Nuprl Lemma : csm-comp-fun-to-comp-op
∀[Gamma,K:j⊢]. ∀[tau:K j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)].
((cfun-to-cop(Gamma;A;cA))tau = cfun-to-cop(K;(A)tau;(cA)tau) ∈ K ⊢ CompOp((A)tau))
Proof
Definitions occuring in Statement :
csm-comp-structure: (cA)tau,
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
composition-structure: Gamma ⊢ Compositon(A),
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
comp-fun-to-comp-op-inverse,
composition-structure_wf,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
composition-op_wf,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-composition_wf,
comp-fun-to-comp-op_wf,
csm-comp-structure_wf,
subtype_rel_self,
iff_weakening_equal,
csm-comp-op-to-comp-fun-sq,
comp-op-to-comp-fun-inverse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
hypothesis,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :memTop
Latex:
\mforall{}[Gamma,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} Compositon(A)].
((cfun-to-cop(Gamma;A;cA))tau = cfun-to-cop(K;(A)tau;(cA)tau))
Date html generated:
2020_05_20-PM-04_36_18
Last ObjectModification:
2020_04_17-PM-08_49_23
Theory : cubical!type!theory
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