Nuprl Lemma : comp_term_wf
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:composition-function{j:l,i:l}(Gamma.𝕀;A)].
∀[u:{Gamma, phi.𝕀 ⊢ _:A}]. ∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
(comp cA [phi ⊢→ u] a0 ∈ {Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ (u)[1(𝕀)]]})
Proof
Definitions occuring in Statement :
comp_term: comp cA [phi ⊢→ u] a0,
composition-function: composition-function{j:l,i:l}(Gamma;A),
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],
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,
comp_term: comp cA [phi ⊢→ u] a0,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T},
squash: ↓T,
prop: ℙ,
true: True,
interval-1: 1(𝕀),
csm-adjoin: (s;u),
csm-ap: (s)x,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-type: 𝕀,
composition-function: composition-function{j:l,i:l}(Gamma;A),
interval-0: 0(𝕀)
Lemmas referenced :
cube-context-adjoin_wf,
interval-type_wf,
context-subset-adjoin-subtype,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
csm-ap-term_wf,
context-subset_wf,
subset-cubical-type,
sub_cubical_set_functionality,
context-subset-is-subset,
cubical-term_wf,
composition-function_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
squash_wf,
true_wf,
thin-context-subset,
cube_set_map_wf,
csm-id-adjoin_wf-interval-1,
csm-ap-id-type,
csm-id_wf,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
sqequalRule,
because_Cache,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
setElimination,
rename,
productElimination,
cumulativity
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, phi.\mBbbI{} \mvdash{} \_:A\}].
\mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}].
(comp cA [phi \mvdash{}\mrightarrow{} u] a0 \mmember{} \{Gamma \mvdash{} \_:(A)[1(\mBbbI{})][phi |{}\mrightarrow{} (u)[1(\mBbbI{})]]\})
Date html generated:
2020_05_20-PM-04_37_07
Last ObjectModification:
2020_04_11-AM-11_24_48
Theory : cubical!type!theory
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