Nuprl Lemma : fiber-comp-exists
∀X:j⊢. ∀T,A:{X ⊢ _}. ∀w:{X ⊢ _:(T ⟶ A)}. ∀a:{X ⊢ _:A}. (X ⊢ CompOp(T) ⇒ X ⊢ CompOp(A) ⇒ X ⊢ CompOp(Fiber(w;a)))
Proof
Definitions occuring in Statement :
composition-op: Gamma ⊢ CompOp(A),
cubical-fiber: Fiber(w;a),
cubical-fun: (A ⟶ B),
cubical-term: {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-op-implies-composition-structure,
composition-structure-implies-composition-op,
cubical-fiber_wf,
fiber-comp_wf,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
istype-cubical-term,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
instantiate,
because_Cache,
isectElimination,
hypothesis,
independent_functionElimination,
universeIsType,
applyEquality,
sqequalRule,
inhabitedIsType
Latex:
\mforall{}X:j\mvdash{}. \mforall{}T,A:\{X \mvdash{} \_\}. \mforall{}w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}. \mforall{}a:\{X \mvdash{} \_:A\}.
(X \mvdash{} CompOp(T) {}\mRightarrow{} X \mvdash{} CompOp(A) {}\mRightarrow{} X \mvdash{} CompOp(Fiber(w;a)))
Date html generated:
2020_05_20-PM-05_13_21
Last ObjectModification:
2020_04_18-PM-00_02_16
Theory : cubical!type!theory
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