Nuprl Lemma : swap-interval_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (swap-interval(Gamma;A) ∈ Gamma.𝕀.(A)p j⟶ Gamma.A.𝕀)
Proof
Definitions occuring in Statement :
swap-interval: swap-interval(G;A),
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
swap-interval: swap-interval(G;A),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
functor-ob: ob(F),
cube-context-adjoin: X.A,
I_cube: A(I),
cubical-type-at: A(a),
interval-type: 𝕀,
constant-cubical-type: (X),
interval-presheaf: 𝕀,
lattice-point: Point(l),
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,
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),
btrue: tt,
csm-ap-type: (AF)s,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
cubical-type-ap-morph: (u a f),
dM-lift: dM-lift(I;J;f),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
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,
subtype_rel: A ⊆r B
Lemmas referenced :
csm-swap_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. (swap-interval(Gamma;A) \mmember{} Gamma.\mBbbI{}.(A)p j{}\mrightarrow{} Gamma.A.\mBbbI{})
Date html generated:
2020_05_20-PM-02_39_50
Last ObjectModification:
2020_04_04-PM-01_36_24
Theory : cubical!type!theory
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