Nuprl Lemma : p+-swap-interval
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[B:{Gamma ⊢ _}]. (((A)p+)swap-interval(Gamma;B) = (A)p ∈ {Gamma.𝕀.(B)p ⊢ _})
Proof
Definitions occuring in Statement :
swap-interval: swap-interval(G;A),
interval-type: 𝕀,
csm+: tau+,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
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,
cubical-type: {X ⊢ _},
pi1: fst(t),
pi2: snd(t),
csm-ap-type: (AF)s,
cube-context-adjoin: X.A,
all: ∀x:A. B[x],
cc-fst: p,
csm-ap: (s)x,
swap-interval: swap-interval(G;A),
csm+: tau+,
csm-swap: csm-swap(G;A;B),
csm-adjoin: (s;u),
interval-type: 𝕀,
constant-cubical-type: (X),
cc-snd: q,
csm-ap-term: (t)s,
csm-comp: G o F,
compose: f o g,
and: P ∧ Q,
cube-set-restriction: f(s),
uimplies: b supposing a
Lemmas referenced :
cubical-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical_set_wf,
I_cube_wf,
csm-ap-type_wf,
cc-fst_wf,
cubical-type-cumulativity2,
fset_wf,
nat_wf,
names-hom_wf,
cube-set-restriction_wf,
I_cube_pair_redex_lemma,
csm-ap_wf,
subtype_rel_self,
cubical-type-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
setElimination,
rename,
productElimination,
dependent_pairEquality_alt,
functionExtensionality,
because_Cache,
functionIsType,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
equalityTransitivity,
independent_isectElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[B:\{Gamma \mvdash{} \_\}]. (((A)p+)swap-interval(Gamma;B) = (A)p)
Date html generated:
2020_05_20-PM-02_40_01
Last ObjectModification:
2020_04_04-PM-06_54_09
Theory : cubical!type!theory
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