Nuprl Lemma : face-zero-as-face-one
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ((i=0) = (1-(i)=1) ∈ {Gamma ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-zero: (i=0),
face-one: (i=1),
face-type: 𝔽,
interval-rev: 1-(r),
interval-type: 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],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
face-zero: (i=0),
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),
cubical-term-at: u(a),
face-one: (i=1),
interval-rev: 1-(r),
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,
subtype_rel: A ⊆r B,
cubical-term: {X ⊢ _:A}
Lemmas referenced :
cubical-term-equal2,
face-type_wf,
face-zero_wf,
face-one_wf,
interval-rev_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
interval-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
independent_isectElimination,
lambdaFormation_alt,
sqequalRule,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
because_Cache,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[i:\{Gamma \mvdash{} \_:\mBbbI{}\}]. ((i=0) = (1-(i)=1))
Date html generated:
2020_05_20-PM-02_43_03
Last ObjectModification:
2020_04_04-PM-04_51_13
Theory : cubical!type!theory
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