Nuprl Lemma : transEquiv-trans-eq-path-trans
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}].
(transEquivFun(p) = (PathTransport(p) o ConstTrans(decode(A))) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Proof
Definitions occuring in Statement :
transEquiv-trans: transEquivFun(p),
path-trans: PathTransport(p),
universe-comp-op: compOp(t),
universe-decode: decode(t),
cubical-universe: c𝕌,
const-transport-fun: ConstTrans(A),
path-type: (Path_A a b),
cubical-fun-comp: (f o g),
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,
uimplies: b supposing a,
cubical-term-at: u(a),
cubical-fun-comp: (f o g),
cubical-app: app(w; u),
const-transport-fun: ConstTrans(A),
csm-ap-term: (t)s,
cubical-lam: cubical-lam(X;b),
transport-const: transport-const(G;cA;a),
cubical-lambda: (λb),
all: ∀x:A. B[x],
cc-adjoin-cube: (v;u),
cc-snd: q,
csm-composition: (comp)sigma,
transport: transport(Gamma;a),
pi2: snd(t),
composition-term: comp cA [phi ⊢→ u] a0,
discrete-cubical-term: discr(t),
face-0: 0(𝔽),
cc-fst: p,
csm-ap: (s)x,
cube-context-adjoin: X.A,
pi1: fst(t),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-universe: c𝕌,
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
fibrant-type: FibrantType(X),
composition-op: Gamma ⊢ CompOp(A),
formal-cube: formal-cube(I),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
cube-set-restriction: f(s),
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
cand: A c∧ B,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-type-at: A(a),
closed-type-to-type: closed-type-to-type(T),
closed-cubical-universe: cc𝕌,
names-hom: I ⟶ J,
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u)
Lemmas referenced :
cubical-term-at_wf,
transEquiv-trans-eq2,
cubical-fun-equal,
universe-decode_wf,
cubical-fun-comp_wf,
const-transport-fun_wf,
universe-comp-op_wf,
path-trans_wf,
cc_fst_adjoin_cube_lemma,
istype-cubical-type-at,
cube-set-restriction_wf,
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-term,
path-type_wf,
cubical-universe_wf,
istype-cubical-universe-term,
cubical_set_wf,
cube_set_restriction_pair_lemma,
path-trans-sq2,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add-name_wf,
new-name_wf,
nc-s_wf,
f-subset-add-name,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
path-type-at,
nh-id_wf,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
I_cube_pair_redex_lemma,
lattice-0_wf,
face_lattice_wf,
face-presheaf_wf2,
member-empty-cubical-subset,
cubical-path-0_wf,
formal-cube_wf1,
pi1_wf_top,
cubical-type_wf,
cubical-type-cumulativity2,
cubical-path-1_wf,
subtype_rel_dep_function,
cubical-type-at_wf,
lattice-point_wf,
dM_wf,
fibrant-type_wf_formal-cube,
nh-comp_wf,
nc-0_wf,
universe-type-at,
universe-path-type-lemma-0,
cube-set-restriction-comp,
nh-id-left,
s-comp-nc-0-new,
equal-wf-T-base,
nh-id-right,
cubical-path-condition-0,
cubical-path-condition_wf,
nc-1_wf,
universe-path-type-lemma-1,
s-comp-if-lemma1,
s-comp-nc-1-new,
nh-comp-assoc,
empty-cubical-subset-term,
cube-set-restriction-when-id,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
hypothesis,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
Error :memTop,
universeIsType,
applyEquality,
lambdaEquality_alt,
imageElimination,
universeEquality,
setElimination,
rename,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
intEquality,
lambdaFormation_alt,
independent_pairEquality,
equalityIstype,
cumulativity,
hyp_replacement,
functionEquality,
equalityElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[p:\{G \mvdash{} \_:(Path\_c\mBbbU{} A B)\}].
(transEquivFun(p) = (PathTransport(p) o ConstTrans(decode(A))))
Date html generated:
2020_05_20-PM-07_39_31
Last ObjectModification:
2020_05_01-AM-10_19_51
Theory : cubical!type!theory
Home
Index