Beginning of Section A28027
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use <= as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
(*** $I sig/OEISPreamble.mgs ***)
(*** Bounty 1 PFG TMFQpNahD6BTBJjpzGvmQhUGvgP17RpBeKZ ***)
L9
Variable F1 : set → set → set
L10
Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L11
Variable G1 : set → set → set
L12
Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L13
Variable H1 : set → set → set
L14
Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int
L15
Variable I1 : set
L16
Hypothesis HI1 : I1 ∈ int
L17
Variable J1 : set
L18
Hypothesis HJ1 : J1 ∈ int
L19
Variable U1 : set → set → set → set
L20
Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int
L21
Variable V1 : set → set → set → set
L22
Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int
L23
Variable W1 : set → set → set
L24
Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 x0 x1 ∈ int
L25
Variable F0 : set → set → set
L26
Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L27
Variable G0 : set → set
L28
Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L29
Variable H0 : set
L30
Hypothesis HH0 : H0 ∈ int
L31
Variable U0 : set → set → set
L32
Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L33
Variable V0 : set → set
L34
Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L35
Variable SMALL : set → set
L36
Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L37
Variable F3 : set → set → set
L38
Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L39
Variable G3 : set → set → set
L40
Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int
L41
Variable H3 : set → set → set
L42
Hypothesis HH3 : ∀x0 ∈ int, ∀x1 ∈ int, H3 x0 x1 ∈ int
L43
Variable I3 : set
L44
Hypothesis HI3 : I3 ∈ int
L45
Variable J3 : set
L46
Hypothesis HJ3 : J3 ∈ int
L47
Variable U3 : set → set → set → set
L48
Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L49
Variable V3 : set → set → set → set
L50
Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L51
Variable W3 : set → set → set
L52
Hypothesis HW3 : ∀x0 ∈ int, ∀x1 ∈ int, W3 x0 x1 ∈ int
L53
Variable F2 : set → set → set
L54
Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L55
Variable G2 : set → set
L56
Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L57
Variable H2 : set
L58
Hypothesis HH2 : H2 ∈ int
L59
Variable U2 : set → set → set
L60
Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L61
Variable V2 : set → set
L62
Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L63
Variable FAST : set → set
L64
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L65
Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = (((((Y + - 1) + Y) * (Y * Y)) + X) + (2 * (X + X))))))
L66
Hypothesis H2 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = (Y + Y))))
L67
Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = Y)))
L68
Hypothesis H4 : (I1 = 1)
L69
Hypothesis H5 : (J1 = 2)
L70
Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
L71
Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
L72
Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 (H1 X Y) I1 J1))))
L73
Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((((W1 X Y) + X) + X) + X))))
L74
Hypothesis H10 : (∀X ∈ int, ((G0 X) = X))
L75
Hypothesis H11 : (H0 = 1)
L76
Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L77
Hypothesis H13 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L78
Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L79
Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((((((Y + - 1) + Y) * (Y * Y)) + X) + X) + X))))
L80
Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = (Y + Y))))
L81
Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((H3 X Y) = Y)))
L82
Hypothesis H18 : (I3 = 1)
L83
Hypothesis H19 : (J3 = 2)
L84
Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U3 X Y Z) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L85
Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V3 X Y Z) = (if (X <= 0) then Z else (G3 (U3 (X + - 1) Y Z) (V3 (X + - 1) Y Z)))))))
L86
Hypothesis H22 : (∀X ∈ int, (∀Y ∈ int, ((W3 X Y) = (U3 (H3 X Y) I3 J3))))
L87
Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (((W3 X Y) + (2 * (X + X))) + X))))
L88
Hypothesis H24 : (∀X ∈ int, ((G2 X) = X))
L89
Hypothesis H25 : (H2 = 1)
L90
Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y) X)))))
L91
Hypothesis H27 : (∀X ∈ int, ((V2 X) = (U2 (G2 X) H2)))
L92
Hypothesis H28 : (∀X ∈ int, ((FAST X) = (V2 X)))
L93
Theorem. (A28027)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
In Proofgold the corresponding term root is b12f19... and proposition id is 9ca01a...
Proof:
Proof not loaded.
End of Section A28027