Beginning of Section A13714
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 TMK2KG4xEzgEVhXMyhX6BioPsyysZYBraMQ ***)
L9
Variable F0 : set → set
L10
Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L11
Variable G0 : set → set
L12
Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L13
Variable H0 : set
L14
Hypothesis HH0 : H0 ∈ int
L15
Variable U0 : set → set → set
L16
Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L17
Variable V0 : set → set
L18
Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L19
Variable SMALL : set → set
L20
Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L21
Variable F1 : set → set
L22
Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L23
Variable G1 : set
L24
Hypothesis HG1 : G1 ∈ int
L25
Variable F2 : set → set → set
L26
Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L27
Variable G2 : set → set → set
L28
Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L29
Variable H2 : set → set
L30
Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L31
Variable I2 : set
L32
Hypothesis HI2 : I2 ∈ int
L33
Variable J2 : set
L34
Hypothesis HJ2 : J2 ∈ int
L35
Variable U2 : set → set → set → set
L36
Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L37
Variable V2 : set → set → set → set
L38
Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L39
Variable W2 : set → set
L40
Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L41
Variable H1 : set → set
L42
Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L43
Variable U1 : set → set → set
L44
Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L45
Variable V1 : set → set
L46
Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L47
Variable FAST : set → set
L48
Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L49
Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((X + X) + X)))
L50
Hypothesis H2 : (∀X ∈ int, ((G0 X) = (2 + (2 * (X + X)))))
L51
Hypothesis H3 : (H0 = 1)
L52
Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L53
Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L54
Hypothesis H6 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L55
Hypothesis H7 : (∀X ∈ int, ((F1 X) = (X * X)))
L56
Hypothesis H8 : (G1 = 1)
L57
Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))
L58
Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L59
Hypothesis H11 : (∀X ∈ int, ((H2 X) = X))
L60
Hypothesis H12 : (I2 = (1 + 2))
L61
Hypothesis H13 : (J2 = (1 + (2 * (2 + 2))))
L62
Hypothesis H14 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U2 X Y Z) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L63
Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L64
Hypothesis H16 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L65
Hypothesis H17 : (∀X ∈ int, ((H1 X) = (W2 X)))
L66
Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L67
Hypothesis H19 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L68
Hypothesis H20 : (∀X ∈ int, ((FAST X) = (V1 X)))
L69
Theorem. (A13714)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
In Proofgold the corresponding term root is 1d7218... and proposition id is b3459c...
Proof:
Proof not loaded.
End of Section A13714