Beginning of Section A81045
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 → set
L14Hypothesis HH1 : ∀x0 ∈ int, ∀x1 ∈ int, H1 x0 x1 ∈ int
L15Variable U1 : set → set → set
L16Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L17Variable V1 : set → set → set
L18Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, V1 x0 x1 ∈ int
L19Variable F0 : set → set → set
L20Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L22Hypothesis HG0 : G0 ∈ int
L23Variable H0 : set → set
L24Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L26Hypothesis HI0 : I0 ∈ int
L27Variable J0 : set → set
L28Hypothesis HJ0 : ∀x0 ∈ int, J0 x0 ∈ 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
L40Hypothesis HG2 : G2 ∈ int
L41Variable H2 : set → set
L42Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L43Variable I2 : set → set
L44Hypothesis HI2 : ∀x0 ∈ int, I2 x0 ∈ int
L45Variable J2 : set → set
L46Hypothesis HJ2 : ∀x0 ∈ int, J2 x0 ∈ 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 FAST : set → set
L54Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L55Hypothesis H1 : (∀X ∈ int, ((F1 X) = ((X + X) + X)))
L56Hypothesis H2 : (G1 = 2)
L57Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H1 X Y) = (X + Y))))
L58Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L59Hypothesis H5 : (∀X ∈ int, (∀Y ∈ int, ((V1 X Y) = (U1 G1 (H1 X Y)))))
L60Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((V1 X Y) + X))))
L61Hypothesis H7 : (G0 = 0)
L62Hypothesis H8 : (∀X ∈ int, ((H0 X) = X))
L63Hypothesis H9 : (I0 = 1)
L64Hypothesis H10 : (∀X ∈ int, ((J0 X) = X))
L65Hypothesis 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)))))))
L66Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V0 X Y Z) = (if (X <= 0) then Z else G0)))))
L67Hypothesis H13 : (∀X ∈ int, ((W0 X) = (U0 (H0 X) I0 (J0 X))))
L68Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (W0 X)))
L69Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((2 * ((2 * (X + X)) + X)) + - Y))))
L70Hypothesis H16 : (G2 = 0)
L71Hypothesis H17 : (∀X ∈ int, ((H2 X) = X))
L72Hypothesis H18 : (∀X ∈ int, ((I2 X) = (1 + X)))
L73Hypothesis H19 : (∀X ∈ int, ((J2 X) = X))
L74Hypothesis 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)))))))
L75Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else G2)))))
L76Hypothesis H22 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) (I2 X) (J2 X))))
L77Hypothesis H23 : (∀X ∈ int, ((FAST X) = (W2 X)))
L78Theorem. (
A81045)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.