Beginning of Section A98249
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 → set
L10Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L12Hypothesis HG1 : G1 ∈ int
L14Hypothesis HH1 : H1 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L18Hypothesis HV1 : V1 ∈ int
L19Variable F0 : set → set → set
L20Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L21Variable G0 : set → set
L22Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ 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 F3 : set → set → set
L38Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L40Hypothesis HG3 : G3 ∈ int
L42Hypothesis HH3 : H3 ∈ int
L43Variable U3 : set → set → set
L44Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L46Hypothesis HV3 : V3 ∈ int
L47Variable F2 : set → set → set
L48Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L49Variable G2 : set → set
L50Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L51Variable H2 : set → set
L52Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L54Hypothesis HI2 : I2 ∈ int
L56Hypothesis HJ2 : J2 ∈ int
L57Variable U2 : set → set → set → set
L58Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L59Variable V2 : set → set → set → set
L60Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L61Variable W2 : set → set
L62Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L63Variable FAST : set → set
L64Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L65Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((X * X) + Y))))
L66Hypothesis H2 : (G1 = 2)
L67Hypothesis H3 : (H1 = (2 + 2))
L68Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y) X)))))
L69Hypothesis H5 : (V1 = (U1 G1 H1))
L70Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 * X) + Y))))
L71Hypothesis H7 : (∀X ∈ int, ((G0 X) = (0 + - X)))
L72Hypothesis H8 : (∀X ∈ int, ((H0 X) = X))
L73Hypothesis H9 : (I0 = 1)
L74Hypothesis H10 : (J0 = 1)
L75Hypothesis 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)))))))
L76Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else (G0 (U0 (X + - 1) Y Z)))))))
L77Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 J0)))
L78Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (W0 X)))
L79Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((X * X) + Y))))
L80Hypothesis H16 : (G3 = 2)
L81Hypothesis H17 : (H3 = (2 + 2))
L82Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L83Hypothesis H19 : (V3 = (U3 G3 H3))
L84Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((V3 * X) + - Y))))
L85Hypothesis H21 : (∀X ∈ int, ((G2 X) = X))
L86Hypothesis H22 : (∀X ∈ int, ((H2 X) = X))
L87Hypothesis H23 : (I2 = 1)
L88Hypothesis H24 : (J2 = (0 + - 1))
L89Hypothesis H25 : (∀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)))))))
L90Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L91Hypothesis H27 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L92Hypothesis H28 : (∀X ∈ int, ((FAST X) = (W2 X)))
L93Theorem. (
A98249)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.