Nuprl Lemma : pscm-ap-comp-term
∀[C:SmallCategory]. ∀[Gamma,Delta,Z:ps_context{j:l}(C)]. ∀[s1:psc_map{j:l}(C; Z; Delta)]. ∀[s2:psc_map{j:l}(C;
Delta;
Gamma)].
∀[A:{Gamma ⊢ _}]. ∀[t:{Gamma ⊢ _:A}].
((t)s2 o s1 = ((t)s2)s1 ∈ {Z ⊢ _:(A)s2 o s1})
Proof
Definitions occuring in Statement :
pscm-ap-term: (t)s,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
pscm-comp: G o F,
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
true: True
Lemmas referenced :
pscm-comp-term,
pscm-ap-term_wf,
pscm-ap-type_wf,
subtype_rel-equal,
presheaf-term_wf,
pscm-comp_wf,
small-category-cumulativity-2,
pscm-ap-comp-type,
presheaf-type-cumulativity2,
psc_map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
hypothesisEquality,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma,Delta,Z:ps\_context\{j:l\}(C)]. \mforall{}[s1:psc\_map\{j:l\}(C; Z; Delta)].
\mforall{}[s2:psc\_map\{j:l\}(C; Delta; Gamma)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[t:\{Gamma \mvdash{} \_:A\}].
((t)s2 o s1 = ((t)s2)s1)
Date html generated:
2020_05_20-PM-01_27_01
Last ObjectModification:
2020_04_01-PM-10_37_35
Theory : presheaf!models!of!type!theory
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