Nuprl Lemma : face-zero-interval-1
∀[H:j⊢]. ((1(𝕀)=0) = 0(𝔽) ∈ {H ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-zero: (i=0),
face-0: 0(𝔽),
face-type: 𝔽,
interval-1: 1(𝕀),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
face-0: 0(𝔽),
interval-1: 1(𝕀),
face-zero: (i=0),
cubical-term-at: u(a),
dM-to-FL: dM-to-FL(I;z),
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,
dm-neg: ¬(x),
dM1: 1,
lattice-1: 1,
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,
fset-singleton: {x},
cons: [a / b],
nil: [],
it: ⋅,
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),
empty-fset: {},
lattice-0: 0,
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),
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
uimplies: b supposing a
Lemmas referenced :
lattice-0_wf,
face_lattice_wf,
subtype_rel_self,
cubical-type-at_wf_face-type,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
face-type_wf,
face-zero_wf,
interval-1_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionExtensionality,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
independent_isectElimination,
universeIsType,
instantiate
Latex:
\mforall{}[H:j\mvdash{}]. ((1(\mBbbI{})=0) = 0(\mBbbF{}))
Date html generated:
2020_05_20-PM-02_43_55
Last ObjectModification:
2020_04_04-PM-04_58_02
Theory : cubical!type!theory
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