Nuprl Lemma : univ-trans-equiv_path
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(univ-trans(G;equiv_path(G;A;B;f)) = trans-equiv-path(G;A;B;f) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Proof
Definitions occuring in Statement :
trans-equiv-path: trans-equiv-path(G;A;B;f),
univ-trans: univ-trans(G;T),
equiv_path: equiv_path(G;A;B;f),
universe-decode: decode(t),
cubical-universe: c𝕌,
cubical-equiv: Equiv(T;A),
cubical-fun: (A ⟶ B),
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,
equiv_path: equiv_path(G;A;B;f),
univ-trans: univ-trans(G;T),
let: let,
all: ∀x:A. B[x],
guard: {T},
prop: ℙ,
uimplies: b supposing a,
transprt-fun: transprt-fun(Gamma;A;cA),
trans-equiv-path: trans-equiv-path(G;A;B;f),
equiv-path2: equiv-path2(G;A;B;cA;cB;f),
transprt: transprt(G;cA;a0),
comp_term: comp cA [phi ⊢→ u] a0,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
implies: P ⇒ Q,
csm-comp-structure: (cA)tau,
csm+: tau+,
csm-comp: G o F,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
universe-decode: decode(t),
csm-adjoin: (s;u),
compose: f o g,
csm-ap: (s)x,
cubical-term-at: u(a),
glue-comp: comp(Glue [phi ⊢→ (T, f)] A) ,
csm-id: 1(X),
pi1: fst(t),
csm-ap-term: (t)s,
interval-1: 1(𝕀),
cubical-lam: cubical-lam(X;b),
squash: ↓T,
true: True,
cubical-equiv-by-cases: cubical-equiv-by-cases(G;B;f),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
same-cubical-term: X ⊢ u=v:A,
cubical-type: {X ⊢ _},
cubical-id-fun: cubical-id-fun(X),
cubical-id-equiv: IdEquiv(X;T),
equiv-fun: equiv-fun(f),
equiv-witness: equiv-witness(f;cntr),
cubical-fst: p.1,
cubical-lambda: (λb),
cubical-pair: cubical-pair(u;v),
same-cubical-type: Gamma ⊢ A = B,
so_lambda: λ2x.t[x],
so_apply: x[s],
transprt-const: transprt-const(G;cA;a),
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),
equiv-contr: equiv-contr(f;a),
contr-center: contr-center(c),
cubical-snd: p.2,
cubical-app: app(w; u),
pi2: snd(t),
cubical-id-is-equiv: cubical-id-is-equiv(X;T),
contr-witness: contr-witness(X;c;p),
cc-adjoin-cube: (v;u),
id-fiber-center: id-fiber-center(X;T),
fiber-member: fiber-member(p),
equiv-path1: equiv-path1(G;A;B;f),
glue-type: Glue [phi ⊢→ (T;w)] 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,
universe-decode-encode,
universe-comp-op-encode,
equal_wf,
cubical-fun_wf,
cubical-term-eqcd,
istype-cubical-term,
cubical-equiv_wf,
istype-cubical-universe-term,
cubical_set_wf,
csm-glue-comp,
csm-ap-type_wf,
cc-fst_wf_interval,
face-or_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
case-type_wf,
thin-context-subset,
same-cubical-type-zero-and-one,
face-0_wf,
csm_id_ap_term_lemma,
cc-snd-1,
cubical-lam_wf,
squash_wf,
true_wf,
cubical-type-cumulativity2,
cc-fst_wf,
cubical-type_wf,
csm-face-or,
csm-face-zero,
csm-face-one,
csm-case-type,
q-csm+,
csm-universe-decode,
p-csm+-type,
csm-id-comp-sq,
cc-snd-0,
istype-top,
csm-case-type-comp,
csm-equiv-fun,
csm-case-term,
p-csm+-term,
composition-structure_wf,
subtype_rel-equal,
csm-cubical-equiv,
istype-universe,
cube_set_map_wf,
subtype_rel_self,
iff_weakening_equal,
csm-ap-term_wf,
face-zero-interval-0,
case-term-1,
equiv-fun_wf,
unglue-term-1,
subset-cubical-type,
context-subset_wf,
interval-0_wf,
context-subset-is-subset,
face-type_wf,
face-1-or,
cubical-fun-subset,
subset-cubical-term,
glue-type-1',
cubical-app_wf_fun,
csm-comp-structure_wf2,
case-type-comp-false-true,
interval-1_wf,
face-one-interval-1,
member_wf,
face-zero-interval-1,
face-term-implies_wf,
face-term-implies-1,
composition-structure-subset,
csm-cubical-id-equiv,
csm+_wf_interval,
csm-id-adjoin_wf-interval-1,
cubical-id-equiv-subset,
case-term-0,
case-type-0,
subtype_rel_product,
fset_wf,
nat_wf,
I_cube_wf,
names-hom_wf,
cube-set-restriction_wf,
top_wf,
equals-transprt,
fiber-comp-subset,
fiber-comp_wf,
equiv-fun-cubical-id-equiv,
equiv-term-0-subset-1,
face-forall_wf,
face-forall-q=0-or-q=1,
face-or-0,
subset-cubical-term2,
sub_cubical_set_self,
cubical-fiber_wf,
face-or-1,
glue-term-1',
fiber-member_wf,
contractible-type_wf,
equiv-contr_wf,
contr-center_wf,
cubical-id-equiv_wf,
transprt-const_wf,
equal_functionality_wrt_subtype_rel2,
cubical-id-fun_wf,
fiber-member-transprt-const-fiber-comp,
cubical-term_wf,
csm-ap-id-term,
equiv-path1-0,
equiv-path1-1,
transprt_wf,
csm-id-adjoin_wf,
csm-interval-type,
csm-comp-structure_wf,
csm+_wf,
transprt-fun_wf,
comp-op-to-comp-fun_wf2,
comp-fun-to-comp-op-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,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
independent_isectElimination,
universeIsType,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
applyLambdaEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
functionEquality,
cumulativity,
functionIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}].
(univ-trans(G;equiv\_path(G;A;B;f)) = trans-equiv-path(G;A;B;f))
Date html generated:
2020_05_20-PM-07_41_45
Last ObjectModification:
2020_05_01-PM-01_45_52
Theory : cubical!type!theory
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