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Type | Label | Description |
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Statement | ||
Theorem | f1resfz0f1d 34401 | 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 34402 | 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 34403 | 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 34404 | 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 34405 | 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 34406* | 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 34407* | 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 34408* | 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 34409* | 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 34410 | 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 34411 | 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 34412 | 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 34413 | The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.) |
β’ (πΉ(WalksβπΊ)π β (reverseβπΉ)(WalksβπΊ)(reverseβπ)) | ||
Theorem | revwlkb 34414 | 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 34415 | 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 34416 | 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 34417 | 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 34418 | 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 34419 | 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 34420 | 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 34421 | 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 34422 | 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 34423 | 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 34424 | 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 34425* | 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 34426* | 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 34427 | 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 34428 | 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 34429* | Lemma for umgr2cycl 34430. (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 34430* | 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 34431 | Extend class notation with acyclic graphs. |
class AcyclicGraph | ||
Definition | df-acycgr 34432* | 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 34433* | 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 34434* | 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 34435* | 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 34436 | 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 34437 | 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 34438 | 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 34439 | 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 34440* | A proper class (representing a null graph, see vtxvalprc 28572) has the property of an acyclic graph (see also acycgr0v 34437). (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 34441* | 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 34442 | 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 34443 | 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 34444 | 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 34445 | 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 34446 | 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 34447 | 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 34448* | Distinct variable version of ax-11 2152. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (βπ₯βπ¦π β βπ¦βπ₯π) | ||
Axiom | ax-8d 34449* | Distinct variable version of ax-7 2009. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (π₯ = π¦ β (π₯ = π§ β π¦ = π§)) | ||
Axiom | ax-9d1 34450 | Distinct variable version of ax-6 1969, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ Β¬ βπ₯ Β¬ π₯ = π₯ | ||
Axiom | ax-9d2 34451* | Distinct variable version of ax-6 1969, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ Β¬ βπ₯ Β¬ π₯ = π¦ | ||
Axiom | ax-10d 34452* | Distinct variable version of axc11n 2423. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (βπ₯ π₯ = π¦ β βπ¦ π¦ = π₯) | ||
Axiom | ax-11d 34453* | Distinct variable version of ax-12 2169. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ (π₯ = π¦ β (βπ¦π β βπ₯(π₯ = π¦ β π))) | ||
Theorem | quartfull 34454 | 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 34455* | 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 34456* | 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 34457* | 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 34458* | 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 34459* | 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 34460* | One half of derangen 34461. (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 34461* | 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 34462* | The subfactorial is defined as the number of derangements (see derangval 34456) 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 34463* | 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 34464* | 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 34465* | 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 34466* | 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 34467* | 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 34468* | Lemma for subfacp1 34475. 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 34469* | Lemma for subfacp1 34475. 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 34470* | Lemma for subfacp1 34475. 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 34471* | Lemma for subfacp1 34475. In subfacp1lem6 34474 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 34472* | Lemma for subfacp1 34475. 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 34473* | Lemma for subfacp1 34475. In subfacp1lem6 34474 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 34474* | Lemma for subfacp1 34475. 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 34471), while the subset with (πβπ) β 1 has size π(π) (by subfacp1lem5 34473). 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 34475* | 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 34466, subfac1 34467. (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 34476* | 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 34477* | 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 34478* | 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 34479* | 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 34480* | Lemma for erdsze 34491. (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 34481* | Lemma for erdsze 34491. (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 34482* | Lemma for erdsze 34491. (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 34483* | Lemma for erdsze 34491. (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 34484* | Lemma for erdsze 34491. (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 34485* | Lemma for erdsze 34491. (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 34486* | Lemma for erdsze 34491. (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 34487* | Lemma for erdsze 34491. (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 34488* | Lemma for erdsze 34491. (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 34489* | Lemma for erdsze 34491. (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 34490* | Lemma for erdsze 34491. (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 34491* | 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 34492* | Lemma for erdsze2 34494. (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 34493* | Lemma for erdsze2 34494. (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 34494* | Generalize the statement of the ErdΕs-Szekeres theorem erdsze 34491 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 34495 | Lemma for kur14 34505. (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 34496 | Lemma for kur14 34505. 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 34497 | Lemma for kur14 34505. 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 34498 | Lemma for kur14 34505. 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΄ | ||
Theorem | kur14lem5 34499 | Lemma for kur14 34505. Closure is an idempotent operation in 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’ (πΎβ(πΎβπ΄)) = (πΎβπ΄) | ||
Theorem | kur14lem6 34500 | Lemma for kur14 34505. If π is the complementation operator and π is the closure operator, this expresses the identity ππππ΄ = ππππππππ΄ for any subset π΄ of the topological space. This is the key result that lets us cut down long enough sequences of ππππ... that arise when applying closure and complement repeatedly to π΄, and explains why we end up with a number as large as 14, yet no larger. (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’ π΅ = (π β (πΎβπ΄)) β β’ (πΎβ(πΌβ(πΎβπ΅))) = (πΎβπ΅) |
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