Nuprl Lemma : ps-sigma-elim-rule
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[T:{X.Σ A B ⊢ _}].
∀[t:{X.A.B ⊢ _:(T)SigmaUnElim}].
((t)SigmaElim ∈ {X.Σ A B ⊢ _:T})
Proof
Definitions occuring in Statement :
sigma-unelim-pscm: SigmaUnElim,
sigma-elim-pscm: SigmaElim,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
pscm-ap-term_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-sigma_wf,
presheaf-type-cumulativity2,
pscm-ap-type_wf,
sigma-unelim-pscm_wf,
sigma-elim-pscm_wf,
presheaf-term_wf2,
squash_wf,
true_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
equal_wf,
istype-universe,
pscm-ap-comp-type,
subtype_rel_self,
iff_weakening_equal,
pscm-ap-id-type,
psc_map_wf,
ps-sigma-elim-unelim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[T:\{X.\mSigma{} A B \mvdash{} \_\}].
\mforall{}[t:\{X.A.B \mvdash{} \_:(T)SigmaUnElim\}].
((t)SigmaElim \mmember{} \{X.\mSigma{} A B \mvdash{} \_:T\})
Date html generated:
2020_05_20-PM-01_33_01
Last ObjectModification:
2020_04_02-PM-06_47_01
Theory : presheaf!models!of!type!theory
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