Nuprl Lemma : csm-filling_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((fill cA [phi ⊢→ u] a0)s+ = fill (cA)s+ [(phi)s ⊢→ (u)s+] (a0)s ∈ {Delta.𝕀 ⊢ _:(A)s+[((phi)s)p |⟶ (u)s+]})
Proof
Definitions occuring in Statement :
filling_term: fill cA [phi ⊢→ u] a0,
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _: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,
filling_term: fill cA [phi ⊢→ u] a0,
guard: {T},
csm-composition: (comp)sigma,
comp-op-to-comp-fun: cop-to-cfun(cA),
csm-comp-structure: (cA)tau,
csm-ap: (s)x,
interval-type: 𝕀,
csm+: tau+,
csm-comp: G o F,
compose: f o g,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-adjoin: (s;u)
Lemmas referenced :
csm-fill_term,
comp-op-to-comp-fun_wf2,
cube-context-adjoin_wf,
interval-type_wf,
cubical-type-cumulativity2,
cube_set_map_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
partial-term-0_wf,
cubical-term_wf,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
thin-context-subset,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
universeIsType,
inhabitedIsType,
Error :memTop,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)].
\mforall{}[u:\{Gamma.\mBbbI{}, (phi)p \mvdash{} \_:A\}]. \mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} u[0]]\}]. \mforall{}[Delta:j\mvdash{}].
\mforall{}[s:Delta j{}\mrightarrow{} Gamma].
((fill cA [phi \mvdash{}\mrightarrow{} u] a0)s+ = fill (cA)s+ [(phi)s \mvdash{}\mrightarrow{} (u)s+] (a0)s)
Date html generated:
2020_05_20-PM-04_53_48
Last ObjectModification:
2020_04_10-AM-11_32_27
Theory : cubical!type!theory
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