Nuprl Lemma : app-trans-equiv-path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. ∀[a:{G ⊢ _:decode(A)}].
(app(trans-equiv-path(G;A;B;f); a)
= transprt-const(G;CompFun(B);transprt-const(G;CompFun(B);app(equiv-fun(f); a)))
∈ {G ⊢ _:decode(B)})
Proof
Definitions occuring in Statement :
trans-equiv-path: trans-equiv-path(G;A;B;f),
universe-comp-fun: CompFun(A),
universe-decode: decode(t),
cubical-universe: c𝕌,
transprt-const: transprt-const(G;cA;a),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
cubical-app: app(w; u),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
trans-equiv-path: trans-equiv-path(G;A;B;f),
uimplies: b supposing a,
cubical-lam: cubical-lam(X;b),
let: let,
all: ∀x:A. B[x],
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe-comp-fun: CompFun(A),
universe-comp-op: compOp(t),
comp-op-to-comp-fun: cop-to-cfun(cA),
csm-comp-structure: (cA)tau,
cubical-term-at: u(a),
cc-fst: p,
csm-id-adjoin: [u],
interval-type: 𝕀,
csm-comp: G o F,
csm-id: 1(X),
csm-adjoin: (s;u),
compose: f o g,
pi1: fst(t),
csm-ap: (s)x,
csm-composition: (comp)sigma,
rev_implies: P ⇐ Q
Lemmas referenced :
csm-ap-term-universe,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cc-fst_wf,
universe-decode_wf,
csm-ap-term_wf,
cubical-equiv-p,
cubical-term-eqcd,
cc-snd_wf,
csm-comp-structure_wf2,
universe-comp-fun_wf,
istype-cubical-term,
cubical-equiv_wf,
istype-cubical-universe-term,
cubical_set_wf,
subtype_rel_self,
composition-structure_wf,
csm-universe-decode,
cubical-app_wf_fun,
csm-ap-type_wf,
equiv-fun_wf,
cubical-beta,
transprt-const_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term_wf,
csm-id-adjoin_wf,
csm-transprt-const,
iff_weakening_equal,
cubical-type_wf,
cubical-lambda_wf,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
cube_set_map_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-cubical-app,
cc_snd_csm_id_adjoin_lemma,
cubical-fun_wf,
csm-equiv-fun,
csm_id_adjoin_fst_term_lemma,
csm-ap-id-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
because_Cache,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
dependent_functionElimination,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
applyLambdaEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}]. \mforall{}[a:\{G \mvdash{} \_:decode(A)\}].
(app(trans-equiv-path(G;A;B;f); a)
= transprt-const(G;CompFun(B);transprt-const(G;CompFun(B);app(equiv-fun(f); a))))
Date html generated:
2020_05_20-PM-07_40_37
Last ObjectModification:
2020_04_30-PM-05_09_19
Theory : cubical!type!theory
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