Beginning of Section A9979
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
L20Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L21Variable G0 : set → set
L22Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L24Hypothesis HH0 : H0 ∈ int
L25Variable U0 : set → set → set
L26Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L27Variable V0 : set → set
L28Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L29Variable SMALL : set → set
L30Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L31Variable F2 : set → set → set
L32Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L33Variable G2 : set → set → set
L34Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L35Variable H2 : set → set
L36Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L38Hypothesis HI2 : I2 ∈ int
L39Variable F3 : set → set
L40Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L42Hypothesis HG3 : G3 ∈ int
L44Hypothesis HH3 : H3 ∈ int
L45Variable U3 : set → set → set
L46Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L48Hypothesis HV3 : V3 ∈ int
L50Hypothesis HJ2 : J2 ∈ int
L51Variable U2 : set → set → set → set
L52Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L53Variable V2 : set → set → set → set
L54Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L55Variable W2 : set → set
L56Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L57Variable FAST : set → set
L58Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L59Hypothesis H1 : (∀X ∈ int, ((F1 X) = (2 * ((X + X) + X))))
L60Hypothesis H2 : (G1 = 2)
L61Hypothesis H3 : (∀X ∈ int, ((H1 X) = X))
L62Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L63Hypothesis H5 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L64Hypothesis H6 : (∀X ∈ int, ((F0 X) = ((V1 X) + - X)))
L65Hypothesis H7 : (∀X ∈ int, ((G0 X) = X))
L66Hypothesis H8 : (H0 = 1)
L67Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L68Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L69Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L70Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))
L71Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L72Hypothesis H14 : (∀X ∈ int, ((H2 X) = X))
L73Hypothesis H15 : (I2 = 1)
L74Hypothesis H16 : (∀X ∈ int, ((F3 X) = (X * X)))
L75Hypothesis H17 : (G3 = 1)
L76Hypothesis H18 : (H3 = (2 + (2 + 2)))
L77Hypothesis H19 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L78Hypothesis H20 : (V3 = (U3 G3 H3))
L79Hypothesis H21 : (J2 = (V3 + - 1))
L80Hypothesis H22 : (∀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)))))))
L81Hypothesis H23 : (∀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)))))))
L82Hypothesis H24 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L83Hypothesis H25 : (∀X ∈ int, ((FAST X) = (W2 X)))
L84Theorem. (
A9979)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.