Nuprl Lemma : cubical-pi-comp-structure
∀X:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. (X ⊢ Compositon(A) ⇒ X.A +⊢ Compositon(B) ⇒ X ⊢ Compositon(ΠA B))
Proof
Definitions occuring in Statement :
composition-structure: Gamma ⊢ Compositon(A),
cubical-pi: ΠA 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 :
composition-structure-implies-composition-op,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
composition-op-implies-composition-structure,
cubical-pi_wf,
pi-comp_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
instantiate,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
universeIsType
Latex:
\mforall{}X:j\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}.
(X \mvdash{} Compositon(A) {}\mRightarrow{} X.A +\mvdash{} Compositon(B) {}\mRightarrow{} X \mvdash{} Compositon(\mPi{}A B))
Date html generated:
2020_05_20-PM-05_12_37
Last ObjectModification:
2020_04_20-AM-10_09_03
Theory : cubical!type!theory
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