Nuprl Lemma : constrained-cubical-term-eqcd
∀[Gamma:j⊢]. ∀[A,B:{Gamma ⊢ _}]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[t:{Gamma, phi ⊢ _:A}]. ∀[t':{Gamma, psi ⊢ _:B}].
({Gamma ⊢ _:A[phi |⟶ t]} = {Gamma ⊢ _:B[psi |⟶ t']} ∈ 𝕌{[i | j']}) supposing
((t' = t ∈ {Gamma, phi ⊢ _:A}) and
(phi = psi ∈ {Gamma ⊢ _:𝔽}) and
(A = B ∈ {Gamma ⊢ _}))
Proof
Definitions occuring in Statement :
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-term_wf,
squash_wf,
true_wf,
equal_wf,
istype-universe,
cubical_set_wf,
context-subset_wf,
istype-cubical-term,
face-type_wf,
subtype_rel_self,
iff_weakening_equal,
subset-cubical-type,
context-subset-is-subset,
equal_functionality_wrt_subtype_rel2,
cubical-type_wf,
cubical-type-cumulativity2,
constrained-cubical-term_wf,
cubical_set_cumulativity-i-j,
thin-context-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
because_Cache,
universeEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType,
hyp_replacement,
equalityIstype
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A,B:\{Gamma \mvdash{} \_\}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[t:\{Gamma, phi \mvdash{} \_:A\}].
\mforall{}[t':\{Gamma, psi \mvdash{} \_:B\}].
(\{Gamma \mvdash{} \_:A[phi |{}\mrightarrow{} t]\} = \{Gamma \mvdash{} \_:B[psi |{}\mrightarrow{} t']\}) supposing
((t' = t) and
(phi = psi) and
(A = B))
Date html generated:
2020_05_20-PM-02_58_16
Last ObjectModification:
2020_04_23-PM-02_26_50
Theory : cubical!type!theory
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