Nuprl Lemma : csm-transprt-const
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[a:{G ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((transprt-const(G;cA;a))s = transprt-const(H;(cA)s;(a)s) ∈ {H ⊢ _:(A)s})
Proof
Definitions occuring in Statement :
transprt-const: transprt-const(G;cA;a),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
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,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
csm-id: 1(X),
csm-ap-type: (AF)s,
cc-fst: p,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-adjoin: (s;u),
pi1: fst(t),
interval-1: 1(𝕀),
uimplies: b supposing a,
transprt-const: transprt-const(G;cA;a),
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
csm+: tau+,
compose: f o g,
cc-snd: q,
constant-cubical-type: (X),
squash: ↓T,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-transprt,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
csm-comp-structure_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-id_wf,
csm-ap-id-type,
cube_set_map_cumulativity-i-j,
cube_set_map_wf,
istype-cubical-term,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
transprt_wf,
equal_wf,
cubical-term_wf,
csm_id_adjoin_fst_type_lemma,
cubical-term-eqcd,
iff_weakening_equal,
csm-ap-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
setElimination,
rename,
productElimination,
independent_isectElimination,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
equalityTransitivity,
lambdaEquality_alt,
imageElimination,
closedConclusion,
universeEquality,
dependent_functionElimination,
Error :memTop,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
hyp_replacement
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)]. \mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((transprt-const(G;cA;a))s = transprt-const(H;(cA)s;(a)s))
Date html generated:
2020_05_20-PM-04_39_18
Last ObjectModification:
2020_04_19-PM-02_08_39
Theory : cubical!type!theory
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