Nuprl Lemma : cubical-fst_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. (p.1 ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
cubical-fst: p.1,
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
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,
cubical-fst: p.1,
cubical-term: {X ⊢ _:AF},
cubical-sigma: Σ A B,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
cubical-type-at: A(a),
prop: ℙ,
cubical-type: {X ⊢ _},
and: P ∧ Q,
cubical-type-ap-morph: (u a f),
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
cubical-term_wf,
cubical-sigma_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf,
list_wf,
coordinate_name_wf,
cubical-type-at_wf,
cc-adjoin-cube_wf,
equal_wf,
I-cube_wf,
name-morph_wf,
all_wf,
cube-set-restriction_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
hypothesisEquality,
isect_memberEquality,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
productEquality,
dependent_functionElimination,
lambdaFormation,
productElimination,
independent_functionElimination,
applyLambdaEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[p:\{X \mvdash{} \_:\mSigma{} A B\}]. (p.1 \mmember{} \{X \mvdash{} \_:A\})
Date html generated:
2017_10_05-AM-10_15_36
Last ObjectModification:
2017_07_28-AM-11_19_39
Theory : cubical!sets
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