Nuprl Lemma : filling_term_1
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[T:{H.𝕀 ⊢ _}]. ∀[u:{H.𝕀, (phi)p ⊢ _:T}]. ∀[a0:{H ⊢ _:(T)[0(𝕀)][phi |⟶ u[0]]}].
∀[cT:H.𝕀 ⊢ CompOp(T)].
((fill cT [phi ⊢→ u] a0)[1(𝕀)] = comp cT [phi ⊢→ u] a0 ∈ {H ⊢ _:(T)[1(𝕀)]})
Proof
Definitions occuring in Statement :
filling_term: fill cA [phi ⊢→ u] a0,
composition-term: comp cA [phi ⊢→ u] a0,
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-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
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 ⊢ _},
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}
Lemmas referenced :
fill_term_1,
comp-op-to-comp-fun_wf2,
cube-context-adjoin_wf,
interval-type_wf,
cubical-type-cumulativity2,
comp_term-composition-term,
composition-op_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,
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,
Error :memTop,
universeIsType,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}[T:\{H.\mBbbI{} \mvdash{} \_\}]. \mforall{}[u:\{H.\mBbbI{}, (phi)p \mvdash{} \_:T\}].
\mforall{}[a0:\{H \mvdash{} \_:(T)[0(\mBbbI{})][phi |{}\mrightarrow{} u[0]]\}]. \mforall{}[cT:H.\mBbbI{} \mvdash{} CompOp(T)].
((fill cT [phi \mvdash{}\mrightarrow{} u] a0)[1(\mBbbI{})] = comp cT [phi \mvdash{}\mrightarrow{} u] a0)
Date html generated:
2020_05_20-PM-04_54_15
Last ObjectModification:
2020_04_10-AM-11_33_31
Theory : cubical!type!theory
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