Nuprl Lemma : equal-fiber-when-discrete
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ T)}]. ∀[z:{X ⊢ _:T}]. ∀[a,b:{X ⊢ _:Fiber(f;z)}].
a = b ∈ {X ⊢ _:Fiber(f;z)} ⇐⇒ a.1 = b.1 ∈ {X ⊢ _:A}
supposing ∀x:{X ⊢ _:Path(T)}. (x = refl(x @ 0(𝕀)) ∈ {X ⊢ _:Path(T)})
Proof
Definitions occuring in Statement :
cubical-fiber: Fiber(w;a),
cubical-refl: refl(a),
cubicalpath-app: pth @ r,
pathtype: Path(A),
interval-0: 0(𝕀),
cubical-fst: p.1,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
cubical-fiber: Fiber(w;a),
true: True,
guard: {T},
rev_implies: P ⇐ Q,
cubical-path-app: pth @ r,
cubical-refl: refl(a),
term-to-path: <>(a),
cc-fst: p,
csm-ap-term: (t)s,
csm-ap: (s)x
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-fun_wf,
cc-fst_wf,
csm-cubical-fun,
cubical-term-eqcd,
cubical-app_wf_fun,
csm-ap-type_wf,
cc-snd_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-fst_wf,
istype-cubical-term,
cubical-sigma_wf,
cubical-type_wf,
path-type_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-sigma-equal,
pathtype_wf,
cubical-refl_wf,
cubicalpath-app_wf,
interval-0_wf,
path-type-subtype,
cubical-fiber_wf,
cubical_set_wf,
csm-path-type,
csm-id-adjoin_wf,
cubical-snd_wf,
cube_set_map_wf,
csm_id_adjoin_fst_type_lemma,
csm_id_adjoin_fst_term_lemma,
subtype_rel_transitivity,
cubical-term_wf,
csm-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-type,
paths-equal,
subtype_rel_wf,
cubical-path-app-0,
csm_id_ap_term_lemma,
subset-cubical-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
independent_pairFormation,
lambdaFormation_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
equalityIstype,
inhabitedIsType,
functionIsType,
cumulativity,
Error :memTop,
applyLambdaEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[f:\{X \mvdash{} \_:(A {}\mrightarrow{} T)\}]. \mforall{}[z:\{X \mvdash{} \_:T\}]. \mforall{}[a,b:\{X \mvdash{} \_:Fiber(f;z)\}].
a = b \mLeftarrow{}{}\mRightarrow{} a.1 = b.1 supposing \mforall{}x:\{X \mvdash{} \_:Path(T)\}. (x = refl(x @ 0(\mBbbI{})))
Date html generated:
2020_05_20-PM-03_37_19
Last ObjectModification:
2020_04_20-AM-11_12_44
Theory : cubical!type!theory
Home
Index