Nuprl Lemma : equiv_term_wf
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cT:G +⊢ Compositon(T)].
(equiv f [phi ⊢→ (t,c)] a ∈ {G ⊢ _:Fiber(equiv-fun(f);a)[phi |⟶ fiber-point(t;c)]})
Proof
Definitions occuring in Statement :
equiv_term: equiv f [phi ⊢→ (t,c)] a,
composition-structure: Gamma ⊢ Compositon(A),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
fiber-point: fiber-point(t;c),
cubical-fiber: Fiber(w;a),
path-type: (Path_A a b),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-app: app(w; u),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
equiv_term: equiv f [phi ⊢→ (t,c)] a,
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
cubical-app_wf_fun,
context-subset_wf,
thin-context-subset,
cubical-fun-subset,
equiv-fun_wf,
subset-cubical-term,
context-subset-is-subset,
cubical-fun_wf,
equiv-term_wf,
cubical-type-cumulativity2,
fiber-comp_wf,
subtype_rel_self,
composition-structure_wf,
istype-cubical-term,
path-type_wf,
cubical_set_cumulativity-i-j,
cubical-equiv_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
Error :memTop,
applyEquality,
independent_isectElimination,
instantiate,
universeIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(T;A)\}]. \mforall{}[t:\{G, phi \mvdash{} \_:T\}].
\mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[c:\{G, phi \mvdash{} \_:(Path\_A a app(equiv-fun(f); t))\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)].
\mforall{}[cT:G +\mvdash{} Compositon(T)].
(equiv f [phi \mvdash{}\mrightarrow{} (t,c)] a \mmember{} \{G \mvdash{} \_:Fiber(equiv-fun(f);a)[phi |{}\mrightarrow{} fiber-point(t;c)]\})
Date html generated:
2020_05_20-PM-05_36_19
Last ObjectModification:
2020_04_19-AM-00_02_10
Theory : cubical!type!theory
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