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PVS-NASALib

statement stringlengths 1 3.6k	proof stringlengths 0 4.33k	type stringclasses 14 values	symbolic_name stringlengths 1 70	library stringclasses 91 values	filename stringlengths 9 66	imports listlengths 0 17	deps listlengths 0 44	source_url stringclasses 1 value	commit stringclasses 1 value
vs_min_lt_max: vsmin < vsmax		assumption	vs_min_lt_max	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_critical(TF,voz) : (vs_fseq?)	= IF TF AND vsmin<=voz AND voz<=vsmax THEN addend(voz,emptyseq) ELSE emptyseq ENDIF	definition	vs_critical	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "addend", "emptyseq", "voz", "vs_fseq?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_critical_rew: vs_critical(TF,voz) = IF TF AND vsmin<=voz AND voz<=vsmax THEN singleton(voz) ELSE emptyseq ENDIF		lemma	vs_critical_rew	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "emptyseq", "singleton", "voz", "vs_critical" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
member_vs_critical: member(x,vs_critical(TF,voz)) IFF (TF AND x = voz AND vsmin<=voz AND voz<=vsmax)		lemma	member_vs_critical	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "IFF", "member", "voz", "vs_critical" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_critical_composition: FORALL (vsfs: (vs_fseq?), TF: bool, voz: real, x:real): member(x,vsfs o vs_critical(TF,voz)) IFF ((TF AND x = voz AND vsmin<=voz AND voz<=vsmax) OR member(x,vsfs))		lemma	vs_critical_composition	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "IFF", "OR", "member", "voz", "vs_critical", "vs_fseq?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands(sz,viz) : {fs: (vs_fseq?) \| increasing?(fs)}	= sort( LET fsB : (vs_fseq?) = IF abs(sz) < H AND B > 0 THEN vs_critical(vs_circle_at(sz,viz,B,-1,1)) o vs_critical(vs_circle_at(sz,viz,B,1,1)) ELSIF abs(sz) >= H AND B > 0 THEN vs_critical(vs_circle_at(sz,viz,B,-sign(sz),1)) ELSE emptyseq ...	definition	vs_bands	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "abs", "addend", "emptyseq", "increasing?", "sign", "sort", "sz", "viz", "vs_circle_at", "vs_critical", "vs_fseq?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_not_empty : vs_bands(sz,viz)`length > 1		lemma	vs_bands_not_empty	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "length", "sz", "viz", "vs_bands" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_critical: (vs_critical?(sz,viz)(vsp) OR vsp = vsmin OR vsp = vsmax) IFF member(vsp,vs_bands(sz,viz))		lemma	vs_bands_critical	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "IFF", "OR", "member", "sz", "viz", "vs_bands", "vs_critical?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_vs_band?(sz,viz,vsb)(i:subrange(0,vsb`length-2)) : bool	= LET mp = (vsb`seq(i)+vsb`seq(i+1))/2 in cd_vertical?(sz,mp-viz)	definition	red_vs_band?	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "cd_vertical?", "length", "sz", "viz" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_red_bands : LET vsb = vs_bands(sz,viz) IN FORALL (i:{ii: subrange(0,vsb`length-2)\| vsb`seq(ii) /= vsb`seq(ii+1)}) : red_vs_band?(sz,viz,vsb)(i) IFF (FORALL (x \| vsb`seq(i) < x AND x < vsb`seq(i+1)): conflict_vertical?(sz,x-viz))		theorem	vs_red_bands	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "IFF", "conflict_vertical?", "length", "red_vs_band?", "sz", "viz", "vs_bands" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_green_bands : LET vsb = vs_bands(sz,viz) IN FORALL (i:{ii: subrange(0,vsb`length-2)\| vsb`seq(ii) /= vsb`seq(ii+1)}) : NOT red_vs_band?(sz,viz,vsb)(i) IFF (FORALL (x \| vsb`seq(i) < x AND x < vsb`seq(i+1)): NOT conflict_vertical?(sz,x-viz))		theorem	vs_green_bands	ACCoRD	ACCoRD/bands_1D.pvs	[ "vs_bands[H,B,T]," ]	[ "AND", "IFF", "NOT", "conflict_vertical?", "length", "red_vs_band?", "sz", "viz", "vs_bands" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_min_lt_max: gsmin < gsmax		assumption	gs_min_lt_max	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_critical(nvo:Vect2) : (trk_fseq?)	= IF nvo /= zero THEN addend(track(nvo),emptyseq) ELSE emptyseq ENDIF	definition	trk_critical	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "Vect2", "addend", "emptyseq", "nvo", "track", "trk_fseq?", "zero" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_critical_rew: trk_critical(nvo) = IF nvo /= zero THEN singleton(track(nvo)) ELSE emptyseq ENDIF		lemma	trk_critical_rew	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "emptyseq", "nvo", "singleton", "track", "trk_critical", "zero" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
member_trk_critical: member(x,trk_critical(nvo)) IMPLIES x = track(nvo)		lemma	member_trk_critical	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "IMPLIES", "member", "nvo", "track", "trk_critical" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_critical_composition: FORALL (trkfs: fseq, nvo:Vect2, x:real): member(x,trkfs o trk_critical(nvo)) IFF ((nvo /= zero AND x = track(nvo)) OR member(x,trkfs))		lemma	trk_critical_composition	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "IFF", "OR", "Vect2", "fseq", "member", "nvo", "track", "trk_critical", "zero" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands(s,vo,vi) : {fs : (trk_fseq?) \| increasing?(fs)}	= sort( LET fs2 : (trk_fseq?) = IF horizontal_sep?(s) THEN trk_critical(trk_line_eps_irt(s,vo,vi,-1,-1)) o trk_critical(trk_line_eps_irt(s,vo,vi,-1, 1)) o trk_critical(trk_line_eps_irt(s,vo,vi, 1,-1)) o trk_critical(trk_line_eps_irt(s,vo,vi, 1, 1)) o trk_critical(trk_...	definition	trk_bands	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "Entry", "Exit", "addend", "emptyseq", "fs2", "horizontal_sep?", "increasing?", "pi", "sort", "trk_critical", "trk_fseq?", "trk_line_eps_irt", "trk_only_circle", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_not_empty : trk_bands(s,vo,vi)`length > 1 AUTO_REWRITE- member		lemma	trk_bands_not_empty	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "length", "member", "trk_bands", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_critical: (trk_critical?(s,vo,vi)(trk2) OR to2pi(trk2) = 0) IFF member(trk2,trk_bands(s,vo,vi))		lemma	trk_bands_critical	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "IFF", "OR", "member", "to2pi", "trk_bands", "trk_critical?", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_critical(nvo:Vect2) : (gs_fseq?)	= IF nvo/=zero AND gsmin<=gs(nvo) AND gs(nvo)<=gsmax THEN addend(gs(nvo),emptyseq) ELSE emptyseq ENDIF	definition	gs_critical	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "Vect2", "addend", "emptyseq", "gs", "gs_fseq?", "nvo", "zero" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_critical_rew: FORALL (nvo:Vect2): gs_critical(nvo) = IF nvo/=zero AND gsmin<=gs(nvo) AND gs(nvo)<=gsmax THEN singleton(gs(nvo)) ELSE emptyseq ENDIF		lemma	gs_critical_rew	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "Vect2", "emptyseq", "gs", "gs_critical", "nvo", "singleton", "zero" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
member_gs_critical: FORALL (nvo:Vect2,x:real): member(x,gs_critical(nvo)) IMPLIES x = gs(nvo)		lemma	member_gs_critical	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "IMPLIES", "Vect2", "gs", "gs_critical", "member", "nvo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_critical_composition: FORALL (gsfs: (gs_fseq?), nvo:Vect2, x:real): member(x,gsfs o gs_critical(nvo)) IFF ((nz_vect2?(nvo) AND x = gs(nvo) AND gsmin<=gs(nvo) AND gs(nvo)<=gsmax) OR member(x,gsfs))		lemma	gs_critical_composition	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "IFF", "OR", "Vect2", "gs", "gs_critical", "gs_fseq?", "member", "nvo", "nz_vect2?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands(s,vo,vi) : {fs: (gs_fseq?) \| increasing?(fs)}	= sort( LET fs2 : (gs_fseq?) = IF horizontal_sep?(s) THEN gs_critical(gs_line_eps(s,vo,vi,-1)) o gs_critical(gs_line_eps(s,vo,vi, 1)) o gs_critical(gs_only_circle(s,vo,vi,T,Entry,-1)) o gs_critical(gs_only_circle(s,vo,vi,T,Entry, 1)) ELSE emptyseq ENDIF, fs3 : (gs...	definition	gs_bands	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "Entry", "Exit", "addend", "emptyseq", "fs2", "gs_critical", "gs_fseq?", "gs_line_eps", "gs_only_circle", "horizontal_sep?", "increasing?", "sort", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_not_empty : gs_bands(s,vo,vi)`length > 1		lemma	gs_bands_not_empty	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "gs_bands", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_critical: (gs_critical?(s,vo,vi)(gsp) OR gsp = gsmin OR gsp = gsmax) IFF member(gsp,gs_bands(s,vo,vi))		lemma	gs_bands_critical	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "IFF", "OR", "gs_bands", "gs_critical?", "member", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_trk_band?(s,vo,vi,trkb)(i:subrange(0,trkb`length-2)) : bool	= LET mp = (trkb`seq(i)+trkb`seq(i+1))/2 in cd2d?(s,Vtrk(vo,vi)(mp))	definition	red_trk_band?	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "Vtrk", "cd2d?", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_red_bands : LET trkb = trk_bands(s,vo,vi) IN FORALL (i:{ii: subrange(0,trkb`length-2)\| trkb`seq(ii) /= trkb`seq(ii+1)}) : red_trk_band?(s,vo,vi,trkb)(i) IFF (FORALL (x \| trkb`seq(i) < x AND x < trkb`seq(i+1)): conflict_2D?(s,Vtrk(vo,vi)(x)))		theorem	trk_red_bands	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "IFF", "Vtrk", "conflict_2D?", "length", "red_trk_band?", "trk_bands", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_green_bands : LET trkb = trk_bands(s,vo,vi) IN FORALL (i:{ii: subrange(0,trkb`length-2)\| trkb`seq(ii) /= trkb`seq(ii+1)}) : NOT red_trk_band?(s,vo,vi,trkb)(i) IFF (FORALL (x \| trkb`seq(i) < x AND x < trkb`seq(i+1)): NOT conflict_2D?(s,Vtrk(vo,vi)(x)))		theorem	trk_green_bands	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "IFF", "NOT", "Vtrk", "conflict_2D?", "length", "red_trk_band?", "trk_bands", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_gs_band?(s,vo,vi,gsb)(i:subrange(0,gsb`length-2)) : bool	= LET mp = (gsb`seq(i)+gsb`seq(i+1))/2 in cd2d?(s,Vgs(vo,vi)(mp))	definition	red_gs_band?	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "Vgs", "cd2d?", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_red_bands : LET gsb = gs_bands(s,vo,vi) IN FORALL (i:{ii: subrange(0,gsb`length-2)\| gsb`seq(ii) /= gsb`seq(ii+1)}) : red_gs_band?(s,vo,vi,gsb)(i) IFF (FORALL (x \| gsb`seq(i) < x AND x < gsb`seq(i+1)): conflict_2D?(s,Vgs(vo,vi)(x)))		theorem	gs_red_bands	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "IFF", "Vgs", "conflict_2D?", "gs_bands", "length", "red_gs_band?", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_green_bands : LET gsb = gs_bands(s,vo,vi) IN FORALL (i:{ii: subrange(0,gsb`length-2)\| gsb`seq(ii) /= gsb`seq(ii+1)}) : NOT red_gs_band?(s,vo,vi,gsb)(i) IFF (FORALL (x \| gsb`seq(i) < x AND x < gsb`seq(i+1)): NOT conflict_2D?(s,Vgs(vo,vi)(x)))		theorem	gs_green_bands	ACCoRD	ACCoRD/bands_2D.pvs	[ "gs_bands_2D[D,B,T]," ]	[ "AND", "IFF", "NOT", "Vgs", "conflict_2D?", "gs_bands", "length", "red_gs_band?", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_3D(s,vo,vi) : {fs : (trk_fseq?[gsmin,gsmax]) \| increasing?(fs)}	= IF vo`z = vi`z AND abs(s`z) < H THEN trk_bands[D,B,T,gsmin,gsmax](s,vo,vi) ELSIF vo`z /= vi`z THEN LET BB = max(Theta_H(s`z,vo`z-vi`z,Entry),B), TT = min(Theta_H(s`z,vo`z-vi`z,Exit),T) IN IF BB < TT THEN trk_bands[D,BB,TT,gsmin,gsmax](s,vo,vi) ELSE add(0,add(2*pi,emptyseq)...	definition	trk_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "Entry", "Exit", "TT", "Theta_H", "abs", "add", "emptyseq", "increasing?", "max", "min", "pi", "trk_bands", "trk_fseq?", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_3D(s,vo,vi) : {fs : (gs_fseq?[gsmin,gsmax]) \| increasing?(fs)}	= IF vo`z = vi`z AND abs(s`z) < H THEN gs_bands[D,B,T,gsmin,gsmax](s,vo,vi) ELSIF vo`z /= vi`z THEN LET BB = max(Theta_H(s`z,vo`z-vi`z,Entry),B), TT = min(Theta_H(s`z,vo`z-vi`z,Exit),T) IN IF BB < TT THEN gs_bands[D,BB,TT,gsmin,gsmax](s,vo,vi) ELSE add(gsmin,add(gsmax,emptys...	definition	gs_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "Entry", "Exit", "TT", "Theta_H", "abs", "add", "emptyseq", "gs_bands", "gs_fseq?", "increasing?", "max", "min", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_3D(s,vo,vi) : {fs : (vs_fseq?[vsmin,vsmax]) \| increasing?(fs)}	= IF vect2(vo-vi) = zero AND sqv(vect2(s)) < sq(D) THEN vs_bands[H,B,T,vsmin,vsmax](s`z,vi`z) ELSIF Delta(s,vo-vi) > 0 THEN LET BB = max(Theta_D(s,vo-vi,Entry),B), TT = min(Theta_D(s,vo-vi,Exit),T) IN IF BB < TT THEN vs_bands[H,BB,TT,vsmin,vsmax](s`z,vi`z) ELSE add(vsmin,add(vsmax,e...	definition	vs_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "Delta", "Entry", "Exit", "TT", "Theta_D", "add", "emptyseq", "increasing?", "max", "min", "sq", "sqv", "vect2", "vi", "vo", "vs_bands", "vs_fseq?", "zero" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_3D_not_empty : trk_bands_3D(s,vo,vi)`length > 1		lemma	trk_bands_3D_not_empty	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "length", "trk_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_last: LET trkb = trk_bands_3D(s,vo,vi) IN trkb`seq(trkb`length-1) = 2*pi		lemma	trk_bands_last	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "length", "pi", "trk_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_first: LET trkb = trk_bands_3D(s,vo,vi) IN trkb`seq(0) = 0		lemma	trk_bands_first	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "trk_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_3D_not_empty : gs_bands_3D(s,vo,vi)`length > 1		lemma	gs_bands_3D_not_empty	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "gs_bands_3D", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_last: LET gsb = gs_bands_3D(s,vo,vi) IN gsb`seq(gsb`length-1) = gsmax		lemma	gs_bands_last	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "gs_bands_3D", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_first: LET gsb = gs_bands_3D(s,vo,vi) IN gsb`seq(0) = gsmin		lemma	gs_bands_first	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "gs_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_3D_not_empty : vs_bands_3D(s,vo,vi)`length > 1		lemma	vs_bands_3D_not_empty	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "length", "vi", "vo", "vs_bands_3D" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_last: LET vsb = vs_bands_3D(s,vo,vi) IN vsb`seq(vsb`length-1) = vsmax		lemma	vs_bands_last	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "length", "vi", "vo", "vs_bands_3D" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_first: LET vsb = vs_bands_3D(s,vo,vi) IN vsb`seq(0) = vsmin		lemma	vs_bands_first	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "vi", "vo", "vs_bands_3D" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_trk_band_3D?(s,vo,vi,trkb)(i:subrange(0,trkb`length-2)) : bool	= LET mp = (trkb`seq(i)+trkb`seq(i+1))/2 in cd3d?(s,Vtrk_3D(vo,vi)(mp))	definition	red_trk_band_3D?	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "Vtrk_3D", "cd3d?", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_gs_band_3D?(s,vo,vi,gsb)(i:subrange(0,gsb`length-2)) : bool	= LET mp = (gsb`seq(i)+gsb`seq(i+1))/2 in cd3d?(s,Vgs_3D(vo,vi)(mp))	definition	red_gs_band_3D?	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "Vgs_3D", "cd3d?", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_vs_band_3D?(s,vo,vi,vsb)(i:subrange(0,vsb`length-2)) : bool	= LET mp = (vsb`seq(i)+vsb`seq(i+1))/2 in cd3d?(s,Vs(vo,vi)(mp))	definition	red_vs_band_3D?	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "Vs", "cd3d?", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_red_bands_3D : LET trkb = trk_bands_3D(s,vo,vi) IN FORALL (i:subrange(0,trkb`length-2) \| trkb`seq(i) /= trkb`seq(i+1)) : red_trk_band_3D?(s,vo,vi,trkb)(i) IFF (FORALL (x \| trkb`seq(i) < x AND x < trkb`seq(i+1)): conflict_3D?(s,Vtrk_3D(vo,vi)(x)))		theorem	trk_red_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "Vtrk_3D", "conflict_3D?", "length", "red_trk_band_3D?", "trk_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_green_bands_3D : LET trkb = trk_bands_3D(s,vo,vi) IN FORALL (i:subrange(0,trkb`length-2) \| trkb`seq(i) /= trkb`seq(i+1)) : NOT red_trk_band_3D?(s,vo,vi,trkb)(i) IFF (FORALL (x \| trkb`seq(i) < x AND x < trkb`seq(i+1)): NOT conflict_3D?(s,Vtrk_3D(vo,vi)(x)))		theorem	trk_green_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "NOT", "Vtrk_3D", "conflict_3D?", "length", "red_trk_band_3D?", "trk_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_equivalence: LET trkb = trk_bands_3D(s,vo,vi) IN FORALL (x,y): (0<=x AND x<=y AND y<=2*pi AND (FORALL (i:subrange(0,trkb`length-1)): trkb`seq(i)<x OR y<trkb`seq(i))) IMPLIES (conflict_3D?(s,Vtrk_3D(vo,vi)(x)) IFF conflict_3D?(s,Vtrk_3D(vo,vi)(y)))		lemma	trk_bands_equivalence	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "IMPLIES", "OR", "Vtrk_3D", "conflict_3D?", "length", "pi", "trk_bands_3D", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_red_bands_3D : LET gsb = gs_bands_3D(s,vo,vi) IN FORALL (i:subrange(0,gsb`length-2) \| gsb`seq(i) /= gsb`seq(i+1)) : red_gs_band_3D?(s,vo,vi,gsb)(i) IFF (FORALL (x \| gsb`seq(i) < x AND x < gsb`seq(i+1)): conflict_3D?(s,Vgs_3D(vo,vi)(x)))		theorem	gs_red_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "Vgs_3D", "conflict_3D?", "gs_bands_3D", "length", "red_gs_band_3D?", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_green_bands_3D : LET gsb = gs_bands_3D(s,vo,vi) IN FORALL (i:subrange(0,gsb`length-2) \| gsb`seq(i) /= gsb`seq(i+1)) : NOT red_gs_band_3D?(s,vo,vi,gsb)(i) IFF (FORALL (x \| gsb`seq(i) < x AND x < gsb`seq(i+1)): NOT conflict_3D?(s,Vgs_3D(vo,vi)(x)))		theorem	gs_green_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "NOT", "Vgs_3D", "conflict_3D?", "gs_bands_3D", "length", "red_gs_band_3D?", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_equivalence: LET gsb = gs_bands_3D(s,vo,vi) IN FORALL (x,y): (gsmin<=x AND x<=y AND y<=gsmax AND (FORALL (i:subrange(0,gsb`length-1)): gsb`seq(i)<x OR y<gsb`seq(i))) IMPLIES (conflict_3D?(s,Vgs_3D(vo,vi)(x)) IFF conflict_3D?(s,Vgs_3D(vo,vi)(y)))		lemma	gs_bands_equivalence	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "IMPLIES", "OR", "Vgs_3D", "conflict_3D?", "gs_bands_3D", "length", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_red_bands_3D : LET vsb = vs_bands_3D(s,vo,vi) IN FORALL (i:subrange(0,vsb`length-2) \| vsb`seq(i) /= vsb`seq(i+1)) : red_vs_band_3D?(s,vo,vi,vsb)(i) IFF (FORALL (x \| vsb`seq(i) < x AND x < vsb`seq(i+1)): conflict_3D?(s,Vs(vo,vi)(x)))		theorem	vs_red_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "Vs", "conflict_3D?", "length", "red_vs_band_3D?", "vi", "vo", "vs_bands_3D" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_green_bands_3D : LET vsb = vs_bands_3D(s,vo,vi) IN FORALL (i:subrange(0,vsb`length-2) \| vsb`seq(i) /= vsb`seq(i+1)) : NOT red_vs_band_3D?(s,vo,vi,vsb)(i) IFF (FORALL (x \| vsb`seq(i) < x AND x < vsb`seq(i+1)): NOT conflict_3D?(s,Vs(vo,vi)(x)))		theorem	vs_green_bands_3D	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "NOT", "Vs", "conflict_3D?", "length", "red_vs_band_3D?", "vi", "vo", "vs_bands_3D" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_equivalence: LET vsb = vs_bands_3D(s,vo,vi) IN FORALL (x,y): (vsmin<=x AND x<=y AND y<=vsmax AND (FORALL (i:subrange(0,vsb`length-1)): vsb`seq(i)<x OR y<vsb`seq(i))) IMPLIES (conflict_3D?(s,Vs(vo,vi)(x)) IFF conflict_3D?(s,Vs(vo,vi)(y)))		lemma	vs_bands_equivalence	ACCoRD	ACCoRD/bands_3D.pvs	[ "bands_2D," ]	[ "AND", "IFF", "IMPLIES", "OR", "Vs", "conflict_3D?", "length", "vi", "vo", "vs_bands_3D" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
FlightPlanRelevant_nz(to): TYPE+	= {flp: FlightPlanRelevant(to) \| FORALL(i:below[N-1]): nz_vect2?(velocity(flp)(i))}	type	FlightPlanRelevant_nz	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlanRelevant", "below", "nz_vect2?", "to", "velocity" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_i(so: Vect3, vo: Nzv2_vect3, to: real, fp: FlightPlan_nz) (i : below[N-1]) : {fs : (trk_fseq?[gsmin,gsmax]) \| increasing?(fs)}	= LET fi_st = fp(i)`time, % assume 0 <= fi_st <= to fi_et = fp(i+1)`time, % assume fist < fi_et vi = velocity(fp)(i), % intruder velocity for segment i Bo = B+to, % absolute range bottom To = T+to % absolute range bottom IN IF fi_et <= Bo OR fi_st >= To THEN empty...	definition	trk_bands_si_i	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlan_nz", "Nzv2_vect3", "OR", "Vect3", "below", "emptyseq", "increasing?", "max", "min", "position", "si", "so", "time", "to", "trk_bands_3D", "trk_fseq?", "velocity", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
expand_trk_bands_si_i : FORALL (i:below[N-1]): fp(i+1)`time > B+to AND fp(i)`time < T+to IMPLIES trk_bands_si_i(so,vo,to,fp)(i) = LET LB = max(fp(i)`time,B+to)-to, UB = min(fp(1+i)`time,T+to)-to IN trk_bands_3D[D,H,LB,UB,gsmin,gsmax,vsmin,vsmax](so-(fp(i)...		lemma	expand_trk_bands_si_i	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "IMPLIES", "below", "max", "min", "position", "so", "time", "to", "trk_bands_3D", "trk_bands_si_i", "velocity", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si(so: Vect3, vo: Nzv2_vect3, to: real, fp: FlightPlan_nz) (i : below[N-1]) : {fs : (trk_fseq?[gsmin,gsmax]) \| increasing?(fs)}	= IF i=0 THEN trk_bands_si_i(so,vo,to,fp)(i) ELSE sort (trk_bands_si_i(so,vo,to,fp)(i) o trk_bands_si(so,vo,to,fp)(i-1)) ENDIF MEASURE i	function	trk_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlan_nz", "Nzv2_vect3", "Vect3", "below", "increasing?", "so", "sort", "to", "trk_bands_si_i", "trk_fseq?", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_si_i(so: Vect3, vo: Nzv2_vect3, to: real, fp: FlightPlan_nz) (i : below[N-1]) : {fs : (gs_fseq?[gsmin,gsmax]) \| increasing?(fs)}	= LET fi_st = fp(i)`time, % assume 0 <= fi_st <= to fi_et = fp(i+1)`time, % assume fist < fi_et vi = velocity(fp)(i), % intruder velocity for segment i Bo = B+to, % absolute range bottom To = T+to % absolute range bottom IN IF fi_et <= Bo OR fi_st >= To THEN empty...	definition	gs_bands_si_i	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlan_nz", "Nzv2_vect3", "OR", "Vect3", "below", "emptyseq", "gs_bands_3D", "gs_fseq?", "increasing?", "max", "min", "position", "si", "so", "time", "to", "velocity", "vi", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_si(so: Vect3, vo: Nzv2_vect3, to: real, fp: FlightPlan_nz) (i : below[N-1]) : {fs : (gs_fseq?[gsmin,gsmax]) \| increasing?(fs)}	= IF i=0 THEN gs_bands_si_i(so,vo,to,fp)(i) ELSE sort (gs_bands_si_i(so,vo,to,fp)(i) o gs_bands_si(so,vo,to,fp)(i-1)) ENDIF MEASURE i	function	gs_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlan_nz", "Nzv2_vect3", "Vect3", "below", "gs_bands_si_i", "gs_fseq?", "increasing?", "so", "sort", "to", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_si_i(so: Vect3, vo: Nzv2_vect3, to: real, fp: FlightPlan_nz) (i : below[N-1]) : {fs : (vs_fseq?[vsmin,vsmax]) \| increasing?(fs)}	= LET fi_st = fp(i)`time, % assume 0 <= fi_st <= to fi_et = fp(i+1)`time, % assume fist < fi_et vi = velocity(fp)(i), % intruder velocity for segment i Bo = B+to, % absolute range bottom To = T+to % absolute range bottom IN IF fi_et <= Bo OR fi_st >= To THEN empty...	definition	vs_bands_si_i	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlan_nz", "Nzv2_vect3", "OR", "Vect3", "below", "emptyseq", "increasing?", "max", "min", "position", "si", "so", "time", "to", "velocity", "vi", "vo", "vs_bands_3D", "vs_fseq?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_si(so: Vect3, vo: Nzv2_vect3, to: real, fp: FlightPlan_nz) (i : below[N-1]) : {fs : (vs_fseq?[vsmin,vsmax]) \| increasing?(fs)}	= IF i=0 THEN vs_bands_si_i(so,vo,to,fp)(i) ELSE sort (vs_bands_si_i(so,vo,to,fp)(i) o vs_bands_si(so,vo,to,fp)(i-1)) ENDIF MEASURE i	function	vs_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "FlightPlan_nz", "Nzv2_vect3", "Vect3", "below", "increasing?", "so", "sort", "to", "vo", "vs_bands_si_i", "vs_fseq?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_trk_band_fp?(so,vo,to,fp,trkb)(i:subrange(0,trkb`length-2)) : bool	= LET mp = (trkb`seq(i)+trkb`seq(i+1))/2, trk = trk2v3(vo)(mp) IN conflict_3D?(so,to,trk,fp)	definition	red_trk_band_fp?	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "conflict_3D?", "length", "so", "to", "trk", "trk2v3", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_gs_band_fp?(so,vo,to,fp,gsb)(i:subrange(0,gsb`length-2)) : bool	= LET mp = (gsb`seq(i)+gsb`seq(i+1))/2, gs = gs2v3(vo)(mp) IN conflict_3D?(so,to,gs,fp)	definition	red_gs_band_fp?	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "conflict_3D?", "gs", "gs2v3", "length", "so", "to", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
red_vs_band_fp?(so,vo,to,fp,vsb)(i:subrange(0,vsb`length-2)) : bool	= LET mp = (vsb`seq(i)+vsb`seq(i+1))/2, vs = vs2v3(vo)(mp) IN conflict_3D?(so,to,vs,fp)	definition	red_vs_band_fp?	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "conflict_3D?", "length", "so", "to", "vo", "vs", "vs2v3" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
segdefs : FORALL (j:subrange(0,N-2)): seg_lh_bottom(fp,to)(j)+fp(j)`time-to = max(fp(j)`time,B+to)-to AND seg_lh_top(fp,to)(j)+fp(j)`time-to = min(fp(1+j)`time,T+to)-to		lemma	segdefs	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "max", "min", "seg_lh_bottom", "seg_lh_top", "time", "to" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_combines : FORALL (i:subrange(0,N-2), x:real): member(x,trk_bands_si(so,vo,to,fp)(i)) IFF EXISTS (j:subrange(0,i)): member(x,trk_bands_si_i(so,vo,to,fp)(j))		lemma	trk_bands_si_combines	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "IFF", "member", "so", "to", "trk_bands_si", "trk_bands_si_i", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
empty_no_members : NOT member(x,emptyseq)		lemma	empty_no_members	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "NOT", "emptyseq", "member" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_component : FORALL (i:subrange(0,N-2), x:real): member(x,trk_bands_si(so,vo,to,fp)(i)) IFF EXISTS (j:subrange(0,i)): LET LB = max(fp(j)`time,B+to)-to, UB = min(fp(1+j)`time,T+to)-to IN fp(j+1)`time > B+to AND fp(j)`time < T+ to AND ...		lemma	trk_bands_si_component	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "IFF", "max", "member", "min", "position", "so", "time", "to", "trk_bands_3D", "trk_bands_si", "velocity", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_length : FORALL (i:subrange(0,N-2), j: subrange(0,i)): trk_bands_si(so,vo,to,fp)(i)`length >= trk_bands_si_i(so,vo,to,fp)(j)`length		lemma	trk_bands_si_length	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "length", "so", "to", "trk_bands_si", "trk_bands_si_i", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_i_relevant : FORALL (i:subrange(0,N-2)): trk_bands_si_i(so,vo,to,fp)(i)`length > 0 IFF fp(i)`time < to+T AND to+B < fp(i+1)`time		lemma	trk_bands_si_i_relevant	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "IFF", "length", "so", "time", "to", "trk_bands_si_i", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_i_relevant_has_2 : FORALL (i:subrange(0,N-2)): trk_bands_si_i(so,vo,to,fp)(i)`length > 0 IFF trk_bands_si_i(so,vo,to,fp)(i)`length > 1		lemma	trk_bands_si_i_relevant_has_2	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "IFF", "length", "so", "to", "trk_bands_si_i", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_relevant : FORALL (i:subrange(0,N-2)): trk_bands_si(so,vo,to,fp)(i)`length > 0 IFF fp(0)`time < to+T AND to+B < fp(i+1)`time		lemma	trk_bands_si_relevant	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "IFF", "length", "so", "time", "to", "trk_bands_si", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_relevant_has_2 : FORALL (i:subrange(0,N-2)): trk_bands_si(so,vo,to,fp)(i)`length > 0 IFF trk_bands_si(so,vo,to,fp)(i)`length > 1		lemma	trk_bands_si_relevant_has_2	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "IFF", "length", "so", "to", "trk_bands_si", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_bands_si_equivalence: FORALL (fpr:FlightPlanRelevant_nz(to)): LET trkb = trk_bands_si(so,vo,to,fpr)(N-2), trk = trk2v3(vo) IN FORALL (x,y): (0<=x AND x<=y AND y<=2*pi AND (FORALL (i:subrange(0,trkb`length-1)): trkb`seq(i) < x OR y < trkb`seq(i))) IMPLIES (...		lemma	trk_bands_si_equivalence	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "IMPLIES", "OR", "conflict_3D?", "length", "pi", "so", "to", "trk", "trk2v3", "trk_bands_si", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_red_bands_si : FORALL (fpr:FlightPlanRelevant_nz(to)): LET trkb = trk_bands_si(so,vo,to,fpr)(N-2) IN FORALL (i:subrange(0,trkb`length-2) \| trkb`seq(i) /= trkb`seq(i+1)) : red_trk_band_fp?(so,vo,to,fpr,trkb)(i) IFF (FORALL (x \| trkb`seq(i) < x AND x < trkb`seq(i+1)): LET trk = trk2v3...		theorem	trk_red_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "conflict_3D?", "length", "red_trk_band_fp?", "so", "to", "trk", "trk2v3", "trk_bands_si", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk_green_bands_si : FORALL (fpr:FlightPlanRelevant_nz(to)): LET trkb = trk_bands_si(so,vo,to,fpr)(N-2) IN FORALL (i:subrange(0,trkb`length-2) \| trkb`seq(i) /= trkb`seq(i+1)) : NOT red_trk_band_fp?(so,vo,to,fpr,trkb)(i) IFF (FORALL (x \| trkb`seq(i) < x AND x < trkb`seq(i+1)): LET trk = ...		theorem	trk_green_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "NOT", "conflict_3D?", "length", "red_trk_band_fp?", "so", "to", "trk", "trk2v3", "trk_bands_si", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_si_combines : FORALL (i:subrange(0,N-2), x:real): member(x,gs_bands_si(so,vo,to,fp)(i)) IFF EXISTS (j:subrange(0,i)): member(x,gs_bands_si_i(so,vo,to,fp)(j))		lemma	gs_bands_si_combines	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "IFF", "gs_bands_si", "gs_bands_si_i", "member", "so", "to", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_si_component : FORALL (i:subrange(0,N-2), x:real): member(x,gs_bands_si(so,vo,to,fp)(i)) IFF EXISTS (j:subrange(0,i)): LET LB = max(fp(j)`time,B+to)-to, UB = min(fp(1+j)`time,T+to)-to IN fp(j+1)`time > B+to AND fp(j)`time < T+ to AND ...		lemma	gs_bands_si_component	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "IFF", "gs_bands_3D", "gs_bands_si", "max", "member", "min", "position", "so", "time", "to", "velocity", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_bands_si_equivalence: FORALL (fpr:FlightPlanRelevant_nz(to)): LET gsb = gs_bands_si(so,vo,to,fpr)(N-2), gs = gs2v3(vo) IN FORALL (x,y): (gsmin<=x AND x<=y AND y<=gsmax AND (FORALL (i:subrange(0,gsb`length-1)): gsb`seq(i) < x OR y < gsb`seq(i))) IMPLIES (con...		lemma	gs_bands_si_equivalence	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "IMPLIES", "OR", "conflict_3D?", "gs", "gs2v3", "gs_bands_si", "length", "so", "to", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_red_bands_si : FORALL (fpr:FlightPlanRelevant_nz(to)): LET gsb = gs_bands_si(so,vo,to,fpr)(N-2) IN FORALL (i:subrange(0,gsb`length-2) \| gsb`seq(i) /= gsb`seq(i+1)) : red_gs_band_fp?(so,vo,to,fpr,gsb)(i) IFF (FORALL (x \| gsb`seq(i) < x AND x < gsb`seq(i+1)): LET gs = gs2v3(vo)(x) IN ...		theorem	gs_red_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "conflict_3D?", "gs", "gs2v3", "gs_bands_si", "length", "red_gs_band_fp?", "so", "to", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs_green_bands_si : FORALL (fpr:FlightPlanRelevant_nz(to)): LET gsb = gs_bands_si(so,vo,to,fpr)(N-2) IN FORALL (i:subrange(0,gsb`length-2) \| gsb`seq(i) /= gsb`seq(i+1)) : NOT red_gs_band_fp?(so,vo,to,fpr,gsb)(i) IFF (FORALL (x \| gsb`seq(i) < x AND x < gsb`seq(i+1)): LET gs = gs2v3(vo)(x...		theorem	gs_green_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "NOT", "conflict_3D?", "gs", "gs2v3", "gs_bands_si", "length", "red_gs_band_fp?", "so", "to", "vo" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_si_combines : FORALL (i:subrange(0,N-2), x:real): member(x,vs_bands_si(so,vo,to,fp)(i)) IFF EXISTS (j:subrange(0,i)): member(x,vs_bands_si_i(so,vo,to,fp)(j))		lemma	vs_bands_si_combines	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "IFF", "member", "so", "to", "vo", "vs_bands_si", "vs_bands_si_i" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_si_component : FORALL (i:subrange(0,N-2), x:real): member(x,vs_bands_si(so,vo,to,fp)(i)) IFF EXISTS (j:subrange(0,i)): LET LB = max(fp(j)`time,B+to)-to, UB = min(fp(1+j)`time,T+to)-to IN fp(j+1)`time > B+to AND fp(j)`time < T+ to AND ...		lemma	vs_bands_si_component	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "IFF", "max", "member", "min", "position", "so", "time", "to", "velocity", "vo", "vs_bands_3D", "vs_bands_si" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_bands_si_equivalence: FORALL (fpr:FlightPlanRelevant_nz(to)): LET vsb = vs_bands_si(so,vo,to,fpr)(N-2), vs = vs2v3(vo) IN FORALL (x,y): (vsmin<=x AND x<=y AND y<=vsmax AND (FORALL (i:subrange(0,vsb`length-1)): vsb`seq(i) < x OR y < vsb`seq(i))) IMPLIES (con...		lemma	vs_bands_si_equivalence	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "IMPLIES", "OR", "conflict_3D?", "length", "so", "to", "vo", "vs", "vs2v3", "vs_bands_si" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_red_bands_si : FORALL (fpr:FlightPlanRelevant_nz(to)): LET vsb = vs_bands_si(so,vo,to,fpr)(N-2) IN FORALL (i:subrange(0,vsb`length-2) \| vsb`seq(i) /= vsb`seq(i+1)) : red_vs_band_fp?(so,vo,to,fpr,vsb)(i) IFF (FORALL (x \| vsb`seq(i) < x AND x < vsb`seq(i+1)): LET vs = vs2v3(vo)(x) IN ...		theorem	vs_red_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "conflict_3D?", "length", "red_vs_band_fp?", "so", "to", "vo", "vs", "vs2v3", "vs_bands_si" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vs_green_bands_si : FORALL (fpr:FlightPlanRelevant_nz(to)): LET vsb = vs_bands_si(so,vo,to,fpr)(N-2) IN FORALL (i:subrange(0,vsb`length-2) \| vsb`seq(i) /= vsb`seq(i+1)) : NOT red_vs_band_fp?(so,vo,to,fpr,vsb)(i) IFF (FORALL (x \| vsb`seq(i) < x AND x < vsb`seq(i+1)): LET vs = vs2v3(vo)(x...		theorem	vs_green_bands_si	ACCoRD	ACCoRD/bands_si.pvs	[ "cd3d_si[D,H,B,T,N]," ]	[ "AND", "FlightPlanRelevant_nz", "IFF", "NOT", "conflict_3D?", "length", "red_vs_band_fp?", "so", "to", "vo", "vs", "vs2v3", "vs_bands_si" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk2v2(vo2)(trk) : (trk_only?(vo2))	= trkgs2vect(trk,norm(vo2))	definition	trk2v2	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "norm", "trk", "trk_only?", "trkgs2vect" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk2v2_Nzv : trk2v2(vo2)(trk) HAS_TYPE Nz_vect2		judgement	trk2v2_Nzv	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "Nz_vect2", "trk", "trk2v2" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
Vtrk(vo2,vi2)(trk:real): Vect2	= trk2v2(vo2)(trk) - vi2	definition	Vtrk	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "Vect2", "trk", "trk2v2" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs2v2(vo2)(k) : Vect2	= k*^(vo2)	definition	gs2v2	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "Vect2" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs2v2_gs_only : gs2v2(vo2)(gsp) HAS_TYPE (gs_only?(vo2))		judgement	gs2v2_gs_only	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "gs2v2", "gs_only?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
gs2v2_id: gs_only?(vo2)(nvo2) IMPLIES gs2v2(vo2)(gs(nvo2)) = nvo2		lemma	gs2v2_id	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "IMPLIES", "gs", "gs2v2", "gs_only?" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk2v3(vo3)(trkz:real) : (trk_only_3D?(vo3))	= trkgs2vect(trkz,norm(vect2(vo3))) WITH [z \|-> vo3`z]	definition	trk2v3	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "norm", "trk_only_3D?", "trkgs2vect", "vect2" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
vect2_trk2v3 : vect2(trk2v3(vo3)(trk)) = trk2v2(vo3)(trk)		lemma	vect2_trk2v3	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "trk", "trk2v2", "trk2v3", "vect2" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk2v3_Nzv : trk2v3(vo3)(trk) HAS_TYPE Nzv2_vect3		judgement	trk2v3_Nzv	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "Nzv2_vect3", "trk", "trk2v3" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
Vtrk_3D(vo3,vi3)(trkz:real):(horizontal_only?(vo3-vi3))	= trk2v3(vo3)(trkz) - vi3	definition	Vtrk_3D	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "horizontal_only?", "trk2v3" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696
trk2v3_trk2v3 : trk2v3(trk2v3(vo3)(trk))(trk)=trk2v3(vo3)(trk)		lemma	trk2v3_trk2v3	ACCoRD	ACCoRD/bands_util.pvs	[ "definitions_3D," ]	[ "trk", "trk2v3" ]	https://github.com/nasa/pvslib	59714e007f6acc7440fe014ae7ff83bfb2ef0696

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PVS-NASALib

Declarations from the NASA PVS Library (NASALib).

Source

Repository: https://github.com/nasa/pvslib
Commit: 59714e007f6acc7440fe014ae7ff83bfb2ef0696
Files: -
License: other

Schema

Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 33,871
With proof: 10,619 (31.4%)
With docstring: 0 (0.0%)
Libraries: 91

By type

Type	Count
lemma	18,703
definition	8,946
theorem	1,305
judgement	1,186
type	1,110
function	910
macro	764
axiom	457
corollary	239
assumption	179
conjecture	42
proposition	19
formula	10
fact	1

Example

vs_critical(TF,voz) : (vs_fseq?)
=
    IF TF AND vsmin<=voz AND voz<=vsmax THEN addend(voz,emptyseq)
    ELSE emptyseq
    ENDIF

type: definition | symbolic_name: vs_critical | ACCoRD/bands_1D.pvs

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{pvs_nasalib_dataset,
  title  = {PVS-NASALib},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/nasa/pvslib, commit 59714e007f6a},
  url    = {https://huggingface.co/datasets/phanerozoic/PVS-NASALib}
}

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