Nuprl Lemma : face-one-and-zero
∀[X:j⊢]. ∀[z:{X ⊢ _:𝕀}]. (((z=1) ∧ (z=0)) = 0(𝔽) ∈ {X ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-zero: (i=0),
face-one: (i=1),
face-and: (a ∧ b),
face-0: 0(𝔽),
face-type: 𝔽,
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,
face-0: 0(𝔽),
face-zero: (i=0),
face-one: (i=1),
face-and: (a ∧ b),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
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],
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),
face-type: 𝔽,
face-presheaf: 𝔽,
true: True,
squash: ↓T,
all: ∀x:A. B[x],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
face-type_wf,
face-and_wf,
face-one_wf,
face-zero_wf,
cubical-term_wf,
interval-type_wf,
cubical_set_wf,
cubical-type-at_wf_face-type,
subtype_rel_self,
lattice-point_wf,
free-DeMorgan-lattice_wf,
names_wf,
names-deq_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
face_lattice_wf,
squash_wf,
true_wf,
istype-universe,
dM-to-FL-neg,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
instantiate,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
Error :memTop,
applyEquality,
setElimination,
rename,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
natural_numberEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[z:\{X \mvdash{} \_:\mBbbI{}\}]. (((z=1) \mwedge{} (z=0)) = 0(\mBbbF{}))
Date html generated:
2020_05_20-PM-02_43_14
Last ObjectModification:
2020_04_04-PM-04_57_17
Theory : cubical!type!theory
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