Beginning of Section A134169
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
L10Hypothesis HF0 : ∀x0 ∈ int, F0 x0 ∈ int
L11Variable G0 : set → set
L12Hypothesis HG0 : ∀x0 ∈ int, G0 x0 ∈ int
L13Variable F1 : set → set
L14Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L15Variable G1 : set → set
L16Hypothesis HG1 : ∀x0 ∈ int, G1 x0 ∈ int
L18Hypothesis HH1 : H1 ∈ int
L19Variable U1 : set → set → set
L20Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L21Variable V1 : set → set
L22Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L23Variable H0 : set → set
L24Hypothesis HH0 : ∀x0 ∈ int, H0 x0 ∈ 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
L32Hypothesis HF2 : ∀x0 ∈ int, F2 x0 ∈ int
L34Hypothesis HG2 : G2 ∈ int
L35Variable F3 : set → set
L36Hypothesis HF3 : ∀x0 ∈ int, F3 x0 ∈ int
L37Variable G3 : set → set
L38Hypothesis HG3 : ∀x0 ∈ int, G3 x0 ∈ int
L40Hypothesis HH3 : H3 ∈ int
L41Variable U3 : set → set → set
L42Hypothesis HU3 : ∀x0 ∈ int, ∀x1 ∈ int, U3 x0 x1 ∈ int
L43Variable V3 : set → set
L44Hypothesis HV3 : ∀x0 ∈ int, V3 x0 ∈ int
L45Variable H2 : set → set
L46Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L47Variable U2 : set → set → set
L48Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, U2 x0 x1 ∈ int
L49Variable V2 : set → set
L50Hypothesis HV2 : ∀x0 ∈ int, V2 x0 ∈ int
L51Variable FAST : set → set
L52Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L53Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((X + - 1) + X)))
L54Hypothesis H2 : (∀X ∈ int, ((G0 X) = (X + - 1)))
L55Hypothesis H3 : (∀X ∈ int, ((F1 X) = (X + X)))
L56Hypothesis H4 : (∀X ∈ int, ((G1 X) = X))
L57Hypothesis H5 : (H1 = 1)
L58Hypothesis H6 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L59Hypothesis H7 : (∀X ∈ int, ((V1 X) = (U1 (G1 X) H1)))
L60Hypothesis H8 : (∀X ∈ int, ((H0 X) = (V1 X)))
L61Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L62Hypothesis H10 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) (H0 X))))
L63Hypothesis H11 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L64Hypothesis H12 : (∀X ∈ int, ((F2 X) = ((2 * (X * X)) + - X)))
L65Hypothesis H13 : (G2 = 1)
L66Hypothesis H14 : (∀X ∈ int, ((F3 X) = (X + X)))
L67Hypothesis H15 : (∀X ∈ int, ((G3 X) = (X + - 1)))
L68Hypothesis H16 : (H3 = 1)
L69Hypothesis H17 : (∀X ∈ int, (∀Y ∈ int, ((U3 X Y) = (if (X <= 0) then Y else (F3 (U3 (X + - 1) Y))))))
L70Hypothesis H18 : (∀X ∈ int, ((V3 X) = (U3 (G3 X) H3)))
L71Hypothesis H19 : (∀X ∈ int, ((H2 X) = (if (X <= 0) then 0 else (V3 X))))
L72Hypothesis H20 : (∀X ∈ int, (∀Y ∈ int, ((U2 X Y) = (if (X <= 0) then Y else (F2 (U2 (X + - 1) Y))))))
L73Hypothesis H21 : (∀X ∈ int, ((V2 X) = (U2 G2 (H2 X))))
L74Hypothesis H22 : (∀X ∈ int, ((FAST X) = (1 + (V2 X))))
L75Theorem. (
A134169)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.