Beginning of Section A98533
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 F2 : set → set → set
L10Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L11Variable G2 : set → set
L12Hypothesis HG2 : ∀x0 ∈ int, G2 x0 ∈ int
L13Variable H2 : set → set → set
L14Hypothesis HH2 : ∀x0 ∈ int, ∀x1 ∈ int, H2 x0 x1 ∈ int
L16Hypothesis HI2 : I2 ∈ int
L17Variable J2 : set → set
L18Hypothesis HJ2 : ∀x0 ∈ int, J2 x0 ∈ int
L19Variable U2 : set → set → set → set
L20Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L21Variable V2 : set → set → set → set
L22Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L23Variable W2 : set → set → set
L24Hypothesis HW2 : ∀x0 ∈ int, ∀x1 ∈ int, W2 x0 x1 ∈ int
L25Variable F1 : set → set → set
L26Hypothesis HF1 : ∀x0 ∈ int, ∀x1 ∈ int, F1 x0 x1 ∈ int
L27Variable G1 : set → set → set
L28Hypothesis HG1 : ∀x0 ∈ int, ∀x1 ∈ int, G1 x0 x1 ∈ int
L30Hypothesis HH1 : H1 ∈ int
L32Hypothesis HI1 : I1 ∈ int
L33Variable J1 : set → set → set
L34Hypothesis HJ1 : ∀x0 ∈ int, ∀x1 ∈ int, J1 x0 x1 ∈ int
L35Variable U1 : set → set → set → set
L36Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U1 x0 x1 x2 ∈ int
L37Variable V1 : set → set → set → set
L38Hypothesis HV1 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V1 x0 x1 x2 ∈ int
L39Variable W1 : set → set → set
L40Hypothesis HW1 : ∀x0 ∈ int, ∀x1 ∈ int, W1 x0 x1 ∈ int
L41Variable F0 : set → set → set
L42Hypothesis HF0 : ∀x0 ∈ int, ∀x1 ∈ int, F0 x0 x1 ∈ int
L43Variable G0 : set → set
L44Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L46Hypothesis HH0 : H0 ∈ int
L47Variable U0 : set → set → set
L48Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L49Variable V0 : set → set
L50Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L51Variable SMALL : set → set
L52Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L53Variable F6 : set → set
L54Hypothesis HF6 : ∀x0 ∈ int, F6 x0 ∈ int
L56Hypothesis HG6 : G6 ∈ int
L57Variable H6 : set → set
L58Hypothesis HH6 : ∀x0 ∈ int, H6 x0 ∈ int
L59Variable U6 : set → set → set
L60Hypothesis HU6 : ∀x0 ∈ int, ∀x1 ∈ int, U6 x0 x1 ∈ int
L61Variable V6 : set → set
L62Hypothesis HV6 : ∀x0 ∈ int, V6 x0 ∈ int
L63Variable F5 : set → set
L64Hypothesis HF5 : ∀x0 ∈ int, F5 x0 ∈ int
L66Hypothesis HG5 : G5 ∈ int
L67Variable F7 : set → set → set
L68Hypothesis HF7 : ∀x0 ∈ int, ∀x1 ∈ int, F7 x0 x1 ∈ int
L69Variable G7 : set → set
L70Hypothesis HG7 : ∀x0 ∈ int, G7 x0 ∈ int
L71Variable H7 : set → set
L72Hypothesis HH7 : ∀x0 ∈ int, H7 x0 ∈ int
L74Hypothesis HI7 : I7 ∈ int
L76Hypothesis HJ7 : J7 ∈ int
L77Variable U7 : set → set → set → set
L78Hypothesis HU7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U7 x0 x1 x2 ∈ int
L79Variable V7 : set → set → set → set
L80Hypothesis HV7 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V7 x0 x1 x2 ∈ int
L81Variable W7 : set → set
L82Hypothesis HW7 : ∀x0 ∈ int, W7 x0 ∈ int
L83Variable H5 : set → set
L84Hypothesis HH5 : ∀x0 ∈ int, H5 x0 ∈ int
L85Variable U5 : set → set → set
L86Hypothesis HU5 : ∀x0 ∈ int, ∀x1 ∈ int, U5 x0 x1 ∈ int
L87Variable V5 : set → set
L88Hypothesis HV5 : ∀x0 ∈ int, V5 x0 ∈ int
L89Variable F4 : set → set
L90Hypothesis HF4 : ∀x0 ∈ int, F4 x0 ∈ int
L92Hypothesis HG4 : G4 ∈ int
L93Variable H4 : set → set → set
L94Hypothesis HH4 : ∀x0 ∈ int, ∀x1 ∈ int, H4 x0 x1 ∈ int
L95Variable U4 : set → set → set
L96Hypothesis HU4 : ∀x0 ∈ int, ∀x1 ∈ int, U4 x0 x1 ∈ int
L97Variable V4 : set → set → set
L98Hypothesis HV4 : ∀x0 ∈ int, ∀x1 ∈ int, V4 x0 x1 ∈ int
L99Variable F3 : set → set → set
L100Hypothesis HF3 : ∀x0 ∈ int, ∀x1 ∈ int, F3 x0 x1 ∈ int
L101Variable G3 : set → set
L102Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L104Hypothesis HH3 : H3 ∈ int
L105Variable U3 : set → set → set
L106Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L107Variable V3 : set → set
L108Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L109Variable FAST : set → set
L110Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L111Hypothesis H1 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X + Y))))
L112Hypothesis H2 : (∀X ∈ int, ((G2 X) = X))
L113Hypothesis H3 : (∀X ∈ int, (∀Y ∈ int, ((H2 X Y) = Y)))
L114Hypothesis H4 : (I2 = 0)
L115Hypothesis H5 : (∀X ∈ int, ((J2 X) = X))
L116Hypothesis H6 : (∀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)))))))
L117Hypothesis H7 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z)))))))
L118Hypothesis H8 : (∀X ∈ int, (∀Y ∈ int, ((W2 X Y) = (U2 (H2 X Y) I2 (J2 X)))))
L119Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F1 X Y) = ((W2 X Y) * X))))
L120Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G1 X Y) = Y)))
L121Hypothesis H11 : (H1 = (1 + 2))
L122Hypothesis H12 : (I1 = 1)
L123Hypothesis H13 : (∀X ∈ int, (∀Y ∈ int, ((J1 X Y) = Y)))
L124Hypothesis H14 : (∀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)))))))
L125Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V1 X Y Z) = (if (X <= 0) then Z else (G1 (U1 (X + - 1) Y Z) (V1 (X + - 1) Y Z)))))))
L126Hypothesis H16 : (∀X ∈ int, (∀Y ∈ int, ((W1 X Y) = (U1 H1 I1 (J1 X Y)))))
L127Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((F0 X Y) = ((W1 X Y) + X))))
L128Hypothesis H18 : (∀X ∈ int, ((G0 X) = X))
L129Hypothesis H19 : (H0 = 0)
L130Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y) X)))))
L131Hypothesis H21 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L132Hypothesis H22 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L133Hypothesis H23 : (∀X ∈ int, ((F6 X) = ((X * X) * X)))
L134Hypothesis H24 : (G6 = 1)
L135Hypothesis H25 : (∀X ∈ int, ((H6 X) = (X * X)))
L136Hypothesis H26 : (∀X ∈ int, (∀Y ∈ int, ((U6 X Y) = (if (X <= 0) then Y else (F6 (U6 (X + - 1) Y))))))
L137Hypothesis H27 : (∀X ∈ int, ((V6 X) = (U6 G6 (H6 X))))
L138Hypothesis H28 : (∀X ∈ int, ((F5 X) = ((V6 X) * X)))
L139Hypothesis H29 : (G5 = 1)
L140Hypothesis H30 : (∀X ∈ int, (∀Y ∈ int, ((F7 X Y) = (X + Y))))
L141Hypothesis H31 : (∀X ∈ int, ((G7 X) = X))
L142Hypothesis H32 : (∀X ∈ int, ((H7 X) = (X + - 2)))
L143Hypothesis H33 : (I7 = 1)
L144Hypothesis H34 : (J7 = 1)
L145Hypothesis H35 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((U7 X Y Z) = (if (X <= 0) then Y else (F7 (U7 (X + - 1) Y Z) (V7 (X + - 1) Y Z)))))))
L146Hypothesis H36 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V7 X Y Z) = (if (X <= 0) then Z else (G7 (U7 (X + - 1) Y Z)))))))
L147Hypothesis H37 : (∀X ∈ int, ((W7 X) = (U7 (H7 X) I7 J7)))
L148Hypothesis H38 : (∀X ∈ int, ((H5 X) = (W7 X)))
L149Hypothesis H39 : (∀X ∈ int, (∀Y ∈ int, ((U5 X Y) = (if (X <= 0) then Y else (F5 (U5 (X + - 1) Y))))))
L150Hypothesis H40 : (∀X ∈ int, ((V5 X) = (U5 G5 (H5 X))))
L151Hypothesis H41 : (∀X ∈ int, ((F4 X) = (V5 X)))
L152Hypothesis H42 : (G4 = 1)
L153Hypothesis H43 : (∀X ∈ int, (∀Y ∈ int, ((H4 X Y) = Y)))
L154Hypothesis H44 : (∀X ∈ int, (∀Y ∈ int, ((U4 X Y) = (if (X <= 0) then Y else (F4 (U4 (X + - 1) Y))))))
L155Hypothesis H45 : (∀X ∈ int, (∀Y ∈ int, ((V4 X Y) = (U4 G4 (H4 X Y)))))
L156Hypothesis H46 : (∀X ∈ int, (∀Y ∈ int, ((F3 X Y) = ((V4 X Y) + X))))
L157Hypothesis H47 : (∀X ∈ int, ((G3 X) = X))
L158Hypothesis H48 : (H3 = 0)
L159Hypothesis H49 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y) X)))))
L160Hypothesis H50 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L161Hypothesis H51 : (∀X ∈ int, ((FAST X) = (V3 X)))
L162Theorem. (
A98533)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.