Nuprl Lemma : equiv_path-1
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. G ⊢ (equiv_path(G;A;B;f))[1(𝕀)]=B:c𝕌
Proof
Definitions occuring in Statement :
equiv_path: equiv_path(G;A;B;f),
universe-decode: decode(t),
cubical-universe: c𝕌,
cubical-equiv: Equiv(T;A),
same-cubical-term: X ⊢ u=v:A,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
same-cubical-term: X ⊢ u=v:A,
equiv_path: equiv_path(G;A;B;f),
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
let: let,
prop: ℙ,
squash: ↓T,
true: True,
csm-comp-structure: (cA)tau,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
interval-type: 𝕀,
csm-comp: G o F,
csm-id: 1(X),
csm-adjoin: (s;u),
compose: f o g,
csm-ap: (s)x,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
universe-comp-fun: CompFun(A)
Lemmas referenced :
equiv-path2_wf,
universe-decode_wf,
universe-comp-fun_wf,
cubical_set_cumulativity-i-j,
comp-fun-to-comp-op_wf,
cube-context-adjoin_wf,
interval-type_wf,
equiv-path1_wf,
composition-structure-cumulativity,
csm-universe-encode,
csm-id-adjoin_wf,
interval-1_wf,
same-cubical-term_wf,
cubical-universe_wf,
cubical-type-cumulativity2,
istype-cubical-term,
cubical-equiv_wf,
istype-cubical-universe-term,
cubical_set_wf,
universe-encode-decode,
squash_wf,
true_wf,
cubical-type_wf,
universe-encode_wf,
composition-op_wf,
equiv-path1-1,
subtype_rel_self,
composition-structure_wf,
csm-comp-structure_wf,
csm-id-adjoin_wf-interval-1,
equiv-path2-1,
csm-comp-fun-to-comp-op,
equal_wf,
istype-universe,
csm-ap-type_wf,
universe-comp-op_wf,
subtype_rel-equal,
iff_weakening_equal,
comp-op-to-comp-fun-inverse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
dependent_functionElimination,
hyp_replacement,
applyLambdaEquality,
equalityIstype,
independent_functionElimination,
universeIsType,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
setElimination,
productElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}].
G \mvdash{} (equiv\_path(G;A;B;f))[1(\mBbbI{})]=B:c\mBbbU{}
Date html generated:
2020_05_20-PM-07_30_15
Last ObjectModification:
2020_04_28-PM-07_05_30
Theory : cubical!type!theory
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