Nuprl Lemma : face-zero-eq-1
∀[H:j⊢]. ∀[z:{H ⊢ _:𝕀}]. ∀[I:fset(ℕ)]. ∀[a:H(I)]. z(a) = 0 ∈ 𝕀(a) supposing (z=0)(a) = 1 ∈ Point(face_lattice(I))
Proof
Definitions occuring in Statement :
face-zero: (i=0),
interval-type: 𝕀,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
face_lattice: face_lattice(I),
I_cube: A(I),
cubical_set: CubicalSet,
dM0: 0,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
lattice-1: 1,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
face-zero: (i=0),
cubical-term-at: u(a),
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uiff: uiff(P;Q),
face-type: 𝔽,
face-presheaf: 𝔽,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
squash: ↓T,
dM0: 0,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
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,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
fset-union: x ⋃ y,
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
lattice-join: a ∨ b,
opposite-lattice: opposite-lattice(L),
so_lambda: λ2x y.t[x; y],
lattice-meet: a ∧ b,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
lattice-fset-meet: /\(s),
true: True
Lemmas referenced :
dM-to-FL-eq-1,
dm-neg_wf,
names_wf,
names-deq_wf,
cubical-term-at_wf,
interval-type_wf,
subtype_rel_self,
lattice-point_wf,
free-DeMorgan-lattice_wf,
subtype_rel_set,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
bounded-lattice-structure_wf,
bounded-lattice-structure-subtype,
face_lattice_wf,
lattice-structure_wf,
face-type_wf,
face-zero_wf,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical_set_wf,
subtype_rel-equal,
dM_wf,
DeMorgan-algebra-axioms_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
squash_wf,
true_wf,
istype-universe,
dm-neg-neg,
cubical-type-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
universeIsType,
independent_isectElimination,
productElimination,
equalityIstype,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hyp_replacement,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[z:\{H \mvdash{} \_:\mBbbI{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:H(I)]. z(a) = 0 supposing (z=0)(a) = 1
Date html generated:
2020_05_20-PM-02_43_34
Last ObjectModification:
2020_04_04-PM-04_57_51
Theory : cubical!type!theory
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