Nuprl Lemma : pscm-swap_wf
∀[C:SmallCategory]. ∀[G:ps_context{j:l}(C)]. ∀[A,B:{G ⊢ _}].
(pscm-swap(G;A;B) ∈ psc_map{[i | j]:l}(C; G.A.(B)p; G.B.(A)p))
Proof
Definitions occuring in Statement :
pscm-swap: pscm-swap(G;A;B),
psc-fst: p,
psc-adjoin: X.A,
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
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,
pscm-swap: pscm-swap(G;A;B),
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
psc-fst: p,
pscm-ap-type: (AF)s,
pscm+: tau+,
pscm-ap: (s)x,
psc-snd: q,
pscm-comp: G o F,
pscm-adjoin: (s;u),
pi1: fst(t),
compose: f o g
Lemmas referenced :
pscm-adjoin_wf,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-ap-type_wf,
psc-fst_wf,
pscm+_wf,
pscm-ap-term_wf,
psc-snd_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
applyEquality,
hypothesis,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[G:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{G \mvdash{} \_\}].
(pscm-swap(G;A;B) \mmember{} psc\_map\{[i | j]:l\}(C; G.A.(B)p; G.B.(A)p))
Date html generated:
2020_05_20-PM-01_28_37
Last ObjectModification:
2020_04_03-AM-11_09_12
Theory : presheaf!models!of!type!theory
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