Nuprl Lemma : pscm-id-adjoin-ap-type
∀C:SmallCategory. ∀Gamma,Delta:ps_context{j:l}(C). ∀A:{Gamma ⊢ _}. ∀B:{Gamma.A ⊢ _}.
∀sigma:psc_map{j:l}(C; Delta; Gamma). ∀u:{Delta ⊢ _:(A)sigma}.
(((B)(sigma o p;q))[u] = (B)(sigma;u) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
pscm-id-adjoin: [u],
pscm-adjoin: (s;u),
psc-snd: q,
psc-fst: p,
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
pscm-comp: G o F,
psc_map: A ⟶ B,
ps_context: __⊢,
all: ∀x:A. B[x],
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
presheaf-type: {X ⊢ _},
pscm-adjoin: (s;u),
pscm-ap-type: (AF)s,
psc-snd: q,
psc-fst: p,
pscm-comp: G o F,
pscm-id-adjoin: [u],
pscm-ap: (s)x,
pscm-id: 1(X),
compose: f o g,
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
cat-comp: cat-comp(C)
Lemmas referenced :
pscm-ap-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
pscm-adjoin_wf,
subtype_rel_self,
psc_map_wf,
presheaf-term_wf,
presheaf-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
hypothesis,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
universeIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}Gamma,Delta:ps\_context\{j:l\}(C). \mforall{}A:\{Gamma \mvdash{} \_\}. \mforall{}B:\{Gamma.A \mvdash{} \_\}.
\mforall{}sigma:psc\_map\{j:l\}(C; Delta; Gamma). \mforall{}u:\{Delta \mvdash{} \_:(A)sigma\}.
(((B)(sigma o p;q))[u] = (B)(sigma;u))
Date html generated:
2020_05_20-PM-01_28_25
Last ObjectModification:
2020_04_02-PM-01_56_09
Theory : presheaf!models!of!type!theory
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