Nuprl Lemma : comp_path_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. ∀[a,b,c:{G ⊢ _:A}]. ∀[pth_a_b:{G ⊢ _:(Path_A a b)}].
∀[pth_b_c:{G ⊢ _:(Path_A b c)}].
(pth_a_b + pth_b_c ∈ {G ⊢ _:(Path_A a c)})
Proof
Definitions occuring in Statement :
comp_path: pth_a_b + pth_b_c,
composition-structure: Gamma ⊢ Compositon(A),
path-type: (Path_A a b),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
comp_path: pth_a_b + pth_b_c,
member: t ∈ T,
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True
Lemmas referenced :
cubical-term_wf,
cubical_set_cumulativity-i-j,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
path-type_wf,
cubical-type-cumulativity2,
cubical-path-app-0,
cubical-type-cumulativity,
subtype_rel_self,
iff_weakening_equal,
comp-path_wf,
cubical-path-app_wf,
interval-0_wf,
subset-cubical-term2,
sub_cubical_set_self,
composition-structure_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
hyp_replacement
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[cA:G \mvdash{} Compositon(A)]. \mforall{}[a,b,c:\{G \mvdash{} \_:A\}]. \mforall{}[pth\_a$_{b}\mbackslash{}ff2\000C4:\{G \mvdash{} \_:(Path\_A a b)\}].
\mforall{}[pth\_b$_{c}$:\{G \mvdash{} \_:(Path\_A b c)\}].
(pth\_a$_{b}$ + pth\_b$_{c}$ \mmember{} \{G \mvdash{} \_:(Path\_A a c)\})
Date html generated:
2020_05_20-PM-04_58_09
Last ObjectModification:
2020_04_13-PM-02_08_22
Theory : cubical!type!theory
Home
Index