%   ORIGINAL: h4/util__prob/IMAGE__IMAGE
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/pred__set/IMAGE__COMPOSE: !s g f. h4/pred__set/IMAGE (h4/combin/o f g) s = h4/pred__set/IMAGE f (h4/pred__set/IMAGE g s)
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/pred__set/SURJ__COMPOSE: !u t s g f. h4/pred__set/SURJ f s t /\ h4/pred__set/SURJ g t u ==> h4/pred__set/SURJ (h4/combin/o g f) s u
% Assm: h4/pred__set/INJ__COMPOSE: !u t s g f. h4/pred__set/INJ f s t /\ h4/pred__set/INJ g t u ==> h4/pred__set/INJ (h4/combin/o g f) s u
% Assm: h4/list/MAP__o: !g f. h4/list/MAP (h4/combin/o f g) = h4/combin/o (h4/list/MAP f) (h4/list/MAP g)
% Assm: h4/pred__set/INJ__DEF: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> f x = f y ==> x = y)
% Assm: h4/quotient/FUN__REL__MP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/combin/GEN__literal__case__RAND: !v f P. P (h4/bool/literal__case f v) = h4/bool/literal__case (h4/combin/o P f) v
% Assm: h4/list/MAP__MAP__o: !l g f. h4/list/MAP f (h4/list/MAP g l) = h4/list/MAP (h4/combin/o f g) l
% Assm: h4/option/OPTION__MAP__COMPOSE: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm: h4/combin/GEN__LET__RAND: !v f P. P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Assm: h4/pred__set/BIJ__COMPOSE: !u t s g f. h4/pred__set/BIJ f s t /\ h4/pred__set/BIJ g t u ==> h4/pred__set/BIJ (h4/combin/o g f) s u
% Assm: h4/sum/sum__Axiom: !g f. ?h. (!x. h (h4/sum/INL x) = f x) /\ (!y. h (h4/sum/INR y) = g y)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/util__prob/PREIMAGE__COMP: !s g f. h4/util__prob/PREIMAGE f (h4/util__prob/PREIMAGE g s) = h4/util__prob/PREIMAGE (h4/combin/o g f) s
% Assm: h4/util__prob/BIJ__TRANS: !u t s. (?f. h4/pred__set/BIJ f s t) /\ (?g. h4/pred__set/BIJ g t u) ==> (?h. h4/pred__set/BIJ h s u)
% Assm: h4/pred__set/BIJ__DEF: !t s f. h4/pred__set/BIJ f s t <=> h4/pred__set/INJ f s t /\ h4/pred__set/SURJ f s t
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/bool/literal__case__THM: !x f. h4/bool/literal__case f x = f x
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/quotient/FUN__REL__EQUALS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) g ==> (h4/quotient/_2D_2D_3E rep1 abs2 f = h4/quotient/_2D_2D_3E rep1 abs2 g <=> (!x y. R1 x y ==> R2 (f x) (g y)))))
% Assm: h4/quotient/FUN__MAP: !g f. h4/quotient/_2D_2D_3E f g = (\h x. g (h (f x)))
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/pred__set/SURJ__DEF: !t s f. h4/pred__set/SURJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (f x) t) /\ (!x. h4/bool/IN x t ==> (?y. h4/bool/IN y s /\ f y = x))
% Assm: h4/util__prob/IN__PREIMAGE: !x s f. h4/bool/IN x (h4/util__prob/PREIMAGE f s) <=> h4/bool/IN (f x) s
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/combin/W__THM: !x f. h4/combin/W f x = f x x
% Assm: h4/combin/S__THM: !x g f. h4/combin/S f g x = f x (g x)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/combin/K__o__THM_c1: !v f. h4/combin/o f (h4/combin/K v) = h4/combin/K (f v)
% Assm: h4/combin/o__ABS__L: !g f. h4/combin/o (\x. f x) g = (\x. f (g x))
% Assm: h4/bool/literal__case__DEF: h4/bool/literal__case = (\f x. f x)
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Assm: h4/quotient/FUN__REL__IMP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) g /\ h4/quotient/_2D_2D_3E rep1 abs2 f = h4/quotient/_2D_2D_3E rep1 abs2 g ==> (!x y. R1 x y ==> R2 (f x) (g y))))
% Assm: h4/pair/FST__PAIR__MAP: !p g f. h4/pair/FST (h4/pair/_23_23 f g p) = f (h4/pair/FST p)
% Assm: h4/combin/S__ABS__L: !g f. h4/combin/S (\x. f x) g = (\x. f x (g x))
% Assm: h4/quotient__sum/INL__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!a1 a2. R1 a1 a2 ==> h4/quotient__sum/_2B_2B_2B R1 R2 (h4/sum/INL a1) (h4/sum/INL a2)))
% Assm: h4/quotient/APPLY__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/quotient__list/FOLDR__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!l1 l2 f1 f2 e1 e2. h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R2 R2) f1 f2 /\ R2 e1 e2 /\ h4/list/LIST__REL R1 l1 l2 ==> R2 (h4/list/FOLDR f1 e1 l1) (h4/list/FOLDR f2 e2 l2)))
% Assm: h4/quotient__option/OPTION__MAP__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!a1 a2 f1 f2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 /\ h4/option/OPTREL R1 a1 a2 ==> h4/option/OPTREL R2 (h4/option/OPTION__MAP f1 a1) (h4/option/OPTION__MAP f2 a2)))
% Assm: h4/quotient__list/MAP__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!l1 l2 f1 f2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 /\ h4/list/LIST__REL R1 l1 l2 ==> h4/list/LIST__REL R2 (h4/list/MAP f1 l1) (h4/list/MAP f2 l2)))
% Assm: h4/quotient/literal__case__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (h4/bool/literal__case f x) (h4/bool/literal__case g y)))
% Assm: h4/quotient/RESPECTS__REP__ABS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 f x. h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ R1 x x ==> R2 (f (rep1 (abs1 x))) (f x))
% Assm: h4/quotient/W__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2 x1 x2. h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R1 R2) f1 f2 /\ R1 x1 x2 ==> R2 (h4/combin/W f1 x1) (h4/combin/W f2 x2)))
% Assm: h4/quotient/LAMBDA__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2. h4/quotient/_3D_3D_3D_3E R1 R2 f1 f2 ==> h4/quotient/_3D_3D_3D_3E R1 R2 (\x. f1 x) (\y. f2 y)))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/quotient/LET__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (h4/bool/LET f x) (h4/bool/LET g y)))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/real__sigma/REAL__SUM__IMAGE__IMAGE: !P. h4/pred__set/FINITE P ==> (!f_27. h4/pred__set/INJ f_27 P (h4/pred__set/IMAGE f_27 P) ==> (!f. h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/IMAGE f_27 P) = h4/real__sigma/REAL__SUM__IMAGE (h4/combin/o f f_27) P))
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/pair/PAIR__MAP: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (f (h4/pair/FST p)) (g (h4/pair/SND p))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/quotient/RESPECTS__MP: !y x f R2 R1. h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2) f /\ R1 x y ==> R2 (f x) (f y)
% Assm: h4/quotient/QUOTIENT__REP__ABS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r. R r r ==> R (rep (abs r)) r)
% Assm: h4/quotient/respects__def: h4/quotient/respects = h4/combin/W
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/quotient/QUOTIENT__REL__ABS__EQ: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r r ==> R s s ==> (R r s <=> abs r = abs s))
% Assm: h4/list/LIST__REL__def_c2: !bs b R. h4/list/LIST__REL R h4/list/NIL (h4/list/CONS b bs) <=> F
% Assm: h4/list/LIST__REL__def_c3: !bs b as a R. h4/list/LIST__REL R (h4/list/CONS a as) (h4/list/CONS b bs) <=> R a b /\ h4/list/LIST__REL R as bs
% Assm: h4/list/LIST__REL__def_c1: !as a R. h4/list/LIST__REL R (h4/list/CONS a as) h4/list/NIL <=> F
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/LIST__REL__def_c0: !R. h4/list/LIST__REL R h4/list/NIL h4/list/NIL <=> T
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/quotient__sum/SUM__REL__tupled__primitive__def: h4/quotient__sum/SUM__REL__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\SUM__REL__tupled a. h4/pair/pair__CASE a (\R1 v1. h4/pair/pair__CASE v1 (\R2 v3. h4/pair/pair__CASE v3 (\v4 v5. h4/sum/sum__CASE v4 (\a1. h4/sum/sum__CASE v5 (\a2. h4/combin/I (R1 a1 a2)) (\b2_27. h4/combin/I F)) (\b1. h4/sum/sum__CASE v5 (\a2_27. h4/combin/I F) (\b2. h4/combin/I (R2 b1 b2)))))))
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/quotient__sum/SUM__REL__curried__def: !x3 x2 x1 x. h4/quotient__sum/_2B_2B_2B x x1 x2 x3 <=> h4/quotient__sum/SUM__REL__tupled (h4/pair/_2C x (h4/pair/_2C x1 (h4/pair/_2C x2 x3)))
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/quotient__option/OPTION__REL__def_c3: !y x R. h4/option/OPTREL R (h4/option/SOME x) (h4/option/SOME y) <=> R x y
% Assm: h4/quotient__option/OPTION__REL__def_c1: !x R. h4/option/OPTREL R (h4/option/SOME x) h4/option/NONE <=> F
% Assm: h4/quotient__option/OPTION__REL__def_c0: !R. h4/option/OPTREL R h4/option/NONE h4/option/NONE <=> T
% Assm: h4/quotient__option/OPTION__REL__def_c2: !y R. h4/option/OPTREL R h4/option/NONE (h4/option/SOME y) <=> F
% Assm: h4/list/FOLDR0_c0: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm: h4/list/FOLDR0_c1: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = f x (h4/list/FOLDR f e l)
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/IMAGE__FINITE: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/INJ__EMPTY_c0: !s f. h4/pred__set/INJ f h4/pred__set/EMPTY s
% Assm: h4/pred__set/IMAGE__INSERT: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (f x) (h4/pred__set/IMAGE f s)
% Assm: h4/pred__set/IMAGE__EMPTY: !f. h4/pred__set/IMAGE f h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm: h4/pred__set/IMAGE__IN: !x s. h4/bool/IN x s ==> (!f. h4/bool/IN (f x) (h4/pred__set/IMAGE f s))
% Assm: h4/pred__set/IN__DELETE: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Goal: !s g f. h4/pred__set/IMAGE f (h4/pred__set/IMAGE g s) = h4/pred__set/IMAGE (h4/combin/o f g) s
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_predu_u_sets_IMAGEu_u_COMPOSE]: !s g f. h4/pred__set/IMAGE (h4/combin/o f g) s = h4/pred__set/IMAGE f (h4/pred__set/IMAGE g s)
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_predu_u_sets_SURJu_u_COMPOSE]: !u t s g f. h4/pred__set/SURJ f s t /\ h4/pred__set/SURJ g t u ==> h4/pred__set/SURJ (h4/combin/o g f) s u
% Assm [h4s_predu_u_sets_INJu_u_COMPOSE]: !u t s g f. h4/pred__set/INJ f s t /\ h4/pred__set/INJ g t u ==> h4/pred__set/INJ (h4/combin/o g f) s u
% Assm [h4s_lists_MAPu_u_o]: !g f. h4/list/MAP (h4/combin/o f g) = h4/combin/o (h4/list/MAP f) (h4/list/MAP g)
% Assm [h4s_predu_u_sets_INJu_u_DEF]: !t s f. h4/pred__set/INJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x y. h4/bool/IN x s /\ h4/bool/IN y s ==> happ f x = happ f y ==> x = y)
% Assm [h4s_quotients_FUNu_u_RELu_u_MP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_combins_GENu_u_literalu_u_caseu_u_RAND]: !v f P. happ P (h4/bool/literal__case f v) = h4/bool/literal__case (h4/combin/o P f) v
% Assm [h4s_lists_MAPu_u_MAPu_u_o]: !l g f. happ (h4/list/MAP f) (happ (h4/list/MAP g) l) = happ (h4/list/MAP (h4/combin/o f g)) l
% Assm [h4s_options_OPTIONu_u_MAPu_u_COMPOSE]: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm [h4s_combins_GENu_u_LETu_u_RAND]: !v f P. happ P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Assm [h4s_predu_u_sets_BIJu_u_COMPOSE]: !u t s g f. h4/pred__set/BIJ f s t /\ h4/pred__set/BIJ g t u ==> h4/pred__set/BIJ (h4/combin/o g f) s u
% Assm [h4s_sums_sumu_u_Axiom]: !g f. ?h. (!x. happ h (h4/sum/INL x) = happ f x) /\ (!y. happ h (h4/sum/INR y) = happ g y)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_utilu_u_probs_PREIMAGEu_u_COMP]: !s g f. h4/util__prob/PREIMAGE f (h4/util__prob/PREIMAGE g s) = h4/util__prob/PREIMAGE (h4/combin/o g f) s
% Assm [h4s_utilu_u_probs_BIJu_u_TRANS]: !u t s. (?f. h4/pred__set/BIJ f s t) /\ (?g. h4/pred__set/BIJ g t u) ==> (?h. h4/pred__set/BIJ h s u)
% Assm [h4s_predu_u_sets_BIJu_u_DEF]: !t s f. h4/pred__set/BIJ f s t <=> h4/pred__set/INJ f s t /\ h4/pred__set/SURJ f s t
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_bools_literalu_u_caseu_u_THM]: !x f. h4/bool/literal__case f x = happ f x
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_quotients_FUNu_u_RELu_u_EQUALS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) g ==> (happ (h4/quotient/_2D_2D_3E rep1 abs2) f = happ (h4/quotient/_2D_2D_3E rep1 abs2) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))))
% Assm [h4s_quotients_FUNu_u_MAP]: !g f x x. happ (happ (h4/quotient/_2D_2D_3E f g) x) x = happ g (happ x (happ f x))
% Assm [h4s_combins_Su_u_DEF]: !x x x. h4/combin/S x x x = happ (happ x x) (happ x x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_predu_u_sets_SURJu_u_DEF]: !t s f. h4/pred__set/SURJ f s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN (happ f x) t) /\ (!x. h4/bool/IN x t ==> (?y. h4/bool/IN y s /\ happ f y = x))
% Assm [h4s_utilu_u_probs_INu_u_PREIMAGE]: !x s f. h4/bool/IN x (h4/util__prob/PREIMAGE f s) <=> h4/bool/IN (happ f x) s
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_combins_Wu_u_THM]: !x f. happ (happ h4/combin/W f) x = happ (happ f x) x
% Assm [h4s_combins_Su_u_THM]: !x g f. h4/combin/S f g x = happ (happ f x) (happ g x)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_combins_Ku_u_ou_u_THMu_c1]: !v f. h4/combin/o f (h4/combin/K v) = h4/combin/K (happ f v)
% Assm [h4s_combins_ou_u_ABSu_u_L]: !_0. (!f x. happ (happ _0 f) x = happ f x) ==> (!g f x. happ (h4/combin/o (happ _0 f) g) x = happ f (happ g x))
% Assm [h4s_bools_literalu_u_caseu_u_DEF]: !x x. h4/bool/literal__case x x = happ x x
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = happ f x
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. happ (happ (h4/quotient/_2D_2D_3E f g) h) x = happ g (happ h (happ f x))
% Assm [h4s_quotients_FUNu_u_RELu_u_IMP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g. happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) g /\ happ (h4/quotient/_2D_2D_3E rep1 abs2) f = happ (h4/quotient/_2D_2D_3E rep1 abs2) g ==> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))))
% Assm [h4s_pairs_FSTu_u_PAIRu_u_MAP]: !p g f. h4/pair/FST (h4/pair/_23_23 f g p) = happ f (h4/pair/FST p)
% Assm [h4s_combins_Su_u_ABSu_u_L]: !_0. (!f x. happ (happ _0 f) x = happ f x) ==> (!g f x. h4/combin/S (happ _0 f) g x = happ (happ f x) (happ g x))
% Assm [h4s_quotientu_u_sums_INLu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!a1 a2. happ (happ R1 a1) a2 ==> h4/quotient__sum/_2B_2B_2B R1 R2 (h4/sum/INL a1) (h4/sum/INL a2)))
% Assm [h4s_quotients_APPLYu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_quotientu_u_lists_FOLDRu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!l1 l2 f1 f2 e1 e2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R2 R2)) f1) f2 /\ happ (happ R2 e1) e2 /\ h4/list/LIST__REL R1 l1 l2 ==> happ (happ R2 (h4/list/FOLDR f1 e1 l1)) (h4/list/FOLDR f2 e2 l2)))
% Assm [h4s_quotientu_u_options_OPTIONu_u_MAPu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!a1 a2 f1 f2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f1) f2 /\ h4/option/OPTREL R1 a1 a2 ==> h4/option/OPTREL R2 (h4/option/OPTION__MAP f1 a1) (h4/option/OPTION__MAP f2 a2)))
% Assm [h4s_quotientu_u_lists_MAPu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!l1 l2 f1 f2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f1) f2 /\ h4/list/LIST__REL R1 l1 l2 ==> h4/list/LIST__REL R2 (happ (h4/list/MAP f1) l1) (happ (h4/list/MAP f2) l2)))
% Assm [h4s_quotients_literalu_u_caseu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (h4/bool/literal__case f x)) (h4/bool/literal__case g y)))
% Assm [h4s_quotients_RESPECTSu_u_REPu_u_ABS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 f x. happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ R1 x) x ==> happ (happ R2 (happ f (happ rep1 (happ abs1 x)))) (happ f x))
% Assm [h4s_quotients_Wu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2 x1 x2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 (h4/quotient/_3D_3D_3D_3E R1 R2)) f1) f2 /\ happ (happ R1 x1) x2 ==> happ (happ R2 (happ (happ h4/combin/W f1) x1)) (happ (happ h4/combin/W f2) x2)))
% Assm [h4s_quotients_LAMBDAu_u_RSP]: !_0. (!f1 x. happ (happ _0 f1) x = happ f1 x) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f1 f2. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f1) f2 ==> happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) (happ _0 f1)) (happ _0 f2))))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_quotients_LETu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (h4/bool/LET f x)) (h4/bool/LET g y)))
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_IMAGE]: !P. h4/pred__set/FINITE P ==> (!f_27. h4/pred__set/INJ f_27 P (h4/pred__set/IMAGE f_27 P) ==> (!f. h4/real__sigma/REAL__SUM__IMAGE f (h4/pred__set/IMAGE f_27 P) = h4/real__sigma/REAL__SUM__IMAGE (h4/combin/o f f_27) P))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_pairs_PAIRu_u_MAP]: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (happ f (h4/pair/FST p)) (happ g (h4/pair/SND p))
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (h4/combin/K x) y = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_quotients_RESPECTSu_u_MP]: !y x f R2 R1. happ (happ h4/quotient/respects (h4/quotient/_3D_3D_3D_3E R1 R2)) f /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ f y)
% Assm [h4s_quotients_QUOTIENTu_u_REPu_u_ABS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r. happ (happ R r) r ==> happ (happ R (happ rep (happ abs r))) r)
% Assm [h4s_quotients_respectsu_u_def]: h4/quotient/respects = h4/combin/W
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_quotients_QUOTIENTu_u_RELu_u_ABSu_u_EQ]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) r ==> happ (happ R s) s ==> (happ (happ R r) s <=> happ abs r = happ abs s))
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c2]: !bs b R. h4/list/LIST__REL R h4/list/NIL (h4/list/CONS b bs) <=> F
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c3]: !bs b as a R. h4/list/LIST__REL R (h4/list/CONS a as) (h4/list/CONS b bs) <=> happ (happ R a) b /\ h4/list/LIST__REL R as bs
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c1]: !as a R. h4/list/LIST__REL R (h4/list/CONS a as) h4/list/NIL <=> F
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c0]: !R. h4/list/LIST__REL R h4/list/NIL h4/list/NIL <=> T
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_quotientu_u_sums_SUMu_u_RELu_u_tupledu_u_primitiveu_u_def]: !_14. (!R2 b1 b2. happ (happ (happ _14 R2) b1) b2 <=> h4/combin/I (happ (happ R2 b1) b2)) ==> (!_13. (!a2_27. happ _13 a2_27 <=> h4/combin/I F) ==> (!_12. (!v5 R2 b1. happ (happ (happ _12 v5) R2) b1 <=> h4/sum/sum__CASE v5 _13 (happ (happ _14 R2) b1)) ==> (!_11. (!b2_27. happ _11 b2_27 <=> h4/combin/I F) ==> (!_10. (!R1 a1 a2. happ (happ (happ _10 R1) a1) a2 <=> h4/combin/I (happ (happ R1 a1) a2)) ==> (!_9. (!v5 R1 a1. happ (happ (happ _9 v5) R1) a1 <=> h4/sum/sum__CASE v5 (happ (happ _10 R1) a1) _11) ==> (!_8. (!v4 R1 R2 v5. happ (happ (happ (happ _8 v4) R1) R2) v5 <=> h4/sum/sum__CASE v4 (happ (happ _9 v5) R1) (happ (happ _12 v5) R2)) ==> (!_7. (!R1 R2 v4. happ (happ (happ _7 R1) R2) v4 = happ (happ (happ _8 v4) R1) R2) ==> (!_6. (!R1 R2 v3. happ (happ (happ _6 R1) R2) v3 <=> h4/pair/pair__CASE v3 (happ (happ _7 R1) R2)) ==> (!_5. (!R1 R2. happ (happ _5 R1) R2 = happ (happ _6 R1) R2) ==> (!_4. (!R1 v1. happ (happ _4 R1) v1 <=> h4/pair/pair__CASE v1 (happ _5 R1)) ==> (!_3. (!R1. happ _3 R1 = happ _4 R1) ==> (!_2. (!a. happ _2 a <=> h4/pair/pair__CASE a _3) ==> (!_1. (!SUM__REL__tupled. happ _1 SUM__REL__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/quotient__sum/SUM__REL__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_quotientu_u_sums_SUMu_u_RELu_u_curriedu_u_def]: !x3 x2 x1 x. h4/quotient__sum/_2B_2B_2B x x1 x2 x3 <=> happ h4/quotient__sum/SUM__REL__tupled (h4/pair/_2C x (h4/pair/_2C x1 (h4/pair/_2C x2 x3)))
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c3]: !y x R. h4/option/OPTREL R (h4/option/SOME x) (h4/option/SOME y) <=> happ (happ R x) y
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c1]: !x R. h4/option/OPTREL R (h4/option/SOME x) h4/option/NONE <=> F
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c0]: !R. h4/option/OPTREL R h4/option/NONE h4/option/NONE <=> T
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c2]: !y R. h4/option/OPTREL R h4/option/NONE (h4/option/SOME y) <=> F
% Assm [h4s_lists_FOLDR0u_c0]: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm [h4s_lists_FOLDR0u_c1]: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = happ (happ f x) (h4/list/FOLDR f e l)
% Assm [h4s_lists_MAP0u_c1]: !t h f. happ (h4/list/MAP f) (h4/list/CONS h t) = h4/list/CONS (happ f h) (happ (h4/list/MAP f) t)
% Assm [h4s_lists_MAP0u_c0]: !f. happ (h4/list/MAP f) h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_IMAGEu_u_FINITE]: !s. h4/pred__set/FINITE s ==> (!f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_INJu_u_EMPTYu_c0]: !s f. h4/pred__set/INJ f h4/pred__set/EMPTY s
% Assm [h4s_predu_u_sets_IMAGEu_u_INSERT]: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (happ f x) (h4/pred__set/IMAGE f s)
% Assm [h4s_predu_u_sets_IMAGEu_u_EMPTY]: !f. h4/pred__set/IMAGE f h4/pred__set/EMPTY = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_IMAGEu_u_IN]: !x s. h4/bool/IN x s ==> (!f. h4/bool/IN (happ f x) (h4/pred__set/IMAGE f s))
% Assm [h4s_predu_u_sets_INu_u_DELETE]: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Goal: !s g f. h4/pred__set/IMAGE f (h4/pred__set/IMAGE g s) = h4/pred__set/IMAGE (h4/combin/o f g) s
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1275125,TV_Q1275121]: ![V_f, V_g]: (![V_x]: s(TV_Q1275121,happ(s(t_fun(TV_Q1275125,TV_Q1275121),V_f),s(TV_Q1275125,V_x))) = s(TV_Q1275121,happ(s(t_fun(TV_Q1275125,TV_Q1275121),V_g),s(TV_Q1275125,V_x))) => s(t_fun(TV_Q1275125,TV_Q1275121),V_f) = s(t_fun(TV_Q1275125,TV_Q1275121),V_g))).
fof(ah4s_predu_u_sets_IMAGEu_u_COMPOSE, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_s, V_g, V_f]: s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27c,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_predu_u_sets_SURJu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_u, V_t, V_s, V_g, V_f]: ((p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(TV_u_27c,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27c,t_bool),V_u)))))).
fof(ah4s_predu_u_sets_INJu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_u, V_t, V_s, V_g, V_f]: ((p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(TV_u_27c,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27c,t_bool),V_u)))))).
fof(ah4s_lists_MAPu_u_o, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c)),h4s_combins_o(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_g)))))).
fof(ah4s_predu_u_sets_INJu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s))))) => (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_quotients_FUNu_u_RELu_u_MP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_combins_GENu_u_literalu_u_caseu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_literalu_u_case(s(t_fun(TV_u_27c,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ah4s_lists_MAPu_u_MAPu_u_o, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_l, V_g, V_f]: s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27c),V_f))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27c)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(t_h4s_options_option(TV_u_27a),V_x)))).
fof(ah4s_combins_GENu_u_LETu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ah4s_predu_u_sets_BIJu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_u, V_t, V_s, V_g, V_f]: ((p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27b,t_bool),V_t),s(t_fun(TV_u_27c,t_bool),V_u))))) => p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27c,t_bool),V_u)))))).
fof(ah4s_sums_sumu_u_Axiom, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_g, V_f]: ?[V_h]: (![V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) & ![V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,V_y))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_utilu_u_probs_PREIMAGEu_u_COMP, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_s, V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27c,t_bool),V_s))))) = s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,t_bool),V_s)))).
fof(ah4s_utilu_u_probs_BIJu_u_TRANS, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_u, V_t, V_s]: ((?[V_f]: p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27c,t_bool),V_t)))) & ?[V_g]: p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27c,TV_u_27b),V_g),s(t_fun(TV_u_27c,t_bool),V_t),s(t_fun(TV_u_27b,t_bool),V_u))))) => ?[V_h]: p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27a,TV_u_27b),V_h),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_u)))))).
fof(ah4s_predu_u_sets_BIJu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_bij(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) & p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_bools_literalu_u_caseu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_quotients_FUNu_u_RELu_u_EQUALS, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) => (s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_g))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))))).
fof(ah4s_quotients_FUNu_u_MAP, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x, V_x0]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_x),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x0)))))))).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_predu_u_sets_SURJu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_s, V_f]: (p(s(t_bool,h4s_predu_u_sets_surj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27b,t_bool),V_t)))) <=> (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_t))))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_bool),V_t)))) => ?[V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,V_x)))))).
fof(ah4s_utilu_u_probs_INu_u_PREIMAGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_s, V_f]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_utilu_u_probs_preimage(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,t_bool),V_s))))) = s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),V_s)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_combins_Wu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))).
fof(ah4s_combins_Su_u_THM, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(TV_u_27c,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_combins_Ku_u_ou_u_THMu_c1, axiom, ![TV_u_27f,TV_u_27e,TV_u_27d]: ![V_v, V_f]: s(t_fun(TV_u_27f,TV_u_27e),h4s_combins_o(s(t_fun(TV_u_27d,TV_u_27e),V_f),s(t_fun(TV_u_27f,TV_u_27d),h4s_combins_k(s(TV_u_27d,V_v))))) = s(t_fun(TV_u_27f,TV_u_27e),h4s_combins_k(s(TV_u_27e,happ(s(t_fun(TV_u_27d,TV_u_27e),V_f),s(TV_u_27d,V_v)))))).
fof(ah4s_combins_ou_u_ABSu_u_L, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) => ![V_g, V_f, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27a),V_g))),s(TV_u_27b,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))))).
fof(ah4s_bools_literalu_u_caseu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_bools_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_h))),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_quotients_FUNu_u_RELu_u_IMP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),V_g))))) => ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))))).
fof(ah4s_pairs_FSTu_u_PAIRu_u_MAP, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27c,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_combins_Su_u_ABSu_u_L, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_f, V_x]: s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(TV_u_27a,V_x))) => ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_quotientu_u_sums_INLu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_a1, V_a2]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_a1))),s(TV_u_27a,V_a2)))) => p(s(t_bool,h4s_quotientu_u_sums_u_2bu_2bu_2b(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a1))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a2)))))))))).
fof(ah4s_quotients_APPLYu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotientu_u_lists_FOLDRu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_l1, V_l2, V_f1, V_f2, V_e1, V_e2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(t_fun(TV_u_27b,TV_u_27b),t_fun(t_fun(TV_u_27b,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f1))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f2)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_e1))),s(TV_u_27b,V_e2)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f1),s(TV_u_27b,V_e1),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f2),s(TV_u_27b,V_e2),s(t_h4s_lists_list(TV_u_27a),V_l2)))))))))).
fof(ah4s_quotientu_u_options_OPTIONu_u_MAPu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_a1, V_a2, V_f1, V_f2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))) & p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_h4s_options_option(TV_u_27a),V_a1),s(t_h4s_options_option(TV_u_27a),V_a2))))) => p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_h4s_options_option(TV_u_27a),V_a1))),s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(t_h4s_options_option(TV_u_27a),V_a2)))))))))).
fof(ah4s_quotientu_u_lists_MAPu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_l1, V_l2, V_f1, V_f2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))))) => p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f2))),s(t_h4s_lists_list(TV_u_27a),V_l2)))))))))).
fof(ah4s_quotients_literalu_u_caseu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotients_RESPECTSu_u_REPu_u_ABS, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_f, V_x]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_x))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))))))).
fof(ah4s_quotients_Wu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f1, V_f2, V_x1, V_x2]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f1))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f1))),s(TV_u_27a,V_x1))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f2))),s(TV_u_27a,V_x2)))))))))).
fof(ah4s_quotients_LAMBDAu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_x))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f1, V_f2]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f2))))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_quotients_LETu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_realu_u_sigmas_REALu_u_SUMu_u_IMAGEu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_P)))) => ![V_fu_27]: (p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_P)))))) => ![V_f]: s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27b,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_fu_27),s(t_fun(TV_u_27a,t_bool),V_P))))) = s(t_h4s_realaxs_real,h4s_realu_u_sigmas_realu_u_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_realaxs_real),h4s_combins_o(s(t_fun(TV_u_27b,t_h4s_realaxs_real),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_fu_27))),s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_pairs_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f0))) <=> p(s(t_bool,f0)))).
fof(ah4s_quotients_RESPECTSu_u_MP, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_R2, V_R1]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_respects),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))))))).
fof(ah4s_quotients_QUOTIENTu_u_REPu_u_ABS, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))))),s(TV_u_27a,V_r))))))).
fof(ah4s_quotients_respectsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_respects) = s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w)).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_quotients_QUOTIENTu_u_RELu_u_ABSu_u_EQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_bs, V_b, V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_b),s(t_h4s_lists_list(TV_u_27b),V_bs))))) = s(t_bool,f0)).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_bs, V_b, V_as, V_a, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a),s(t_h4s_lists_list(TV_u_27a),V_as))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_b),s(t_h4s_lists_list(TV_u_27b),V_bs)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_a))),s(TV_u_27b,V_b)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_as),s(t_h4s_lists_list(TV_u_27b),V_bs))))))).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_as, V_a, V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a),s(t_h4s_lists_list(TV_u_27a),V_as))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_bool,f0)).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_quotientu_u_sums_SUMu_u_RELu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_14]: (![V_R2, V_b1, V_b2]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),V_uu_14),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_b1))),s(TV_u_27b,V_b2))) = s(t_bool,h4s_combins_i(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_b1))),s(TV_u_27b,V_b2))))) => ![V_uu_13]: (![V_a2u_27]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_13),s(TV_u_27a,V_a2u_27))) = s(t_bool,h4s_combins_i(s(t_bool,f0))) => ![V_uu_12]: (![V_v5, V_R2, V_b1]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool))),V_uu_12),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_b1))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5),s(t_fun(TV_u_27a,t_bool),V_uu_13),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),V_uu_14),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_b1))))) => ![V_uu_11]: (![V_b2u_27]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_uu_11),s(TV_u_27b,V_b2u_27))) = s(t_bool,h4s_combins_i(s(t_bool,f0))) => ![V_uu_10]: (![V_R1, V_a1, V_a2]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_uu_10),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_a1))),s(TV_u_27a,V_a2))) = s(t_bool,h4s_combins_i(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_a1))),s(TV_u_27a,V_a2))))) => ![V_uu_9]: (![V_v5, V_R1, V_a1]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_9),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_a1))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_uu_10),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_a1))),s(t_fun(TV_u_27b,t_bool),V_uu_11))) => ![V_uu_8]: (![V_v4, V_R1, V_R2, V_v5]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_8),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_9),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_bool))),V_uu_12),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v5))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))) => ![V_uu_7]: (![V_R1, V_R2, V_v4]: s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_7),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4))) = s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_8),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_v4))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) => ![V_uu_6]: (![V_R1, V_R2, V_v3]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_6),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_v3))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),V_v3),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool)))),V_uu_7),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))))) => ![V_uu_5]: (![V_R1, V_R2]: s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_5),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) = s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_6),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) => ![V_uu_4]: (![V_R1, V_v1]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_4),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_v1))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),V_v1),s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_bool))),V_uu_5),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))))) => ![V_uu_3]: (![V_R1]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_4),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))) => ![V_uu_2]: (![V_a]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),V_uu_2),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_a))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),V_a),s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),t_bool)),V_uu_3))) => ![V_uu_1]: (![V_SUMu_u_RELu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),V_SUMu_u_RELu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),h4s_quotientu_u_sums_sumu_u_relu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool)),V_uu_1))))))))))))))))))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_quotientu_u_sums_SUMu_u_RELu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x3, V_x2, V_x1, V_x]: s(t_bool,h4s_quotientu_u_sums_u_2bu_2bu_2b(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_x),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_x1),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x2),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x3))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),t_bool),h4s_quotientu_u_sums_sumu_u_relu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_x),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_x1),s(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x2),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x3)))))))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c3, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_R]: s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f0)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c0, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,t)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c2, axiom, ![TV_u_27a]: ![V_y, V_R]: s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))))) = s(t_bool,f0)).
fof(ah4s_lists_FOLDR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_FOLDR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_IMAGEu_u_FINITE, axiom, ![TV_u_27b,TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_INJu_u_EMPTYu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_s, V_f]: p(s(t_bool,h4s_predu_u_sets_inj(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27b,t_bool),V_s))))).
fof(ah4s_predu_u_sets_IMAGEu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))).
fof(ah4s_predu_u_sets_IMAGEu_u_EMPTY, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_IMAGEu_u_IN, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_f]: p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_predu_u_sets_INu_u_DELETE, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ch4s_utilu_u_probs_IMAGEu_u_IMAGE, conjecture, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_s, V_g, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(t_fun(TV_u_27c,t_bool),V_s))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(t_fun(TV_u_27c,t_bool),V_s)))).
