Nuprl Lemma : csm-path-type
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}.
(((Path_A a b))s = (Delta ⊢ Path_(A)s (a)s (b)s) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
path-type: (Path_A a b),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
squash: ↓T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
pathtype: Path(A),
csm-ap-type: (AF)s,
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
and: P ∧ Q,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
interval-presheaf: 𝕀,
true: True,
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
csm-cubical-subset,
cubical-subset_wf,
squash_wf,
cubical-type-restriction_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
istype-cubical-type-at,
I_cube_wf,
fset_wf,
nat_wf,
cubical-type_wf,
csm-pathtype,
cubical_type_at_pair_lemma,
csm-ap-term-at,
csm-ap-type-at,
equal_wf,
cubical-type-at_wf,
csm-ap_wf,
nh-id_wf,
dM0_wf,
subtype_rel_self,
interval-type_wf,
cube-set-restriction_wf,
subtype_rel-equal,
cube-set-restriction-id,
cubical-term-at_wf,
dM1_wf,
csm-ap-type_wf,
pathtype_wf,
cubical-term_wf,
cube_set_map_wf,
cubical_set_wf,
cubical-type-restriction-and,
cubical-type-restriction-eq,
csm-ap-term_wf,
names-hom_wf,
interval-type-at,
I_cube_pair_redex_lemma,
interval-type-ap-morph,
cubical_type_ap_morph_pair_lemma,
true_wf,
istype-universe,
csm-ap-restriction,
iff_weakening_equal,
cubical-type-ap-morph_wf,
nh-id-left,
nh-comp_wf,
dM-lift_wf2,
nh-id-right,
dM-lift-0-sq,
csm-cubical-type-ap-morph,
dM-lift-1-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
applyEquality,
lambdaEquality_alt,
imageElimination,
hypothesisEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
cumulativity,
functionIsType,
universeEquality,
because_Cache,
dependent_functionElimination,
productEquality,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
independent_functionElimination,
hyp_replacement,
productElimination
Latex:
\mforall{}X,Delta:j\mvdash{}. \mforall{}s:Delta j{}\mrightarrow{} X. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}.
(((Path\_A a b))s = (Delta \mvdash{} Path\_(A)s (a)s (b)s))
Date html generated:
2020_05_20-PM-03_15_21
Last ObjectModification:
2020_04_08-AM-11_25_44
Theory : cubical!type!theory
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