Nuprl Lemma : system-type-same
∀[Gamma:j⊢]. ∀[sys:(phi:{Gamma ⊢ _:𝔽} × {Gamma, phi ⊢ _}) List].
∀i:ℕ||sys||. Gamma, fst(sys[i]) ⊢ system-type(sys) = snd(sys[i]) supposing compatible-system{i:l}(Gamma;sys)
Proof
Definitions occuring in Statement :
system-type: system-type(sys),
compatible-system: compatible-system{i:l}(Gamma;sys),
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
product: x:A × B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
same-cubical-type: Gamma ⊢ A = B,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
system-type-properties,
int_seg_wf,
length_wf,
cubical-term_wf,
face-type_wf,
cubical-type_wf,
context-subset_wf,
compatible-system_wf,
list_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
universeIsType,
isectElimination,
natural_numberEquality,
instantiate,
productEquality,
cumulativity
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[sys:(phi:\{Gamma \mvdash{} \_:\mBbbF{}\} \mtimes{} \{Gamma, phi \mvdash{} \_\}) List].
\mforall{}i:\mBbbN{}||sys||. Gamma, fst(sys[i]) \mvdash{} system-type(sys) = snd(sys[i])
supposing compatible-system\{i:l\}(Gamma;sys)
Date html generated:
2020_05_20-PM-03_09_58
Last ObjectModification:
2020_04_06-PM-11_56_52
Theory : cubical!type!theory
Home
Index