Nuprl Lemma : equiv-term-0-subset-1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
∀[psi:{G ⊢ _:𝔽}]
∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[a:{G ⊢ _:A}]. ∀[t,c:Top]. ∀[cF:G +⊢ Compositon(Fiber(equiv-fun(f);a))].
(equiv f [phi ⊢→ (t, c)] a = transprt(G;(cF)p;contr-center(equiv-contr(f;a))) ∈ {G ⊢ _:Fiber(equiv-fun(f);a)})
supposing psi = 1(𝔽) ∈ {G ⊢ _:𝔽}
supposing phi = 0(𝔽) ∈ {G ⊢ _:𝔽}
Proof
Definitions occuring in Statement :
equiv-term: equiv f [phi ⊢→ (t, c)] a,
transprt: transprt(G;cA;a0),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
equiv-contr: equiv-contr(f;a),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
cubical-fiber: Fiber(w;a),
contr-center: contr-center(c),
context-subset: Gamma, phi,
face-1: 1(𝔽),
face-0: 0(𝔽),
face-type: 𝔽,
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
path-type: (Path_A a b),
cubical-subset: cubical-subset
Lemmas referenced :
equiv-term-0,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-fiber_wf,
equiv-fun_wf,
cubical-term-eqcd,
empty-context-subset-lemma3,
subtype_rel-equal,
thin-context-subset,
face-0_wf,
context-subset_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
equiv-term-subset,
composition-structure_wf,
istype-top,
istype-cubical-term,
cubical-equiv_wf,
cubical-type_wf,
face-1_wf,
face-type_wf,
cubical_set_wf,
subset-cubical-term,
sub_cubical_set_self,
sub_cubical_set_wf,
iff_weakening_equal,
empty-context-subset-lemma6,
composition-structure-cumulativity,
subset-cubical-type,
context-subset-is-subset,
context-1-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
hyp_replacement,
equalitySymmetry,
sqequalRule,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeIsType,
universeEquality,
Error :memTop,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}].
\mforall{}[psi:\{G \mvdash{} \_:\mBbbF{}\}]
\mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(T;A)\}]. \mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[t,c:Top].
\mforall{}[cF:G +\mvdash{} Compositon(Fiber(equiv-fun(f);a))].
(equiv f [phi \mvdash{}\mrightarrow{} (t, c)] a = transprt(G;(cF)p;contr-center(equiv-contr(f;a))))
supposing psi = 1(\mBbbF{})
supposing phi = 0(\mBbbF{})
Date html generated:
2020_05_20-PM-05_36_04
Last ObjectModification:
2020_04_21-PM-01_46_22
Theory : cubical!type!theory
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