Nuprl Lemma : pathtype-comp_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. (pathtype-comp(G;A;cA) ∈ G ⊢ Compositon(Path(A)))
Proof
Definitions occuring in Statement :
pathtype-comp: pathtype-comp(G;A;cA),
composition-structure: Gamma ⊢ Compositon(A),
pathtype: Path(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pathtype-comp: pathtype-comp(G;A;cA),
composition-function: composition-function{j:l,i:l}(Gamma;A),
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
csm+: tau+,
csm-comp: G o F,
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
compose: f o g,
cc-fst: p,
cc-snd: q,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
cubical-type: {X ⊢ _},
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
pi2: snd(t),
pi1: fst(t),
csm-ap-term: (t)s,
interval-0: 0(𝕀),
path-elim: path-elim(pth),
cubicalpath-app: pth @ r,
cubical-app: app(w; u),
cubical-path-app: pth @ r,
term-to-pathtype: <>a,
pathtype: Path(A),
cubical-fun: (A ⟶ B),
composition-structure: Gamma ⊢ Compositon(A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp)
Lemmas referenced :
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-context-adjoin_wf,
context-subset_wf,
interval-type_wf,
csm-pathtype,
subset-cubical-type,
sub_cubical_set_functionality,
context-subset-is-subset,
pathtype_wf,
csm-ap-type_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-term-eqcd,
cubicalpath-app_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
interval-0_wf,
csm+_wf,
cc-fst_wf_interval,
csm-interval-type,
comp_term_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
csm-comp-structure-composition-function,
csm+_wf_interval,
csm-comp-structure_wf,
constrained-cubical-term_wf,
cubical-type-cumulativity2,
csm-context-subset-subtype3,
cube_set_map_wf,
csm-context-subset-subtype2,
context-subset-map,
subset-cubical-term,
sub_cubical_set_transitivity,
sub_cubical_set_self,
context-adjoin-subset1,
context-adjoin-subset4,
cc-snd_wf,
csm-id-adjoin_wf,
path-elim_wf,
thin-context-subset-adjoin,
istype-cubical-term,
context-subset-term-subtype,
csm-cubicalpath-app,
csm-id-adjoin_wf-interval-1,
interval-1_wf,
csm-context-subset-subtype,
csm-id-adjoin-subset,
term-to-pathtype_wf,
term-to-pathtype-eta,
thin-context-subset,
term-to-path-subset,
subset-cubical-term2,
uniform-comp-function_wf,
csm-constrained-cubical-term,
csm-comp_term,
subtype_rel-equal,
cubical-term_wf,
csm_ap_term_fst_adjoin_lemma,
csm-term-to-pathtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
functionExtensionality,
sqequalRule,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
independent_isectElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
cumulativity,
setElimination,
rename,
hyp_replacement,
Error :memTop,
inhabitedIsType,
independent_pairFormation,
lambdaFormation_alt,
equalityIstype,
dependent_set_memberEquality_alt,
applyLambdaEquality,
functionEquality,
setEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[cA:G \mvdash{} Compositon(A)]. (pathtype-comp(G;A;cA) \mmember{} G \mvdash{} Compositon(Path(A)))
Date html generated:
2020_05_20-PM-05_08_56
Last ObjectModification:
2020_04_18-PM-00_02_45
Theory : cubical!type!theory
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