Nuprl Lemma : fst-transprt-const-sigma
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)]. ∀[pr:{X ⊢ _:Σ A B}].
(transprt-const(X;sigma_comp(cA;cB);pr).1 = transprt-const(X;cA;pr.1) ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
sigma_comp: sigma_comp(cA;cB),
transprt-const: transprt-const(G;cA;a),
composition-structure: Gamma ⊢ Compositon(A),
cubical-fst: p.1,
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
transprt-const: transprt-const(G;cA;a),
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
csm-comp-structure: (cA)tau,
cubical-type: {X ⊢ _},
cc-fst: p,
interval-type: 𝕀,
csm+: tau+,
csm-ap-type: (AF)s,
csm-comp: G o F,
cc-snd: q,
constant-cubical-type: (X),
csm-adjoin: (s;u),
compose: f o g,
csm-ap: (s)x,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
cubical-term-eqcd,
fst-transprt-sigma,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm+_wf,
csm-comp-structure_wf,
csm-sigma_comp,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
istype-cubical-term,
cubical-sigma_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-cubical-sigma,
csm-id-adjoin_wf-interval-0,
subtype_rel_self,
iff_weakening_equal,
csm-adjoin-p-q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
sqequalRule,
hyp_replacement,
instantiate,
because_Cache,
setElimination,
rename,
productElimination,
Error :memTop,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[cA:X +\mvdash{} Compositon(A)]. \mforall{}[cB:X.A +\mvdash{} Compositon(B)].
\mforall{}[pr:\{X \mvdash{} \_:\mSigma{} A B\}].
(transprt-const(X;sigma\_comp(cA;cB);pr).1 = transprt-const(X;cA;pr.1))
Date html generated:
2020_05_20-PM-05_00_55
Last ObjectModification:
2020_04_18-PM-00_29_09
Theory : cubical!type!theory
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