Nuprl Lemma : interval-presheaf_wf
𝕀 ∈ SmallCubicalSet
Proof
Definitions occuring in Statement :
interval-presheaf: 𝕀,
small_cubical_set: SmallCubicalSet,
member: t ∈ T
Definitions unfolded in proof :
small_cubical_set: SmallCubicalSet,
interval-presheaf: 𝕀,
member: t ∈ T,
small_ps_context: small_ps_context{i:l}(C),
cat-functor: Functor(C1;C2),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
cat-arrow: cat-arrow(C),
pi2: snd(t),
names-hom: I ⟶ J,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
type-cat: TypeCat,
cat-id: cat-id(C),
cat-comp: cat-comp(C),
uimplies: b supposing a,
nh-id: 1,
dM_inc: <x>,
dminc: <i>,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
dma-neg: ¬(x),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
btrue: tt,
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
dma-hom: dma-hom(dma1;dma2),
lattice-0: 0,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
compose: f o g,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
dM-lift: dM-lift(I;J;f),
implies: P ⇒ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice-point_wf,
dM_wf,
subtype_rel_self,
fset_wf,
nat_wf,
cat-ob_wf,
op-cat_wf,
cube-cat_wf,
dM-lift_wf2,
names-hom_wf,
cat-arrow_wf,
type-cat_wf,
dM-lift-unique-fun,
nh-comp_wf,
cat-id_wf,
cat-comp_wf,
nh-id_wf,
dM_inc_wf,
names_wf,
dma-neg_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
lattice-0_wf,
lattice-1_wf,
compose-dma-hom,
dM-lift_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
lambdaEquality_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
universeIsType,
inhabitedIsType,
functionIsType,
instantiate,
lambdaFormation_alt,
independent_pairFormation,
independent_isectElimination,
productElimination,
productIsType,
equalityIstype,
isect_memberFormation_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
isectEquality,
independent_pairEquality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
isectIsType,
imageElimination,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mBbbI{} \mmember{} SmallCubicalSet
Date html generated:
2020_05_20-PM-01_39_59
Last ObjectModification:
2019_12_09-PM-07_29_58
Theory : cubical!type!theory
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