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Type | Label | Description |
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Statement | ||
Theorem | nn0ltp1ne 34101 | Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.) |
β’ ((π΄ β β0 β§ π΅ β β0) β ((π΄ + 1) < π΅ β (π΄ < π΅ β§ π΅ β (π΄ + 1)))) | ||
Theorem | 0nn0m1nnn0 34102 | A number is zero if and only if it's a nonnegative integer that becomes negative after subtracting 1. (Contributed by BTernaryTau, 30-Sep-2023.) |
β’ (π = 0 β (π β β0 β§ Β¬ (π β 1) β β0)) | ||
Theorem | f1resfz0f1d 34103 | If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.) |
β’ (π β πΎ β β0) & β’ (π β πΉ:(0...πΎ)βΆπ) & β’ (π β (πΉ βΎ (1...πΎ)):(1...πΎ)β1-1βπ) & β’ (π β ((πΉ β {0}) β© (πΉ β (1...πΎ))) = β ) β β’ (π β πΉ:(0...πΎ)β1-1βπ) | ||
Theorem | fisshasheq 34104 | A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.) |
β’ ((π΅ β Fin β§ π΄ β π΅ β§ (β―βπ΄) = (β―βπ΅)) β π΄ = π΅) | ||
Theorem | hashf1dmcdm 34105 | The size of the domain of a one-to-one set function is less than or equal to the size of its codomain, if it exists. (Contributed by BTernaryTau, 1-Oct-2023.) |
β’ ((πΉ β π β§ π΅ β π β§ πΉ:π΄β1-1βπ΅) β (β―βπ΄) β€ (β―βπ΅)) | ||
Theorem | revpfxsfxrev 34106 | The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.) |
β’ ((π β Word π β§ πΏ β (0...(β―βπ))) β (reverseβ(π prefix πΏ)) = ((reverseβπ) substr β¨((β―βπ) β πΏ), (β―βπ)β©)) | ||
Theorem | swrdrevpfx 34107 | A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.) |
β’ ((π β Word π β§ πΉ β (0...πΏ) β§ πΏ β (0...(β―βπ))) β (π substr β¨πΉ, πΏβ©) = (reverseβ((reverseβ(π prefix πΏ)) prefix (πΏ β πΉ)))) | ||
Theorem | lfuhgr 34108* | A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β UHGraph β (πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β βπ₯ β (EdgβπΊ)2 β€ (β―βπ₯))) | ||
Theorem | lfuhgr2 34109* | A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β UHGraph β (πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β βπ₯ β (EdgβπΊ)(β―βπ₯) β 1)) | ||
Theorem | lfuhgr3 34110* | A hypergraph is loop-free if and only if none of its edges connect to only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β UHGraph β (πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β Β¬ βπ{π} β (EdgβπΊ))) | ||
Theorem | cplgredgex 34111* | Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β ComplGraph β ((π΄ β π β§ π΅ β (π β {π΄})) β βπ β πΈ {π΄, π΅} β π)) | ||
Theorem | cusgredgex 34112 | Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β ComplUSGraph β ((π΄ β π β§ π΅ β (π β {π΄})) β {π΄, π΅} β πΈ)) | ||
Theorem | cusgredgex2 34113 | Any two distinct vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 4-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β ComplUSGraph β ((π΄ β π β§ π΅ β π β§ π΄ β π΅) β {π΄, π΅} β πΈ)) | ||
Theorem | pfxwlk 34114 | A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.) |
β’ ((πΉ(WalksβπΊ)π β§ πΏ β (0...(β―βπΉ))) β (πΉ prefix πΏ)(WalksβπΊ)(π prefix (πΏ + 1))) | ||
Theorem | revwlk 34115 | The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.) |
β’ (πΉ(WalksβπΊ)π β (reverseβπΉ)(WalksβπΊ)(reverseβπ)) | ||
Theorem | revwlkb 34116 | Two words represent a walk if and only if their reverses also represent a walk. (Contributed by BTernaryTau, 4-Dec-2023.) |
β’ ((πΉ β Word π β§ π β Word π) β (πΉ(WalksβπΊ)π β (reverseβπΉ)(WalksβπΊ)(reverseβπ))) | ||
Theorem | swrdwlk 34117 | Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.) |
β’ ((πΉ(WalksβπΊ)π β§ π΅ β (0...πΏ) β§ πΏ β (0...(β―βπΉ))) β (πΉ substr β¨π΅, πΏβ©)(WalksβπΊ)(π substr β¨π΅, (πΏ + 1)β©)) | ||
Theorem | pthhashvtx 34118 | A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ (πΉ(PathsβπΊ)π β (β―βπΉ) β€ (β―βπ)) | ||
Theorem | pthisspthorcycl 34119 | A path is either a simple path or a cycle (or both). (Contributed by BTernaryTau, 20-Oct-2023.) |
β’ (πΉ(PathsβπΊ)π β (πΉ(SPathsβπΊ)π β¨ πΉ(CyclesβπΊ)π)) | ||
Theorem | spthcycl 34120 | A walk is a trivial path if and only if it is both a simple path and a cycle. (Contributed by BTernaryTau, 8-Oct-2023.) |
β’ ((πΉ(PathsβπΊ)π β§ πΉ = β ) β (πΉ(SPathsβπΊ)π β§ πΉ(CyclesβπΊ)π)) | ||
Theorem | usgrgt2cycl 34121 | A non-trivial cycle in a simple graph has a length greater than 2. (Contributed by BTernaryTau, 24-Sep-2023.) |
β’ ((πΊ β USGraph β§ πΉ(CyclesβπΊ)π β§ πΉ β β ) β 2 < (β―βπΉ)) | ||
Theorem | usgrcyclgt2v 34122 | A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β USGraph β§ πΉ(CyclesβπΊ)π β§ πΉ β β ) β 2 < (β―βπ)) | ||
Theorem | subgrwlk 34123 | If a walk exists in a subgraph of a graph πΊ, then that walk also exists in πΊ. (Contributed by BTernaryTau, 22-Oct-2023.) |
β’ (π SubGraph πΊ β (πΉ(Walksβπ)π β πΉ(WalksβπΊ)π)) | ||
Theorem | subgrtrl 34124 | If a trail exists in a subgraph of a graph πΊ, then that trail also exists in πΊ. (Contributed by BTernaryTau, 22-Oct-2023.) |
β’ (π SubGraph πΊ β (πΉ(Trailsβπ)π β πΉ(TrailsβπΊ)π)) | ||
Theorem | subgrpth 34125 | If a path exists in a subgraph of a graph πΊ, then that path also exists in πΊ. (Contributed by BTernaryTau, 22-Oct-2023.) |
β’ (π SubGraph πΊ β (πΉ(Pathsβπ)π β πΉ(PathsβπΊ)π)) | ||
Theorem | subgrcycl 34126 | If a cycle exists in a subgraph of a graph πΊ, then that cycle also exists in πΊ. (Contributed by BTernaryTau, 23-Oct-2023.) |
β’ (π SubGraph πΊ β (πΉ(Cyclesβπ)π β πΉ(CyclesβπΊ)π)) | ||
Theorem | cusgr3cyclex 34127* | Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β ComplUSGraph β§ 2 < (β―βπ)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3)) | ||
Theorem | loop1cycl 34128* | A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.) |
β’ (πΊ β UHGraph β (βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1 β§ (πβ0) = π΄) β {π΄} β (EdgβπΊ))) | ||
Theorem | 2cycld 34129 | Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.) |
β’ π = β¨βπ΄π΅πΆββ© & β’ πΉ = β¨βπ½πΎββ© & β’ (π β (π΄ β π β§ π΅ β π β§ πΆ β π)) & β’ (π β (π΄ β π΅ β§ π΅ β πΆ)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β π½ β πΎ) & β’ (π β π΄ = πΆ) β β’ (π β πΉ(CyclesβπΊ)π) | ||
Theorem | 2cycl2d 34130 | Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.) |
β’ π = β¨βπ΄π΅π΄ββ© & β’ πΉ = β¨βπ½πΎββ© & β’ (π β (π΄ β π β§ π΅ β π)) & β’ (π β π΄ β π΅) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΄, π΅} β (πΌβπΎ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β π½ β πΎ) β β’ (π β πΉ(CyclesβπΊ)π) | ||
Theorem | umgr2cycllem 34131* | Lemma for umgr2cycl 34132. (Contributed by BTernaryTau, 17-Oct-2023.) |
β’ πΉ = β¨βπ½πΎββ© & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UMGraph) & β’ (π β π½ β dom πΌ) & β’ (π β π½ β πΎ) & β’ (π β (πΌβπ½) = (πΌβπΎ)) β β’ (π β βπ πΉ(CyclesβπΊ)π) | ||
Theorem | umgr2cycl 34132* | A multigraph with two distinct edges that connect the same vertices has a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UMGraph β§ βπ β dom πΌβπ β dom πΌ((πΌβπ) = (πΌβπ) β§ π β π)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 2)) | ||
Syntax | cacycgr 34133 | Extend class notation with acyclic graphs. |
class AcyclicGraph | ||
Definition | df-acycgr 34134* | Define the class of all acyclic graphs. A graph is called acyclic if it has no (non-trivial) cycles. (Contributed by BTernaryTau, 11-Oct-2023.) |
β’ AcyclicGraph = {π β£ Β¬ βπβπ(π(Cyclesβπ)π β§ π β β )} | ||
Theorem | dfacycgr1 34135* | An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.) |
β’ AcyclicGraph = {π β£ βπβπ(π(Cyclesβπ)π β π = β )} | ||
Theorem | isacycgr 34136* | The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
β’ (πΊ β π β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) | ||
Theorem | isacycgr1 34137* | The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
β’ (πΊ β π β (πΊ β AcyclicGraph β βπβπ(π(CyclesβπΊ)π β π = β ))) | ||
Theorem | acycgrcycl 34138 | Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.) |
β’ ((πΊ β AcyclicGraph β§ πΉ(CyclesβπΊ)π) β πΉ = β ) | ||
Theorem | acycgr0v 34139 | A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β π β§ π = β ) β πΊ β AcyclicGraph) | ||
Theorem | acycgr1v 34140 | A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UMGraph β§ (β―βπ) = 1) β πΊ β AcyclicGraph) | ||
Theorem | acycgr2v 34141 | A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β USGraph β§ (β―βπ) = 2) β πΊ β AcyclicGraph) | ||
Theorem | prclisacycgr 34142* | A proper class (representing a null graph, see vtxvalprc 28305) has the property of an acyclic graph (see also acycgr0v 34139). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.) |
β’ π = (VtxβπΊ) β β’ (Β¬ πΊ β V β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) | ||
Theorem | acycgrislfgr 34143* | An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)}) | ||
Theorem | upgracycumgr 34144 | An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.) |
β’ ((πΊ β UPGraph β§ πΊ β AcyclicGraph) β πΊ β UMGraph) | ||
Theorem | umgracycusgr 34145 | An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.) |
β’ ((πΊ β UMGraph β§ πΊ β AcyclicGraph) β πΊ β USGraph) | ||
Theorem | upgracycusgr 34146 | An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.) |
β’ ((πΊ β UPGraph β§ πΊ β AcyclicGraph) β πΊ β USGraph) | ||
Theorem | cusgracyclt3v 34147 | A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.) |
β’ π = (VtxβπΊ) β β’ (πΊ β ComplUSGraph β (πΊ β AcyclicGraph β (β―βπ) < 3)) | ||
Theorem | pthacycspth 34148 | A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, 21-Oct-2023.) |
β’ ((πΊ β AcyclicGraph β§ πΉ(PathsβπΊ)π) β πΉ(SPathsβπΊ)π) | ||
Theorem | acycgrsubgr 34149 | The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.) |
β’ ((πΊ β AcyclicGraph β§ π SubGraph πΊ) β π β AcyclicGraph) | ||
Axiom | ax-7d 34150* | Distinct variable version of ax-11 2155. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (βπ₯βπ¦π β βπ¦βπ₯π) | ||
Axiom | ax-8d 34151* | Distinct variable version of ax-7 2012. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (π₯ = π¦ β (π₯ = π§ β π¦ = π§)) | ||
Axiom | ax-9d1 34152 | Distinct variable version of ax-6 1972, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ Β¬ βπ₯ Β¬ π₯ = π₯ | ||
Axiom | ax-9d2 34153* | Distinct variable version of ax-6 1972, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ Β¬ βπ₯ Β¬ π₯ = π¦ | ||
Axiom | ax-10d 34154* | Distinct variable version of axc11n 2426. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (βπ₯ π₯ = π¦ β βπ¦ π¦ = π₯) | ||
Axiom | ax-11d 34155* | Distinct variable version of ax-12 2172. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (π₯ = π¦ β (βπ¦π β βπ₯(π₯ = π¦ β π))) | ||
Theorem | quartfull 34156 | The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π· β β) & β’ (π β π β β) & β’ (π β (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)) β 0) & β’ (π β -((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3) β 0) β β’ (π β ((((πβ4) + (π΄ Β· (πβ3))) + ((π΅ Β· (πβ2)) + ((πΆ Β· π) + π·))) = 0 β ((π = ((-(π΄ / 4) β ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)) + (ββ((-(((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)β2) β ((π΅ β ((3 / 8) Β· (π΄β2))) / 2)) + ((((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8)) / 4) / ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2))))) β¨ π = ((-(π΄ / 4) β ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)) β (ββ((-(((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)β2) β ((π΅ β ((3 / 8) Β· (π΄β2))) / 2)) + ((((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8)) / 4) / ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)))))) β¨ (π = ((-(π΄ / 4) + ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)) + (ββ((-(((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)β2) β ((π΅ β ((3 / 8) Β· (π΄β2))) / 2)) β ((((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8)) / 4) / ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2))))) β¨ π = ((-(π΄ / 4) + ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)) β (ββ((-(((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2)β2) β ((π΅ β ((3 / 8) Β· (π΄β2))) / 2)) β ((((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8)) / 4) / ((ββ-((((2 Β· (π΅ β ((3 / 8) Β· (π΄β2)))) + (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3))) + ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))) / (((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))) + (ββ((((-(2 Β· ((π΅ β ((3 / 8) Β· (π΄β2)))β3)) β (;27 Β· (((πΆ β ((π΄ Β· π΅) / 2)) + ((π΄β3) / 8))β2))) + (;72 Β· ((π΅ β ((3 / 8) Β· (π΄β2))) Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4)))))))β2) β (4 Β· ((((π΅ β ((3 / 8) Β· (π΄β2)))β2) + (;12 Β· ((π· β ((πΆ Β· π΄) / 4)) + ((((π΄β2) Β· π΅) / ;16) β ((3 / ;;256) Β· (π΄β4))))))β3))))) / 2)βπ(1 / 3)))) / 3)) / 2))))))))) | ||
Theorem | deranglem 34157* | Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ (π΄ β Fin β {π β£ (π:π΄β1-1-ontoβπ΄ β§ π)} β Fin) | ||
Theorem | derangval 34158* | Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ (π΄ β Fin β (π·βπ΄) = (β―β{π β£ (π:π΄β1-1-ontoβπ΄ β§ βπ¦ β π΄ (πβπ¦) β π¦)})) | ||
Theorem | derangf 34159* | The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ π·:FinβΆβ0 | ||
Theorem | derang0 34160* | The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ (π·ββ ) = 1 | ||
Theorem | derangsn 34161* | The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ (π΄ β π β (π·β{π΄}) = 0) | ||
Theorem | derangenlem 34162* | One half of derangen 34163. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ ((π΄ β π΅ β§ π΅ β Fin) β (π·βπ΄) β€ (π·βπ΅)) | ||
Theorem | derangen 34163* | The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ ((π΄ β π΅ β§ π΅ β Fin) β (π·βπ΄) = (π·βπ΅)) | ||
Theorem | subfacval 34164* | The subfactorial is defined as the number of derangements (see derangval 34158) of the set (1...π). (Contributed by Mario Carneiro, 21-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π β β0 β (πβπ) = (π·β(1...π))) | ||
Theorem | derangen2 34165* | Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π΄ β Fin β (π·βπ΄) = (πβ(β―βπ΄))) | ||
Theorem | subfacf 34166* | The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ π:β0βΆβ0 | ||
Theorem | subfaclefac 34167* | The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π β β0 β (πβπ) β€ (!βπ)) | ||
Theorem | subfac0 34168* | The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (πβ0) = 1 | ||
Theorem | subfac1 34169* | The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (πβ1) = 0 | ||
Theorem | subfacp1lem1 34170* | Lemma for subfacp1 34177. The set πΎ together with {1, π} partitions the set 1...(π + 1). (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} & β’ (π β π β β) & β’ (π β π β (2...(π + 1))) & β’ π β V & β’ πΎ = ((2...(π + 1)) β {π}) β β’ (π β ((πΎ β© {1, π}) = β β§ (πΎ βͺ {1, π}) = (1...(π + 1)) β§ (β―βπΎ) = (π β 1))) | ||
Theorem | subfacp1lem2a 34171* | Lemma for subfacp1 34177. Properties of a bijection on πΎ augmented with the two-element flip to get a bijection on πΎ βͺ {1, π}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} & β’ (π β π β β) & β’ (π β π β (2...(π + 1))) & β’ π β V & β’ πΎ = ((2...(π + 1)) β {π}) & β’ πΉ = (πΊ βͺ {β¨1, πβ©, β¨π, 1β©}) & β’ (π β πΊ:πΎβ1-1-ontoβπΎ) β β’ (π β (πΉ:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ (πΉβ1) = π β§ (πΉβπ) = 1)) | ||
Theorem | subfacp1lem2b 34172* | Lemma for subfacp1 34177. Properties of a bijection on πΎ augmented with the two-element flip to get a bijection on πΎ βͺ {1, π}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} & β’ (π β π β β) & β’ (π β π β (2...(π + 1))) & β’ π β V & β’ πΎ = ((2...(π + 1)) β {π}) & β’ πΉ = (πΊ βͺ {β¨1, πβ©, β¨π, 1β©}) & β’ (π β πΊ:πΎβ1-1-ontoβπΎ) β β’ ((π β§ π β πΎ) β (πΉβπ) = (πΊβπ)) | ||
Theorem | subfacp1lem3 34173* | Lemma for subfacp1 34177. In subfacp1lem6 34176 we cut up the set of all derangements on 1...(π + 1) first according to the value at 1, and then by whether or not (πβ(πβ1)) = 1. In this lemma, we show that the subset of all π + 1 derangements that satisfy this for fixed π = (πβ1) is in bijection with π β 1 derangements, by simply dropping the π₯ = 1 and π₯ = π points from the function to get a derangement on πΎ = (1...(π β 1)) β {1, π}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} & β’ (π β π β β) & β’ (π β π β (2...(π + 1))) & β’ π β V & β’ πΎ = ((2...(π + 1)) β {π}) & β’ π΅ = {π β π΄ β£ ((πβ1) = π β§ (πβπ) = 1)} & β’ πΆ = {π β£ (π:πΎβ1-1-ontoβπΎ β§ βπ¦ β πΎ (πβπ¦) β π¦)} β β’ (π β (β―βπ΅) = (πβ(π β 1))) | ||
Theorem | subfacp1lem4 34174* | Lemma for subfacp1 34177. The function πΉ, which swaps 1 with π and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} & β’ (π β π β β) & β’ (π β π β (2...(π + 1))) & β’ π β V & β’ πΎ = ((2...(π + 1)) β {π}) & β’ π΅ = {π β π΄ β£ ((πβ1) = π β§ (πβπ) β 1)} & β’ πΉ = (( I βΎ πΎ) βͺ {β¨1, πβ©, β¨π, 1β©}) β β’ (π β β‘πΉ = πΉ) | ||
Theorem | subfacp1lem5 34175* | Lemma for subfacp1 34177. In subfacp1lem6 34176 we cut up the set of all derangements on 1...(π + 1) first according to the value at 1, and then by whether or not (πβ(πβ1)) = 1. In this lemma, we show that the subset of all π + 1 derangements with (πβ(πβ1)) β 1 for fixed π = (πβ1) is in bijection with derangements of 2...(π + 1), because pre-composing with the function πΉ swaps 1 and π and turns the function into a bijection with (πβ1) = 1 and (πβπ₯) β π₯ for all other π₯, so dropping the point at 1 yields a derangement on the π remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} & β’ (π β π β β) & β’ (π β π β (2...(π + 1))) & β’ π β V & β’ πΎ = ((2...(π + 1)) β {π}) & β’ π΅ = {π β π΄ β£ ((πβ1) = π β§ (πβπ) β 1)} & β’ πΉ = (( I βΎ πΎ) βͺ {β¨1, πβ©, β¨π, 1β©}) & β’ πΆ = {π β£ (π:(2...(π + 1))β1-1-ontoβ(2...(π + 1)) β§ βπ¦ β (2...(π + 1))(πβπ¦) β π¦)} β β’ (π β (β―βπ΅) = (πβπ)) | ||
Theorem | subfacp1lem6 34176* | Lemma for subfacp1 34177. By induction, we cut up the set of all derangements on π + 1 according to the π possible values of (πβ1) (since (πβ1) β 1), and for each set for fixed π = (πβ1), the subset of derangements with (πβπ) = 1 has size π(π β 1) (by subfacp1lem3 34173), while the subset with (πβπ) β 1 has size π(π) (by subfacp1lem5 34175). Adding it all up yields the desired equation π(π(π) + π(π β 1)) for the number of derangements on π + 1. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) & β’ π΄ = {π β£ (π:(1...(π + 1))β1-1-ontoβ(1...(π + 1)) β§ βπ¦ β (1...(π + 1))(πβπ¦) β π¦)} β β’ (π β β β (πβ(π + 1)) = (π Β· ((πβπ) + (πβ(π β 1))))) | ||
Theorem | subfacp1 34177* | A two-term recurrence for the subfactorial. This theorem allows to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 34168, subfac1 34169. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π β β β (πβ(π + 1)) = (π Β· ((πβπ) + (πβ(π β 1))))) | ||
Theorem | subfacval2 34178* | A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π β β0 β (πβπ) = ((!βπ) Β· Ξ£π β (0...π)((-1βπ) / (!βπ)))) | ||
Theorem | subfaclim 34179* | The subfactorial converges rapidly to π! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π β β β (absβ(((!βπ) / e) β (πβπ))) < (1 / π)) | ||
Theorem | subfacval3 34180* | Another closed form expression for the subfactorial. The expression ββ(π₯ + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) & β’ π = (π β β0 β¦ (π·β(1...π))) β β’ (π β β β (πβπ) = (ββ(((!βπ) / e) + (1 / 2)))) | ||
Theorem | derangfmla 34181* | The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression ββ(π₯ + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.) |
β’ π· = (π₯ β Fin β¦ (β―β{π β£ (π:π₯β1-1-ontoβπ₯ β§ βπ¦ β π₯ (πβπ¦) β π¦)})) β β’ ((π΄ β Fin β§ π΄ β β ) β (π·βπ΄) = (ββ(((!β(β―βπ΄)) / e) + (1 / 2)))) | ||
Theorem | erdszelem1 34182* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ π = {π¦ β π« (1...π΄) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π΄ β π¦)} β β’ (π β π β (π β (1...π΄) β§ (πΉ βΎ π) Isom < , π (π, (πΉ β π)) β§ π΄ β π)) | ||
Theorem | erdszelem2 34183* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ π = {π¦ β π« (1...π΄) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π΄ β π¦)} β β’ ((β― β π) β Fin β§ (β― β π) β β) | ||
Theorem | erdszelem3 34184* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΎ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) β β’ (π΄ β (1...π) β (πΎβπ΄) = sup((β― β {π¦ β π« (1...π΄) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π΄ β π¦)}), β, < )) | ||
Theorem | erdszelem4 34185* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΎ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π Or β β β’ ((π β§ π΄ β (1...π)) β {π΄} β {π¦ β π« (1...π΄) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π΄ β π¦)}) | ||
Theorem | erdszelem5 34186* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΎ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π Or β β β’ ((π β§ π΄ β (1...π)) β (πΎβπ΄) β (β― β {π¦ β π« (1...π΄) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π΄ β π¦)})) | ||
Theorem | erdszelem6 34187* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΎ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π Or β β β’ (π β πΎ:(1...π)βΆβ) | ||
Theorem | erdszelem7 34188* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΎ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π Or β & β’ (π β π΄ β (1...π)) & β’ (π β π β β) & β’ (π β Β¬ (πΎβπ΄) β (1...(π β 1))) β β’ (π β βπ β π« (1...π)(π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , π (π , (πΉ β π )))) | ||
Theorem | erdszelem8 34189* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΎ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , π (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π Or β & β’ (π β π΄ β (1...π)) & β’ (π β π΅ β (1...π)) & β’ (π β π΄ < π΅) β β’ (π β ((πΎβπ΄) = (πΎβπ΅) β Β¬ (πΉβπ΄)π(πΉβπ΅))) | ||
Theorem | erdszelem9 34190* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΌ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , < (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π½ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , β‘ < (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π = (π β (1...π) β¦ β¨(πΌβπ), (π½βπ)β©) β β’ (π β π:(1...π)β1-1β(β Γ β)) | ||
Theorem | erdszelem10 34191* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΌ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , < (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π½ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , β‘ < (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π = (π β (1...π) β¦ β¨(πΌβπ), (π½βπ)β©) & β’ (π β π β β) & β’ (π β π β β) & β’ (π β ((π β 1) Β· (π β 1)) < π) β β’ (π β βπ β (1...π)(Β¬ (πΌβπ) β (1...(π β 1)) β¨ Β¬ (π½βπ) β (1...(π β 1)))) | ||
Theorem | erdszelem11 34192* | Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ πΌ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , < (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π½ = (π₯ β (1...π) β¦ sup((β― β {π¦ β π« (1...π₯) β£ ((πΉ βΎ π¦) Isom < , β‘ < (π¦, (πΉ β π¦)) β§ π₯ β π¦)}), β, < )) & β’ π = (π β (1...π) β¦ β¨(πΌβπ), (π½βπ)β©) & β’ (π β π β β) & β’ (π β π β β) & β’ (π β ((π β 1) Β· (π β 1)) < π) β β’ (π β βπ β π« (1...π)((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) | ||
Theorem | erdsze 34193* | The ErdΕs-Szekeres theorem. For any injective sequence πΉ on the reals of length at least (π β 1) Β· (π β 1) + 1, there is either a subsequence of length at least π on which πΉ is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least π on which πΉ is decreasing (i.e. a < , β‘ < order isomorphism, recalling that β‘ < is the "greater than" relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β πΉ:(1...π)β1-1ββ) & β’ (π β π β β) & β’ (π β π β β) & β’ (π β ((π β 1) Β· (π β 1)) < π) β β’ (π β βπ β π« (1...π)((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) | ||
Theorem | erdsze2lem1 34194* | Lemma for erdsze2 34196. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β π β β) & β’ (π β πΉ:π΄β1-1ββ) & β’ (π β π΄ β β) & β’ π = ((π β 1) Β· (π β 1)) & β’ (π β π < (β―βπ΄)) β β’ (π β βπ(π:(1...(π + 1))β1-1βπ΄ β§ π Isom < , < ((1...(π + 1)), ran π))) | ||
Theorem | erdsze2lem2 34195* | Lemma for erdsze2 34196. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β π β β) & β’ (π β πΉ:π΄β1-1ββ) & β’ (π β π΄ β β) & β’ π = ((π β 1) Β· (π β 1)) & β’ (π β π < (β―βπ΄)) & β’ (π β πΊ:(1...(π + 1))β1-1βπ΄) & β’ (π β πΊ Isom < , < ((1...(π + 1)), ran πΊ)) β β’ (π β βπ β π« π΄((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) | ||
Theorem | erdsze2 34196* | Generalize the statement of the ErdΕs-Szekeres theorem erdsze 34193 to "sequences" indexed by an arbitrary subset of β, which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ (π β π β β) & β’ (π β π β β) & β’ (π β πΉ:π΄β1-1ββ) & β’ (π β π΄ β β) & β’ (π β ((π β 1) Β· (π β 1)) < (β―βπ΄)) β β’ (π β βπ β π« π΄((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) | ||
Theorem | kur14lem1 34197 | Lemma for kur14 34207. (Contributed by Mario Carneiro, 17-Feb-2015.) |
β’ π΄ β π & β’ (π β π΄) β π & β’ (πΎβπ΄) β π β β’ (π = π΄ β (π β π β§ {(π β π), (πΎβπ)} β π)) | ||
Theorem | kur14lem2 34198 | Lemma for kur14 34207. Write interior in terms of closure and complement: ππ΄ = ππππ΄ where π is complement and π is closure. (Contributed by Mario Carneiro, 11-Feb-2015.) |
β’ π½ β Top & β’ π = βͺ π½ & β’ πΎ = (clsβπ½) & β’ πΌ = (intβπ½) & β’ π΄ β π β β’ (πΌβπ΄) = (π β (πΎβ(π β π΄))) | ||
Theorem | kur14lem3 34199 | Lemma for kur14 34207. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.) |
β’ π½ β Top & β’ π = βͺ π½ & β’ πΎ = (clsβπ½) & β’ πΌ = (intβπ½) & β’ π΄ β π β β’ (πΎβπ΄) β π | ||
Theorem | kur14lem4 34200 | Lemma for kur14 34207. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
β’ π½ β Top & β’ π = βͺ π½ & β’ πΎ = (clsβπ½) & β’ πΌ = (intβπ½) & β’ π΄ β π β β’ (π β (π β π΄)) = π΄ |
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