Nuprl Lemma : csm-fiber-comp
∀[G:j⊢]. ∀[A,T:{G ⊢ _}]. ∀[a:{G ⊢ _:A}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cT:G ⊢ Compositon(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
∀[f:{G ⊢ _:(T ⟶ A)}].
((fiber-comp(G;T;A;f;a;cT;cA))s = fiber-comp(H;(T)s;(A)s;(f)s;(a)s;(cT)s;(cA)s) ∈ H ⊢ Compositon(Fiber((f)s;(a)s)))
Proof
Definitions occuring in Statement :
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
cubical-fiber: Fiber(w;a),
cubical-fun: (A ⟶ B),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
compose: f o g
Lemmas referenced :
csm-fiber-comp-sq,
fiber-comp_wf,
csm-ap-type_wf,
csm-ap-term_wf,
cubical-fun_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-cubical-fun,
csm-comp-structure_wf,
cube_set_map_cumulativity-i-j,
istype-cubical-term,
cubical-type-cumulativity2,
cube_set_map_wf,
composition-structure_wf,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeIsType,
inhabitedIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)]. \mforall{}[cT:G \mvdash{} Compositon(T)]. \mforall{}[H:j\mvdash{}].
\mforall{}[s:H j{}\mrightarrow{} G]. \mforall{}[f:\{G \mvdash{} \_:(T {}\mrightarrow{} A)\}].
((fiber-comp(G;T;A;f;a;cT;cA))s = fiber-comp(H;(T)s;(A)s;(f)s;(a)s;(cT)s;(cA)s))
Date html generated:
2020_05_20-PM-05_13_35
Last ObjectModification:
2020_04_18-AM-10_02_55
Theory : cubical!type!theory
Home
Index