Nuprl Lemma : sigma-comp-exists
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}. ∀B:{Gamma.A ⊢ _}. (Gamma ⊢ CompOp(A) ⇒ Gamma.A ⊢ CompOp(B) ⇒ Gamma ⊢ CompOp(Σ A B))
Proof
Definitions occuring in Statement :
composition-op: Gamma ⊢ CompOp(A),
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
sigmacomp_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
rename,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
universeIsType
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}A:\{Gamma \mvdash{} \_\}. \mforall{}B:\{Gamma.A \mvdash{} \_\}.
(Gamma \mvdash{} CompOp(A) {}\mRightarrow{} Gamma.A \mvdash{} CompOp(B) {}\mRightarrow{} Gamma \mvdash{} CompOp(\mSigma{} A B))
Date html generated:
2020_05_20-PM-04_06_32
Last ObjectModification:
2020_04_10-AM-03_42_27
Theory : cubical!type!theory
Home
Index