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,
subtype_rel: A ⊆r B,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
guard: {T},
composition-structure: Gamma ⊢ Compositon(A),
uimplies: b supposing a,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
same-cubical-term: X ⊢ u=v:A,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
pi1: fst(t),
csm-ap-term: (t)s,
face-zero: (i=0),
cubical-refl: refl(a),
cubicalpath-app: pth @ r,
path-eta: path-eta(pth),
term-to-path: <>(a),
cubical-app: app(w; u),
cubical-lambda: (λb),
cc-adjoin-cube: (v;u),
csm+: tau+,
csm-comp: G o F,
compose: f o g,
pi2: snd(t),
cubical-path-app: pth @ r,
prop: ℙ,
squash: ↓T,
true: True,
interval-1: 1(𝕀),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
path-eta_wf,
path-type-subtype,
cubical-refl_wf,
comp_term_wf,
cube-context-adjoin_wf,
interval-type_wf,
face-or_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
csm-ap-type_wf,
cc-fst_wf_interval,
csm-comp-structure_wf,
istype-cubical-term,
path-type_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
csm-face-or,
cc-fst_wf,
subset-cubical-type,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
context-subset-is-subset,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm+_wf,
subtype_rel-equal,
cubical-term_wf,
p-csm+-type,
csm-interval-type,
subset-cubical-term,
case-term_wf2,
csm-face-zero,
csm-face-one,
same-cubical-type-trivial_1,
context-subset-map,
context-iterated-subset0,
sub_cubical_set_transitivity,
context-subset-swap,
sub_cubical_set_functionality2,
thin-context-subset,
context-adjoin-subset2,
sub_cubical_set_self,
csm-case-term,
same-cubical-term-by-cases,
context-subset-term-subtype,
empty-context-subset-lemma3',
case-term-equal-right',
face-and_wf,
context-iterated-subset,
cubical-path-app-1,
csm-cubicalpath-app,
csm-interval-1,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubicalpath-app_wf,
pathtype_wf,
csm-path-type-sub-pathtype,
csm-pathtype,
pathtype-subset,
face-one-context-implies,
case-term-equal-left',
cubical-path-app-0,
csm-interval-0,
face-zero-context-implies,
cubical-term-eqcd,
context-subset-subtype-or,
constrained-cubical-term_wf,
subset-cubical-term2,
context-iterated-subset1,
context-subset-subtype-or2,
cube_set_map_wf,
cube_set_map_subtype3,
csm-context-subset-subtype2,
case-term_wf,
interval-0_wf,
empty-context-subset-lemma3,
face-0_wf,
face-zero-and-one,
term-to-path-wf,
csm-id-adjoin_wf,
interval-1_wf,
cc-snd-1,
face-1-implies-subset,
csm-id_wf,
csm-ap-id-type,
face-one-interval-1,
face-term-implies-or2,
face-term-implies_wf,
subtype_rel_self,
iff_weakening_equal,
case-term-0',
cubical-path-app_wf,
face-zero-interval-1,
cc-snd-0,
face-zero-interval-0,
face-term-implies-or1,
case-term-1'
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
instantiate,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
universeIsType,
Error :memTop,
independent_isectElimination,
cumulativity,
dependent_set_memberEquality_alt,
equalityIstype,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
productElimination,
applyLambdaEquality,
hyp_replacement,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
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_57_55
Last ObjectModification:
2020_04_20-AM-10_08_48
Theory : cubical!type!theory
Home
Index