Nuprl Lemma : equiv-term-0
∀[G:j⊢]. ∀[phi:{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 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),
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,
equiv-term: equiv f [phi ⊢→ (t, c)] a,
let: let,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
guard: {T},
true: True,
and: P ∧ Q,
composition-structure: Gamma ⊢ Compositon(A),
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-term: {X ⊢ _:A},
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
contr-center_wf,
cubical-fiber_wf,
equiv-fun_wf,
equiv-contr_wf,
equals-transprt,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
csm-comp-structure_wf,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
cubical-term-eqcd,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subset-cubical-term2,
sub_cubical_set_self,
csm-id-adjoin_wf-interval-1,
composition-structure_wf,
cubical_set_cumulativity-i-j,
istype-top,
istype-cubical-term,
cubical-equiv_wf,
cubical-type_wf,
face-0_wf,
face-type_wf,
cubical_set_wf,
comp_term_wf,
subtype_rel_set,
composition-function_wf,
uniform-comp-function_wf,
composition-function-cumulativity,
empty-context-subset-lemma4,
cubical-term_wf,
context-subset_wf,
subset-cubical-type,
context-subset-is-subset,
cubical-type-cumulativity2,
empty-context-subset-lemma3,
csm-id-adjoin_wf,
interval-0_wf,
csm-context-subset-subtype2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
Error :memTop,
because_Cache,
instantiate,
applyEquality,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
independent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{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 phi = 0(\mBbbF{})
Date html generated:
2020_05_20-PM-05_35_48
Last ObjectModification:
2020_04_21-AM-09_41_55
Theory : cubical!type!theory
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