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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rmxyelxp 41101* | Lemma for frmx 41102 and frmy 41103. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (β‘(π β (β0 Γ β€) β¦ ((1st βπ) + ((ββ((π΄β2) β 1)) Β· (2nd βπ))))β((π΄ + (ββ((π΄β2) β 1)))βπ)) β (β0 Γ β€)) | ||
Theorem | frmx 41102 | The X sequence is a nonnegative integer. See rmxnn 41140 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | ||
Theorem | frmy 41103 | The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | ||
Theorem | rmxyval 41104 | Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm π))) = ((π΄ + (ββ((π΄β2) β 1)))βπ)) | ||
Theorem | rmspecpos 41105 | The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β+) | ||
Theorem | rmxycomplete 41106* | The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. This is Metamath 100 proof #39. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β€) β (((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1 β βπ β β€ (π = (π΄ Xrm π) β§ π = (π΄ Yrm π)))) | ||
Theorem | rmxynorm 41107 | The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Xrm π)β2) β (((π΄β2) β 1) Β· ((π΄ Yrm π)β2))) = 1) | ||
Theorem | rmbaserp 41108 | The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ + (ββ((π΄β2) β 1))) β β+) | ||
Theorem | rmxyneg 41109 | Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain β0 or β€; we use β€ consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm -π) = (π΄ Xrm π) β§ (π΄ Yrm -π) = -(π΄ Yrm π))) | ||
Theorem | rmxyadd 41110 | Addition formula for X and Y sequences. See rmxadd 41116 and rmyadd 41120 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β ((π΄ Xrm (π + π)) = (((π΄ Xrm π) Β· (π΄ Xrm π)) + (((π΄β2) β 1) Β· ((π΄ Yrm π) Β· (π΄ Yrm π)))) β§ (π΄ Yrm (π + π)) = (((π΄ Yrm π) Β· (π΄ Xrm π)) + ((π΄ Xrm π) Β· (π΄ Yrm π))))) | ||
Theorem | rmxy1 41111 | Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 1) = π΄ β§ (π΄ Yrm 1) = 1)) | ||
Theorem | rmxy0 41112 | Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 0) = 1 β§ (π΄ Yrm 0) = 0)) | ||
Theorem | rmxneg 41113 | Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 41109, rmxyadd 41110, rmxy0 41112, and rmxy1 41111 via qirropth 41096 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm -π) = (π΄ Xrm π)) | ||
Theorem | rmx0 41114 | Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Xrm 0) = 1) | ||
Theorem | rmx1 41115 | Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Xrm 1) = π΄) | ||
Theorem | rmxadd 41116 | Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π΄ Xrm (π + π)) = (((π΄ Xrm π) Β· (π΄ Xrm π)) + (((π΄β2) β 1) Β· ((π΄ Yrm π) Β· (π΄ Yrm π))))) | ||
Theorem | rmyneg 41117 | Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm -π) = -(π΄ Yrm π)) | ||
Theorem | rmy0 41118 | Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) = 0) | ||
Theorem | rmy1 41119 | Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ (π΄ β (β€β₯β2) β (π΄ Yrm 1) = 1) | ||
Theorem | rmyadd 41120 | Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π΄ Yrm (π + π)) = (((π΄ Yrm π) Β· (π΄ Xrm π)) + ((π΄ Xrm π) Β· (π΄ Yrm π)))) | ||
Theorem | rmxp1 41121 | Special addition-of-1 formula for X sequence. Part 1 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (π + 1)) = (((π΄ Xrm π) Β· π΄) + (((π΄β2) β 1) Β· (π΄ Yrm π)))) | ||
Theorem | rmyp1 41122 | Special addition of 1 formula for Y sequence. Part 2 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + 1)) = (((π΄ Yrm π) Β· π΄) + (π΄ Xrm π))) | ||
Theorem | rmxm1 41123 | Subtraction of 1 formula for X sequence. Part 1 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (π β 1)) = ((π΄ Β· (π΄ Xrm π)) β (((π΄β2) β 1) Β· (π΄ Yrm π)))) | ||
Theorem | rmym1 41124 | Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π β 1)) = (((π΄ Yrm π) Β· π΄) β (π΄ Xrm π))) | ||
Theorem | rmxluc 41125 | The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (π + 1)) = (((2 Β· π΄) Β· (π΄ Xrm π)) β (π΄ Xrm (π β 1)))) | ||
Theorem | rmyluc 41126 | The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 41118 and rmy1 41119. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain (β€ Γ β€), which simplifies some later theorems. It may shorten the derivation to use this as our initial definition. Incidentally, the X sequence satisfies the exact same recurrence. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + 1)) = ((2 Β· ((π΄ Yrm π) Β· π΄)) β (π΄ Yrm (π β 1)))) | ||
Theorem | rmyluc2 41127 | Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + 1)) = (((2 Β· π΄) Β· (π΄ Yrm π)) β (π΄ Yrm (π β 1)))) | ||
Theorem | rmxdbl 41128 | "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm (2 Β· π)) = ((2 Β· ((π΄ Xrm π)β2)) β 1)) | ||
Theorem | rmydbl 41129 | "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (2 Β· π)) = ((2 Β· (π΄ Xrm π)) Β· (π΄ Yrm π))) | ||
Theorem | monotuz 41130* | A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π β§ π¦ β π») β πΉ < πΊ) & β’ ((π β§ π₯ β π») β πΆ β β) & β’ π» = (β€β₯βπΌ) & β’ (π₯ = (π¦ + 1) β πΆ = πΊ) & β’ (π₯ = π¦ β πΆ = πΉ) & β’ (π₯ = π΄ β πΆ = π·) & β’ (π₯ = π΅ β πΆ = πΈ) β β’ ((π β§ (π΄ β π» β§ π΅ β π»)) β (π΄ < π΅ β π· < πΈ)) | ||
Theorem | monotoddzzfi 41131* | A function which is odd and monotonic on β0 is monotonic on β€. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
β’ ((π β§ π₯ β β€) β (πΉβπ₯) β β) & β’ ((π β§ π₯ β β€) β (πΉβ-π₯) = -(πΉβπ₯)) & β’ ((π β§ π₯ β β0 β§ π¦ β β0) β (π₯ < π¦ β (πΉβπ₯) < (πΉβπ¦))) β β’ ((π β§ π΄ β β€ β§ π΅ β β€) β (π΄ < π΅ β (πΉβπ΄) < (πΉβπ΅))) | ||
Theorem | monotoddzz 41132* | A function (given implicitly) which is odd and monotonic on β0 is monotonic on β€. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
β’ ((π β§ π₯ β β0 β§ π¦ β β0) β (π₯ < π¦ β πΈ < πΉ)) & β’ ((π β§ π₯ β β€) β πΈ β β) & β’ ((π β§ π¦ β β€) β πΊ = -πΉ) & β’ (π₯ = π΄ β πΈ = πΆ) & β’ (π₯ = π΅ β πΈ = π·) & β’ (π₯ = π¦ β πΈ = πΉ) & β’ (π₯ = -π¦ β πΈ = πΊ) β β’ ((π β§ π΄ β β€ β§ π΅ β β€) β (π΄ < π΅ β πΆ < π·)) | ||
Theorem | oddcomabszz 41133* | An odd function which takes nonnegative values on nonnegative arguments commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π β§ π₯ β β€) β π΄ β β) & β’ ((π β§ π₯ β β€ β§ 0 β€ π₯) β 0 β€ π΄) & β’ ((π β§ π¦ β β€) β πΆ = -π΅) & β’ (π₯ = π¦ β π΄ = π΅) & β’ (π₯ = -π¦ β π΄ = πΆ) & β’ (π₯ = π· β π΄ = πΈ) & β’ (π₯ = (absβπ·) β π΄ = πΉ) β β’ ((π β§ π· β β€) β (absβπΈ) = πΉ) | ||
Theorem | 2nn0ind 41134* | Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ π & β’ π & β’ (π¦ β β β ((π β§ π) β π)) & β’ (π₯ = 0 β (π β π)) & β’ (π₯ = 1 β (π β π)) & β’ (π₯ = (π¦ β 1) β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = π΄ β (π β π)) β β’ (π΄ β β0 β π) | ||
Theorem | zindbi 41135* | Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (π¦ β β€ β (π β π)) & β’ (π₯ = π¦ β (π β π)) & β’ (π₯ = (π¦ + 1) β (π β π)) & β’ (π₯ = 0 β (π β π)) & β’ (π₯ = π΄ β (π β π)) β β’ (π΄ β β€ β (π β π)) | ||
Theorem | rmxypos 41136 | For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (0 < (π΄ Xrm π) β§ 0 β€ (π΄ Yrm π))) | ||
Theorem | ltrmynn0 41137 | The Y-sequence is strictly monotonic on β0. Strengthened by ltrmy 41141. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β0) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) | ||
Theorem | ltrmxnn0 41138 | The X-sequence is strictly monotonic on β0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β0) β (π < π β (π΄ Xrm π) < (π΄ Xrm π))) | ||
Theorem | lermxnn0 41139 | The X-sequence is monotonic on β0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0 β§ π β β0) β (π β€ π β (π΄ Xrm π) β€ (π΄ Xrm π))) | ||
Theorem | rmxnn 41140 | The X-sequence is defined to range over β0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β) | ||
Theorem | ltrmy 41141 | The Y-sequence is strictly monotonic over β€. (Contributed by Stefan O'Rear, 25-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) | ||
Theorem | rmyeq0 41142 | Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π = 0 β (π΄ Yrm π) = 0)) | ||
Theorem | rmyeq 41143 | Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π = π β (π΄ Yrm π) = (π΄ Yrm π))) | ||
Theorem | lermy 41144 | Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π β€ π β (π΄ Yrm π) β€ (π΄ Yrm π))) | ||
Theorem | rmynn 41145 | Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β) β (π΄ Yrm π) β β) | ||
Theorem | rmynn0 41146 | Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (π΄ Yrm π) β β0) | ||
Theorem | rmyabs 41147 | Yrm commutes with abs. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π΅ β β€) β (absβ(π΄ Yrm π΅)) = (π΄ Yrm (absβπ΅))) | ||
Theorem | jm2.24nn 41148 | X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to β. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β) β ((π΄ Yrm (π β 1)) + (π΄ Yrm π)) < (π΄ Xrm π)) | ||
Theorem | jm2.17a 41149 | First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (((2 Β· π΄) β 1)βπ) β€ (π΄ Yrm (π + 1))) | ||
Theorem | jm2.17b 41150 | Weak form of the second half of lemma 2.17 of [JonesMatijasevic] p. 696, allowing induction to start lower. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (π΄ Yrm (π + 1)) β€ ((2 Β· π΄)βπ)) | ||
Theorem | jm2.17c 41151 | Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β) β (π΄ Yrm ((π + 1) + 1)) < ((2 Β· π΄)β(π + 1))) | ||
Theorem | jm2.24 41152 | Lemma 2.24 of [JonesMatijasevic] p. 697 extended to β€. Could be eliminated with a more careful proof of jm2.26lem3 41190. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Yrm (π β 1)) + (π΄ Yrm π)) < (π΄ Xrm π)) | ||
Theorem | rmygeid 41153 | Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β π β€ (π΄ Yrm π)) | ||
Theorem | congtr 41154 | A wff of the form π΄ β₯ (π΅ β πΆ) is interpreted as a congruential equation. This is similar to (π΅ mod π΄) = (πΆ mod π΄), but is defined such that behavior is regular for zero and negative values of π΄. To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€) β§ (πΆ β β€ β§ π· β β€) β§ (π΄ β₯ (π΅ β πΆ) β§ π΄ β₯ (πΆ β π·))) β π΄ β₯ (π΅ β π·)) | ||
Theorem | congadd 41155 | If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β§ (π· β β€ β§ πΈ β β€) β§ (π΄ β₯ (π΅ β πΆ) β§ π΄ β₯ (π· β πΈ))) β π΄ β₯ ((π΅ + π·) β (πΆ + πΈ))) | ||
Theorem | congmul 41156 | If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β§ (π· β β€ β§ πΈ β β€) β§ (π΄ β₯ (π΅ β πΆ) β§ π΄ β₯ (π· β πΈ))) β π΄ β₯ ((π΅ Β· π·) β (πΆ Β· πΈ))) | ||
Theorem | congsym 41157 | Congruence mod π΄ is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€) β§ (πΆ β β€ β§ π΄ β₯ (π΅ β πΆ))) β π΄ β₯ (πΆ β π΅)) | ||
Theorem | congneg 41158 | If two integers are congruent mod π΄, so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€) β§ (πΆ β β€ β§ π΄ β₯ (π΅ β πΆ))) β π΄ β₯ (-π΅ β -πΆ)) | ||
Theorem | congsub 41159 | If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β§ (π· β β€ β§ πΈ β β€) β§ (π΄ β₯ (π΅ β πΆ) β§ π΄ β₯ (π· β πΈ))) β π΄ β₯ ((π΅ β π·) β (πΆ β πΈ))) | ||
Theorem | congid 41160 | Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ ((π΄ β β€ β§ π΅ β β€) β π΄ β₯ (π΅ β π΅)) | ||
Theorem | mzpcong 41161* | Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.) |
β’ ((πΉ β (mzPolyβπ) β§ (π β (β€ βm π) β§ π β (β€ βm π)) β§ (π β β€ β§ βπ β π π β₯ ((πβπ) β (πβπ)))) β π β₯ ((πΉβπ) β (πΉβπ))) | ||
Theorem | congrep 41162* | Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β β β§ π β β€) β βπ β (0...(π΄ β 1))π΄ β₯ (π β π)) | ||
Theorem | congabseq 41163 | If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ (((π΄ β β β§ π΅ β β€ β§ πΆ β β€) β§ π΄ β₯ (π΅ β πΆ)) β ((absβ(π΅ β πΆ)) < π΄ β π΅ = πΆ)) | ||
Theorem | acongid 41164 |
A wff like that in this theorem will be known as an "alternating
congruence". A special symbol might be considered if more uses come
up.
They have many of the same properties as normal congruences, starting with
reflexivity.
JonesMatijasevic uses "a β‘ Β± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β β€ β§ π΅ β β€) β (π΄ β₯ (π΅ β π΅) β¨ π΄ β₯ (π΅ β -π΅))) | ||
Theorem | acongsym 41165 | Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β§ (π΄ β₯ (π΅ β πΆ) β¨ π΄ β₯ (π΅ β -πΆ))) β (π΄ β₯ (πΆ β π΅) β¨ π΄ β₯ (πΆ β -π΅))) | ||
Theorem | acongneg2 41166 | Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β§ (π΄ β₯ (π΅ β -πΆ) β¨ π΄ β₯ (π΅ β --πΆ))) β (π΄ β₯ (π΅ β πΆ) β¨ π΄ β₯ (π΅ β -πΆ))) | ||
Theorem | acongtr 41167 | Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€) β§ (πΆ β β€ β§ π· β β€) β§ ((π΄ β₯ (π΅ β πΆ) β¨ π΄ β₯ (π΅ β -πΆ)) β§ (π΄ β₯ (πΆ β π·) β¨ π΄ β₯ (πΆ β -π·)))) β (π΄ β₯ (π΅ β π·) β¨ π΄ β₯ (π΅ β -π·))) | ||
Theorem | acongeq12d 41168 | Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ (π β π΅ = πΆ) & β’ (π β π· = πΈ) β β’ (π β ((π΄ β₯ (π΅ β π·) β¨ π΄ β₯ (π΅ β -π·)) β (π΄ β₯ (πΆ β πΈ) β¨ π΄ β₯ (πΆ β -πΈ)))) | ||
Theorem | acongrep 41169* | Every integer is alternating-congruent to some number in the first half of the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β β β§ π β β€) β βπ β (0...π΄)((2 Β· π΄) β₯ (π β π) β¨ (2 Β· π΄) β₯ (π β -π))) | ||
Theorem | fzmaxdif 41170 | Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ (((πΆ β β€ β§ π΄ β (π΅...πΆ)) β§ (πΉ β β€ β§ π· β (πΈ...πΉ)) β§ (πΆ β πΈ) β€ (πΉ β π΅)) β (absβ(π΄ β π·)) β€ (πΉ β π΅)) | ||
Theorem | fzneg 41171 | Reflection of a finite range of integers about 0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β (π΄ β (π΅...πΆ) β -π΄ β (-πΆ...-π΅))) | ||
Theorem | acongeq 41172 | Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 41191. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β β β§ π΅ β (0...π΄) β§ πΆ β (0...π΄)) β (π΅ = πΆ β ((2 Β· π΄) β₯ (π΅ β πΆ) β¨ (2 Β· π΄) β₯ (π΅ β -πΆ)))) | ||
Theorem | dvdsacongtr 41173 | Alternating congruence passes from a base to a dividing base. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€) β§ (πΆ β β€ β§ π· β β€) β§ (π· β₯ π΄ β§ (π΄ β₯ (π΅ β πΆ) β¨ π΄ β₯ (π΅ β -πΆ)))) β (π· β₯ (π΅ β πΆ) β¨ π· β₯ (π΅ β -πΆ))) | ||
Theorem | coprmdvdsb 41174 | Multiplication by a coprime number does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
β’ ((πΎ β β€ β§ π β β€ β§ (π β β€ β§ (πΎ gcd π) = 1)) β (πΎ β₯ π β πΎ β₯ (π Β· π))) | ||
Theorem | modabsdifz 41175 | Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π β β β§ π β β β§ π β 0) β ((π β (π mod (absβπ))) / π) β β€) | ||
Theorem | dvdsabsmod0 41176 | Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014.) (Proof shortened by OpenAI, 3-Jul-2020.) |
β’ ((π β β€ β§ π β β€ β§ π β 0) β (π β₯ π β (π mod (absβπ)) = 0)) | ||
Theorem | jm2.18 41177 | Theorem 2.18 of [JonesMatijasevic] p. 696. Direct relationship of the exponential function to X and Y sequences. (Contributed by Stefan O'Rear, 14-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ πΎ β β0 β§ π β β0) β ((((2 Β· π΄) Β· πΎ) β (πΎβ2)) β 1) β₯ (((π΄ Xrm π) β ((π΄ β πΎ) Β· (π΄ Yrm π))) β (πΎβπ))) | ||
Theorem | jm2.19lem1 41178 | Lemma for jm2.19 41182. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) gcd (π΄ Yrm π)) = 1) | ||
Theorem | jm2.19lem2 41179 | Lemma for jm2.19 41182. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β ((π΄ Yrm π) β₯ (π΄ Yrm π) β (π΄ Yrm π) β₯ (π΄ Yrm (π + π)))) | ||
Theorem | jm2.19lem3 41180 | Lemma for jm2.19 41182. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ (π β β€ β§ π β β€) β§ πΌ β β0) β ((π΄ Yrm π) β₯ (π΄ Yrm π) β (π΄ Yrm π) β₯ (π΄ Yrm (π + (πΌ Β· π))))) | ||
Theorem | jm2.19lem4 41181 | Lemma for jm2.19 41182. Extend to ZZ by symmetry. TODO: use zindbi 41135. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ (π β β€ β§ π β β€) β§ πΌ β β€) β ((π΄ Yrm π) β₯ (π΄ Yrm π) β (π΄ Yrm π) β₯ (π΄ Yrm (π + (πΌ Β· π))))) | ||
Theorem | jm2.19 41182 | Lemma 2.19 of [JonesMatijasevic] p. 696. Transfer divisibility constraints between Y-values and their indices. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π β₯ π β (π΄ Yrm π) β₯ (π΄ Yrm π))) | ||
Theorem | jm2.21 41183 | Lemma for jm2.20nn 41186. Express X and Y values as a binomial. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π½ β β€) β ((π΄ Xrm (π Β· π½)) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm (π Β· π½)))) = (((π΄ Xrm π) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm π)))βπ½)) | ||
Theorem | jm2.22 41184* | Lemma for jm2.20nn 41186. Applying binomial theorem and taking irrational part. (Contributed by Stefan O'Rear, 26-Sep-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π½ β β0) β (π΄ Yrm (π Β· π½)) = Ξ£π β {π₯ β (0...π½) β£ Β¬ 2 β₯ π₯} ((π½Cπ) Β· (((π΄ Xrm π)β(π½ β π)) Β· (((π΄ Yrm π)βπ) Β· (((π΄β2) β 1)β((π β 1) / 2)))))) | ||
Theorem | jm2.23 41185 | Lemma for jm2.20nn 41186. Truncate binomial expansion p-adicly. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π½ β β) β ((π΄ Yrm π)β3) β₯ ((π΄ Yrm (π Β· π½)) β (π½ Β· (((π΄ Xrm π)β(π½ β 1)) Β· (π΄ Yrm π))))) | ||
Theorem | jm2.20nn 41186 | Lemma 2.20 of [JonesMatijasevic] p. 696, the "first step down lemma". (Contributed by Stefan O'Rear, 27-Sep-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β β§ π β β) β (((π΄ Yrm π)β2) β₯ (π΄ Yrm π) β (π Β· (π΄ Yrm π)) β₯ π)) | ||
Theorem | jm2.25lem1 41187 | Lemma for jm2.26 41191. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ (((π΄ β β€ β§ π΅ β β€) β§ (πΆ β β€ β§ π· β β€) β§ (π΄ β₯ (πΆ β π·) β¨ π΄ β₯ (πΆ β -π·))) β ((π΄ β₯ (π· β π΅) β¨ π΄ β₯ (π· β -π΅)) β (π΄ β₯ (πΆ β π΅) β¨ π΄ β₯ (πΆ β -π΅)))) | ||
Theorem | jm2.25 41188 | Lemma for jm2.26 41191. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ (π β β€ β§ π β β€) β§ πΌ β β€) β ((π΄ Xrm π) β₯ ((π΄ Yrm (π + (πΌ Β· (2 Β· π)))) β (π΄ Yrm π)) β¨ (π΄ Xrm π) β₯ ((π΄ Yrm (π + (πΌ Β· (2 Β· π)))) β -(π΄ Yrm π)))) | ||
Theorem | jm2.26a 41189 | Lemma for jm2.26 41191. Reverse direction is required to prove forward direction, so do it separately. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ (((π΄ β (β€β₯β2) β§ π β β€) β§ (πΎ β β€ β§ π β β€)) β (((2 Β· π) β₯ (πΎ β π) β¨ (2 Β· π) β₯ (πΎ β -π)) β ((π΄ Xrm π) β₯ ((π΄ Yrm πΎ) β (π΄ Yrm π)) β¨ (π΄ Xrm π) β₯ ((π΄ Yrm πΎ) β -(π΄ Yrm π))))) | ||
Theorem | jm2.26lem3 41190 | Lemma for jm2.26 41191. Use acongrep 41169 to find K', M' ~ K, M in [ 0,N ]. Thus Y(K') ~ Y(M') and both are small; K' = M' on pain of contradicting 2.24, so K ~ M. (Contributed by Stefan O'Rear, 3-Oct-2014.) |
β’ (((π΄ β (β€β₯β2) β§ π β β) β§ (πΎ β (0...π) β§ π β (0...π)) β§ ((π΄ Xrm π) β₯ ((π΄ Yrm πΎ) β (π΄ Yrm π)) β¨ (π΄ Xrm π) β₯ ((π΄ Yrm πΎ) β -(π΄ Yrm π)))) β πΎ = π) | ||
Theorem | jm2.26 41191 | Lemma 2.26 of [JonesMatijasevic] p. 697, the "second step down lemma". (Contributed by Stefan O'Rear, 2-Oct-2014.) |
β’ (((π΄ β (β€β₯β2) β§ π β β) β§ (πΎ β β€ β§ π β β€)) β (((π΄ Xrm π) β₯ ((π΄ Yrm πΎ) β (π΄ Yrm π)) β¨ (π΄ Xrm π) β₯ ((π΄ Yrm πΎ) β -(π΄ Yrm π))) β ((2 Β· π) β₯ (πΎ β π) β¨ (2 Β· π) β₯ (πΎ β -π)))) | ||
Theorem | jm2.15nn0 41192 | Lemma 2.15 of [JonesMatijasevic] p. 695. Yrm is a polynomial for fixed N, so has the expected congruence property. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π΅ β (β€β₯β2) β§ π β β0) β (π΄ β π΅) β₯ ((π΄ Yrm π) β (π΅ Yrm π))) | ||
Theorem | jm2.16nn0 41193 | Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 41192 if Yrm is redefined as described in rmyluc 41126. (Contributed by Stefan O'Rear, 1-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π β β0) β (π΄ β 1) β₯ ((π΄ Yrm π) β π)) | ||
Theorem | jm2.27a 41194 | Lemma for jm2.27 41197. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ (π β π΄ β (β€β₯β2)) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π· β β0) & β’ (π β πΈ β β0) & β’ (π β πΉ β β0) & β’ (π β πΊ β β0) & β’ (π β π» β β0) & β’ (π β πΌ β β0) & β’ (π β π½ β β0) & β’ (π β ((π·β2) β (((π΄β2) β 1) Β· (πΆβ2))) = 1) & β’ (π β ((πΉβ2) β (((π΄β2) β 1) Β· (πΈβ2))) = 1) & β’ (π β πΊ β (β€β₯β2)) & β’ (π β ((πΌβ2) β (((πΊβ2) β 1) Β· (π»β2))) = 1) & β’ (π β πΈ = ((π½ + 1) Β· (2 Β· (πΆβ2)))) & β’ (π β πΉ β₯ (πΊ β π΄)) & β’ (π β (2 Β· πΆ) β₯ (πΊ β 1)) & β’ (π β πΉ β₯ (π» β πΆ)) & β’ (π β (2 Β· πΆ) β₯ (π» β π΅)) & β’ (π β π΅ β€ πΆ) & β’ (π β π β β€) & β’ (π β π· = (π΄ Xrm π)) & β’ (π β πΆ = (π΄ Yrm π)) & β’ (π β π β β€) & β’ (π β πΉ = (π΄ Xrm π)) & β’ (π β πΈ = (π΄ Yrm π)) & β’ (π β π β β€) & β’ (π β πΌ = (πΊ Xrm π )) & β’ (π β π» = (πΊ Yrm π )) β β’ (π β πΆ = (π΄ Yrm π΅)) | ||
Theorem | jm2.27b 41195 | Lemma for jm2.27 41197. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ (π β π΄ β (β€β₯β2)) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π· β β0) & β’ (π β πΈ β β0) & β’ (π β πΉ β β0) & β’ (π β πΊ β β0) & β’ (π β π» β β0) & β’ (π β πΌ β β0) & β’ (π β π½ β β0) & β’ (π β ((π·β2) β (((π΄β2) β 1) Β· (πΆβ2))) = 1) & β’ (π β ((πΉβ2) β (((π΄β2) β 1) Β· (πΈβ2))) = 1) & β’ (π β πΊ β (β€β₯β2)) & β’ (π β ((πΌβ2) β (((πΊβ2) β 1) Β· (π»β2))) = 1) & β’ (π β πΈ = ((π½ + 1) Β· (2 Β· (πΆβ2)))) & β’ (π β πΉ β₯ (πΊ β π΄)) & β’ (π β (2 Β· πΆ) β₯ (πΊ β 1)) & β’ (π β πΉ β₯ (π» β πΆ)) & β’ (π β (2 Β· πΆ) β₯ (π» β π΅)) & β’ (π β π΅ β€ πΆ) β β’ (π β πΆ = (π΄ Yrm π΅)) | ||
Theorem | jm2.27c 41196 | Lemma for jm2.27 41197. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ (π β π΄ β (β€β₯β2)) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β πΆ = (π΄ Yrm π΅)) & β’ π· = (π΄ Xrm π΅) & β’ π = (π΅ Β· (π΄ Yrm π΅)) & β’ πΈ = (π΄ Yrm (2 Β· π)) & β’ πΉ = (π΄ Xrm (2 Β· π)) & β’ πΊ = (π΄ + ((πΉβ2) Β· ((πΉβ2) β π΄))) & β’ π» = (πΊ Yrm π΅) & β’ πΌ = (πΊ Xrm π΅) & β’ π½ = ((πΈ / (2 Β· (πΆβ2))) β 1) β β’ (π β (((π· β β0 β§ πΈ β β0 β§ πΉ β β0) β§ (πΊ β β0 β§ π» β β0 β§ πΌ β β0)) β§ (π½ β β0 β§ (((((π·β2) β (((π΄β2) β 1) Β· (πΆβ2))) = 1 β§ ((πΉβ2) β (((π΄β2) β 1) Β· (πΈβ2))) = 1 β§ πΊ β (β€β₯β2)) β§ (((πΌβ2) β (((πΊβ2) β 1) Β· (π»β2))) = 1 β§ πΈ = ((π½ + 1) Β· (2 Β· (πΆβ2))) β§ πΉ β₯ (πΊ β π΄))) β§ (((2 Β· πΆ) β₯ (πΊ β 1) β§ πΉ β₯ (π» β πΆ)) β§ ((2 Β· πΆ) β₯ (π» β π΅) β§ π΅ β€ πΆ)))))) | ||
Theorem | jm2.27 41197* | Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 41194 and jm2.27c 41196. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine". (Contributed by Stefan O'Rear, 4-Oct-2014.) |
β’ ((π΄ β (β€β₯β2) β§ π΅ β β β§ πΆ β β) β (πΆ = (π΄ Yrm π΅) β βπ β β0 βπ β β0 βπ β β0 βπ β β0 ββ β β0 βπ β β0 βπ β β0 (((((πβ2) β (((π΄β2) β 1) Β· (πΆβ2))) = 1 β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1 β§ π β (β€β₯β2)) β§ (((πβ2) β (((πβ2) β 1) Β· (ββ2))) = 1 β§ π = ((π + 1) Β· (2 Β· (πΆβ2))) β§ π β₯ (π β π΄))) β§ (((2 Β· πΆ) β₯ (π β 1) β§ π β₯ (β β πΆ)) β§ ((2 Β· πΆ) β₯ (β β π΅) β§ π΅ β€ πΆ))))) | ||
Theorem | jm2.27dlem1 41198* | Lemma for rmydioph 41203. Substitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
β’ π΄ β (1...π΅) β β’ (π = (π βΎ (1...π΅)) β (πβπ΄) = (πβπ΄)) | ||
Theorem | jm2.27dlem2 41199 | Lemma for rmydioph 41203. This theorem is used along with the next three to efficiently infer steps like 7 β (1...;10). (Contributed by Stefan O'Rear, 11-Oct-2014.) |
β’ π΄ β (1...π΅) & β’ πΆ = (π΅ + 1) & β’ π΅ β β β β’ π΄ β (1...πΆ) | ||
Theorem | jm2.27dlem3 41200 | Lemma for rmydioph 41203. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.) |
β’ π΄ β β β β’ π΄ β (1...π΄) |
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