Beginning of Section A155657
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.
L9Variable F1 : set → set
L10Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L12Hypothesis HG1 : G1 ∈ int
L13Variable H1 : set → set
L14Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L17Variable V1 : set → set
L18Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L19Variable F0 : set → set → set
L20Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L21Variable G0 : set → set → set
L22Hypothesis HG0 : ∀x0 ∈ int, ∀x1 ∈ int, G0 x0 x1 ∈ int
L23Variable H0 : set → set
L24Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L26Hypothesis HI0 : I0 ∈ int
L28Hypothesis HJ0 : J0 ∈ int
L29Variable U0 : set → set → set → set
L30Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U0 x0 x1 x2 ∈ int
L31Variable V0 : set → set → set → set
L32Hypothesis HV0 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V0 x0 x1 x2 ∈ int
L33Variable W0 : set → set
L34Hypothesis HW0 : ∀x0 ∈ int, W0 x0 ∈ int
L35Variable SMALL : set → set
L36Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L37Variable F2 : set → set → set
L38Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L39Variable G2 : set → set → set
L40Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L41Variable H2 : set → set
L42Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L44Hypothesis HI2 : I2 ∈ int
L46Hypothesis HJ2 : J2 ∈ int
L47Variable U2 : set → set → set → set
L48Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L49Variable V2 : set → set → set → set
L50Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L51Variable W2 : set → set
L52Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L53Variable F3 : set → set → set
L54Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L55Variable G3 : set → set → set
L56Hypothesis HG3 : ∀x0 ∈ int, ∀x1 ∈ int, G3 x0 x1 ∈ int
L57Variable H3 : set → set
L58Hypothesis HH3 : ∀x0 ∈ int, H3 x0 ∈ int
L60Hypothesis HI3 : I3 ∈ int
L62Hypothesis HJ3 : J3 ∈ int
L63Variable U3 : set → set → set → set
L64Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U3 x0 x1 x2 ∈ int
L65Variable V3 : set → set → set → set
L66Hypothesis HV3 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V3 x0 x1 x2 ∈ int
L67Variable W3 : set → set
L68Hypothesis HW3 : ∀x0 ∈ int, W3 x0 ∈ int
L69Variable FAST : set → set
L70Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L71Hypothesis H1 : (∀X ∈ int, ((F1 X) = ((X + X) + X)))
L72Hypothesis H2 : (G1 = 2)
L73Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L74Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L75Hypothesis H5 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L76Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X) + - Y))))
L77Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, ((G0 X Y) = (2 * ((2 * (Y + Y)) + - X)))))
L78Hypothesis H8 : (∀X ∈ int, ((H0 X) = X))
L79Hypothesis H9 : (I0 = 2)
L80Hypothesis H10 : (J0 = 1)
L81Hypothesis H11 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U0 X Y Z) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
L82Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z) (V0 (X + - 1) Y Z)))))))
L83Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))
L84Hypothesis H14 : (∀X ∈ int, ((SMALL X) = ((W0 X) + - 1)))
L85Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))
L86Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L87Hypothesis H17 : (∀X ∈ int, ((H2 X) = X))
L88Hypothesis H18 : (I2 = 1)
L89Hypothesis H19 : (J2 = (1 + (2 + (2 + 2))))
L90Hypothesis H20 : (∀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)))))))
L91Hypothesis H21 : (∀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)))))))
L92Hypothesis H22 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L93Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))
L94Hypothesis H24 : (∀X ∈ int, (∀Y ∈ int, ((G3 X Y) = Y)))
L95Hypothesis H25 : (∀X ∈ int, ((H3 X) = X))
L96Hypothesis H26 : (I3 = 1)
L97Hypothesis H27 : (J3 = (2 + (2 * (2 + 2))))
L98Hypothesis H28 : (∀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)))))))
L99Hypothesis H29 : (∀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)))))))
L100Hypothesis H30 : (∀X ∈ int, ((W3 X) = (U3 (H3 X) I3 J3)))
L101Hypothesis H31 : (∀X ∈ int, ((FAST X) = (((W2 X) + - 1) + (W3 X))))
L102Theorem. (
A155657)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.