Beginning of Section A52677
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 F0 : set → set → set
L10Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L11Variable G0 : set → set
L12Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L13Variable F1 : set → set → set
L14Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L15Variable G1 : set → set
L16Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ int
L17Variable H1 : set → set
L18Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L20Hypothesis HI1 : I1 ∈ int
L22Hypothesis HJ1 : J1 ∈ int
L23Variable U1 : set → set → set → set
L24Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int
L25Variable V1 : set → set → set → set
L26Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int
L27Variable W1 : set → set
L28Hypothesis HW1 : ∀x0 ∈ int, W1 x0 ∈ int
L29Variable H0 : set → set
L30Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ int
L31Variable U0 : set → set → set
L32Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L33Variable V0 : set → set
L34Hypothesis HV0 : ∀x0 ∈ int, V0 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
L40Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ 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
L56Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L58Hypothesis HH3 : H3 ∈ int
L59Variable U3 : set → set → set
L60Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L61Variable V3 : set → set
L62Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L63Variable FAST : set → set
L64Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L65Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = (X * Y))))
L66Hypothesis H2 : (∀X ∈ int, ((G0 X) = X))
L67Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((2 * (X + X)) + - Y))))
L68Hypothesis H4 : (∀X ∈ int, ((G1 X) = X))
L69Hypothesis H5 : (∀X ∈ int, ((H1 X) = X))
L70Hypothesis H6 : (I1 = 1)
L71Hypothesis H7 : (J1 = 1)
L72Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U1 X Y Z) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
L73Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z)))))))
L74Hypothesis H10 : (∀X ∈ int, ((W1 X) = (U1 (H1 X) I1 J1)))
L75Hypothesis H11 : (∀X ∈ int, ((H0 X) = (W1 X)))
L76Hypothesis H12 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L77Hypothesis H13 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) (H0 X))))
L78Hypothesis H14 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L79Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = ((2 * (X + X)) + - Y))))
L80Hypothesis H16 : (∀X ∈ int, ((G2 X) = X))
L81Hypothesis H17 : (∀X ∈ int, ((H2 X) = (X + - 1)))
L82Hypothesis H18 : (I2 = (1 + 2))
L83Hypothesis H19 : (J2 = 1)
L84Hypothesis 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)))))))
L85Hypothesis H21 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L86Hypothesis H22 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L87Hypothesis H23 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = (X * Y))))
L88Hypothesis H24 : (∀X ∈ int, ((G3 X) = X))
L89Hypothesis H25 : (H3 = 1)
L90Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L91Hypothesis H27 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L92Hypothesis H28 : (∀X ∈ int, ((FAST X) = ((if (X <= 0) then 1 else (W2 X)) * (V3 X))))
L93Theorem. (
A52677)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.