Nuprl Lemma : fill_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:composition-function{j:l,i:l}(Gamma.𝕀;A)].
∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}]. ∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}].
(fill cA [phi ⊢→ u] a0 ∈ {Gamma.𝕀 ⊢ _:A[(phi)p |⟶ u]})
Proof
Definitions occuring in Statement :
fill_term: fill cA [phi ⊢→ u] a0,
composition-function: composition-function{j:l,i:l}(Gamma;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-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],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fill_term: fill cA [phi ⊢→ u] a0,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
filling-function: filling-function{j:l, i:l}(Gamma;A),
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
interval-type: 𝕀,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x
Lemmas referenced :
comp-to-fill_wf,
cube-context-adjoin_wf,
interval-type_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
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-function_wf,
cubical-type_wf,
cubical_set_wf,
squash_wf,
true_wf,
csm-ap-id-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-id_wf,
equal_wf,
istype-universe,
subset-cubical-type,
context-subset-is-subset,
subtype_rel_self,
iff_weakening_equal,
cubical-type-cumulativity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
inhabitedIsType,
lambdaFormation_alt,
applyEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
sqequalRule,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
Error :memTop,
imageElimination,
independent_isectElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
setElimination,
rename
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}].
\mforall{}[cA:composition-function\{j:l,i:l\}(Gamma.\mBbbI{};A)]. \mforall{}[u:\{Gamma.\mBbbI{}, (phi)p \mvdash{} \_:A\}].
\mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} u[0]]\}].
(fill cA [phi \mvdash{}\mrightarrow{} u] a0 \mmember{} \{Gamma.\mBbbI{} \mvdash{} \_:A[(phi)p |{}\mrightarrow{} u]\})
Date html generated:
2020_05_20-PM-04_48_00
Last ObjectModification:
2020_04_13-PM-06_30_15
Theory : cubical!type!theory
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