Nuprl Lemma : equiv-path1-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}]. ((equiv-path1(G;A;B;f))[1(𝕀)] = B ∈ {G ⊢ _})
Proof
Definitions occuring in Statement :
equiv-path1: equiv-path1(G;A;B;f),
cubical-equiv: Equiv(T;A),
interval-1: 1(𝕀),
csm-id-adjoin: [u],
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
same-cubical-type: Gamma ⊢ A = B,
subtype_rel: A ⊆r B,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
all: ∀x:A. B[x],
uimplies: b supposing a,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
pi2: snd(t),
cubical-type: {X ⊢ _},
pi1: fst(t),
face-1: 1(𝔽)
Lemmas referenced :
equiv-path1-constraint,
istype-cubical-term,
cubical-equiv_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-type_wf,
context-subset_wf,
cube-context-adjoin_wf,
interval-type_wf,
face-or_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
csm-ap-term_wf,
face-type_wf,
csm-id-adjoin_wf,
interval-1_wf,
csm-face-type,
context-subset-map,
cc_snd_csm_id_adjoin_lemma,
csm-interval-1,
csm-face-or,
csm-face-zero,
csm-face-one,
subset-cubical-type,
face-one-interval-1,
sub_cubical_set_wf,
squash_wf,
true_wf,
iff_weakening_equal,
face-or-1,
context-1-subset,
equal_wf,
istype-universe,
csm-case-type,
case-type-same2,
thin-context-subset,
csm-id_wf,
face-1_wf,
empty-context-subset-lemma6,
face-0_wf,
face-and_wf,
face-term-implies-subset,
face-zero-interval-1,
face-term-and-implies1,
face-term-implies_wf,
subtype_rel_self,
csm-ap-id-type,
context-subset-is-subset
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyLambdaEquality,
inhabitedIsType,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
because_Cache,
Error :memTop,
dependent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
hyp_replacement,
universeEquality,
setElimination,
rename,
independent_pairEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(A;B)\}]. ((equiv-path1(G;A;B;f))[1(\mBbbI{})] = B)
Date html generated:
2020_05_20-PM-07_27_25
Last ObjectModification:
2020_04_28-PM-01_13_00
Theory : cubical!type!theory
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