Beginning of Section A9969
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
L14Hypothesis HH0 : H0 ∈ int
L15Variable U0 : set → set → set
L16Hypothesis HU0 : ∀x0 ∈ int, ∀x1 ∈ int, U0 x0 x1 ∈ int
L17Variable V0 : set → set
L18Hypothesis HV0 : ∀x0 ∈ int, V0 x0 ∈ int
L19Variable SMALL : set → set
L20Hypothesis HSMALL : ∀x0 ∈ int, SMALL x0 ∈ int
L21Variable F1 : set → set
L22Hypothesis HF1 : ∀x0 ∈ int, F1 x0 ∈ int
L24Hypothesis HG1 : G1 ∈ int
L25Variable F2 : set → set → set
L26Hypothesis HF2 : ∀x0 ∈ int, ∀x1 ∈ int, F2 x0 x1 ∈ int
L27Variable G2 : set → set → set
L28Hypothesis HG2 : ∀x0 ∈ int, ∀x1 ∈ int, G2 x0 x1 ∈ int
L29Variable H2 : set → set
L30Hypothesis HH2 : ∀x0 ∈ int, H2 x0 ∈ int
L32Hypothesis HI2 : I2 ∈ int
L34Hypothesis HJ2 : J2 ∈ int
L35Variable U2 : set → set → set → set
L36Hypothesis HU2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, U2 x0 x1 x2 ∈ int
L37Variable V2 : set → set → set → set
L38Hypothesis HV2 : ∀x0 ∈ int, ∀x1 ∈ int, ∀x2 ∈ int, V2 x0 x1 x2 ∈ int
L39Variable W2 : set → set
L40Hypothesis HW2 : ∀x0 ∈ int, W2 x0 ∈ int
L41Variable H1 : set → set
L42Hypothesis HH1 : ∀x0 ∈ int, H1 x0 ∈ int
L43Variable U1 : set → set → set
L44Hypothesis HU1 : ∀x0 ∈ int, ∀x1 ∈ int, U1 x0 x1 ∈ int
L45Variable V1 : set → set
L46Hypothesis HV1 : ∀x0 ∈ int, V1 x0 ∈ int
L47Variable FAST : set → set
L48Hypothesis HFAST : ∀x0 ∈ int, FAST x0 ∈ int
L49Hypothesis H1 : (∀X ∈ int, ((F0 X) = ((2 * (X + X)) + X)))
L50Hypothesis H2 : (∀X ∈ int, ((G0 X) = (X + X)))
L51Hypothesis H3 : (H0 = 1)
L52Hypothesis H4 : (∀X ∈ int, (∀Y ∈ int, ((U0 X Y) = (if (X <= 0) then Y else (F0 (U0 (X + - 1) Y))))))
L53Hypothesis H5 : (∀X ∈ int, ((V0 X) = (U0 (G0 X) H0)))
L54Hypothesis H6 : (∀X ∈ int, ((SMALL X) = (V0 X)))
L55Hypothesis H7 : (∀X ∈ int, ((F1 X) = (X * X)))
L56Hypothesis H8 : (G1 = 1)
L57Hypothesis H9 : (∀X ∈ int, (∀Y ∈ int, ((F2 X Y) = (X * Y))))
L58Hypothesis H10 : (∀X ∈ int, (∀Y ∈ int, ((G2 X Y) = Y)))
L59Hypothesis H11 : (∀X ∈ int, ((H2 X) = X))
L60Hypothesis H12 : (I2 = 1)
L61Hypothesis H13 : (J2 = (1 + (2 + 2)))
L62Hypothesis H14 : (∀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)))))))
L63Hypothesis H15 : (∀X ∈ int, (∀Y ∈ int, (∀Z ∈ int, ((V2 X Y Z) = (if (X <= 0) then Z else (G2 (U2 (X + - 1) Y Z) (V2 (X + - 1) Y Z)))))))
L64Hypothesis H16 : (∀X ∈ int, ((W2 X) = (U2 (H2 X) I2 J2)))
L65Hypothesis H17 : (∀X ∈ int, ((H1 X) = (W2 X)))
L66Hypothesis H18 : (∀X ∈ int, (∀Y ∈ int, ((U1 X Y) = (if (X <= 0) then Y else (F1 (U1 (X + - 1) Y))))))
L67Hypothesis H19 : (∀X ∈ int, ((V1 X) = (U1 G1 (H1 X))))
L68Hypothesis H20 : (∀X ∈ int, ((FAST X) = (V1 X)))
L69Theorem. (
A9969)
(∀N ∈ int, ((0 <= N) → ((SMALL N) = (FAST N))))
Proof: Proof not loaded.