∀X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) &
(∀X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) &
(∀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))) &
(∀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),universal_class),null_class) | member(Z::'a,domain_of(X))) &
(∀X. subclass(rot(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
(∀V W U X. member(ordered_pair(ordered_pair(U::'a,V),W),rot(X)) --> member(ordered_pair(ordered_pair(V::'a,W),U),X)) &
(∀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_class),universal_class))) &
(∀V U W X. member(ordered_pair(ordered_pair(U::'a,V),W),flip(X)) --> member(ordered_pair(ordered_pair(V::'a,U),W),X)) &
(∀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))) &
(∀Z. equal(domain_of(INVERSE(Z)),range_of(Z))) &
(∀Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) &
(∀Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) &
(∀Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image'(Xr::'a,X))) &
(∀X. equal(union(X::'a,singleton(X)),successor(X))) &
(subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) &
(∀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))) &
(∀X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &
(∀X. equal(complement(image'(element_relation::'a,complement(X))),powerClass(X))) &
(∀U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &
(∀Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) &
(∀Z Yr Xr Y. member(ordered_pair(Y::'a,Z),compos(Yr::'a,Xr)) --> member(Z::'a,image'(Yr::'a,image'(Xr::'a,singleton(Y))))) &
(∀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_relation)) &
(∀X. subclass(compos(X::'a,INVERSE(X)),identity_relation) --> single_valued_class(X)) &
(∀Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) &
(∀Xf. function(Xf) --> subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation)) &
(∀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(choice::'a,Y),Y)) &
(∀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),intersection(cross_product(universal_class::'a,universal_class),complement(compos(complement(element_relation),INVERSE(element_relation))))),subset_relation)) &
(equal(intersection(INVERSE(subset_relation),subset_relation),identity_relation)) &
(∀Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) &
(∀X. equal(intersection(domain_of(X),diagonalise(compos(INVERSE(element_relation),X))),cantor(X))) &
(∀Xf. operation(Xf) --> function(Xf)) &
(∀Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) &
(∀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))) --> operation(Xf)) &
(∀Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) &
(∀Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) &
(∀Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) &
(∀Xh Xh1 Xf1 Xf2. function(Xh) & equal(domain_of(domain_of(Xf1)),domain_of(Xh)) & subclass(range_of(Xh),domain_of(domain_of(Xf2))) --> compatible(Xh1::'a,Xf1,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(not_homomorphism1(Xh::'a,Xf1,Xf2),not_homomorphism2(Xh::'a,Xf1,Xf2)),domain_of(Xf1)) | homomorphism(Xh::'a,Xf1,Xf2)) &
(∀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,V))) &
(∀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,D1))) &
(∀E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) &
(∀H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) &
(∀L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) &
(∀P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) &
(∀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))) &
(∀C2 E2 D2. equal(C2::'a,D2) --> equal(image'(E2::'a,C2),image'(E2::'a,D2))) &
(∀F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) &
(∀I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) &
(∀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_homomorphism1(O2::'a,P2,Q2))) &
(∀R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) &
(∀V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) &
(∀Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) &
(∀D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) &
(∀H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) &
(∀L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) &
(∀O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) &
(∀R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) &
(∀U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) &
(∀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,R4,S4))) &
(∀T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) &
(∀X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) &
(∀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(S5::'a,T5))) &
(∀U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) &
(∀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),Z)) &
(∀Y Z X. member(ordered_pair(Y::'a,Z),cross_product(universal_class::'a,universal_class)) & equal(compos(X::'a,Y),Z) --> member(ordered_pair(Y::'a,Z),compose_class(X))) &
(subclass(composition_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
(∀X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) &
(∀X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) --> member(ordered_pair(X::'a,ordered_pair(Y::'a,compos(X::'a,Y))),composition_function)) &
(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)),domain_relation)) &
(∀X. equal(first(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued1(X))) &
(∀X. equal(second(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued2(X))) &
(∀X. equal(domain(X::'a,image'(INVERSE(X),singleton(single_valued1(X))),single_valued2(X)),single_valued3(X))) &
(equal(intersection(complement(compos(element_relation::'a,complement(identity_relation))),element_relation),singleton_relation)) &
(subclass(application_function::'a,cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class)))) &
(∀Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) &
(∀X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) &
(∀Z X Y. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),cross_product(universal_class::'a,cross_product(universal_class::'a,universal_class))) & member(Y::'a,domain_of(X)) --> member(ordered_pair(X::'a,ordered_pair(Y::'a,apply(X::'a,Y))),application_function)) &
(∀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 compose_class equal ≡
(∀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))) &
(∀P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &
(∀R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &
(∀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_relation))) &
(∀X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) &
(∀Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) &
(∀X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) &
(∀Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) &
(∀Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) &
(∀Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class)) &
(∀Xr Y. equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) &
(∀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),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_ordering(Xr::'a,Y)) &
(∀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) --> well_ordering(Xr::'a,Y)) &
(∀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)) &
(∀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)) &
(∀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) --> member(X::'a,ordinal_numbers)) &
(∀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)),kind_1_ordinals)) &
(equal(intersection(complement(kind_1_ordinals),ordinal_numbers),limit_ordinals)) &
(∀X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) &
(∀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,universal_class),V)) &
(∀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)) &
(∀X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) &
(∀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),ordinal_numbers)) &
(∀Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) &
(∀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_class))) &
(∀X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) &
(∀X Y. member(ordered_pair(X::'a,Y),cross_product(universal_class::'a,universal_class)) & equal(sum_class(range_of(X)),Y) --> member(ordered_pair(X::'a,Y),union_of_range_map)) &
(∀X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) &
(∀X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) &
(∀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::'a,N))) &
(∀O' Q P. equal(O'::'a,P) --> equal(not_well_ordering(Q::'a,O'),not_well_ordering(Q::'a,P))) &
(∀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))) &
(∀X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) &
(∀A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) &
(∀F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) &
(∀J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) &
(∀N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) &
(∀R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) &
(∀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,X1,Y1))) &
(∀Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) &
(∀D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) &
(∀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 equal &
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 equal &
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 and' holds ≡
(∀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(pickup(X),State))) &
(∀Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(EMPTY::'a,State) --> holds(clear(Y),do(pickup(X),State))) &
(∀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,W),State))) &
(∀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 &
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 &
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 &
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 &
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),a(Y))) &
(∀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),a(Y))) &
(~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_truth(liar(Liar))) &
(∀Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) &
(∀Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a,Y))) &
(∀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"

oops


(*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_identity sum ≡
(∀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)) &
(∀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)) &
(∀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"

oops

abbreviation "RNG_other_ax multiply add equal product additive_identity additive_inverse sum ≡
(∀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)) &
(∀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)) &
(∀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 &
(~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'))) &
(∀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',H))) &
(∀I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a,L,J))) &
(∀M N O'. equal(M::'a,N) --> equal(commutator(M::'a,O'),commutator(N::'a,O'))) &
(∀P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) &
(∀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_inverse(Y)))) &
(∀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,Y)))) &
(∀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,Z)))) &
(∀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)))) &
(∀X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
(∀X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
(∀X Y. equal(multiply(multiply(associator(X::'a,X,Y),X),associator(X::'a,X,Y)),additive_identity)) &
(~equal(multiply(multiply(associator(a::'a,a,b),a),associator(a::'a,a,b)),additive_identity)) --> False"

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,Z)))) &
(∀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)))) &
(∀X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
(∀X Y Z. equal(associator(X::'a,Y,Z),add(multiply(multiply(X::'a,Y),Z),additive_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
(∀X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(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'))) &
(∀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,U))) &
(∀X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) &
(∀A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B1))) &
(∀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_inverse(Y)))) &
(∀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,Z)))) &
(∀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)))) &
(∀X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
(∀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,Y)))) &
(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'))) &
(∀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_inverse(multiply(X::'a,multiply(Y::'a,Z)))))) &
(∀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,U))) &
(∀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,multiply(cy::'a,multiply(cx::'a,cz))))) --> False"

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 &
(∀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)) &
(∀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)) &
(∀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)) &
(∀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 &
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)) &
(product(a::'a,b,additive_identity)) &
(~equal(a::'a,additive_identity)) &
(~equal(b::'a,additive_identity)) --> False"

oops

(*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"

oops


(*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(V::'a,X)))) &
(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))))))),negate(e))) --> False"

oops

(*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)) --> False"

oops

(*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(Subset::'a,Superset),Subset)) &
(∀Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) &
(∀Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) &
(∀Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) &
(∀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) & member(Element::'a,Set1) --> member(Element::'a,Intersection)) &
(∀Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set1)) &
(∀Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set2)) &
(∀Set1 Set2 Intersection. member(h(Set1::'a,Set2,Intersection),Intersection) & member(h(Set1::'a,Set2,Intersection),Set2) & member(h(Set1::'a,Set2,Intersection),Set1) --> intersection(Set1::'a,Set2,Intersection)) &
(intersection(a::'a,b,aIb)) &
(intersection(b::'a,c,bIc)) &
(intersection(a::'a,bIc,aIbIc)) &
(~intersection(aIb::'a,c,aIbIc)) --> False"

oops


(*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(Subset::'a,Superset),Subset)) &
(∀Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> ssubset(Subset::'a,Superset)) &
(∀Subset Superset. equal_sets(Subset::'a,Superset) --> ssubset(Subset::'a,Superset)) &
(∀Subset Superset. equal_sets(Superset::'a,Subset) --> ssubset(Subset::'a,Superset)) &
(∀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,Set2))) &
(∀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,Set2,Difference),Set1) | member(k(Set1::'a,Set2,Difference),Difference)) &
(∀Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Set1::'a,Set2,Difference),Difference) | difference(Set1::'a,Set2,Difference)) &
(∀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::'a,Y)) &
(∀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_pair(X::'a,Y)))) &
(∀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_predicate(X)) &
(∀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,X)))) &
(∀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,X)))) &
(∀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)) &
(∀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)) &
(∀Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::'a,inv1 X)) &
(∀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),ordered_pair(f10(Z::'a,X),f11(Z::'a,X))))) &
(∀Z X. member(Z::'a,rot_right(X)) --> member(ordered_pair(f10(Z::'a,X),ordered_pair(f11(Z::'a,X),f9(Z::'a,X))),X)) &
(∀Z V W U X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordered_pair(V::'a,W))) & member(ordered_pair(V::'a,ordered_pair(W::'a,U)),X) --> member(Z::'a,rot_right(X))) &
(∀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),ordered_pair(f13(Z::'a,X),f14(Z::'a,X))))) &
(∀Z X. member(Z::'a,flip_range_of(X)) --> member(ordered_pair(f12(Z::'a,X),ordered_pair(f14(Z::'a,X),f13(Z::'a,X))),X)) &
(∀Z U W V X. little_set(Z) & little_set(U) & little_set(V) & little_set(W) & equal(Z::'a,ordered_pair(U::'a,ordered_pair(V::'a,W))) & member(ordered_pair(U::'a,ordered_pair(W::'a,V)),X) --> member(Z::'a,flip_range_of(X))) &
(∀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,Xf))) &
(∀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)) &
(∀X Xf Z. member(Z::'a,image'(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) &
(∀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,image'(X::'a,Xf))) &
(∀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)),f25)) &
(∀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_set)))) &
(∀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,Y))) &
(∀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)) &
(∀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::'a,Y)))) &
(∀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::'a,Xf)) &
(∀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::'a,Xf,Xg),f30(Z::'a,Xf,Xg)))) &
(∀Z Xg Xf. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f29(Z::'a,Xf,Xg),f31(Z::'a,Xf,Xg)),Xf)) &
(∀Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f31(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)),Xg)) &
(∀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)) & member(ordered_pair(X::'a,W),Xf) & member(ordered_pair(W::'a,Y),Xg) --> member(Z::'a,composition(Xf::'a,Xg))) &
(∀Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf1)) &
(∀Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf2)) &
(∀Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,Xs2)) &
(∀Xs2 Xs1 Xf1 Xf2 X Xh Y. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) & member(X::'a,Xs1) & member(Y::'a,Xs1) --> equal(apply(Xh::'a,apply_to_two_arguments(Xf1::'a,X,Y)),apply_to_two_arguments(Xf2::'a,apply(Xh::'a,X),apply(Xh::'a,Y)))) &
(∀Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) | member(f32(Xh::'a,Xs1,Xf1,Xs2,Xf2),Xs1)) &
(∀Xh Xf1 Xs2 Xf2 Xs1. closed(Xs1::'a,Xf1) & closed(Xs2::'a,Xf2) & maps(Xh::'a,Xs1,Xs2) --> homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) | member(f33(Xh::'a,Xs1,Xf1,Xs2,Xf2),Xs1)) &
(∀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::'a,I5,J5,K5,L5))) &
(∀M5 O5 N5 P5 Q5 R5. equal(M5::'a,N5) --> equal(f32(O5::'a,M5,P5,Q5,R5),f32(O5::'a,N5,P5,Q5,R5))) &
(∀S5 U5 V5 T5 W5 X5. equal(S5::'a,T5) --> equal(f32(U5::'a,V5,S5,W5,X5),f32(U5::'a,V5,T5,W5,X5))) &
(∀Y5 A6 B6 C6 Z5 D6. equal(Y5::'a,Z5) --> equal(f32(A6::'a,B6,C6,Y5,D6),f32(A6::'a,B6,C6,Z5,D6))) &
(∀E6 G6 H6 I6 J6 F6. equal(E6::'a,F6) --> equal(f32(G6::'a,H6,I6,J6,E6),f32(G6::'a,H6,I6,J6,F6))) &
(∀K6 L6 M6 N6 O6 P6. equal(K6::'a,L6) --> equal(f33(K6::'a,M6,N6,O6,P6),f33(L6::'a,M6,N6,O6,P6))) &
(∀Q6 S6 R6 T6 U6 V6. equal(Q6::'a,R6) --> equal(f33(S6::'a,Q6,T6,U6,V6),f33(S6::'a,R6,T6,U6,V6))) &
(∀W6 Y6 Z6 X6 A7 B7. equal(W6::'a,X6) --> equal(f33(Y6::'a,Z6,W6,A7,B7),f33(Y6::'a,Z6,X6,A7,B7))) &
(∀C7 E7 F7 G7 D7 H7. equal(C7::'a,D7) --> equal(f33(E7::'a,F7,G7,C7,H7),f33(E7::'a,F7,G7,D7,H7))) &
(∀I7 K7 L7 M7 N7 J7. equal(I7::'a,J7) --> equal(f33(K7::'a,L7,M7,N7,I7),f33(K7::'a,L7,M7,N7,J7))) &
(∀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_two_arguments(H::'a,I',J))) &
(∀K' M L N. equal(K'::'a,L) --> equal(apply_to_two_arguments(M::'a,K',N),apply_to_two_arguments(M::'a,L,N))) &
(∀O' Q R P. equal(O'::'a,P) --> equal(apply_to_two_arguments(Q::'a,R,O'),apply_to_two_arguments(Q::'a,R,P))) &
(∀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))) &
(∀X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) &
(∀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::'a,E1))) &
(∀F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1::'a,G1))) &
(∀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,U10))) &
(∀V10 X10 W10. equal(V10::'a,W10) --> equal(image'(X10::'a,V10),image'(X10::'a,W10))) &
(∀Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersection(Z10::'a,A11))) &
(∀B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersection(D11::'a,C11))) &
(∀E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_ordered_pair(F11::'a,G11))) &
(∀H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_ordered_pair(J11::'a,I11))) &
(∀K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pair(L11::'a,M11))) &
(∀N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pair(P11::'a,O11))) &
(∀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,W11))) &
(∀X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a,Y11))) &
(∀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),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),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))) &
(∀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(Vf::'a,U))) &
(∀U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
(∀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::'a,U),U)) &
(∀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(Vf::'a,U))) &
(∀U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
(∀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_collection(f2(Vf::'a,U),Vf)) &
(∀Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection_of_members(Vf))) &
(∀Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) &
(∀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) --> element_of_collection(intersection_of_sets(Y::'a,Z),Vt)) &
(∀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) --> topological_space(X::'a,Vt) | element_of_collection(f3(X::'a,Vt),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(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) --> topological_space(X::'a,Vt) | element_of_collection(f4(X::'a,Vt),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(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) & element_of_collection(union_of_members(f5(X::'a,Vt)),Vt) --> topological_space(X::'a,Vt)) &
(∀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)) &
(∀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(Vt::'a,Vs,X)) &
(∀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_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_set(Y::'a,f6(X::'a,Vf,Y,Vb1,Vb2))) &
(∀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_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_collection(f6(X::'a,Vf,Y,Vb1,Vb2),Vf)) &
(∀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_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> subset_sets(f6(X::'a,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1::'a,Vb2))) &
(∀Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),X)) &
(∀X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f8(X::'a,Vf),Vf)) &
(∀X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f9(X::'a,Vf),Vf)) &
(∀X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),intersection_of_sets(f8(X::'a,Vf),f9(X::'a,Vf)))) &
(∀Uu9 X Vf. equal_sets(union_of_members(Vf),X) & element_of_set(f7(X::'a,Vf),Uu9) & element_of_collection(Uu9::'a,Vf) & subset_sets(Uu9::'a,intersection_of_sets(f8(X::'a,Vf),f9(X::'a,Vf))) --> basis(X::'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::'a,U),U)) &
(∀Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
(∀U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topological_space(X::'a,Vt)) &
(∀U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subset_sets(Y::'a,X)) &
(∀X Y U Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> element_of_collection(f12(X::'a,Vt,Y,U),Vt)) &
(∀X Vt Y U. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> equal_sets(U::'a,intersection_of_sets(Y::'a,f12(X::'a,Vt,Y,U)))) &
(∀X Vt U Y Uu12. topological_space(X::'a,Vt) & subset_sets(Y::'a,X) & element_of_collection(Uu12::'a,Vt) & equal_sets(U::'a,intersection_of_sets(Y::'a,Uu12)) --> element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y))) &
(∀U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
(∀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,f13(Y::'a,X,Vt,U))) &
(∀X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a,X,Vt,U),Y)) &
(∀Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U),X,Vt)) &
(∀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(Uu13::'a,X,Vt) --> element_of_set(U::'a,interior(Y::'a,X,Vt))) &
(∀U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
(∀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) --> element_of_set(U::'a,V)) &
(∀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)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
(∀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) --> neighborhood(U::'a,Y,X,Vt)) &
(∀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,intersection_of_sets(f16(Z::'a,Y,X,Vt),Y)) --> limit_point(Z::'a,Y,X,Vt) | eq_p(Uu16::'a,Z)) &
(∀U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
(∀U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
(∀U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(relative_complement_sets(Y::'a,X),X,Vt))) &
(∀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_complement_sets(Y::'a,X),X,Vt)) --> element_of_set(U::'a,boundary(Y::'a,X,Vt))) &
(∀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_2) | neighborhood(f17(X::'a,Vt,X_1,X_2),X_1,X,Vt)) &
(∀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_2) | neighborhood(f18(X::'a,Vt,X_1,X_2),X_2,X,Vt)) &
(∀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,X_2) | disjoint_s(f17(X::'a,Vt,X_1,X_2),f18(X::'a,Vt,X_1,X_2))) &
(∀Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19(X::'a,Vt),X)) &
(∀Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20(X::'a,Vt),X)) &
(∀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,Vt),X,Vt) & disjoint_s(Uu19::'a,Uu20) --> hausdorff(X::'a,Vt)) &
(∀Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) &
(∀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,Vt)) &
(∀Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,Vt)) &
(∀Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::'a,Va2),X)) &
(∀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) & disjoint_s(Va1::'a,Va2) --> separation(Va1::'a,Va2,X,Vt) | equal_sets(Va1::'a,empty_set) | equal_sets(Va2::'a,empty_set)) &
(∀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(f21(X::'a,Vt),f22(X::'a,Vt),X,Vt)) &
(∀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_topology(X::'a,Vt,Va))) &
(∀X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & connected_space(Va::'a,subspace_topology(X::'a,Vt,Va)) --> connected_set(Va::'a,X,Vt)) &
(∀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)) &
(∀Vt Vf X. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) & equal_sets(union_of_members(Vf),X) --> open_covering(Vf::'a,X,Vt)) &
(∀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(f24(X::'a,Vt),X,Vt)) &
(∀Uu24 X Vt. topological_space(X::'a,Vt) & finite'(Uu24) & subset_collections(Uu24::'a,f24(X::'a,Vt)) & open_covering(Uu24::'a,X,Vt) --> compact_space(X::'a,Vt)) &
(∀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::'a,Vt,Va))) &
(∀X Vt Va. topological_space(X::'a,Vt) & subset_sets(Va::'a,X) & compact_space(Va::'a,subspace_topology(X::'a,Vt,Va)) --> compact_set(Va::'a,X,Vt)) &
(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))) --> False"

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_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_set(Y::'a,f6(X::'a,Vf,Y,Vb1,Vb2))) &
(∀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_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> element_of_collection(f6(X::'a,Vf,Y,Vb1,Vb2),Vf)) &
(∀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_set(Y::'a,intersection_of_sets(Vb1::'a,Vb2)) --> subset_sets(f6(X::'a,Vf,Y,Vb1,Vb2),intersection_of_sets(Vb1::'a,Vb2))) &
(∀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::'a,U),U)) &
(∀Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
(∀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(Z::'a,X)) &
(∀X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,Y)) &
(∀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))) --> False"

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(Vf::'a,U))) &
(∀U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),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::'a,U),U)) &
(∀Vf Uu11 U. element_of_set(f11(Vf::'a,U),Uu11) & element_of_collection(Uu11::'a,Vf) & subset_sets(Uu11::'a,U) --> element_of_collection(U::'a,top_of_basis(Vf))) &
(∀X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a,Y)) &
(∀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"

oops

end