no longer declare .psimps rules as [simp].
This regularly caused confusion (e.g., they show up in simp traces
when the regular simp rules are disabled). In the few places where the
rules are used, explicitly mentioning them actually clarifies the
proof text.
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 sum quotient
text {*
Test data for the MESON proof procedure
(Excludes the equality problems 51, 52, 56, 58)
*}
subsection {* Interactive examples *}
lemma problem_25:
"(\<exists>x. P x) & (\<forall>x. L x --> ~ (M x & R x)) & (\<forall>x. P x --> (M x & L x)) & ((\<forall>x. P x --> Q x) | (\<exists>x. P x & R x)) --> (\<exists>x. Q x & P x)"
apply (rule ccontr)
ML_prf {*
val prem25 = Thm.assume @{cprop "\<not> ?thesis"};
val nnf25 = Meson.make_nnf @{context} prem25;
val xsko25 = Meson.skolemize @{context} nnf25;
*}
apply (tactic {* cut_facts_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 [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:
"((\<exists>x. p x) = (\<exists>x. q x)) & (\<forall>x. \<forall>y. p x & q y --> (r x = s y)) --> ((\<forall>x. p x --> r x) = (\<forall>x. q x --> s x))"
apply (rule ccontr)
ML_prf {*
val prem26 = Thm.assume @{cprop "\<not> ?thesis"}
val nnf26 = Meson.make_nnf @{context} prem26;
val xsko26 = Meson.skolemize @{context} nnf26;
*}
apply (tactic {* cut_facts_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 [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*)
"(\<forall>x. \<forall>y. q x y = (\<forall>z. p z x = (p z y::bool))) --> (\<forall>x. (\<forall>y. q x y = (q y x::bool)))"
apply (rule ccontr)
ML_prf {*
val prem43 = Thm.assume @{cprop "\<not> ?thesis"};
val nnf43 = Meson.make_nnf @{context} prem43;
val xsko43 = Meson.skolemize @{context} nnf43;
*}
apply (tactic {* cut_facts_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 [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 \<equiv> (\<forall>X. equal(X::'a,X)) &
(\<forall>Y X. equal(X::'a,Y) --> equal(Y::'a,X)) &
(\<forall>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 \<equiv>
(\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
(\<forall>Y X Z. product(X::'a,Y,Z) --> product(Y::'a,X,Z)) &
(\<forall>X. sum(additive_identity::'a,X,X)) &
(\<forall>X. sum(X::'a,additive_identity,X)) &
(\<forall>X. product(multiplicative_identity::'a,X,X)) &
(\<forall>X. product(X::'a,multiplicative_identity,X)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X. sum(INVERSE(X),X,multiplicative_identity)) &
(\<forall>X. sum(X::'a,INVERSE(X),multiplicative_identity)) &
(\<forall>X. product(INVERSE(X),X,additive_identity)) &
(\<forall>X. product(X::'a,INVERSE(X),additive_identity)) &
(\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
(\<forall>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 \<equiv>
(\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
(\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & product(W::'a,Z,X) --> product(W::'a,Z,Y)) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
(\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
(\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
(\<forall>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 \<equiv>
(\<forall>Y X. equivalent(X::'a,Y) --> there_exists(X)) &
(\<forall>X Y. equivalent(X::'a,Y) --> equal(X::'a,Y)) &
(\<forall>X Y. there_exists(X) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
(\<forall>X. there_exists(domain(X)) --> there_exists(X)) &
(\<forall>X. there_exists(codomain(X)) --> there_exists(X)) &
(\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(domain(X))) &
(\<forall>X Y. there_exists(compos(X::'a,Y)) --> equal(domain(X),codomain(Y))) &
(\<forall>X Y. there_exists(domain(X)) & equal(domain(X),codomain(Y)) --> there_exists(compos(X::'a,Y))) &
(\<forall>X Y Z. equal(compos(X::'a,compos(Y::'a,Z)),compos(compos(X::'a,Y),Z))) &
(\<forall>X. equal(compos(X::'a,domain(X)),X)) &
(\<forall>X. equal(compos(codomain(X),X),X)) &
(\<forall>X Y. equivalent(X::'a,Y) --> there_exists(Y)) &
(\<forall>X Y. there_exists(X) & there_exists(Y) & equal(X::'a,Y) --> equivalent(X::'a,Y)) &
(\<forall>Y X. there_exists(compos(X::'a,Y)) --> there_exists(codomain(X))) &
(\<forall>X Y. there_exists(f1(X::'a,Y)) | equal(X::'a,Y)) &
(\<forall>X Y. equal(X::'a,f1(X::'a,Y)) | equal(Y::'a,f1(X::'a,Y)) | equal(X::'a,Y)) &
(\<forall>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 \<equiv>
(\<forall>X Y. equal(X::'a,Y) & there_exists(X) --> there_exists(Y)) &
(\<forall>X Y Z. equal(X::'a,Y) & equivalent(X::'a,Z) --> equivalent(Y::'a,Z)) &
(\<forall>X Z Y. equal(X::'a,Y) & equivalent(Z::'a,X) --> equivalent(Z::'a,Y)) &
(\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
(\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
(\<forall>X Y Z. equal(X::'a,Y) --> equal(compos(X::'a,Z),compos(Y::'a,Z))) &
(\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
(\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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 \<equiv>
(\<forall>X Y. defined(X::'a,Y) --> product(X::'a,Y,compos(X::'a,Y))) &
(\<forall>Z X Y. product(X::'a,Y,Z) --> defined(X::'a,Y)) &
(\<forall>X Xy Y Z. product(X::'a,Y,Xy) & defined(Xy::'a,Z) --> defined(Y::'a,Z)) &
(\<forall>Y Xy Z X Yz. product(X::'a,Y,Xy) & product(Y::'a,Z,Yz) & defined(Xy::'a,Z) --> defined(X::'a,Yz)) &
(\<forall>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)) &
(\<forall>Z Yz X Y. product(Y::'a,Z,Yz) & defined(X::'a,Yz) --> defined(X::'a,Y)) &
(\<forall>Y X Yz Xy Z. product(Y::'a,Z,Yz) & product(X::'a,Y,Xy) & defined(X::'a,Yz) --> defined(Xy::'a,Z)) &
(\<forall>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)) &
(\<forall>Y X Z. defined(X::'a,Y) & defined(Y::'a,Z) & identity_map(Y) --> defined(X::'a,Z)) &
(\<forall>X. identity_map(domain(X))) &
(\<forall>X. identity_map(codomain(X))) &
(\<forall>X. defined(X::'a,domain(X))) &
(\<forall>X. defined(codomain(X),X)) &
(\<forall>X. product(X::'a,domain(X),X)) &
(\<forall>X. product(codomain(X),X,X)) &
(\<forall>X Y. defined(X::'a,Y) & identity_map(X) --> product(X::'a,Y,Y)) &
(\<forall>Y X. defined(X::'a,Y) & identity_map(Y) --> product(X::'a,Y,X)) &
(\<forall>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 \<equiv>
(\<forall>X Y Z W. equal(X::'a,Y) & product(X::'a,Z,W) --> product(Y::'a,Z,W)) &
(\<forall>X Z Y W. equal(X::'a,Y) & product(Z::'a,X,W) --> product(Z::'a,Y,W)) &
(\<forall>X Z W Y. equal(X::'a,Y) & product(Z::'a,W,X) --> product(Z::'a,W,Y)) &
(\<forall>X Y. equal(X::'a,Y) --> equal(domain(X),domain(Y))) &
(\<forall>X Y. equal(X::'a,Y) --> equal(codomain(X),codomain(Y))) &
(\<forall>X Y. equal(X::'a,Y) & identity_map(X) --> identity_map(Y)) &
(\<forall>X Y Z. equal(X::'a,Y) & defined(X::'a,Z) --> defined(Y::'a,Z)) &
(\<forall>X Z Y. equal(X::'a,Y) & defined(Z::'a,X) --> defined(Z::'a,Y)) &
(\<forall>X Z Y. equal(X::'a,Y) --> equal(compos(Z::'a,X),compos(Z::'a,Y))) &
(\<forall>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 &
(\<forall>X Y Z. equal(apply(apply(apply(s::'a,X),Y),Z),apply(apply(X::'a,Z),apply(Y::'a,Z)))) &
(\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
(\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
(\<forall>X. equal(apply(i::'a,X),X)) &
(\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
(\<forall>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))))) &
(\<forall>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 &
(\<forall>X Y Z. equal(apply(apply(apply(b::'a,X),Y),Z),apply(X::'a,apply(Y::'a,Z)))) &
(\<forall>X Y Z. equal(apply(apply(apply(n::'a,X),Y),Z),apply(apply(apply(X::'a,Z),Y),Z))) &
(\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
(\<forall>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 &
(\<forall>X. equal(apply(m::'a,X),apply(X::'a,X))) &
(\<forall>Y X Z. equal(apply(apply(apply(q::'a,X),Y),Z),apply(Y::'a,apply(X::'a,Z)))) &
(\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
(\<forall>G H. equal(G::'a,H) --> equal(f(G),f(H))) &
(\<forall>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 &
(\<forall>X Y W. equal(response(compos(X::'a,Y),W),response(X::'a,response(Y::'a,W)))) &
(\<forall>X Y. agreeable(X) --> equal(response(X::'a,common_bird(Y)),response(Y::'a,common_bird(Y)))) &
(\<forall>Z X. equal(response(X::'a,Z),response(compatible(X),Z)) --> agreeable(X)) &
(\<forall>A B. equal(A::'a,B) --> equal(common_bird(A),common_bird(B))) &
(\<forall>C D. equal(C::'a,D) --> equal(compatible(C),compatible(D))) &
(\<forall>Q R. equal(Q::'a,R) & agreeable(Q) --> agreeable(R)) &
(\<forall>A B C. equal(A::'a,B) --> equal(compos(A::'a,C),compos(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(compos(F'::'a,D),compos(F'::'a,E))) &
(\<forall>G H I'. equal(G::'a,H) --> equal(response(G::'a,I'),response(H::'a,I'))) &
(\<forall>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 &
(\<forall>Y X. equal(apply(apply(k::'a,X),Y),X)) &
(\<forall>X Y Z. equal(apply(apply(apply(abstraction::'a,X),Y),Z),apply(apply(X::'a,apply(k::'a,Z)),apply(Y::'a,Z)))) &
(\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
(\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
(\<forall>A B. equal(A::'a,B) --> equal(b(A),b(B))) &
(\<forall>C D. equal(C::'a,D) --> equal(c(C),c(D))) &
(\<forall>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:
"(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
(\<forall>Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
(\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
(\<forall>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:
"(\<forall>Goal_state Start_state. follows(Goal_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
(\<forall>Goal_state Intermediate_state Start_state. succeeds(Goal_state::'a,Intermediate_state) & succeeds(Intermediate_state::'a,Start_state) --> succeeds(Goal_state::'a,Start_state)) &
(\<forall>Start_state Label Goal_state. has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state) --> succeeds(Goal_state::'a,Start_state)) &
(\<forall>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:
"(\<forall>Goal_state Start_state. ~(fails(Goal_state::'a,Start_state) & follows(Goal_state::'a,Start_state))) &
(\<forall>Goal_state Intermediate_state Start_state. fails(Goal_state::'a,Start_state) --> fails(Goal_state::'a,Intermediate_state) | fails(Intermediate_state::'a,Start_state)) &
(\<forall>Start_state Label Goal_state. ~(fails(Goal_state::'a,Start_state) & has(Start_state::'a,goto(Label)) & labels(Label::'a,Goal_state))) &
(\<forall>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:
"(\<forall>X Y Z. program_decides(X) & program(Y) --> decides(X::'a,Y,Z)) &
(\<forall>X. program_decides(X) | program(f2(X))) &
(\<forall>X. decides(X::'a,f2(X),f1(X)) --> program_decides(X)) &
(\<forall>X. program_program_decides(X) --> program(X)) &
(\<forall>X. program_program_decides(X) --> program_decides(X)) &
(\<forall>X. program(X) & program_decides(X) --> program_program_decides(X)) &
(\<forall>X. algorithm_program_decides(X) --> algorithm(X)) &
(\<forall>X. algorithm_program_decides(X) --> program_decides(X)) &
(\<forall>X. algorithm(X) & program_decides(X) --> algorithm_program_decides(X)) &
(\<forall>Y X. program_halts2(X::'a,Y) --> program(X)) &
(\<forall>X Y. program_halts2(X::'a,Y) --> halts2(X::'a,Y)) &
(\<forall>X Y. program(X) & halts2(X::'a,Y) --> program_halts2(X::'a,Y)) &
(\<forall>W X Y Z. halts3_outputs(X::'a,Y,Z,W) --> halts3(X::'a,Y,Z)) &
(\<forall>Y Z X W. halts3_outputs(X::'a,Y,Z,W) --> outputs(X::'a,W)) &
(\<forall>Y Z X W. halts3(X::'a,Y,Z) & outputs(X::'a,W) --> halts3_outputs(X::'a,Y,Z,W)) &
(\<forall>Y X. program_not_halts2(X::'a,Y) --> program(X)) &
(\<forall>X Y. ~(program_not_halts2(X::'a,Y) & halts2(X::'a,Y))) &
(\<forall>X Y. program(X) --> program_not_halts2(X::'a,Y) | halts2(X::'a,Y)) &
(\<forall>W X Y. halts2_outputs(X::'a,Y,W) --> halts2(X::'a,Y)) &
(\<forall>Y X W. halts2_outputs(X::'a,Y,W) --> outputs(X::'a,W)) &
(\<forall>Y X W. halts2(X::'a,Y) & outputs(X::'a,W) --> halts2_outputs(X::'a,Y,W)) &
(\<forall>X W Y Z. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_halts2(Y::'a,Z)) &
(\<forall>X Y Z W. program_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
(\<forall>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)) &
(\<forall>X W Y Z. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> program_not_halts2(Y::'a,Z)) &
(\<forall>X Y Z W. program_not_halts2_halts3_outputs(X::'a,Y,Z,W) --> halts3_outputs(X::'a,Y,Z,W)) &
(\<forall>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)) &
(\<forall>X W Y. program_halts2_halts2_outputs(X::'a,Y,W) --> program_halts2(Y::'a,Y)) &
(\<forall>X Y W. program_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
(\<forall>X Y W. program_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_halts2_halts2_outputs(X::'a,Y,W)) &
(\<forall>X W Y. program_not_halts2_halts2_outputs(X::'a,Y,W) --> program_not_halts2(Y::'a,Y)) &
(\<forall>X Y W. program_not_halts2_halts2_outputs(X::'a,Y,W) --> halts2_outputs(X::'a,Y,W)) &
(\<forall>X Y W. program_not_halts2(Y::'a,Y) & halts2_outputs(X::'a,Y,W) --> program_not_halts2_halts2_outputs(X::'a,Y,W)) &
(\<forall>X. algorithm_program_decides(X) --> program_program_decides(c1)) &
(\<forall>W Y Z. program_program_decides(W) --> program_halts2_halts3_outputs(W::'a,Y,Z,good)) &
(\<forall>W Y Z. program_program_decides(W) --> program_not_halts2_halts3_outputs(W::'a,Y,Z,bad)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>V. program(V) & program_halts2_halts2_outputs(V::'a,f4(V),good) & program_not_halts2_halts2_outputs(V::'a,f4(V),bad) --> program(c3)) &
(\<forall>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)) &
(\<forall>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 &
(\<forall>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))) &
(\<forall>X. contradictory(negate(X),X)) &
(\<forall>X. contradictory(X::'a,negate(X))) &
(\<forall>X. siblings(left_child_of(X),right_child_of(X))) &
(\<forall>D E. equal(D::'a,E) --> equal(left_child_of(D),left_child_of(E))) &
(\<forall>F' G. equal(F'::'a,G) --> equal(negate(F'),negate(G))) &
(\<forall>H I' J. equal(H::'a,I') --> equal(or(H::'a,J),or(I'::'a,J))) &
(\<forall>K' M L. equal(K'::'a,L) --> equal(or(M::'a,K'),or(M::'a,L))) &
(\<forall>N O' P. equal(N::'a,O') --> equal(parent_of(N::'a,P),parent_of(O'::'a,P))) &
(\<forall>Q S' R. equal(Q::'a,R) --> equal(parent_of(S'::'a,Q),parent_of(S'::'a,R))) &
(\<forall>T' U. equal(T'::'a,U) --> equal(right_child_of(T'),right_child_of(U))) &
(\<forall>V W X. equal(V::'a,W) & contradictory(V::'a,X) --> contradictory(W::'a,X)) &
(\<forall>Y A1 Z. equal(Y::'a,Z) & contradictory(A1::'a,Y) --> contradictory(A1::'a,Z)) &
(\<forall>B1 C1 D1. equal(B1::'a,C1) & failure_node(B1::'a,D1) --> failure_node(C1::'a,D1)) &
(\<forall>E1 G1 F1. equal(E1::'a,F1) & failure_node(G1::'a,E1) --> failure_node(G1::'a,F1)) &
(\<forall>H1 I1 J1. equal(H1::'a,I1) & siblings(H1::'a,J1) --> siblings(I1::'a,J1)) &
(\<forall>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))) &
(\<forall>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 \<equiv>
(\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
(\<forall>V X Y Z. between(X::'a,Y,V) & between(Y::'a,Z,V) --> between(X::'a,Y,Z)) &
(\<forall>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)) &
(\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
(\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
(\<forall>X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>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 \<equiv>
(\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
(\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
(\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
(\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
(\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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'))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
(\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
(\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
(\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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 \<equiv>
(\<forall>Y X. equidistant(X::'a,Y,Y,X)) &
(\<forall>X Y Z V V2 W. equidistant(X::'a,Y,Z,V) & equidistant(X::'a,Y,V2,W) --> equidistant(Z::'a,V,V2,W)) &
(\<forall>Z X Y. equidistant(X::'a,Y,Z,Z) --> equal(X::'a,Y)) &
(\<forall>X Y W V. between(X::'a,Y,extension(X::'a,Y,W,V))) &
(\<forall>X Y W V. equidistant(Y::'a,extension(X::'a,Y,W,V),W,V)) &
(\<forall>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)) &
(\<forall>X Y. between(X::'a,Y,X) --> equal(X::'a,Y)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>X Y W Z. equal(X::'a,Y) & between(X::'a,W,Z) --> between(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & between(W::'a,X,Z) --> between(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & between(W::'a,Z,X) --> between(W::'a,Z,Y)) &
(\<forall>X Y V W Z. equal(X::'a,Y) & equidistant(X::'a,V,W,Z) --> equidistant(Y::'a,V,W,Z)) &
(\<forall>X V Y W Z. equal(X::'a,Y) & equidistant(V::'a,X,W,Z) --> equidistant(V::'a,Y,W,Z)) &
(\<forall>X V W Y Z. equal(X::'a,Y) & equidistant(V::'a,W,X,Z) --> equidistant(V::'a,W,Y,Z)) &
(\<forall>X V W Z Y. equal(X::'a,Y) & equidistant(V::'a,W,Z,X) --> equidistant(V::'a,W,Z,Y)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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'))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>X Y V1 V2 V3. equal(X::'a,Y) --> equal(extension(X::'a,V1,V2,V3),extension(Y::'a,V1,V2,V3))) &
(\<forall>X V1 Y V2 V3. equal(X::'a,Y) --> equal(extension(V1::'a,X,V2,V3),extension(V1::'a,Y,V2,V3))) &
(\<forall>X V1 V2 Y V3. equal(X::'a,Y) --> equal(extension(V1::'a,V2,X,V3),extension(V1::'a,V2,Y,V3))) &
(\<forall>X V1 V2 V3 Y. equal(X::'a,Y) --> equal(extension(V1::'a,V2,V3,X),extension(V1::'a,V2,V3,Y))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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 &
(\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
(\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
(\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(reflection(C1::'a,A1),reflection(C1::'a,B1))) &
(\<forall>U V. equidistant(U::'a,V,U,V)) &
(\<forall>W X U V. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,U,V)) &
(\<forall>V U W X. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,W,X)) &
(\<forall>U V X W. equidistant(U::'a,V,W,X) --> equidistant(U::'a,V,X,W)) &
(\<forall>V U X W. equidistant(U::'a,V,W,X) --> equidistant(V::'a,U,X,W)) &
(\<forall>W X V U. equidistant(U::'a,V,W,X) --> equidistant(W::'a,X,V,U)) &
(\<forall>X W U V. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,U,V)) &
(\<forall>X W V U. equidistant(U::'a,V,W,X) --> equidistant(X::'a,W,V,U)) &
(\<forall>W X U V Y Z. equidistant(U::'a,V,W,X) & equidistant(W::'a,X,Y,Z) --> equidistant(U::'a,V,Y,Z)) &
(\<forall>U V W. equal(V::'a,extension(U::'a,V,W,W))) &
(\<forall>W X U V Y. equal(Y::'a,extension(U::'a,V,W,X)) --> between(U::'a,V,Y)) &
(\<forall>U V. between(U::'a,V,reflection(U::'a,V))) &
(\<forall>U V. equidistant(V::'a,reflection(U::'a,V),U,V)) &
(\<forall>U V. equal(U::'a,V) --> equal(V::'a,reflection(U::'a,V))) &
(\<forall>U. equal(U::'a,reflection(U::'a,U))) &
(\<forall>U V. equal(V::'a,reflection(U::'a,V)) --> equal(U::'a,V)) &
(\<forall>U V. equidistant(U::'a,U,V,V)) &
(\<forall>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)) &
(\<forall>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 &
(\<forall>U V. equal(reflection(U::'a,V),extension(U::'a,V,U,V))) &
(\<forall>X Y Z. equal(X::'a,Y) --> equal(reflection(X::'a,Z),reflection(Y::'a,Z))) &
(\<forall>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:
"(\<forall>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)) &
(\<forall>U V W X Y Z. CONGRUENT(U::'a,V,W,X,Y,Z) --> eq(U::'a,V,W,X,Y,Z)) &
(\<forall>V W U X. trapezoid(U::'a,V,W,X) --> parallel(V::'a,W,U,X)) &
(\<forall>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 \<equiv>
(\<forall>X. product(identity::'a,X,X)) &
(\<forall>X. product(X::'a,identity,X)) &
(\<forall>X. product(INVERSE(X),X,identity)) &
(\<forall>X. product(X::'a,INVERSE(X),identity)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
(\<forall>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)) &
(\<forall>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 \<equiv>
(\<forall>X Y. equal(X::'a,Y) --> equal(INVERSE(X),INVERSE(Y))) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
(\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
(\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
(\<forall>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 &
(\<forall>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 &
(\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
(\<forall>C D. equal(C::'a,D) --> equal(j(C),j(D))) &
(\<forall>A B. equal(A::'a,B) & q(A) --> q(B)) &
(\<forall>B A C. q(A) & product(A::'a,B,C) --> product(B::'a,A,C)) &
(\<forall>A. product(j(A),A,h(A)) | product(A::'a,j(A),h(A)) | q(A)) &
(\<forall>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 &
(\<forall>A. product(A::'a,A,identity)) &
(product(a::'a,b,c)) &
(product(INVERSE(a),INVERSE(b),d)) &
(\<forall>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 &
(\<forall>A B C. subgroup_member(A) & subgroup_member(B) & product(A::'a,INVERSE(B),C) --> subgroup_member(C)) &
(\<forall>A B. equal(A::'a,B) & subgroup_member(A) --> subgroup_member(B)) &
(\<forall>A. subgroup_member(A) --> product(Gidentity::'a,A,A)) &
(\<forall>A. subgroup_member(A) --> product(A::'a,Gidentity,A)) &
(\<forall>A. subgroup_member(A) --> product(A::'a,Ginverse(A),Gidentity)) &
(\<forall>A. subgroup_member(A) --> product(Ginverse(A),A,Gidentity)) &
(\<forall>A. subgroup_member(A) --> subgroup_member(Ginverse(A))) &
(\<forall>A B. equal(A::'a,B) --> equal(Ginverse(A),Ginverse(B))) &
(\<forall>A C D B. product(A::'a,B,C) & product(A::'a,D,C) --> equal(D::'a,B)) &
(\<forall>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:
"(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>A. product(A::'a,INVERSE(A),identity)) &
(\<forall>A. product(A::'a,identity,A)) &
(\<forall>A. ~product(A::'a,a,identity)) --> False"
by meson
(*47 inferences so far. Searching to depth 11. 0.2 secs*)
lemma GRP034_4:
"(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X. product(identity::'a,X,X)) &
(\<forall>X. product(X::'a,identity,X)) &
(\<forall>X. product(X::'a,INVERSE(X),identity)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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:
"(\<forall>X. product(identity::'a,X,X)) &
(\<forall>X. product(INVERSE(X),X,identity)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y Z W. product(X::'a,Y,Z) & product(X::'a,Y,W) --> equal(Z::'a,W)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X Y. group_element(X) & group_element(Y) --> product(X::'a,Y,e_1) | product(X::'a,Y,e_2)) &
(\<forall>X Y W Z. product(X::'a,Y,W) & product(X::'a,Y,Z) --> equal(W::'a,Z)) &
(\<forall>X Y W Z. product(X::'a,W,Y) & product(X::'a,Z,Y) --> equal(W::'a,Z)) &
(\<forall>Y X W Z. product(W::'a,Y,X) & product(Z::'a,Y,X) --> equal(W::'a,Z)) &
(\<forall>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 \<equiv>
(\<forall>X. equal(multiply(identity::'a,X),X)) &
(\<forall>X. equal(multiply(INVERSE(X),X),identity)) &
(\<forall>X Y Z. equal(multiply(multiply(X::'a,Y),Z),multiply(X::'a,multiply(Y::'a,Z)))) &
(\<forall>A B. equal(A::'a,B) --> equal(INVERSE(A),INVERSE(B))) &
(\<forall>C D E. equal(C::'a,D) --> equal(multiply(C::'a,E),multiply(D::'a,E))) &
(\<forall>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 \<equiv>
(\<forall>Y X. equal(greatest_lower_bound(X::'a,Y),greatest_lower_bound(Y::'a,X))) &
(\<forall>Y X. equal(least_upper_bound(X::'a,Y),least_upper_bound(Y::'a,X))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>X. equal(least_upper_bound(X::'a,X),X)) &
(\<forall>X. equal(greatest_lower_bound(X::'a,X),X)) &
(\<forall>Y X. equal(least_upper_bound(X::'a,greatest_lower_bound(X::'a,Y)),X)) &
(\<forall>Y X. equal(greatest_lower_bound(X::'a,least_upper_bound(X::'a,Y)),X)) &
(\<forall>Y X Z. equal(multiply(X::'a,least_upper_bound(Y::'a,Z)),least_upper_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(\<forall>Y X Z. equal(multiply(X::'a,greatest_lower_bound(Y::'a,Z)),greatest_lower_bound(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(\<forall>Y Z X. equal(multiply(least_upper_bound(Y::'a,Z),X),least_upper_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
(\<forall>Y Z X. equal(multiply(greatest_lower_bound(Y::'a,Z),X),greatest_lower_bound(multiply(Y::'a,X),multiply(Z::'a,X)))) &
(\<forall>A B C. equal(A::'a,B) --> equal(greatest_lower_bound(A::'a,C),greatest_lower_bound(B::'a,C))) &
(\<forall>A C B. equal(A::'a,B) --> equal(greatest_lower_bound(C::'a,A),greatest_lower_bound(C::'a,B))) &
(\<forall>A B C. equal(A::'a,B) --> equal(least_upper_bound(A::'a,C),least_upper_bound(B::'a,C))) &
(\<forall>A C B. equal(A::'a,B) --> equal(least_upper_bound(C::'a,A),least_upper_bound(C::'a,B))) &
(\<forall>A B C. equal(A::'a,B) --> equal(multiply(A::'a,C),multiply(B::'a,C))) &
(\<forall>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 \<equiv>
(\<forall>X Y. mless_equal(X::'a,Y) --> equal(Divide(X::'a,Y),Zero)) &
(\<forall>X Y. equal(Divide(X::'a,Y),Zero) --> mless_equal(X::'a,Y)) &
(\<forall>Y X. mless_equal(Divide(X::'a,Y),X)) &
(\<forall>X Y Z. mless_equal(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z))) &
(\<forall>X. mless_equal(Zero::'a,X)) &
(\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
(\<forall>X. mless_equal(X::'a,identity))"
abbreviation "HEN002_0_eq mless_equal Divide equal \<equiv>
(\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
(\<forall>G H I'. equal(G::'a,H) & mless_equal(G::'a,I') --> mless_equal(H::'a,I')) &
(\<forall>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 &
(\<forall>X Y. mless_equal(X::'a,Y) --> quotient(X::'a,Y,Zero)) &
(\<forall>X Y. quotient(X::'a,Y,Zero) --> mless_equal(X::'a,Y)) &
(\<forall>Y Z X. quotient(X::'a,Y,Z) --> mless_equal(Z::'a,X)) &
(\<forall>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)) &
(\<forall>X. mless_equal(Zero::'a,X)) &
(\<forall>X Y. mless_equal(X::'a,Y) & mless_equal(Y::'a,X) --> equal(X::'a,Y)) &
(\<forall>X. mless_equal(X::'a,identity)) &
(\<forall>X Y. quotient(X::'a,Y,Divide(X::'a,Y))) &
(\<forall>X Y Z W. quotient(X::'a,Y,Z) & quotient(X::'a,Y,W) --> equal(Z::'a,W)) &
(\<forall>X Y W Z. equal(X::'a,Y) & quotient(X::'a,W,Z) --> quotient(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & quotient(W::'a,X,Z) --> quotient(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & quotient(W::'a,Z,X) --> quotient(W::'a,Z,Y)) &
(\<forall>X Z Y. equal(X::'a,Y) & mless_equal(Z::'a,X) --> mless_equal(Z::'a,Y)) &
(\<forall>X Y Z. equal(X::'a,Y) & mless_equal(X::'a,Z) --> mless_equal(Y::'a,Z)) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(Divide(X::'a,W),Divide(Y::'a,W))) &
(\<forall>X W Y. equal(X::'a,Y) --> equal(Divide(W::'a,X),Divide(W::'a,Y))) &
(\<forall>X. quotient(X::'a,identity,Zero)) &
(\<forall>X. quotient(Zero::'a,X,Zero)) &
(\<forall>X. quotient(X::'a,X,Zero)) &
(\<forall>X. quotient(X::'a,Zero,X)) &
(\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
(\<forall>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 &
(\<forall>X. equal(Divide(X::'a,identity),Zero)) &
(\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
(\<forall>X. equal(Divide(X::'a,X),Zero)) &
(equal(Divide(a::'a,Zero),a)) &
(\<forall>Y X Z. mless_equal(X::'a,Y) & mless_equal(Y::'a,Z) --> mless_equal(X::'a,Z)) &
(\<forall>X Z Y. mless_equal(Divide(X::'a,Y),Z) --> mless_equal(Divide(X::'a,Z),Y)) &
(\<forall>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 &
(\<forall>Y X. equal(Divide(Divide(X::'a,Y),X),Zero)) &
(\<forall>X Y Z. equal(Divide(Divide(Divide(X::'a,Z),Divide(Y::'a,Z)),Divide(Divide(X::'a,Y),Z)),Zero)) &
(\<forall>X. equal(Divide(Zero::'a,X),Zero)) &
(\<forall>X Y. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,X),Zero) --> equal(X::'a,Y)) &
(\<forall>X. equal(Divide(X::'a,identity),Zero)) &
(\<forall>A B C. equal(A::'a,B) --> equal(Divide(A::'a,C),Divide(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(Divide(F'::'a,D),Divide(F'::'a,E))) &
(\<forall>Y X Z. equal(Divide(X::'a,Y),Zero) & equal(Divide(Y::'a,Z),Zero) --> equal(Divide(X::'a,Z),Zero)) &
(\<forall>X Z Y. equal(Divide(Divide(X::'a,Y),Z),Zero) --> equal(Divide(Divide(X::'a,Z),Y),Zero)) &
(\<forall>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:
"(\<forall>X Y. is_a_theorem(equivalent(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
(\<forall>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:
"(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
(\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
(\<forall>Y X Z. is_a_theorem(implies(implies(X,implies(Y,Z)),implies(implies(X,Y),implies(X,Z))))) &
(\<forall>Y X. is_a_theorem(implies(implies(not(X),not(Y)),implies(Y,X)))) &
(\<forall>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:
"(\<forall>X Y. is_a_theorem(implies(X::'a,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
(\<forall>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:
"(\<forall>X Y. is_a_theorem(implies(X,Y)) & is_a_theorem(X) --> is_a_theorem(Y)) &
(\<forall>Y X. is_a_theorem(implies(X,implies(Y,X)))) &
(\<forall>Y X Z. is_a_theorem(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))))) &
(\<forall>Y X. is_a_theorem(implies(implies(implies(X,Y),Y),implies(implies(Y,X),X)))) &
(\<forall>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:
"(\<forall>X. equal(X,X)) &
(\<forall>Y X. equal(X,Y) --> equal(Y,X)) &
(\<forall>Y X Z. equal(X,Y) & equal(Y,Z) --> equal(X,Z)) &
(\<forall>X. equal(implies(true,X),X)) &
(\<forall>Y X Z. equal(implies(implies(X,Y),implies(implies(Y,Z),implies(X,Z))),true)) &
(\<forall>Y X. equal(implies(implies(X,Y),Y),implies(implies(Y,X),X))) &
(\<forall>Y X. equal(implies(implies(not(X),not(Y)),implies(Y,X)),true)) &
(\<forall>A B C. equal(A,B) --> equal(implies(A,C),implies(B,C))) &
(\<forall>D F' E. equal(D,E) --> equal(implies(F',D),implies(F',E))) &
(\<forall>G H. equal(G,H) --> equal(not(G),not(H))) &
(\<forall>X Y. equal(big_V(X,Y),implies(implies(X,Y),Y))) &
(\<forall>X Y. equal(big_hat(X,Y),not(big_V(not(X),not(Y))))) &
(\<forall>X Y. ordered(X,Y) --> equal(implies(X,Y),true)) &
(\<forall>X Y. equal(implies(X,Y),true) --> ordered(X,Y)) &
(\<forall>A B C. equal(A,B) --> equal(big_V(A,C),big_V(B,C))) &
(\<forall>D F' E. equal(D,E) --> equal(big_V(F',D),big_V(F',E))) &
(\<forall>G H I'. equal(G,H) --> equal(big_hat(G,I'),big_hat(H,I'))) &
(\<forall>J L K'. equal(J,K') --> equal(big_hat(L,J),big_hat(L,K'))) &
(\<forall>M N O'. equal(M,N) & ordered(M,O') --> ordered(N,O')) &
(\<forall>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:
"(\<forall>A. axiom(or(not(or(A,A)),A))) &
(\<forall>B A. axiom(or(not(A),or(B,A)))) &
(\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
(\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
(\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
(\<forall>X. axiom(X) --> theorem(X)) &
(\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
(\<forall>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:
"(\<forall>A. axiom(or(not(or(A,A)),A))) &
(\<forall>B A. axiom(or(not(A),or(B,A)))) &
(\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
(\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
(\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
(\<forall>X. axiom(X) --> theorem(X)) &
(\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
(\<forall>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:
"(\<forall>A. axiom(or(not(or(A,A)),A))) &
(\<forall>B A. axiom(or(not(A),or(B,A)))) &
(\<forall>B A. axiom(or(not(or(A,B)),or(B,A)))) &
(\<forall>B A C. axiom(or(not(or(A,or(B,C))),or(B,or(A,C))))) &
(\<forall>A C B. axiom(or(not(or(not(A),B)),or(not(or(C,A)),or(C,B))))) &
(\<forall>X. axiom(X) --> theorem(X)) &
(\<forall>X Y. axiom(or(not(Y),X)) & theorem(Y) --> theorem(X)) &
(\<forall>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 &
(\<forall>Y X Z. equal(f(X::'a,f(Y::'a,Z)),f(f(X::'a,Y),f(X::'a,Z)))) &
(\<forall>X Y. left(X::'a,f(X::'a,Y))) &
(\<forall>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))) &
(\<forall>A B C. equal(A::'a,B) --> equal(f(A::'a,C),f(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(f(F'::'a,D),f(F'::'a,E))) &
(\<forall>G H I'. equal(G::'a,H) & left(G::'a,I') --> left(H::'a,I')) &
(\<forall>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)) &
(\<forall>Place Situation. hand_at(Place::'a,Situation) --> hand_at(Place::'a,let_go(Situation))) &
(\<forall>Place Another_place Situation. hand_at(Place::'a,Situation) --> hand_at(Another_place::'a,go(Another_place::'a,Situation))) &
(\<forall>Thing Situation. ~held(Thing::'a,let_go(Situation))) &
(\<forall>Situation Thing. at(Thing::'a,here,Situation) --> red(Thing)) &
(\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,let_go(Situation))) &
(\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> at(Thing::'a,Place,pick_up(Situation))) &
(\<forall>Thing Place Situation. at(Thing::'a,Place,Situation) --> grabbed(Thing::'a,pick_up(go(Place::'a,let_go(Situation))))) &
(\<forall>Thing Situation. red(Thing) & put(Thing::'a,there,Situation) --> answer(Situation)) &
(\<forall>Place Thing Another_place Situation. at(Thing::'a,Place,Situation) & grabbed(Thing::'a,Situation) --> put(Thing::'a,Another_place,go(Another_place::'a,Situation))) &
(\<forall>Thing Place Another_place Situation. at(Thing::'a,Place,Situation) --> held(Thing::'a,Situation) | at(Thing::'a,Place,go(Another_place::'a,Situation))) &
(\<forall>One_place Thing Place Situation. hand_at(One_place::'a,Situation) & held(Thing::'a,Situation) --> at(Thing::'a,Place,go(Place::'a,Situation))) &
(\<forall>Place Thing Situation. hand_at(Place::'a,Situation) & at(Thing::'a,Place,Situation) --> held(Thing::'a,pick_up(Situation))) &
(\<forall>Situation. ~answer(Situation)) --> False"
by meson
(*73 inferences so far. Searching to depth 10. 0.2 secs*)
lemma MSC003_1:
"(\<forall>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)) &
(\<forall>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)) &
(\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
(\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
(\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
(\<forall>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:
"(\<forall>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)) &
(\<forall>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)) &
(\<forall>X. in'(X::'a,boy) --> in'(X::'a,human)) &
(\<forall>X. in'(X::'a,hand) --> has_parts(X::'a,num5,fingers)) &
(\<forall>X. in'(X::'a,human) --> has_parts(X::'a,num2,arm)) &
(\<forall>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)) &
(\<forall>X Y. value(X::'a,truth) & value(Y::'a,truth) --> value(xor(X::'a,Y),falsity)) &
(\<forall>X Y. value(X::'a,truth) & value(Y::'a,falsity) --> value(xor(X::'a,Y),truth)) &
(\<forall>X Y. value(X::'a,falsity) & value(Y::'a,truth) --> value(xor(X::'a,Y),truth)) &
(\<forall>X Y. value(X::'a,falsity) & value(Y::'a,falsity) --> value(xor(X::'a,Y),falsity)) &
(\<forall>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:
"(\<forall>Y X Z. p(X::'a,Y) & p(Y::'a,Z) --> p(X::'a,Z)) &
(\<forall>Y X Z. q(X::'a,Y) & q(Y::'a,Z) --> q(X::'a,Z)) &
(\<forall>Y X. q(X::'a,Y) --> q(Y::'a,X)) &
(\<forall>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:
"(\<forall>A. equal(A::'a,A)) &
(\<forall>B A C. equal(A::'a,B) & equal(B::'a,C) --> equal(A::'a,C)) &
(\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
(\<forall>A B C. equal(add(A::'a,add(B::'a,C)),add(add(A::'a,B),C))) &
(\<forall>B A. equal(subtract(add(A::'a,B),B),A)) &
(\<forall>A B. equal(A::'a,subtract(add(A::'a,B),B))) &
(\<forall>A C B. equal(add(subtract(A::'a,B),C),subtract(add(A::'a,C),B))) &
(\<forall>A C B. equal(subtract(add(A::'a,B),C),add(subtract(A::'a,C),B))) &
(\<forall>A C B D. equal(A::'a,B) & equal(C::'a,add(A::'a,D)) --> equal(C::'a,add(B::'a,D))) &
(\<forall>A C D B. equal(A::'a,B) & equal(C::'a,add(D::'a,A)) --> equal(C::'a,add(D::'a,B))) &
(\<forall>A C B D. equal(A::'a,B) & equal(C::'a,subtract(A::'a,D)) --> equal(C::'a,subtract(B::'a,D))) &
(\<forall>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 \<equiv>
(\<forall>A. equal(add(A::'a,num0),A)) &
(\<forall>A B. equal(add(A::'a,successor(B)),successor(add(A::'a,B)))) &
(\<forall>A. equal(multiply(A::'a,num0),num0)) &
(\<forall>B A. equal(multiply(A::'a,successor(B)),add(multiply(A::'a,B),A))) &
(\<forall>A B. equal(successor(A),successor(B)) --> equal(A::'a,B)) &
(\<forall>A B. equal(A::'a,B) --> equal(successor(A),successor(B)))"
abbreviation "NUM001_1_ax predecessor_of_1st_minus_2nd successor add equal mless \<equiv>
(\<forall>A C B. mless(A::'a,B) & mless(C::'a,A) --> mless(C::'a,B)) &
(\<forall>A B C. equal(add(successor(A),B),C) --> mless(B::'a,C)) &
(\<forall>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 \<equiv>
(\<forall>A B. divides(A::'a,B) --> mless(A::'a,B) | equal(A::'a,B)) &
(\<forall>A B. mless(A::'a,B) --> divides(A::'a,B)) &
(\<forall>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)) &
(\<forall>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 &
(\<forall>B A. equal(add(A::'a,B),add(B::'a,A))) &
(\<forall>B A C. equal(add(A::'a,B),add(C::'a,B)) --> equal(A::'a,C)) &
(mless(a::'a,a)) &
(\<forall>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 \<equiv>
(\<forall>X U Y. subclass(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
(\<forall>X Y. member(not_subclass_element(X::'a,Y),X) | subclass(X::'a,Y)) &
(\<forall>X Y. member(not_subclass_element(X::'a,Y),Y) --> subclass(X::'a,Y)) &
(\<forall>X. subclass(X::'a,universal_class)) &
(\<forall>X Y. equal(X::'a,Y) --> subclass(X::'a,Y)) &
(\<forall>Y X. equal(X::'a,Y) --> subclass(Y::'a,X)) &
(\<forall>X Y. subclass(X::'a,Y) & subclass(Y::'a,X) --> equal(X::'a,Y)) &
(\<forall>X U Y. member(U::'a,unordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
(\<forall>X Y. member(X::'a,universal_class) --> member(X::'a,unordered_pair(X::'a,Y))) &
(\<forall>X Y. member(Y::'a,universal_class) --> member(Y::'a,unordered_pair(X::'a,Y))) &
(\<forall>X Y. member(unordered_pair(X::'a,Y),universal_class)) &
(\<forall>X. equal(unordered_pair(X::'a,X),singleton(X))) &
(\<forall>X Y. equal(unordered_pair(singleton(X),unordered_pair(X::'a,singleton(Y))),ordered_pair(X::'a,Y))) &
(\<forall>V Y U X. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(U::'a,X)) &
(\<forall>U X V Y. member(ordered_pair(U::'a,V),cross_product(X::'a,Y)) --> member(V::'a,Y)) &
(\<forall>U V X Y. member(U::'a,X) & member(V::'a,Y) --> member(ordered_pair(U::'a,V),cross_product(X::'a,Y))) &
(\<forall>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))) &
(\<forall>X Y. member(ordered_pair(X::'a,Y),element_relation) --> member(X::'a,Y)) &
(\<forall>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)) &
(\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
(\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
(\<forall>Z X Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
(\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,universal_class) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
(\<forall>X Y. equal(complement(intersection(complement(X),complement(Y))),union(X::'a,Y))) &
(\<forall>X Y. equal(intersection(complement(intersection(X::'a,Y)),complement(intersection(complement(X),complement(Y)))),difference(X::'a,Y))) &
(\<forall>Xr X Y. equal(intersection(Xr::'a,cross_product(X::'a,Y)),restrct(Xr::'a,X,Y))) &
(\<forall>Xr X Y. equal(intersection(cross_product(X::'a,Y),Xr),restrct(Xr::'a,X,Y))) &
(\<forall>Z X. ~(equal(restrct(X::'a,singleton(Z),universal_class),null_class) & member(Z::'a,domain_of(X)))) &
(\<forall>Z X. member(Z::'a,universal_class) --> equal(restrct(X::'a,singleton(Z),universal_class),null_class) | member(Z::'a,domain_of(X))) &
(\<forall>X. subclass(rot(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>X. subclass(flip(X),cross_product(cross_product(universal_class::'a,universal_class),universal_class))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>Y. equal(domain_of(flip(cross_product(Y::'a,universal_class))),INVERSE(Y))) &
(\<forall>Z. equal(domain_of(INVERSE(Z)),range_of(Z))) &
(\<forall>Z X Y. equal(first(not_subclass_element(restrct(Z::'a,X,singleton(Y)),null_class)),domain(Z::'a,X,Y))) &
(\<forall>Z X Y. equal(second(not_subclass_element(restrct(Z::'a,singleton(X),Y),null_class)),rng(Z::'a,X,Y))) &
(\<forall>Xr X. equal(range_of(restrct(Xr::'a,X,universal_class)),image'(Xr::'a,X))) &
(\<forall>X. equal(union(X::'a,singleton(X)),successor(X))) &
(subclass(successor_relation::'a,cross_product(universal_class::'a,universal_class))) &
(\<forall>X Y. member(ordered_pair(X::'a,Y),successor_relation) --> equal(successor(X),Y)) &
(\<forall>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)) &
(\<forall>X. inductive(X) --> member(null_class::'a,X)) &
(\<forall>X. inductive(X) --> subclass(image'(successor_relation::'a,X),X)) &
(\<forall>X. member(null_class::'a,X) & subclass(image'(successor_relation::'a,X),X) --> inductive(X)) &
(inductive(omega)) &
(\<forall>Y. inductive(Y) --> subclass(omega::'a,Y)) &
(member(omega::'a,universal_class)) &
(\<forall>X. equal(domain_of(restrct(element_relation::'a,universal_class,X)),sum_class(X))) &
(\<forall>X. member(X::'a,universal_class) --> member(sum_class(X),universal_class)) &
(\<forall>X. equal(complement(image'(element_relation::'a,complement(X))),powerClass(X))) &
(\<forall>U. member(U::'a,universal_class) --> member(powerClass(U),universal_class)) &
(\<forall>Yr Xr. subclass(compos(Yr::'a,Xr),cross_product(universal_class::'a,universal_class))) &
(\<forall>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))))) &
(\<forall>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))) &
(\<forall>X. single_valued_class(X) --> subclass(compos(X::'a,INVERSE(X)),identity_relation)) &
(\<forall>X. subclass(compos(X::'a,INVERSE(X)),identity_relation) --> single_valued_class(X)) &
(\<forall>Xf. function(Xf) --> subclass(Xf::'a,cross_product(universal_class::'a,universal_class))) &
(\<forall>Xf. function(Xf) --> subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation)) &
(\<forall>Xf. subclass(Xf::'a,cross_product(universal_class::'a,universal_class)) & subclass(compos(Xf::'a,INVERSE(Xf)),identity_relation) --> function(Xf)) &
(\<forall>Xf X. function(Xf) & member(X::'a,universal_class) --> member(image'(Xf::'a,X),universal_class)) &
(\<forall>X. equal(X::'a,null_class) | member(regular(X),X)) &
(\<forall>X. equal(X::'a,null_class) | equal(intersection(X::'a,regular(X)),null_class)) &
(\<forall>Xf Y. equal(sum_class(image'(Xf::'a,singleton(Y))),apply(Xf::'a,Y))) &
(function(choice)) &
(\<forall>Y. member(Y::'a,universal_class) --> equal(Y::'a,null_class) | member(apply(choice::'a,Y),Y)) &
(\<forall>Xf. one_to_one(Xf) --> function(Xf)) &
(\<forall>Xf. one_to_one(Xf) --> function(INVERSE(Xf))) &
(\<forall>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)) &
(\<forall>Xr. equal(complement(domain_of(intersection(Xr::'a,identity_relation))),diagonalise(Xr))) &
(\<forall>X. equal(intersection(domain_of(X),diagonalise(compos(INVERSE(element_relation),X))),cantor(X))) &
(\<forall>Xf. operation(Xf) --> function(Xf)) &
(\<forall>Xf. operation(Xf) --> equal(cross_product(domain_of(domain_of(Xf)),domain_of(domain_of(Xf))),domain_of(Xf))) &
(\<forall>Xf. operation(Xf) --> subclass(range_of(Xf),domain_of(domain_of(Xf)))) &
(\<forall>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)) &
(\<forall>Xf1 Xf2 Xh. compatible(Xh::'a,Xf1,Xf2) --> function(Xh)) &
(\<forall>Xf2 Xf1 Xh. compatible(Xh::'a,Xf1,Xf2) --> equal(domain_of(domain_of(Xf1)),domain_of(Xh))) &
(\<forall>Xf1 Xh Xf2. compatible(Xh::'a,Xf1,Xf2) --> subclass(range_of(Xh),domain_of(domain_of(Xf2)))) &
(\<forall>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)) &
(\<forall>Xh Xf2 Xf1. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf1)) &
(\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> operation(Xf2)) &
(\<forall>Xh Xf1 Xf2. homomorphism(Xh::'a,Xf1,Xf2) --> compatible(Xh::'a,Xf1,Xf2)) &
(\<forall>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))))) &
(\<forall>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)) &
(\<forall>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 \<equiv>
(\<forall>D E F'. equal(D::'a,E) --> equal(apply(D::'a,F'),apply(E::'a,F'))) &
(\<forall>G I' H. equal(G::'a,H) --> equal(apply(I'::'a,G),apply(I'::'a,H))) &
(\<forall>J K'. equal(J::'a,K') --> equal(cantor(J),cantor(K'))) &
(\<forall>L M. equal(L::'a,M) --> equal(complement(L),complement(M))) &
(\<forall>N O' P. equal(N::'a,O') --> equal(compos(N::'a,P),compos(O'::'a,P))) &
(\<forall>Q S' R. equal(Q::'a,R) --> equal(compos(S'::'a,Q),compos(S'::'a,R))) &
(\<forall>T' U V. equal(T'::'a,U) --> equal(cross_product(T'::'a,V),cross_product(U::'a,V))) &
(\<forall>W Y X. equal(W::'a,X) --> equal(cross_product(Y::'a,W),cross_product(Y::'a,X))) &
(\<forall>Z A1. equal(Z::'a,A1) --> equal(diagonalise(Z),diagonalise(A1))) &
(\<forall>B1 C1 D1. equal(B1::'a,C1) --> equal(difference(B1::'a,D1),difference(C1::'a,D1))) &
(\<forall>E1 G1 F1. equal(E1::'a,F1) --> equal(difference(G1::'a,E1),difference(G1::'a,F1))) &
(\<forall>H1 I1 J1 K1. equal(H1::'a,I1) --> equal(domain(H1::'a,J1,K1),domain(I1::'a,J1,K1))) &
(\<forall>L1 N1 M1 O1. equal(L1::'a,M1) --> equal(domain(N1::'a,L1,O1),domain(N1::'a,M1,O1))) &
(\<forall>P1 R1 S1 Q1. equal(P1::'a,Q1) --> equal(domain(R1::'a,S1,P1),domain(R1::'a,S1,Q1))) &
(\<forall>T1 U1. equal(T1::'a,U1) --> equal(domain_of(T1),domain_of(U1))) &
(\<forall>V1 W1. equal(V1::'a,W1) --> equal(first(V1),first(W1))) &
(\<forall>X1 Y1. equal(X1::'a,Y1) --> equal(flip(X1),flip(Y1))) &
(\<forall>Z1 A2 B2. equal(Z1::'a,A2) --> equal(image'(Z1::'a,B2),image'(A2::'a,B2))) &
(\<forall>C2 E2 D2. equal(C2::'a,D2) --> equal(image'(E2::'a,C2),image'(E2::'a,D2))) &
(\<forall>F2 G2 H2. equal(F2::'a,G2) --> equal(intersection(F2::'a,H2),intersection(G2::'a,H2))) &
(\<forall>I2 K2 J2. equal(I2::'a,J2) --> equal(intersection(K2::'a,I2),intersection(K2::'a,J2))) &
(\<forall>L2 M2. equal(L2::'a,M2) --> equal(INVERSE(L2),INVERSE(M2))) &
(\<forall>N2 O2 P2 Q2. equal(N2::'a,O2) --> equal(not_homomorphism1(N2::'a,P2,Q2),not_homomorphism1(O2::'a,P2,Q2))) &
(\<forall>R2 T2 S2 U2. equal(R2::'a,S2) --> equal(not_homomorphism1(T2::'a,R2,U2),not_homomorphism1(T2::'a,S2,U2))) &
(\<forall>V2 X2 Y2 W2. equal(V2::'a,W2) --> equal(not_homomorphism1(X2::'a,Y2,V2),not_homomorphism1(X2::'a,Y2,W2))) &
(\<forall>Z2 A3 B3 C3. equal(Z2::'a,A3) --> equal(not_homomorphism2(Z2::'a,B3,C3),not_homomorphism2(A3::'a,B3,C3))) &
(\<forall>D3 F3 E3 G3. equal(D3::'a,E3) --> equal(not_homomorphism2(F3::'a,D3,G3),not_homomorphism2(F3::'a,E3,G3))) &
(\<forall>H3 J3 K3 I3. equal(H3::'a,I3) --> equal(not_homomorphism2(J3::'a,K3,H3),not_homomorphism2(J3::'a,K3,I3))) &
(\<forall>L3 M3 N3. equal(L3::'a,M3) --> equal(not_subclass_element(L3::'a,N3),not_subclass_element(M3::'a,N3))) &
(\<forall>O3 Q3 P3. equal(O3::'a,P3) --> equal(not_subclass_element(Q3::'a,O3),not_subclass_element(Q3::'a,P3))) &
(\<forall>R3 S3 T3. equal(R3::'a,S3) --> equal(ordered_pair(R3::'a,T3),ordered_pair(S3::'a,T3))) &
(\<forall>U3 W3 V3. equal(U3::'a,V3) --> equal(ordered_pair(W3::'a,U3),ordered_pair(W3::'a,V3))) &
(\<forall>X3 Y3. equal(X3::'a,Y3) --> equal(powerClass(X3),powerClass(Y3))) &
(\<forall>Z3 A4 B4 C4. equal(Z3::'a,A4) --> equal(rng(Z3::'a,B4,C4),rng(A4::'a,B4,C4))) &
(\<forall>D4 F4 E4 G4. equal(D4::'a,E4) --> equal(rng(F4::'a,D4,G4),rng(F4::'a,E4,G4))) &
(\<forall>H4 J4 K4 I4. equal(H4::'a,I4) --> equal(rng(J4::'a,K4,H4),rng(J4::'a,K4,I4))) &
(\<forall>L4 M4. equal(L4::'a,M4) --> equal(range_of(L4),range_of(M4))) &
(\<forall>N4 O4. equal(N4::'a,O4) --> equal(regular(N4),regular(O4))) &
(\<forall>P4 Q4 R4 S4. equal(P4::'a,Q4) --> equal(restrct(P4::'a,R4,S4),restrct(Q4::'a,R4,S4))) &
(\<forall>T4 V4 U4 W4. equal(T4::'a,U4) --> equal(restrct(V4::'a,T4,W4),restrct(V4::'a,U4,W4))) &
(\<forall>X4 Z4 A5 Y4. equal(X4::'a,Y4) --> equal(restrct(Z4::'a,A5,X4),restrct(Z4::'a,A5,Y4))) &
(\<forall>B5 C5. equal(B5::'a,C5) --> equal(rot(B5),rot(C5))) &
(\<forall>D5 E5. equal(D5::'a,E5) --> equal(second(D5),second(E5))) &
(\<forall>F5 G5. equal(F5::'a,G5) --> equal(singleton(F5),singleton(G5))) &
(\<forall>H5 I5. equal(H5::'a,I5) --> equal(successor(H5),successor(I5))) &
(\<forall>J5 K5. equal(J5::'a,K5) --> equal(sum_class(J5),sum_class(K5))) &
(\<forall>L5 M5 N5. equal(L5::'a,M5) --> equal(union(L5::'a,N5),union(M5::'a,N5))) &
(\<forall>O5 Q5 P5. equal(O5::'a,P5) --> equal(union(Q5::'a,O5),union(Q5::'a,P5))) &
(\<forall>R5 S5 T5. equal(R5::'a,S5) --> equal(unordered_pair(R5::'a,T5),unordered_pair(S5::'a,T5))) &
(\<forall>U5 W5 V5. equal(U5::'a,V5) --> equal(unordered_pair(W5::'a,U5),unordered_pair(W5::'a,V5))) &
(\<forall>X5 Y5 Z5 A6. equal(X5::'a,Y5) & compatible(X5::'a,Z5,A6) --> compatible(Y5::'a,Z5,A6)) &
(\<forall>B6 D6 C6 E6. equal(B6::'a,C6) & compatible(D6::'a,B6,E6) --> compatible(D6::'a,C6,E6)) &
(\<forall>F6 H6 I6 G6. equal(F6::'a,G6) & compatible(H6::'a,I6,F6) --> compatible(H6::'a,I6,G6)) &
(\<forall>J6 K6. equal(J6::'a,K6) & function(J6) --> function(K6)) &
(\<forall>L6 M6 N6 O6. equal(L6::'a,M6) & homomorphism(L6::'a,N6,O6) --> homomorphism(M6::'a,N6,O6)) &
(\<forall>P6 R6 Q6 S6. equal(P6::'a,Q6) & homomorphism(R6::'a,P6,S6) --> homomorphism(R6::'a,Q6,S6)) &
(\<forall>T6 V6 W6 U6. equal(T6::'a,U6) & homomorphism(V6::'a,W6,T6) --> homomorphism(V6::'a,W6,U6)) &
(\<forall>X6 Y6. equal(X6::'a,Y6) & inductive(X6) --> inductive(Y6)) &
(\<forall>Z6 A7 B7. equal(Z6::'a,A7) & member(Z6::'a,B7) --> member(A7::'a,B7)) &
(\<forall>C7 E7 D7. equal(C7::'a,D7) & member(E7::'a,C7) --> member(E7::'a,D7)) &
(\<forall>F7 G7. equal(F7::'a,G7) & one_to_one(F7) --> one_to_one(G7)) &
(\<forall>H7 I7. equal(H7::'a,I7) & operation(H7) --> operation(I7)) &
(\<forall>J7 K7. equal(J7::'a,K7) & single_valued_class(J7) --> single_valued_class(K7)) &
(\<forall>L7 M7 N7. equal(L7::'a,M7) & subclass(L7::'a,N7) --> subclass(M7::'a,N7)) &
(\<forall>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 \<equiv>
(\<forall>X. subclass(compose_class(X),cross_product(universal_class::'a,universal_class))) &
(\<forall>X Y Z. member(ordered_pair(Y::'a,Z),compose_class(X)) --> equal(compos(X::'a,Y),Z)) &
(\<forall>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)))) &
(\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),composition_function) --> equal(compos(X::'a,Y),Z)) &
(\<forall>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))) &
(\<forall>X Y. member(ordered_pair(X::'a,Y),domain_relation) --> equal(domain_of(X),Y)) &
(\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,domain_of(X)),domain_relation)) &
(\<forall>X. equal(first(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued1(X))) &
(\<forall>X. equal(second(not_subclass_element(compos(X::'a,INVERSE(X)),identity_relation)),single_valued2(X))) &
(\<forall>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)))) &
(\<forall>Z Y X. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> member(Y::'a,domain_of(X))) &
(\<forall>X Y Z. member(ordered_pair(X::'a,ordered_pair(Y::'a,Z)),application_function) --> equal(apply(X::'a,Y),Z)) &
(\<forall>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)) &
(\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
(\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
(\<forall>X Xf Y. maps(Xf::'a,X,Y) --> subclass(range_of(Xf),Y)) &
(\<forall>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 \<equiv>
(\<forall>L M. equal(L::'a,M) --> equal(compose_class(L),compose_class(M))) &
(\<forall>N2 O2. equal(N2::'a,O2) --> equal(single_valued1(N2),single_valued1(O2))) &
(\<forall>P2 Q2. equal(P2::'a,Q2) --> equal(single_valued2(P2),single_valued2(Q2))) &
(\<forall>R2 S2. equal(R2::'a,S2) --> equal(single_valued3(R2),single_valued3(S2))) &
(\<forall>X2 Y2 Z2 A3. equal(X2::'a,Y2) & maps(X2::'a,Z2,A3) --> maps(Y2::'a,Z2,A3)) &
(\<forall>B3 D3 C3 E3. equal(B3::'a,C3) & maps(D3::'a,B3,E3) --> maps(D3::'a,C3,E3)) &
(\<forall>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 \<equiv>
(\<forall>X. equal(union(X::'a,INVERSE(X)),symmetrization_of(X))) &
(\<forall>X Y. irreflexive(X::'a,Y) --> subclass(restrct(X::'a,Y,Y),complement(identity_relation))) &
(\<forall>X Y. subclass(restrct(X::'a,Y,Y),complement(identity_relation)) --> irreflexive(X::'a,Y)) &
(\<forall>Y X. connected(X::'a,Y) --> subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X)))) &
(\<forall>X Y. subclass(cross_product(Y::'a,Y),union(identity_relation::'a,symmetrization_of(X))) --> connected(X::'a,Y)) &
(\<forall>Xr Y. transitive(Xr::'a,Y) --> subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y))) &
(\<forall>Xr Y. subclass(compos(restrct(Xr::'a,Y,Y),restrct(Xr::'a,Y,Y)),restrct(Xr::'a,Y,Y)) --> transitive(Xr::'a,Y)) &
(\<forall>Xr Y. asymmetric(Xr::'a,Y) --> equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class)) &
(\<forall>Xr Y. equal(restrct(intersection(Xr::'a,INVERSE(Xr)),Y,Y),null_class) --> asymmetric(Xr::'a,Y)) &
(\<forall>Xr Y Z. equal(segment(Xr::'a,Y,Z),domain_of(restrct(Xr::'a,Y,singleton(Z))))) &
(\<forall>X Y. well_ordering(X::'a,Y) --> connected(X::'a,Y)) &
(\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(U::'a,null_class) | member(least(Xr::'a,U),U)) &
(\<forall>Y V Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) & member(V::'a,U) --> member(least(Xr::'a,U),U)) &
(\<forall>Y Xr U. well_ordering(Xr::'a,Y) & subclass(U::'a,Y) --> equal(segment(Xr::'a,U,least(Xr::'a,U)),null_class)) &
(\<forall>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))) &
(\<forall>Xr Y. connected(Xr::'a,Y) & equal(not_well_ordering(Xr::'a,Y),null_class) --> well_ordering(Xr::'a,Y)) &
(\<forall>Xr Y. connected(Xr::'a,Y) --> subclass(not_well_ordering(Xr::'a,Y),Y) | well_ordering(Xr::'a,Y)) &
(\<forall>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)) &
(\<forall>Xr Y Z. section(Xr::'a,Y,Z) --> subclass(Y::'a,Z)) &
(\<forall>Xr Z Y. section(Xr::'a,Y,Z) --> subclass(domain_of(restrct(Xr::'a,Z,Y)),Y)) &
(\<forall>Xr Y Z. subclass(Y::'a,Z) & subclass(domain_of(restrct(Xr::'a,Z,Y)),Y) --> section(Xr::'a,Y,Z)) &
(\<forall>X. member(X::'a,ordinal_numbers) --> well_ordering(element_relation::'a,X)) &
(\<forall>X. member(X::'a,ordinal_numbers) --> subclass(sum_class(X),X)) &
(\<forall>X. well_ordering(element_relation::'a,X) & subclass(sum_class(X),X) & member(X::'a,universal_class) --> member(X::'a,ordinal_numbers)) &
(\<forall>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)) &
(\<forall>X. subclass(rest_of(X),cross_product(universal_class::'a,universal_class))) &
(\<forall>V U X. member(ordered_pair(U::'a,V),rest_of(X)) --> member(U::'a,domain_of(X))) &
(\<forall>X U V. member(ordered_pair(U::'a,V),rest_of(X)) --> equal(restrct(X::'a,U,universal_class),V)) &
(\<forall>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))) &
(\<forall>X Y. member(ordered_pair(X::'a,Y),rest_relation) --> equal(rest_of(X),Y)) &
(\<forall>X. member(X::'a,universal_class) --> member(ordered_pair(X::'a,rest_of(X)),rest_relation)) &
(\<forall>X Z. member(X::'a,recursion_equation_functions(Z)) --> function(Z)) &
(\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> function(X)) &
(\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> member(domain_of(X),ordinal_numbers)) &
(\<forall>Z X. member(X::'a,recursion_equation_functions(Z)) --> equal(compos(Z::'a,rest_of(X)),X)) &
(\<forall>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))) &
(\<forall>X Y. member(ordered_pair(X::'a,Y),union_of_range_map) --> equal(sum_class(range_of(X)),Y)) &
(\<forall>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)) &
(\<forall>X Y. equal(apply(recursion(X::'a,successor_relation,union_of_range_map),Y),ordinal_add(X::'a,Y))) &
(\<forall>X Y. equal(recursion(null_class::'a,apply(add_relation::'a,X),union_of_range_map),ordinal_multiply(X::'a,Y))) &
(\<forall>X. member(X::'a,omega) --> equal(integer_of(X),X)) &
(\<forall>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 \<equiv>
(\<forall>D E. equal(D::'a,E) --> equal(integer_of(D),integer_of(E))) &
(\<forall>F' G H. equal(F'::'a,G) --> equal(least(F'::'a,H),least(G::'a,H))) &
(\<forall>I' K' J. equal(I'::'a,J) --> equal(least(K'::'a,I'),least(K'::'a,J))) &
(\<forall>L M N. equal(L::'a,M) --> equal(not_well_ordering(L::'a,N),not_well_ordering(M::'a,N))) &
(\<forall>O' Q P. equal(O'::'a,P) --> equal(not_well_ordering(Q::'a,O'),not_well_ordering(Q::'a,P))) &
(\<forall>R S' T'. equal(R::'a,S') --> equal(ordinal_add(R::'a,T'),ordinal_add(S'::'a,T'))) &
(\<forall>U W V. equal(U::'a,V) --> equal(ordinal_add(W::'a,U),ordinal_add(W::'a,V))) &
(\<forall>X Y Z. equal(X::'a,Y) --> equal(ordinal_multiply(X::'a,Z),ordinal_multiply(Y::'a,Z))) &
(\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(ordinal_multiply(C1::'a,A1),ordinal_multiply(C1::'a,B1))) &
(\<forall>F1 G1 H1 I1. equal(F1::'a,G1) --> equal(recursion(F1::'a,H1,I1),recursion(G1::'a,H1,I1))) &
(\<forall>J1 L1 K1 M1. equal(J1::'a,K1) --> equal(recursion(L1::'a,J1,M1),recursion(L1::'a,K1,M1))) &
(\<forall>N1 P1 Q1 O1. equal(N1::'a,O1) --> equal(recursion(P1::'a,Q1,N1),recursion(P1::'a,Q1,O1))) &
(\<forall>R1 S1. equal(R1::'a,S1) --> equal(recursion_equation_functions(R1),recursion_equation_functions(S1))) &
(\<forall>T1 U1. equal(T1::'a,U1) --> equal(rest_of(T1),rest_of(U1))) &
(\<forall>V1 W1 X1 Y1. equal(V1::'a,W1) --> equal(segment(V1::'a,X1,Y1),segment(W1::'a,X1,Y1))) &
(\<forall>Z1 B2 A2 C2. equal(Z1::'a,A2) --> equal(segment(B2::'a,Z1,C2),segment(B2::'a,A2,C2))) &
(\<forall>D2 F2 G2 E2. equal(D2::'a,E2) --> equal(segment(F2::'a,G2,D2),segment(F2::'a,G2,E2))) &
(\<forall>H2 I2. equal(H2::'a,I2) --> equal(symmetrization_of(H2),symmetrization_of(I2))) &
(\<forall>J2 K2 L2. equal(J2::'a,K2) & asymmetric(J2::'a,L2) --> asymmetric(K2::'a,L2)) &
(\<forall>M2 O2 N2. equal(M2::'a,N2) & asymmetric(O2::'a,M2) --> asymmetric(O2::'a,N2)) &
(\<forall>P2 Q2 R2. equal(P2::'a,Q2) & connected(P2::'a,R2) --> connected(Q2::'a,R2)) &
(\<forall>S2 U2 T2. equal(S2::'a,T2) & connected(U2::'a,S2) --> connected(U2::'a,T2)) &
(\<forall>V2 W2 X2. equal(V2::'a,W2) & irreflexive(V2::'a,X2) --> irreflexive(W2::'a,X2)) &
(\<forall>Y2 A3 Z2. equal(Y2::'a,Z2) & irreflexive(A3::'a,Y2) --> irreflexive(A3::'a,Z2)) &
(\<forall>B3 C3 D3 E3. equal(B3::'a,C3) & section(B3::'a,D3,E3) --> section(C3::'a,D3,E3)) &
(\<forall>F3 H3 G3 I3. equal(F3::'a,G3) & section(H3::'a,F3,I3) --> section(H3::'a,G3,I3)) &
(\<forall>J3 L3 M3 K3. equal(J3::'a,K3) & section(L3::'a,M3,J3) --> section(L3::'a,M3,K3)) &
(\<forall>N3 O3 P3. equal(N3::'a,O3) & transitive(N3::'a,P3) --> transitive(O3::'a,P3)) &
(\<forall>Q3 S3 R3. equal(Q3::'a,R3) & transitive(S3::'a,Q3) --> transitive(S3::'a,R3)) &
(\<forall>T3 U3 V3. equal(T3::'a,U3) & well_ordering(T3::'a,V3) --> well_ordering(U3::'a,V3)) &
(\<forall>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:
"(\<forall>Situation1 Situation2. warm(Situation1) | cold(Situation2)) &
(\<forall>Situation. at(a::'a,Situation) --> at(b::'a,walk(b::'a,Situation))) &
(\<forall>Situation. at(a::'a,Situation) --> at(b::'a,drive(b::'a,Situation))) &
(\<forall>Situation. at(b::'a,Situation) --> at(a::'a,walk(a::'a,Situation))) &
(\<forall>Situation. at(b::'a,Situation) --> at(a::'a,drive(a::'a,Situation))) &
(\<forall>Situation. cold(Situation) & at(b::'a,Situation) --> at(c::'a,skate(c::'a,Situation))) &
(\<forall>Situation. cold(Situation) & at(c::'a,Situation) --> at(b::'a,skate(b::'a,Situation))) &
(\<forall>Situation. warm(Situation) & at(b::'a,Situation) --> at(d::'a,climb(d::'a,Situation))) &
(\<forall>Situation. warm(Situation) & at(d::'a,Situation) --> at(b::'a,climb(b::'a,Situation))) &
(\<forall>Situation. at(c::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
(\<forall>Situation. at(d::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
(\<forall>Situation. at(c::'a,Situation) --> at(e::'a,go(e::'a,Situation))) &
(\<forall>Situation. at(e::'a,Situation) --> at(c::'a,go(c::'a,Situation))) &
(\<forall>Situation. at(d::'a,Situation) --> at(f::'a,go(f::'a,Situation))) &
(\<forall>Situation. at(f::'a,Situation) --> at(d::'a,go(d::'a,Situation))) &
(at(f::'a,s0)) &
(\<forall>S'. ~at(a::'a,S')) --> False"
by meson
abbreviation "PLA001_0_ax putdown on pickup do holding table differ clear EMPTY and' holds \<equiv>
(\<forall>X Y State. holds(X::'a,State) & holds(Y::'a,State) --> holds(and'(X::'a,Y),State)) &
(\<forall>State X. holds(EMPTY::'a,State) & holds(clear(X),State) & differ(X::'a,table) --> holds(holding(X),do(pickup(X),State))) &
(\<forall>Y X State. holds(on(X::'a,Y),State) & holds(clear(X),State) & holds(EMPTY::'a,State) --> holds(clear(Y),do(pickup(X),State))) &
(\<forall>Y State X Z. holds(on(X::'a,Y),State) & differ(X::'a,Z) --> holds(on(X::'a,Y),do(pickup(Z),State))) &
(\<forall>State X Z. holds(clear(X),State) & differ(X::'a,Z) --> holds(clear(X),do(pickup(Z),State))) &
(\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(EMPTY::'a,do(putdown(X::'a,Y),State))) &
(\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(on(X::'a,Y),do(putdown(X::'a,Y),State))) &
(\<forall>X Y State. holds(holding(X),State) & holds(clear(Y),State) --> holds(clear(X),do(putdown(X::'a,Y),State))) &
(\<forall>Z W X Y State. holds(on(X::'a,Y),State) --> holds(on(X::'a,Y),do(putdown(Z::'a,W),State))) &
(\<forall>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 \<equiv>
(\<forall>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)) &
(\<forall>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 &
(\<forall>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 &
(\<forall>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 &
(\<forall>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 &
(\<forall>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:
"(\<forall>X Y Z. q1(X::'a,Y,Z) & mless_or_equal(X::'a,Y) --> q2(X::'a,Y,Z)) &
(\<forall>X Y Z. q1(X::'a,Y,Z) --> mless_or_equal(X::'a,Y) | q3(X::'a,Y,Z)) &
(\<forall>Z X Y. q2(X::'a,Y,Z) --> q4(X::'a,Y,Y)) &
(\<forall>Z Y X. q3(X::'a,Y,Z) --> q4(X::'a,Y,X)) &
(\<forall>X. mless_or_equal(X::'a,X)) &
(\<forall>X Y. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,X) --> equal(X::'a,Y)) &
(\<forall>Y X Z. mless_or_equal(X::'a,Y) & mless_or_equal(Y::'a,Z) --> mless_or_equal(X::'a,Z)) &
(\<forall>Y X. mless_or_equal(X::'a,Y) | mless_or_equal(Y::'a,X)) &
(\<forall>X Y. equal(X::'a,Y) --> mless_or_equal(X::'a,Y)) &
(\<forall>X Y Z. equal(X::'a,Y) & mless_or_equal(X::'a,Z) --> mless_or_equal(Y::'a,Z)) &
(\<forall>X Z Y. equal(X::'a,Y) & mless_or_equal(Z::'a,X) --> mless_or_equal(Z::'a,Y)) &
(q1(a::'a,b,c)) &
(\<forall>W. ~(q4(a::'a,b,W) & mless_or_equal(a::'a,W) & mless_or_equal(b::'a,W) & mless_or_equal(W::'a,a))) &
(\<forall>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 \<equiv>
(\<forall>X. equal(predecessor(successor(X)),X)) &
(\<forall>X. equal(successor(predecessor(X)),X)) &
(\<forall>X Y. equal(predecessor(X),predecessor(Y)) --> equal(X::'a,Y)) &
(\<forall>X Y. equal(successor(X),successor(Y)) --> equal(X::'a,Y)) &
(\<forall>X. mless_THAN(predecessor(X),X)) &
(\<forall>X. mless_THAN(X::'a,successor(X))) &
(\<forall>X Y Z. mless_THAN(X::'a,Y) & mless_THAN(Y::'a,Z) --> mless_THAN(X::'a,Z)) &
(\<forall>X Y. mless_THAN(X::'a,Y) | mless_THAN(Y::'a,X) | equal(X::'a,Y)) &
(\<forall>X. ~mless_THAN(X::'a,X)) &
(\<forall>Y X. ~(mless_THAN(X::'a,Y) & mless_THAN(Y::'a,X))) &
(\<forall>Y X Z. equal(X::'a,Y) & mless_THAN(X::'a,Z) --> mless_THAN(Y::'a,Z)) &
(\<forall>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 \<equiv>
(\<forall>X Y. equal(X::'a,Y) --> equal(predecessor(X),predecessor(Y))) &
(\<forall>X Y. equal(X::'a,Y) --> equal(successor(X),successor(Y))) &
(\<forall>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))) &
(\<forall>X. mless_THAN(X::'a,j) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,i)) &
(\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
(\<forall>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)))) &
(\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(k)) --> mless_THAN(X::'a,l)) &
(\<forall>X. mless_THAN(One::'a,l) & mless_THAN(a(X),a(predecessor(l))) --> mless_THAN(X::'a,l) | mless_THAN(n::'a,X)) &
(\<forall>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))) &
(\<forall>X. mless_THAN(X::'a,successor(n)) & mless_THAN(a(X),a(m)) --> mless_THAN(X::'a,i)) &
(\<forall>X. mless_THAN(One::'a,i) & mless_THAN(a(X),a(predecessor(i))) --> mless_THAN(X::'a,i) | mless_THAN(n::'a,X)) &
(\<forall>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:
"(\<forall>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)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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:
"(\<forall>X. equal_fruits(X::'a,X)) &
(\<forall>X. equal_boxes(X::'a,X)) &
(\<forall>X Y. ~(label(X::'a,Y) & contains(X::'a,Y))) &
(\<forall>X. contains(boxa::'a,X) | contains(boxb::'a,X) | contains(boxc::'a,X)) &
(\<forall>X. contains(X::'a,apples) | contains(X::'a,bananas) | contains(X::'a,oranges)) &
(\<forall>X Y Z. contains(X::'a,Y) & contains(X::'a,Z) --> equal_fruits(Y::'a,Z)) &
(\<forall>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 &
(\<forall>A B. equal(A::'a,B) --> equal(statement_by(A),statement_by(B))) &
(\<forall>X. person(X) --> knight(X) | knave(X)) &
(\<forall>X. ~(person(X) & knight(X) & knave(X))) &
(\<forall>X Y. says(X::'a,Y) & a_truth(Y) --> a_truth(Y)) &
(\<forall>X Y. ~(says(X::'a,Y) & equal(X::'a,Y))) &
(\<forall>Y X. says(X::'a,Y) --> equal(Y::'a,statement_by(X))) &
(\<forall>X Y. ~(person(X) & equal(X::'a,statement_by(Y)))) &
(\<forall>X. person(X) & a_truth(statement_by(X)) --> knight(X)) &
(\<forall>X. person(X) --> a_truth(statement_by(X)) | knave(X)) &
(\<forall>X Y. equal(X::'a,Y) & knight(X) --> knight(Y)) &
(\<forall>X Y. equal(X::'a,Y) & knave(X) --> knave(Y)) &
(\<forall>X Y. equal(X::'a,Y) & person(X) --> person(Y)) &
(\<forall>X Y Z. equal(X::'a,Y) & says(X::'a,Z) --> says(Y::'a,Z)) &
(\<forall>X Z Y. equal(X::'a,Y) & says(Z::'a,X) --> says(Z::'a,Y)) &
(\<forall>X Y. equal(X::'a,Y) & a_truth(X) --> a_truth(Y)) &
(\<forall>X Y. knight(X) & says(X::'a,Y) --> a_truth(Y)) &
(\<forall>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:
"(\<forall>X. a_truth(truthteller(X)) | a_truth(liar(X))) &
(\<forall>X. ~(a_truth(truthteller(X)) & a_truth(liar(X)))) &
(\<forall>Truthteller Statement. a_truth(truthteller(Truthteller)) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(Statement)) &
(\<forall>Liar Statement. ~(a_truth(liar(Liar)) & a_truth(says(Liar::'a,Statement)) & a_truth(Statement))) &
(\<forall>Statement Truthteller. a_truth(Statement) & a_truth(says(Truthteller::'a,Statement)) --> a_truth(truthteller(Truthteller))) &
(\<forall>Statement Liar. a_truth(says(Liar::'a,Statement)) --> a_truth(Statement) | a_truth(liar(Liar))) &
(\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(liar(X)) & a_truth(liar(Y)) --> a_truth(equal_type(X::'a,Y))) &
(\<forall>Z X Y. people(X::'a,Y,Z) & a_truth(truthteller(X)) & a_truth(truthteller(Y)) --> a_truth(equal_type(X::'a,Y))) &
(\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(truthteller(X)) --> a_truth(truthteller(Y))) &
(\<forall>X Y. a_truth(equal_type(X::'a,Y)) & a_truth(liar(X)) --> a_truth(liar(Y))) &
(\<forall>X Y. a_truth(truthteller(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(liar(Y))) &
(\<forall>X Y. a_truth(liar(X)) --> a_truth(equal_type(X::'a,Y)) | a_truth(truthteller(Y))) &
(\<forall>Y X. a_truth(equal_type(X::'a,Y)) --> a_truth(equal_type(Y::'a,X))) &
(\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) & a_truth(Y) --> answer(yes)) &
(\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(truthteller(X)) --> a_truth(Y) | answer(no)) &
(\<forall>X Y. ask_1_if_2(X::'a,Y) & a_truth(liar(X)) & a_truth(Y) --> answer(no)) &
(\<forall>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))) &
(\<forall>Answer. ~answer(Answer)) --> False"
oops
(*621 inferences so far. Searching to depth 18. 0.2 secs*)
lemma PUZ029_1:
"(\<forall>X. dances_on_tightropes(X) | eats_pennybuns(X) | old(X)) &
(\<forall>X. pig(X) & liable_to_giddiness(X) --> treated_with_respect(X)) &
(\<forall>X. wise(X) & balloonist(X) --> has_umbrella(X)) &
(\<forall>X. ~(looks_ridiculous(X) & eats_pennybuns(X) & eats_lunch_in_public(X))) &
(\<forall>X. balloonist(X) & young(X) --> liable_to_giddiness(X)) &
(\<forall>X. fat(X) & looks_ridiculous(X) --> dances_on_tightropes(X) | eats_lunch_in_public(X)) &
(\<forall>X. ~(liable_to_giddiness(X) & wise(X) & dances_on_tightropes(X))) &
(\<forall>X. pig(X) & has_umbrella(X) --> looks_ridiculous(X)) &
(\<forall>X. treated_with_respect(X) --> dances_on_tightropes(X) | fat(X)) &
(\<forall>X. young(X) | old(X)) &
(\<forall>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 \<equiv>
(\<forall>X. sum(additive_identity::'a,X,X)) &
(\<forall>X. sum(X::'a,additive_identity,X)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
(\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
(\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
(\<forall>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 \<equiv>
(\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
(\<forall>X W Y. equal(X::'a,Y) --> equal(add(W::'a,X),add(W::'a,Y))) &
(\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
(\<forall>X W Y. equal(X::'a,Y) --> equal(multiply(W::'a,X),multiply(W::'a,Y))) &
(\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
(\<forall>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:
"(\<forall>X. sum(additive_identity::'a,X,X)) &
(\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>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)) &
(\<forall>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 \<equiv>
(\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
(\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
(\<forall>X Y. equal(X::'a,Y) --> equal(additive_inverse(X),additive_inverse(Y))) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(add(X::'a,W),add(Y::'a,W))) &
(\<forall>X Y W Z. equal(X::'a,Y) & sum(X::'a,W,Z) --> sum(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & sum(W::'a,X,Z) --> sum(W::'a,Y,Z)) &
(\<forall>X W Z Y. equal(X::'a,Y) & sum(W::'a,Z,X) --> sum(W::'a,Z,Y)) &
(\<forall>X Y W. equal(X::'a,Y) --> equal(multiply(X::'a,W),multiply(Y::'a,W))) &
(\<forall>X Y W Z. equal(X::'a,Y) & product(X::'a,W,Z) --> product(Y::'a,W,Z)) &
(\<forall>X W Y Z. equal(X::'a,Y) & product(W::'a,X,Z) --> product(W::'a,Y,Z)) &
(\<forall>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 &
(\<forall>X. sum(additive_identity::'a,X,X)) &
(\<forall>X. sum(X::'a,additive_identity,X)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
(\<forall>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 &
(\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
(\<forall>G H. equal(G::'a,H) --> equal(additive_inverse(G),additive_inverse(H))) &
(\<forall>I' J K'. equal(I'::'a,J) --> equal(multiply(I'::'a,K'),multiply(J::'a,K'))) &
(\<forall>L N M. equal(L::'a,M) --> equal(multiply(N::'a,L),multiply(N::'a,M))) &
(\<forall>A B C D. equal(A::'a,B) --> equal(associator(A::'a,C,D),associator(B::'a,C,D))) &
(\<forall>E G F' H. equal(E::'a,F') --> equal(associator(G::'a,E,H),associator(G::'a,F',H))) &
(\<forall>I' K' L J. equal(I'::'a,J) --> equal(associator(K'::'a,L,I'),associator(K'::'a,L,J))) &
(\<forall>M N O'. equal(M::'a,N) --> equal(commutator(M::'a,O'),commutator(N::'a,O'))) &
(\<forall>P R Q. equal(P::'a,Q) --> equal(commutator(R::'a,P),commutator(R::'a,Q))) &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
(\<forall>X. equal(add(X::'a,additive_identity),X)) &
(\<forall>X. equal(add(additive_identity::'a,X),X)) &
(\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
(\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
(equal(additive_inverse(additive_identity),additive_identity)) &
(\<forall>X Y. equal(add(X::'a,add(additive_inverse(X),Y)),Y)) &
(\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
(\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
(\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
(\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
(\<forall>X Y. equal(multiply(additive_inverse(X),additive_inverse(Y)),multiply(X::'a,Y))) &
(\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
(\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
(\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
(\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
(\<forall>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)))))) &
(\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
(\<forall>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 &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
(\<forall>X. equal(add(additive_identity::'a,X),X)) &
(\<forall>X. equal(add(X::'a,additive_identity),X)) &
(\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
(\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
(\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
(\<forall>X. equal(add(X::'a,additive_inverse(X)),additive_identity)) &
(\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
(\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
(\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
(\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
(\<forall>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)))))) &
(\<forall>X Y. equal(commutator(X::'a,Y),add(multiply(Y::'a,X),additive_inverse(multiply(X::'a,Y))))) &
(\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
(\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
(\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
(\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
(\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
(\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
(\<forall>X Y Z. equal(X::'a,Y) --> equal(commutator(X::'a,Z),commutator(Y::'a,Z))) &
(\<forall>A1 C1 B1. equal(A1::'a,B1) --> equal(commutator(C1::'a,A1),commutator(C1::'a,B1))) &
(\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
(\<forall>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 &
(\<forall>X. equal(add(additive_identity::'a,X),X)) &
(\<forall>X. equal(multiply(additive_identity::'a,X),additive_identity)) &
(\<forall>X. equal(multiply(X::'a,additive_identity),additive_identity)) &
(\<forall>X. equal(add(additive_inverse(X),X),additive_identity)) &
(\<forall>X Y. equal(additive_inverse(add(X::'a,Y)),add(additive_inverse(X),additive_inverse(Y)))) &
(\<forall>X. equal(additive_inverse(additive_inverse(X)),X)) &
(\<forall>Y X Z. equal(multiply(X::'a,add(Y::'a,Z)),add(multiply(X::'a,Y),multiply(X::'a,Z)))) &
(\<forall>X Y Z. equal(multiply(add(X::'a,Y),Z),add(multiply(X::'a,Z),multiply(Y::'a,Z)))) &
(\<forall>X Y. equal(multiply(multiply(X::'a,Y),Y),multiply(X::'a,multiply(Y::'a,Y)))) &
(\<forall>X Y. equal(multiply(multiply(X::'a,X),Y),multiply(X::'a,multiply(X::'a,Y)))) &
(\<forall>X Y. equal(multiply(additive_inverse(X),Y),additive_inverse(multiply(X::'a,Y)))) &
(\<forall>X Y. equal(multiply(X::'a,additive_inverse(Y)),additive_inverse(multiply(X::'a,Y)))) &
(equal(additive_inverse(additive_identity),additive_identity)) &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(X::'a,add(Y::'a,Z)),add(add(X::'a,Y),Z))) &
(\<forall>Z X Y. equal(add(X::'a,Z),add(Y::'a,Z)) --> equal(X::'a,Y)) &
(\<forall>Z X Y. equal(add(Z::'a,X),add(Z::'a,Y)) --> equal(X::'a,Y)) &
(\<forall>D E F'. equal(D::'a,E) --> equal(add(D::'a,F'),add(E::'a,F'))) &
(\<forall>G I' H. equal(G::'a,H) --> equal(add(I'::'a,G),add(I'::'a,H))) &
(\<forall>J K'. equal(J::'a,K') --> equal(additive_inverse(J),additive_inverse(K'))) &
(\<forall>D1 E1 F1. equal(D1::'a,E1) --> equal(multiply(D1::'a,F1),multiply(E1::'a,F1))) &
(\<forall>G1 I1 H1. equal(G1::'a,H1) --> equal(multiply(I1::'a,G1),multiply(I1::'a,H1))) &
(\<forall>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)))))) &
(\<forall>L M N O'. equal(L::'a,M) --> equal(associator(L::'a,N,O'),associator(M::'a,N,O'))) &
(\<forall>P R Q S'. equal(P::'a,Q) --> equal(associator(R::'a,P,S'),associator(R::'a,Q,S'))) &
(\<forall>T' V W U. equal(T'::'a,U) --> equal(associator(V::'a,W,T'),associator(V::'a,W,U))) &
(\<forall>X Y. ~equal(multiply(multiply(Y::'a,X),Y),multiply(Y::'a,multiply(X::'a,Y)))) &
(\<forall>X Y Z. ~equal(associator(Y::'a,X,Z),additive_inverse(associator(X::'a,Y,Z)))) &
(\<forall>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:
"(\<forall>X. sum(X::'a,additive_identity,X)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
RNG_other_ax multiply add equal product additive_identity additive_inverse sum &
(\<forall>X. product(additive_identity::'a,X,additive_identity)) &
(\<forall>X. product(X::'a,additive_identity,additive_identity)) &
(\<forall>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 &
(\<forall>X. sum(additive_identity::'a,X,X)) &
(\<forall>X. sum(X::'a,additive_identity,X)) &
(\<forall>X Y. product(X::'a,Y,multiply(X::'a,Y))) &
(\<forall>X Y. sum(X::'a,Y,add(X::'a,Y))) &
(\<forall>X. sum(additive_inverse(X),X,additive_identity)) &
(\<forall>X. sum(X::'a,additive_inverse(X),additive_identity)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>Y X Z. sum(X::'a,Y,Z) --> sum(Y::'a,X,Z)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X Y U V. sum(X::'a,Y,U) & sum(X::'a,Y,V) --> equal(U::'a,V)) &
(\<forall>X Y U V. product(X::'a,Y,U) & product(X::'a,Y,V) --> equal(U::'a,V)) &
(\<forall>A. product(A::'a,multiplicative_identity,A)) &
(\<forall>A. product(multiplicative_identity::'a,A,A)) &
(\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
(\<forall>A. product(h(A),A,multiplicative_identity) | equal(A::'a,additive_identity)) &
(\<forall>B A C. product(A::'a,B,C) --> product(B::'a,A,C)) &
(\<forall>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 &
(\<forall>A B. equal(A::'a,B) --> equal(h(A),h(B))) &
(\<forall>A. product(additive_identity::'a,A,additive_identity)) &
(\<forall>A. product(A::'a,additive_identity,additive_identity)) &
(\<forall>A. product(A::'a,multiplicative_identity,A)) &
(\<forall>A. product(multiplicative_identity::'a,A,A)) &
(\<forall>A. product(A::'a,h(A),multiplicative_identity) | equal(A::'a,additive_identity)) &
(\<forall>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 &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
(\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
(\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
(\<forall>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 &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
(\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
(\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
(\<forall>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 &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
(\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
(\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
(\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
(\<forall>J K' L. equal(J::'a,K') --> equal(multiply(J::'a,L),multiply(K'::'a,L))) &
(\<forall>M O' N. equal(M::'a,N) --> equal(multiply(O'::'a,M),multiply(O'::'a,N))) &
(\<forall>P Q. equal(P::'a,Q) --> equal(successor(P),successor(Q))) &
(\<forall>R S'. equal(R::'a,S') & positive_integer(R) --> positive_integer(S')) &
(\<forall>X. equal(multiply(One::'a,X),X)) &
(\<forall>V X. positive_integer(X) --> equal(multiply(successor(V),X),add(X::'a,multiply(V::'a,X)))) &
(positive_integer(One)) &
(\<forall>X. positive_integer(X) --> positive_integer(successor(X))) &
(equal(negate(add(d::'a,e)),negate(e))) &
(positive_integer(k)) &
(\<forall>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 &
(\<forall>Y X. equal(add(X::'a,Y),add(Y::'a,X))) &
(\<forall>X Y Z. equal(add(add(X::'a,Y),Z),add(X::'a,add(Y::'a,Z)))) &
(\<forall>Y X. equal(negate(add(negate(add(X::'a,Y)),negate(add(X::'a,negate(Y))))),X)) &
(\<forall>A B C. equal(A::'a,B) --> equal(add(A::'a,C),add(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(add(F'::'a,D),add(F'::'a,E))) &
(\<forall>G H. equal(G::'a,H) --> equal(negate(G),negate(H))) &
(\<forall>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:
"(\<forall>Subset Element Superset. member(Element::'a,Subset) & subset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
(\<forall>Superset Subset. subset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
(\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> subset(Subset::'a,Superset)) &
(\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> subset(Subset::'a,Superset)) &
(\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> subset(Subset::'a,Superset)) &
(\<forall>Set2 Set1. subset(Set1::'a,Set2) & subset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
(\<forall>Set2 Intersection Element Set1. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set1)) &
(\<forall>Set1 Intersection Element Set2. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Intersection) --> member(Element::'a,Set2)) &
(\<forall>Set2 Set1 Element Intersection. intersection(Set1::'a,Set2,Intersection) & member(Element::'a,Set2) & member(Element::'a,Set1) --> member(Element::'a,Intersection)) &
(\<forall>Set2 Intersection Set1. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set1)) &
(\<forall>Set1 Intersection Set2. member(h(Set1::'a,Set2,Intersection),Intersection) | intersection(Set1::'a,Set2,Intersection) | member(h(Set1::'a,Set2,Intersection),Set2)) &
(\<forall>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:
"(\<forall>Subset Element Superset. member(Element::'a,Subset) & ssubset(Subset::'a,Superset) --> member(Element::'a,Superset)) &
(\<forall>Superset Subset. ssubset(Subset::'a,Superset) | member(member_of_1_not_of_2(Subset::'a,Superset),Subset)) &
(\<forall>Subset Superset. member(member_of_1_not_of_2(Subset::'a,Superset),Superset) --> ssubset(Subset::'a,Superset)) &
(\<forall>Subset Superset. equal_sets(Subset::'a,Superset) --> ssubset(Subset::'a,Superset)) &
(\<forall>Subset Superset. equal_sets(Superset::'a,Subset) --> ssubset(Subset::'a,Superset)) &
(\<forall>Set2 Set1. ssubset(Set1::'a,Set2) & ssubset(Set2::'a,Set1) --> equal_sets(Set2::'a,Set1)) &
(\<forall>Set2 Difference Element Set1. difference(Set1::'a,Set2,Difference) & member(Element::'a,Difference) --> member(Element::'a,Set1)) &
(\<forall>Element A_set Set1 Set2. ~(member(Element::'a,Set1) & member(Element::'a,Set2) & difference(A_set::'a,Set1,Set2))) &
(\<forall>Set1 Difference Element Set2. member(Element::'a,Set1) & difference(Set1::'a,Set2,Difference) --> member(Element::'a,Difference) | member(Element::'a,Set2)) &
(\<forall>Set1 Set2 Difference. difference(Set1::'a,Set2,Difference) | member(k(Set1::'a,Set2,Difference),Set1) | member(k(Set1::'a,Set2,Difference),Difference)) &
(\<forall>Set1 Set2 Difference. member(k(Set1::'a,Set2,Difference),Set2) --> member(k(Set1::'a,Set2,Difference),Difference) | difference(Set1::'a,Set2,Difference)) &
(\<forall>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 &
(\<forall>Y X. member(X::'a,Y) --> little_set(X)) &
(\<forall>X Y. little_set(f1(X::'a,Y)) | equal(X::'a,Y)) &
(\<forall>X Y. member(f1(X::'a,Y),X) | member(f1(X::'a,Y),Y) | equal(X::'a,Y)) &
(\<forall>X Y. member(f1(X::'a,Y),X) & member(f1(X::'a,Y),Y) --> equal(X::'a,Y)) &
(\<forall>X U Y. member(U::'a,non_ordered_pair(X::'a,Y)) --> equal(U::'a,X) | equal(U::'a,Y)) &
(\<forall>Y U X. little_set(U) & equal(U::'a,X) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
(\<forall>X U Y. little_set(U) & equal(U::'a,Y) --> member(U::'a,non_ordered_pair(X::'a,Y))) &
(\<forall>X Y. little_set(non_ordered_pair(X::'a,Y))) &
(\<forall>X. equal(singleton_set(X),non_ordered_pair(X::'a,X))) &
(\<forall>X Y. equal(ordered_pair(X::'a,Y),non_ordered_pair(singleton_set(X),non_ordered_pair(X::'a,Y)))) &
(\<forall>X. ordered_pair_predicate(X) --> little_set(f2(X))) &
(\<forall>X. ordered_pair_predicate(X) --> little_set(f3(X))) &
(\<forall>X. ordered_pair_predicate(X) --> equal(X::'a,ordered_pair(f2(X),f3(X)))) &
(\<forall>X Y Z. little_set(Y) & little_set(Z) & equal(X::'a,ordered_pair(Y::'a,Z)) --> ordered_pair_predicate(X)) &
(\<forall>Z X. member(Z::'a,first(X)) --> little_set(f4(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,first(X)) --> little_set(f5(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,first(X)) --> equal(X::'a,ordered_pair(f4(Z::'a,X),f5(Z::'a,X)))) &
(\<forall>Z X. member(Z::'a,first(X)) --> member(Z::'a,f4(Z::'a,X))) &
(\<forall>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))) &
(\<forall>Z X. member(Z::'a,second(X)) --> little_set(f6(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,second(X)) --> little_set(f7(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,second(X)) --> equal(X::'a,ordered_pair(f6(Z::'a,X),f7(Z::'a,X)))) &
(\<forall>Z X. member(Z::'a,second(X)) --> member(Z::'a,f7(Z::'a,X))) &
(\<forall>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))) &
(\<forall>Z. member(Z::'a,estin) --> ordered_pair_predicate(Z)) &
(\<forall>Z. member(Z::'a,estin) --> member(first(Z),second(Z))) &
(\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & member(first(Z),second(Z)) --> member(Z::'a,estin)) &
(\<forall>Y Z X. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,X)) &
(\<forall>X Z Y. member(Z::'a,intersection(X::'a,Y)) --> member(Z::'a,Y)) &
(\<forall>X Z Y. member(Z::'a,X) & member(Z::'a,Y) --> member(Z::'a,intersection(X::'a,Y))) &
(\<forall>Z X. ~(member(Z::'a,complement(X)) & member(Z::'a,X))) &
(\<forall>Z X. little_set(Z) --> member(Z::'a,complement(X)) | member(Z::'a,X)) &
(\<forall>X Y. equal(union(X::'a,Y),complement(intersection(complement(X),complement(Y))))) &
(\<forall>Z X. member(Z::'a,domain_of(X)) --> ordered_pair_predicate(f8(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,domain_of(X)) --> member(f8(Z::'a,X),X)) &
(\<forall>Z X. member(Z::'a,domain_of(X)) --> equal(Z::'a,first(f8(Z::'a,X)))) &
(\<forall>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))) &
(\<forall>X Y Z. member(Z::'a,cross_product(X::'a,Y)) --> ordered_pair_predicate(Z)) &
(\<forall>Y Z X. member(Z::'a,cross_product(X::'a,Y)) --> member(first(Z),X)) &
(\<forall>X Z Y. member(Z::'a,cross_product(X::'a,Y)) --> member(second(Z),Y)) &
(\<forall>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))) &
(\<forall>X Z. member(Z::'a,inv1 X) --> ordered_pair_predicate(Z)) &
(\<forall>Z X. member(Z::'a,inv1 X) --> member(ordered_pair(second(Z),first(Z)),X)) &
(\<forall>Z X. little_set(Z) & ordered_pair_predicate(Z) & member(ordered_pair(second(Z),first(Z)),X) --> member(Z::'a,inv1 X)) &
(\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f9(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f10(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,rot_right(X)) --> little_set(f11(Z::'a,X))) &
(\<forall>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))))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f12(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f13(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,flip_range_of(X)) --> little_set(f14(Z::'a,X))) &
(\<forall>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))))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>X. equal(successor(X),union(X::'a,singleton_set(X)))) &
(\<forall>Z. ~member(Z::'a,empty_set)) &
(\<forall>Z. little_set(Z) --> member(Z::'a,universal_set)) &
(little_set(infinity)) &
(member(empty_set::'a,infinity)) &
(\<forall>X. member(X::'a,infinity) --> member(successor(X),infinity)) &
(\<forall>Z X. member(Z::'a,sigma(X)) --> member(f16(Z::'a,X),X)) &
(\<forall>Z X. member(Z::'a,sigma(X)) --> member(Z::'a,f16(Z::'a,X))) &
(\<forall>X Z Y. member(Y::'a,X) & member(Z::'a,Y) --> member(Z::'a,sigma(X))) &
(\<forall>U. little_set(U) --> little_set(sigma(U))) &
(\<forall>X U Y. ssubset(X::'a,Y) & member(U::'a,X) --> member(U::'a,Y)) &
(\<forall>Y X. ssubset(X::'a,Y) | member(f17(X::'a,Y),X)) &
(\<forall>X Y. member(f17(X::'a,Y),Y) --> ssubset(X::'a,Y)) &
(\<forall>X Y. proper_subset(X::'a,Y) --> ssubset(X::'a,Y)) &
(\<forall>X Y. ~(proper_subset(X::'a,Y) & equal(X::'a,Y))) &
(\<forall>X Y. ssubset(X::'a,Y) --> proper_subset(X::'a,Y) | equal(X::'a,Y)) &
(\<forall>Z X. member(Z::'a,powerset(X)) --> ssubset(Z::'a,X)) &
(\<forall>Z X. little_set(Z) & ssubset(Z::'a,X) --> member(Z::'a,powerset(X))) &
(\<forall>U. little_set(U) --> little_set(powerset(U))) &
(\<forall>Z X. relation(Z) & member(X::'a,Z) --> ordered_pair_predicate(X)) &
(\<forall>Z. relation(Z) | member(f18(Z),Z)) &
(\<forall>Z. ordered_pair_predicate(f18(Z)) --> relation(Z)) &
(\<forall>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)) &
(\<forall>X. single_valued_set(X) | little_set(f19(X))) &
(\<forall>X. single_valued_set(X) | little_set(f20(X))) &
(\<forall>X. single_valued_set(X) | little_set(f21(X))) &
(\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f20(X)),X)) &
(\<forall>X. single_valued_set(X) | member(ordered_pair(f19(X),f21(X)),X)) &
(\<forall>X. equal(f20(X),f21(X)) --> single_valued_set(X)) &
(\<forall>Xf. function(Xf) --> relation(Xf)) &
(\<forall>Xf. function(Xf) --> single_valued_set(Xf)) &
(\<forall>Xf. relation(Xf) & single_valued_set(Xf) --> function(Xf)) &
(\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> ordered_pair_predicate(f22(Z::'a,X,Xf))) &
(\<forall>Z X Xf. member(Z::'a,image'(X::'a,Xf)) --> member(f22(Z::'a,X,Xf),Xf)) &
(\<forall>Z Xf X. member(Z::'a,image'(X::'a,Xf)) --> member(first(f22(Z::'a,X,Xf)),X)) &
(\<forall>X Xf Z. member(Z::'a,image'(X::'a,Xf)) --> equal(second(f22(Z::'a,X,Xf)),Z)) &
(\<forall>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))) &
(\<forall>X Xf. little_set(X) & function(Xf) --> little_set(image'(X::'a,Xf))) &
(\<forall>X U Y. ~(disjoint(X::'a,Y) & member(U::'a,X) & member(U::'a,Y))) &
(\<forall>Y X. disjoint(X::'a,Y) | member(f23(X::'a,Y),X)) &
(\<forall>X Y. disjoint(X::'a,Y) | member(f23(X::'a,Y),Y)) &
(\<forall>X. equal(X::'a,empty_set) | member(f24(X),X)) &
(\<forall>X. equal(X::'a,empty_set) | disjoint(f24(X),X)) &
(function(f25)) &
(\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(f26(X),X)) &
(\<forall>X. little_set(X) --> equal(X::'a,empty_set) | member(ordered_pair(X::'a,f26(X)),f25)) &
(\<forall>Z X. member(Z::'a,range_of(X)) --> ordered_pair_predicate(f27(Z::'a,X))) &
(\<forall>Z X. member(Z::'a,range_of(X)) --> member(f27(Z::'a,X),X)) &
(\<forall>Z X. member(Z::'a,range_of(X)) --> equal(Z::'a,second(f27(Z::'a,X)))) &
(\<forall>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))) &
(\<forall>Z. member(Z::'a,identity_relation) --> ordered_pair_predicate(Z)) &
(\<forall>Z. member(Z::'a,identity_relation) --> equal(first(Z),second(Z))) &
(\<forall>Z. little_set(Z) & ordered_pair_predicate(Z) & equal(first(Z),second(Z)) --> member(Z::'a,identity_relation)) &
(\<forall>X Y. equal(restrct(X::'a,Y),intersection(X::'a,cross_product(Y::'a,universal_set)))) &
(\<forall>Xf. one_to_one_function(Xf) --> function(Xf)) &
(\<forall>Xf. one_to_one_function(Xf) --> function(inv1 Xf)) &
(\<forall>Xf. function(Xf) & function(inv1 Xf) --> one_to_one_function(Xf)) &
(\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> ordered_pair_predicate(f28(Z::'a,Xf,Y))) &
(\<forall>Z Y Xf. member(Z::'a,apply(Xf::'a,Y)) --> member(f28(Z::'a,Xf,Y),Xf)) &
(\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> equal(first(f28(Z::'a,Xf,Y)),Y)) &
(\<forall>Z Xf Y. member(Z::'a,apply(Xf::'a,Y)) --> member(Z::'a,second(f28(Z::'a,Xf,Y)))) &
(\<forall>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))) &
(\<forall>Xf X Y. equal(apply_to_two_arguments(Xf::'a,X,Y),apply(Xf::'a,ordered_pair(X::'a,Y)))) &
(\<forall>X Y Xf. maps(Xf::'a,X,Y) --> function(Xf)) &
(\<forall>Y Xf X. maps(Xf::'a,X,Y) --> equal(domain_of(Xf),X)) &
(\<forall>X Xf Y. maps(Xf::'a,X,Y) --> ssubset(range_of(Xf),Y)) &
(\<forall>X Xf Y. function(Xf) & equal(domain_of(Xf),X) & ssubset(range_of(Xf),Y) --> maps(Xf::'a,X,Y)) &
(\<forall>Xf Xs. closed(Xs::'a,Xf) --> little_set(Xs)) &
(\<forall>Xs Xf. closed(Xs::'a,Xf) --> little_set(Xf)) &
(\<forall>Xf Xs. closed(Xs::'a,Xf) --> maps(Xf::'a,cross_product(Xs::'a,Xs),Xs)) &
(\<forall>Xf Xs. little_set(Xs) & little_set(Xf) & maps(Xf::'a,cross_product(Xs::'a,Xs),Xs) --> closed(Xs::'a,Xf)) &
(\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f29(Z::'a,Xf,Xg))) &
(\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f30(Z::'a,Xf,Xg))) &
(\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> little_set(f31(Z::'a,Xf,Xg))) &
(\<forall>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)))) &
(\<forall>Z Xg Xf. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f29(Z::'a,Xf,Xg),f31(Z::'a,Xf,Xg)),Xf)) &
(\<forall>Z Xf Xg. member(Z::'a,composition(Xf::'a,Xg)) --> member(ordered_pair(f31(Z::'a,Xf,Xg),f30(Z::'a,Xf,Xg)),Xg)) &
(\<forall>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))) &
(\<forall>Xh Xs2 Xf2 Xs1 Xf1. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs1::'a,Xf1)) &
(\<forall>Xh Xs1 Xf1 Xs2 Xf2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> closed(Xs2::'a,Xf2)) &
(\<forall>Xf1 Xf2 Xh Xs1 Xs2. homomorphism(Xh::'a,Xs1,Xf1,Xs2,Xf2) --> maps(Xh::'a,Xs1,Xs2)) &
(\<forall>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)))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>A B C. equal(A::'a,B) --> equal(f1(A::'a,C),f1(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(f1(F'::'a,D),f1(F'::'a,E))) &
(\<forall>A2 B2. equal(A2::'a,B2) --> equal(f2(A2),f2(B2))) &
(\<forall>G4 H4. equal(G4::'a,H4) --> equal(f3(G4),f3(H4))) &
(\<forall>O7 P7 Q7. equal(O7::'a,P7) --> equal(f4(O7::'a,Q7),f4(P7::'a,Q7))) &
(\<forall>R7 T7 S7. equal(R7::'a,S7) --> equal(f4(T7::'a,R7),f4(T7::'a,S7))) &
(\<forall>U7 V7 W7. equal(U7::'a,V7) --> equal(f5(U7::'a,W7),f5(V7::'a,W7))) &
(\<forall>X7 Z7 Y7. equal(X7::'a,Y7) --> equal(f5(Z7::'a,X7),f5(Z7::'a,Y7))) &
(\<forall>A8 B8 C8. equal(A8::'a,B8) --> equal(f6(A8::'a,C8),f6(B8::'a,C8))) &
(\<forall>D8 F8 E8. equal(D8::'a,E8) --> equal(f6(F8::'a,D8),f6(F8::'a,E8))) &
(\<forall>G8 H8 I8. equal(G8::'a,H8) --> equal(f7(G8::'a,I8),f7(H8::'a,I8))) &
(\<forall>J8 L8 K8. equal(J8::'a,K8) --> equal(f7(L8::'a,J8),f7(L8::'a,K8))) &
(\<forall>M8 N8 O8. equal(M8::'a,N8) --> equal(f8(M8::'a,O8),f8(N8::'a,O8))) &
(\<forall>P8 R8 Q8. equal(P8::'a,Q8) --> equal(f8(R8::'a,P8),f8(R8::'a,Q8))) &
(\<forall>S8 T8 U8. equal(S8::'a,T8) --> equal(f9(S8::'a,U8),f9(T8::'a,U8))) &
(\<forall>V8 X8 W8. equal(V8::'a,W8) --> equal(f9(X8::'a,V8),f9(X8::'a,W8))) &
(\<forall>G H I'. equal(G::'a,H) --> equal(f10(G::'a,I'),f10(H::'a,I'))) &
(\<forall>J L K'. equal(J::'a,K') --> equal(f10(L::'a,J),f10(L::'a,K'))) &
(\<forall>M N O'. equal(M::'a,N) --> equal(f11(M::'a,O'),f11(N::'a,O'))) &
(\<forall>P R Q. equal(P::'a,Q) --> equal(f11(R::'a,P),f11(R::'a,Q))) &
(\<forall>S' T' U. equal(S'::'a,T') --> equal(f12(S'::'a,U),f12(T'::'a,U))) &
(\<forall>V X W. equal(V::'a,W) --> equal(f12(X::'a,V),f12(X::'a,W))) &
(\<forall>Y Z A1. equal(Y::'a,Z) --> equal(f13(Y::'a,A1),f13(Z::'a,A1))) &
(\<forall>B1 D1 C1. equal(B1::'a,C1) --> equal(f13(D1::'a,B1),f13(D1::'a,C1))) &
(\<forall>E1 F1 G1. equal(E1::'a,F1) --> equal(f14(E1::'a,G1),f14(F1::'a,G1))) &
(\<forall>H1 J1 I1. equal(H1::'a,I1) --> equal(f14(J1::'a,H1),f14(J1::'a,I1))) &
(\<forall>K1 L1 M1. equal(K1::'a,L1) --> equal(f16(K1::'a,M1),f16(L1::'a,M1))) &
(\<forall>N1 P1 O1. equal(N1::'a,O1) --> equal(f16(P1::'a,N1),f16(P1::'a,O1))) &
(\<forall>Q1 R1 S1. equal(Q1::'a,R1) --> equal(f17(Q1::'a,S1),f17(R1::'a,S1))) &
(\<forall>T1 V1 U1. equal(T1::'a,U1) --> equal(f17(V1::'a,T1),f17(V1::'a,U1))) &
(\<forall>W1 X1. equal(W1::'a,X1) --> equal(f18(W1),f18(X1))) &
(\<forall>Y1 Z1. equal(Y1::'a,Z1) --> equal(f19(Y1),f19(Z1))) &
(\<forall>C2 D2. equal(C2::'a,D2) --> equal(f20(C2),f20(D2))) &
(\<forall>E2 F2. equal(E2::'a,F2) --> equal(f21(E2),f21(F2))) &
(\<forall>G2 H2 I2 J2. equal(G2::'a,H2) --> equal(f22(G2::'a,I2,J2),f22(H2::'a,I2,J2))) &
(\<forall>K2 M2 L2 N2. equal(K2::'a,L2) --> equal(f22(M2::'a,K2,N2),f22(M2::'a,L2,N2))) &
(\<forall>O2 Q2 R2 P2. equal(O2::'a,P2) --> equal(f22(Q2::'a,R2,O2),f22(Q2::'a,R2,P2))) &
(\<forall>S2 T2 U2. equal(S2::'a,T2) --> equal(f23(S2::'a,U2),f23(T2::'a,U2))) &
(\<forall>V2 X2 W2. equal(V2::'a,W2) --> equal(f23(X2::'a,V2),f23(X2::'a,W2))) &
(\<forall>Y2 Z2. equal(Y2::'a,Z2) --> equal(f24(Y2),f24(Z2))) &
(\<forall>A3 B3. equal(A3::'a,B3) --> equal(f26(A3),f26(B3))) &
(\<forall>C3 D3 E3. equal(C3::'a,D3) --> equal(f27(C3::'a,E3),f27(D3::'a,E3))) &
(\<forall>F3 H3 G3. equal(F3::'a,G3) --> equal(f27(H3::'a,F3),f27(H3::'a,G3))) &
(\<forall>I3 J3 K3 L3. equal(I3::'a,J3) --> equal(f28(I3::'a,K3,L3),f28(J3::'a,K3,L3))) &
(\<forall>M3 O3 N3 P3. equal(M3::'a,N3) --> equal(f28(O3::'a,M3,P3),f28(O3::'a,N3,P3))) &
(\<forall>Q3 S3 T3 R3. equal(Q3::'a,R3) --> equal(f28(S3::'a,T3,Q3),f28(S3::'a,T3,R3))) &
(\<forall>U3 V3 W3 X3. equal(U3::'a,V3) --> equal(f29(U3::'a,W3,X3),f29(V3::'a,W3,X3))) &
(\<forall>Y3 A4 Z3 B4. equal(Y3::'a,Z3) --> equal(f29(A4::'a,Y3,B4),f29(A4::'a,Z3,B4))) &
(\<forall>C4 E4 F4 D4. equal(C4::'a,D4) --> equal(f29(E4::'a,F4,C4),f29(E4::'a,F4,D4))) &
(\<forall>I4 J4 K4 L4. equal(I4::'a,J4) --> equal(f30(I4::'a,K4,L4),f30(J4::'a,K4,L4))) &
(\<forall>M4 O4 N4 P4. equal(M4::'a,N4) --> equal(f30(O4::'a,M4,P4),f30(O4::'a,N4,P4))) &
(\<forall>Q4 S4 T4 R4. equal(Q4::'a,R4) --> equal(f30(S4::'a,T4,Q4),f30(S4::'a,T4,R4))) &
(\<forall>U4 V4 W4 X4. equal(U4::'a,V4) --> equal(f31(U4::'a,W4,X4),f31(V4::'a,W4,X4))) &
(\<forall>Y4 A5 Z4 B5. equal(Y4::'a,Z4) --> equal(f31(A5::'a,Y4,B5),f31(A5::'a,Z4,B5))) &
(\<forall>C5 E5 F5 D5. equal(C5::'a,D5) --> equal(f31(E5::'a,F5,C5),f31(E5::'a,F5,D5))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>A B C. equal(A::'a,B) --> equal(apply(A::'a,C),apply(B::'a,C))) &
(\<forall>D F' E. equal(D::'a,E) --> equal(apply(F'::'a,D),apply(F'::'a,E))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>S' T'. equal(S'::'a,T') --> equal(complement(S'),complement(T'))) &
(\<forall>U V W. equal(U::'a,V) --> equal(composition(U::'a,W),composition(V::'a,W))) &
(\<forall>X Z Y. equal(X::'a,Y) --> equal(composition(Z::'a,X),composition(Z::'a,Y))) &
(\<forall>A1 B1. equal(A1::'a,B1) --> equal(inv1 A1,inv1 B1)) &
(\<forall>C1 D1 E1. equal(C1::'a,D1) --> equal(cross_product(C1::'a,E1),cross_product(D1::'a,E1))) &
(\<forall>F1 H1 G1. equal(F1::'a,G1) --> equal(cross_product(H1::'a,F1),cross_product(H1::'a,G1))) &
(\<forall>I1 J1. equal(I1::'a,J1) --> equal(domain_of(I1),domain_of(J1))) &
(\<forall>I10 J10. equal(I10::'a,J10) --> equal(first(I10),first(J10))) &
(\<forall>Q10 R10. equal(Q10::'a,R10) --> equal(flip_range_of(Q10),flip_range_of(R10))) &
(\<forall>S10 T10 U10. equal(S10::'a,T10) --> equal(image'(S10::'a,U10),image'(T10::'a,U10))) &
(\<forall>V10 X10 W10. equal(V10::'a,W10) --> equal(image'(X10::'a,V10),image'(X10::'a,W10))) &
(\<forall>Y10 Z10 A11. equal(Y10::'a,Z10) --> equal(intersection(Y10::'a,A11),intersection(Z10::'a,A11))) &
(\<forall>B11 D11 C11. equal(B11::'a,C11) --> equal(intersection(D11::'a,B11),intersection(D11::'a,C11))) &
(\<forall>E11 F11 G11. equal(E11::'a,F11) --> equal(non_ordered_pair(E11::'a,G11),non_ordered_pair(F11::'a,G11))) &
(\<forall>H11 J11 I11. equal(H11::'a,I11) --> equal(non_ordered_pair(J11::'a,H11),non_ordered_pair(J11::'a,I11))) &
(\<forall>K11 L11 M11. equal(K11::'a,L11) --> equal(ordered_pair(K11::'a,M11),ordered_pair(L11::'a,M11))) &
(\<forall>N11 P11 O11. equal(N11::'a,O11) --> equal(ordered_pair(P11::'a,N11),ordered_pair(P11::'a,O11))) &
(\<forall>Q11 R11. equal(Q11::'a,R11) --> equal(powerset(Q11),powerset(R11))) &
(\<forall>S11 T11. equal(S11::'a,T11) --> equal(range_of(S11),range_of(T11))) &
(\<forall>U11 V11 W11. equal(U11::'a,V11) --> equal(restrct(U11::'a,W11),restrct(V11::'a,W11))) &
(\<forall>X11 Z11 Y11. equal(X11::'a,Y11) --> equal(restrct(Z11::'a,X11),restrct(Z11::'a,Y11))) &
(\<forall>A12 B12. equal(A12::'a,B12) --> equal(rot_right(A12),rot_right(B12))) &
(\<forall>C12 D12. equal(C12::'a,D12) --> equal(second(C12),second(D12))) &
(\<forall>K12 L12. equal(K12::'a,L12) --> equal(sigma(K12),sigma(L12))) &
(\<forall>M12 N12. equal(M12::'a,N12) --> equal(singleton_set(M12),singleton_set(N12))) &
(\<forall>O12 P12. equal(O12::'a,P12) --> equal(successor(O12),successor(P12))) &
(\<forall>Q12 R12 S12. equal(Q12::'a,R12) --> equal(union(Q12::'a,S12),union(R12::'a,S12))) &
(\<forall>T12 V12 U12. equal(T12::'a,U12) --> equal(union(V12::'a,T12),union(V12::'a,U12))) &
(\<forall>W12 X12 Y12. equal(W12::'a,X12) & closed(W12::'a,Y12) --> closed(X12::'a,Y12)) &
(\<forall>Z12 B13 A13. equal(Z12::'a,A13) & closed(B13::'a,Z12) --> closed(B13::'a,A13)) &
(\<forall>C13 D13 E13. equal(C13::'a,D13) & disjoint(C13::'a,E13) --> disjoint(D13::'a,E13)) &
(\<forall>F13 H13 G13. equal(F13::'a,G13) & disjoint(H13::'a,F13) --> disjoint(H13::'a,G13)) &
(\<forall>I13 J13. equal(I13::'a,J13) & function(I13) --> function(J13)) &
(\<forall>K13 L13 M13 N13 O13 P13. equal(K13::'a,L13) & homomorphism(K13::'a,M13,N13,O13,P13) --> homomorphism(L13::'a,M13,N13,O13,P13)) &
(\<forall>Q13 S13 R13 T13 U13 V13. equal(Q13::'a,R13) & homomorphism(S13::'a,Q13,T13,U13,V13) --> homomorphism(S13::'a,R13,T13,U13,V13)) &
(\<forall>W13 Y13 Z13 X13 A14 B14. equal(W13::'a,X13) & homomorphism(Y13::'a,Z13,W13,A14,B14) --> homomorphism(Y13::'a,Z13,X13,A14,B14)) &
(\<forall>C14 E14 F14 G14 D14 H14. equal(C14::'a,D14) & homomorphism(E14::'a,F14,G14,C14,H14) --> homomorphism(E14::'a,F14,G14,D14,H14)) &
(\<forall>I14 K14 L14 M14 N14 J14. equal(I14::'a,J14) & homomorphism(K14::'a,L14,M14,N14,I14) --> homomorphism(K14::'a,L14,M14,N14,J14)) &
(\<forall>O14 P14. equal(O14::'a,P14) & little_set(O14) --> little_set(P14)) &
(\<forall>Q14 R14 S14 T14. equal(Q14::'a,R14) & maps(Q14::'a,S14,T14) --> maps(R14::'a,S14,T14)) &
(\<forall>U14 W14 V14 X14. equal(U14::'a,V14) & maps(W14::'a,U14,X14) --> maps(W14::'a,V14,X14)) &
(\<forall>Y14 A15 B15 Z14. equal(Y14::'a,Z14) & maps(A15::'a,B15,Y14) --> maps(A15::'a,B15,Z14)) &
(\<forall>C15 D15 E15. equal(C15::'a,D15) & member(C15::'a,E15) --> member(D15::'a,E15)) &
(\<forall>F15 H15 G15. equal(F15::'a,G15) & member(H15::'a,F15) --> member(H15::'a,G15)) &
(\<forall>I15 J15. equal(I15::'a,J15) & one_to_one_function(I15) --> one_to_one_function(J15)) &
(\<forall>K15 L15. equal(K15::'a,L15) & ordered_pair_predicate(K15) --> ordered_pair_predicate(L15)) &
(\<forall>M15 N15 O15. equal(M15::'a,N15) & proper_subset(M15::'a,O15) --> proper_subset(N15::'a,O15)) &
(\<forall>P15 R15 Q15. equal(P15::'a,Q15) & proper_subset(R15::'a,P15) --> proper_subset(R15::'a,Q15)) &
(\<forall>S15 T15. equal(S15::'a,T15) & relation(S15) --> relation(T15)) &
(\<forall>U15 V15. equal(U15::'a,V15) & single_valued_set(U15) --> single_valued_set(V15)) &
(\<forall>W15 X15 Y15. equal(W15::'a,X15) & ssubset(W15::'a,Y15) --> ssubset(X15::'a,Y15)) &
(\<forall>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:
"(\<forall>Y X. ~(element(X::'a,a) & element(X::'a,Y) & element(Y::'a,X))) &
(\<forall>X. element(X::'a,f(X)) | element(X::'a,a)) &
(\<forall>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:
"(\<forall>X Z Y. set_equal(X::'a,Y) & element(Z::'a,X) --> element(Z::'a,Y)) &
(\<forall>Y Z X. set_equal(X::'a,Y) & element(Z::'a,Y) --> element(Z::'a,X)) &
(\<forall>X Y. element(f(X::'a,Y),X) | element(f(X::'a,Y),Y) | set_equal(X::'a,Y)) &
(\<forall>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:
"(\<forall>A. p(A::'a,a) | p(A::'a,f(A))) &
(\<forall>A. p(A::'a,a) | p(f(A),A)) &
(\<forall>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:
"(\<forall>X Y. f(w(X),g(X::'a,Y)) --> f(X::'a,g(X::'a,Y))) &
(\<forall>X Y. f(X::'a,g(X::'a,Y)) --> f(w(X),g(X::'a,Y))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>X Y. f(X::'a,Y) --> f(b::'a,z(X::'a,Y)) | f(Y::'a,z(X::'a,Y))) &
(\<forall>X Y. f(X::'a,Y) | f(z(X::'a,Y),z(X::'a,Y))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>X Y. ~(f(X::'a,Y) & f(X::'a,z(X::'a,Y)) & f(Y::'a,z(X::'a,Y)))) &
(\<forall>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:
"(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
(\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
(\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
(\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>X. subset_sets(X::'a,X)) &
(\<forall>X U Y. subset_sets(X::'a,Y) & element_of_set(U::'a,X) --> element_of_set(U::'a,Y)) &
(\<forall>X Y. equal_sets(X::'a,Y) --> subset_sets(X::'a,Y)) &
(\<forall>Y X. subset_sets(X::'a,Y) | element_of_set(in_1st_set(X::'a,Y),X)) &
(\<forall>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:
"(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
(\<forall>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:
"(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
(\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
(\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
(\<forall>Vf U Va. element_of_set(U::'a,intersection_of_members(Vf)) & element_of_collection(Va::'a,Vf) --> element_of_set(U::'a,Va)) &
(\<forall>U Vf. element_of_set(U::'a,intersection_of_members(Vf)) | element_of_collection(f2(Vf::'a,U),Vf)) &
(\<forall>Vf U. element_of_set(U::'a,f2(Vf::'a,U)) --> element_of_set(U::'a,intersection_of_members(Vf))) &
(\<forall>Vt X. topological_space(X::'a,Vt) --> equal_sets(union_of_members(Vt),X)) &
(\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(empty_set::'a,Vt)) &
(\<forall>X Vt. topological_space(X::'a,Vt) --> element_of_collection(X::'a,Vt)) &
(\<forall>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)) &
(\<forall>X Vf Vt. topological_space(X::'a,Vt) & subset_collections(Vf::'a,Vt) --> element_of_collection(union_of_members(Vf),Vt)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>U X Vt. open(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>X U Vt. open(U::'a,X,Vt) --> element_of_collection(U::'a,Vt)) &
(\<forall>X U Vt. topological_space(X::'a,Vt) & element_of_collection(U::'a,Vt) --> open(U::'a,X,Vt)) &
(\<forall>U X Vt. closed(U::'a,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>U X Vt. closed(U::'a,X,Vt) --> open(relative_complement_sets(U::'a,X),X,Vt)) &
(\<forall>U X Vt. topological_space(X::'a,Vt) & open(relative_complement_sets(U::'a,X),X,Vt) --> closed(U::'a,X,Vt)) &
(\<forall>Vs X Vt. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vt)) &
(\<forall>Vt X Vs. finer(Vt::'a,Vs,X) --> topological_space(X::'a,Vs)) &
(\<forall>X Vs Vt. finer(Vt::'a,Vs,X) --> subset_collections(Vs::'a,Vt)) &
(\<forall>X Vs Vt. topological_space(X::'a,Vt) & topological_space(X::'a,Vs) & subset_collections(Vs::'a,Vt) --> finer(Vt::'a,Vs,X)) &
(\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>Vf X. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_set(f7(X::'a,Vf),X)) &
(\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f8(X::'a,Vf),Vf)) &
(\<forall>X Vf. equal_sets(union_of_members(Vf),X) --> basis(X::'a,Vf) | element_of_collection(f9(X::'a,Vf),Vf)) &
(\<forall>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)))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
(\<forall>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))) &
(\<forall>U Y X Vt. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> topological_space(X::'a,Vt)) &
(\<forall>U Vt Y X. element_of_collection(U::'a,subspace_topology(X::'a,Vt,Y)) --> subset_sets(Y::'a,X)) &
(\<forall>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)) &
(\<forall>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)))) &
(\<forall>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))) &
(\<forall>U Y X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
(\<forall>U Vt Y X. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
(\<forall>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))) &
(\<forall>X Vt U Y. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> subset_sets(f13(Y::'a,X,Vt,U),Y)) &
(\<forall>Y U X Vt. element_of_set(U::'a,interior(Y::'a,X,Vt)) --> open(f13(Y::'a,X,Vt,U),X,Vt)) &
(\<forall>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))) &
(\<forall>U Y X Vt. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
(\<forall>U Vt Y X. element_of_set(U::'a,closure(Y::'a,X,Vt)) --> subset_sets(Y::'a,X)) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>U Y X Vt. neighborhood(U::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>Y U X Vt. neighborhood(U::'a,Y,X,Vt) --> open(U::'a,X,Vt)) &
(\<forall>X Vt Y U. neighborhood(U::'a,Y,X,Vt) --> element_of_set(Y::'a,U)) &
(\<forall>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)) &
(\<forall>Z Y X Vt. limit_point(Z::'a,Y,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>Z Vt Y X. limit_point(Z::'a,Y,X,Vt) --> subset_sets(Y::'a,X)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> topological_space(X::'a,Vt)) &
(\<forall>U Y X Vt. element_of_set(U::'a,boundary(Y::'a,X,Vt)) --> element_of_set(U::'a,closure(Y::'a,X,Vt))) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>X Vt. hausdorff(X::'a,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f19(X::'a,Vt),X)) &
(\<forall>Vt X. topological_space(X::'a,Vt) --> hausdorff(X::'a,Vt) | element_of_set(f20(X::'a,Vt),X)) &
(\<forall>X Vt. topological_space(X::'a,Vt) & eq_p(f19(X::'a,Vt),f20(X::'a,Vt)) --> hausdorff(X::'a,Vt)) &
(\<forall>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)) &
(\<forall>Va1 Va2 X Vt. separation(Va1::'a,Va2,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>Va2 X Vt Va1. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va1::'a,empty_set))) &
(\<forall>Va1 X Vt Va2. ~(separation(Va1::'a,Va2,X,Vt) & equal_sets(Va2::'a,empty_set))) &
(\<forall>Va2 X Va1 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va1::'a,Vt)) &
(\<forall>Va1 X Va2 Vt. separation(Va1::'a,Va2,X,Vt) --> element_of_collection(Va2::'a,Vt)) &
(\<forall>Vt Va1 Va2 X. separation(Va1::'a,Va2,X,Vt) --> equal_sets(union_of_sets(Va1::'a,Va2),X)) &
(\<forall>X Vt Va1 Va2. separation(Va1::'a,Va2,X,Vt) --> disjoint_s(Va1::'a,Va2)) &
(\<forall>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)) &
(\<forall>X Vt. connected_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>Va1 Va2 X Vt. ~(connected_space(X::'a,Vt) & separation(Va1::'a,Va2,X,Vt))) &
(\<forall>X Vt. topological_space(X::'a,Vt) --> connected_space(X::'a,Vt) | separation(f21(X::'a,Vt),f22(X::'a,Vt),X,Vt)) &
(\<forall>Va X Vt. connected_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>Vt Va X. connected_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
(\<forall>X Vt Va. connected_set(Va::'a,X,Vt) --> connected_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
(\<forall>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)) &
(\<forall>Vf X Vt. open_covering(Vf::'a,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>X Vf Vt. open_covering(Vf::'a,X,Vt) --> subset_collections(Vf::'a,Vt)) &
(\<forall>Vt Vf X. open_covering(Vf::'a,X,Vt) --> equal_sets(union_of_members(Vf),X)) &
(\<forall>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)) &
(\<forall>X Vt. compact_space(X::'a,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> finite'(f23(X::'a,Vt,Vf1))) &
(\<forall>X Vt Vf1. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> subset_collections(f23(X::'a,Vt,Vf1),Vf1)) &
(\<forall>Vf1 X Vt. compact_space(X::'a,Vt) & open_covering(Vf1::'a,X,Vt) --> open_covering(f23(X::'a,Vt,Vf1),X,Vt)) &
(\<forall>X Vt. topological_space(X::'a,Vt) --> compact_space(X::'a,Vt) | open_covering(f24(X::'a,Vt),X,Vt)) &
(\<forall>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)) &
(\<forall>Va X Vt. compact_set(Va::'a,X,Vt) --> topological_space(X::'a,Vt)) &
(\<forall>Vt Va X. compact_set(Va::'a,X,Vt) --> subset_sets(Va::'a,X)) &
(\<forall>X Vt Va. compact_set(Va::'a,X,Vt) --> compact_space(Va::'a,subspace_topology(X::'a,Vt,Va))) &
(\<forall>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)) &
(\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
(\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
(\<forall>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:
"(\<forall>U Uu1 Vf. element_of_set(U::'a,Uu1) & element_of_collection(Uu1::'a,Vf) --> element_of_set(U::'a,union_of_members(Vf))) &
(\<forall>Vf X. basis(X::'a,Vf) --> equal_sets(union_of_members(Vf),X)) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
(\<forall>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))) &
(\<forall>Y X Z. subset_sets(X::'a,Y) & subset_sets(Y::'a,Z) --> subset_sets(X::'a,Z)) &
(\<forall>Y Z X. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,X)) &
(\<forall>X Z Y. element_of_set(Z::'a,intersection_of_sets(X::'a,Y)) --> element_of_set(Z::'a,Y)) &
(\<forall>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))) &
(\<forall>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))) &
(\<forall>X Z Y. equal_sets(X::'a,Y) & element_of_set(Z::'a,X) --> element_of_set(Z::'a,Y)) &
(\<forall>Y X. equal_sets(intersection_of_sets(X::'a,Y),intersection_of_sets(Y::'a,X))) &
(basis(cx::'a,f)) &
(\<forall>U. element_of_collection(U::'a,top_of_basis(f))) &
(\<forall>V. element_of_collection(V::'a,top_of_basis(f))) &
(\<forall>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:
"(\<forall>Vf U. element_of_set(U::'a,union_of_members(Vf)) --> element_of_set(U::'a,f1(Vf::'a,U))) &
(\<forall>U Vf. element_of_set(U::'a,union_of_members(Vf)) --> element_of_collection(f1(Vf::'a,U),Vf)) &
(\<forall>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))) &
(\<forall>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)) &
(\<forall>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)) &
(\<forall>Vf U. element_of_collection(U::'a,top_of_basis(Vf)) | element_of_set(f11(Vf::'a,U),U)) &
(\<forall>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))) &
(\<forall>X U Y. element_of_set(U::'a,X) --> subset_sets(X::'a,Y) | element_of_set(U::'a,Y)) &
(\<forall>Y X Z. subset_sets(X::'a,Y) & element_of_collection(Y::'a,Z) --> subset_sets(X::'a,union_of_members(Z))) &
(\<forall>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