Commit d80932bc by nipkow

### New entry No_FTL_observers

parent 1fdda5d6f2be
 ... ... @@ -3634,3 +3634,15 @@ abstract = recursive fashion, following the Shannon decomposition of the argument functions. The implementation mixes and adapts known techniques and is built with efficiency in mind. [No_FTL_observers] title = No Faster-Than-Light Observers author = Mike Stannett , István Németi date = 2016-04-28 topic = Mathematics/Physics abstract = We provide a formal proof within First Order Relativity Theory that no observer can travel faster than the speed of light. Originally reported in Stannett & Németi (2014) "Using Isabelle/HOL to verify first-order relativity theory", Journal of Automated Reasoning 52(4), pp. 361-378.
 ... ... @@ -33,5 +33,6 @@ Mathematics Graph Theory Combinatorics Category Theory Physics Misc Tools
 (* Author: Mike Stannett Date: 22 October 2012 m.stannett@sheffield.ac.uk Updated 28 April 2016 to run under Isabelle2016. *) theory Axioms imports SpaceTime SomeFunc begin record Body = Ph :: "bool" IOb :: "bool" class WorldView = SpaceTime + fixes (* Worldview relation *) W :: "Body \ Body \ 'a Point \ bool" ("_ sees _ at _") and (* Worldview transformation *) wvt :: "Body \ Body \ 'a Point \ 'a Point" assumes AxWVT: "\ IOb m; IOb k \ \ (W k b x \ W m b (wvt m k x))" and AxWVTSym: "\ IOb m; IOb k \ \ (y = wvt k m x \ x = wvt m k y)" begin end (* THE BASIC AXIOMS *) (* ================ *) class AxiomPreds = WorldView begin fun sqrtTest :: "'a \ 'a \ bool" where "sqrtTest x r = ((r \ 0) \ (r*r = x))" fun cTest :: "Body \ 'a \ bool" where "cTest m v = ( (v > 0) \ ( \x y . ( (\p. (Ph p \ W m p x \ W m p y)) \ (space2 x y = (v * v)*(time2 x y)) )))" end (* AxEuclidean Quantities form a Euclidean field, ie positive quantities have square roots. We introduce a function, sqrt, that determines them. *) class AxEuclidean = AxiomPreds + Quantities + assumes AxEuclidean: "(x \ Groups.zero_class.zero) \ (\r. sqrtTest x r)" begin abbreviation sqrt :: "'a \ 'a" where "sqrt \ someFunc sqrtTest" lemma lemSqrt: assumes "x \ 0" and "r = sqrt x" shows "r \ 0 \ r*r = x" proof - have rootExists: "(\r. sqrtTest x r)" by (metis AxEuclidean assms(1)) hence "sqrtTest x (sqrt x)" by (metis lemSomeFunc) thus ?thesis using assms(2) by simp qed end (* AxLight There is an inertial observer, according to whom, any light signal moves with the same velocity in any direction *) class AxLight = WorldView + assumes AxLight: "\m v.( IOb m \ (v > (0::'a)) \ ( \x y.( (\p.(Ph p \ W m p x \ W m p y)) \ (space2 x y = (v * v)*time2 x y) )))" begin end (* AxPh For any inertial observer, the speed of light is the same in every direction everywhere, and it is finite. Furthermore, it is possible to send out a light signal in any direction. *) class AxPh = WorldView + AxiomPreds + assumes AxPh: "IOb m \ (\v. cTest m v)" begin abbreviation c :: "Body \ 'a" where "c \ someFunc cTest" fun lightcone :: "Body \ 'a Point \ 'a Cone" where "lightcone m v = mkCone v (c m)" lemma lemCProps: assumes "IOb m" and "v = c m" shows "(v > 0) \ (\x y.((\p. (Ph p \ W m p x \ W m p y)) \ ( space2 x y = (c m * c m)*time2 x y )))" proof - have vExists: "(\v. cTest m v)" by (metis AxPh assms(1)) hence "cTest m (c m)" by (metis lemSomeFunc) thus ?thesis using assms(2) by simp qed lemma lemCCone: assumes "IOb m" and "onCone y (lightcone m x)" shows "\p. (Ph p \ W m p x \ W m p y)" proof - have "(\p.(Ph p \ W m p x \ W m p y)) \ ( space2 x y = (c m * c m)*time2 x y )" by (smt assms(1) lemCProps) hence ph_exists: "(space2 x y = (c m * c m)*time2 x y) \ (\p.(Ph p \ W m p x \ W m p y))" by metis def lcmx \ "lightcone m x" have lcmx_vertex: "vertex lcmx = x" by (simp add: lcmx_def) have lcmx_slope: "slope lcmx = c m" by (simp add: lcmx_def) have "onCone y lcmx \ (space2 x y = (c m * c m)*time2 x y)" by (metis lcmx_vertex lcmx_slope onCone.simps) hence "space2 x y = (c m * c m)*time2 x y" by (metis lcmx_def assms(2)) thus ?thesis by (metis ph_exists) qed lemma lemCPos: assumes "IOb m" shows "c m > 0" by (metis assms(1) lemCProps) lemma lemCPhoton: assumes "IOb m" shows "\x y. (\p. (Ph p \ W m p x \ W m p y)) \ (space2 x y = (c m * c m)*(time2 x y))" by (metis assms(1) lemCProps) end (* AxEv Inertial observers see the same events (meetings of bodies). This also enables us to discuss the worldview transformation. *) class AxEv = WorldView + assumes AxEv: "\ IOb m; IOb k\ \ (\y. (\b. (W m b x \ W k b y)))" begin end (* Inertial observers can move with any speed slower than that of light *) class AxThExp = WorldView + AxPh + assumes AxThExp: "IOb m \ (\x y .( (\k.(IOb k \ W m k x \ W m k y)) \ (space2 x y < (c m * c m) * time2 x y) ))" begin end (* Every inertial observer is stationary according to himself *) class AxSelf = WorldView + assumes AxSelf: "IOb m \ (W m m x) \ (onAxisT x)" begin end (* All inertial observers agree that the speed of light is 1 *) class AxC = WorldView + AxPh + assumes AxC: "IOb m \ c m = 1" begin end (* Inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them. *) class AxSym = WorldView + assumes AxSym: "\ IOb m; IOb k \ \ (W m e x \ W m f y \ W k e x'\ W k f y' \ tval x = tval y \ tval x' = tval y' ) \ (space2 x y = space2 x' y')" begin end (* AxLines All observers agree about lines *) class AxLines = WorldView + assumes AxLines: "\ IOb m; IOb k; collinear x p q \ \ collinear (wvt k m x) (wvt k m p) (wvt k m q)" begin end (* AxPlanes All observers agree about planes *) class AxPlanes = WorldView + assumes AxPlanes: "\ IOb m; IOb k \ \ (coplanar e x y z \ coplanar (wvt k m e) (wvt k m x) (wvt k m y) (wvt k m z))" begin end (* AxCones All observers agree about lightcones *) class AxCones = WorldView + AxPh + assumes AxCones: "\ IOb m; IOb k \ \ ( onCone x (lightCone m v) \ onCone (wvt k m x) (lightcone k (wvt k m v)))" begin end (* All inertial observers see time travelling in the same direction. That is, if m thinks that k reached y after he reached x, then k should also think that he reached y after he reached x. *) class AxTime = WorldView + assumes AxTime: "\ IOb m; IOb k \ \( x \ y \ wvt k m x \ wvt k m y )" begin end end \ No newline at end of file
 chapter AFP session No_FTL_observers (AFP) = HOL + options [timeout = 600] theories SpecRel document_files "root.tex" "root.bib"
 (* Author: Mike Stannett Date: 27 April 2016 m.stannett@sheffield.ac.uk *) theory SomeFunc imports Main begin fun someFunc :: "('a \ 'b \ bool) \ 'a \ 'b" where "someFunc P x = (SOME y. (P x y))" lemma lemSomeFunc: assumes "\y . P x y" and "f = someFunc P" shows "P x (f x)" proof - have "f x = (SOME y. (P x y))" using assms(2) by simp thus ?thesis using assms(1) by (simp add: someI_ex) qed end \ No newline at end of file
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