Nuprl Lemma : extend-face-term-uniqueness
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[a,b:Point(face_lattice(I))].
a = b ∈ Point(face_lattice(I))
supposing a ≤ phi
∧ b ≤ phi
∧ (∀[g:{f:I ⟶ I| (phi f) = 1} ]. ((a)<g> = u(g) ∈ Point(face_lattice(I))))
∧ (∀[g:{f:I ⟶ I| (phi f) = 1} ]. ((b)<g> = u(g) ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
name-morph-satisfies: (psi f) = 1,
fl-morph: <f>,
face_lattice: face_lattice(I),
names-hom: I ⟶ J,
lattice-le: a ≤ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-subset: I,psi,
cube-cat: CubeCat,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-point: Point(l),
record-select: r.x,
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),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
names-hom: I ⟶ J,
cat-arrow: cat-arrow(C),
pi2: snd(t),
name-morph-satisfies: (psi f) = 1,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
so_apply: x[s],
ext-eq: A ≡ B,
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
implies: P ⇒ Q,
uiff: uiff(P;Q),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
I_cube_pair_redex_lemma,
cat_arrow_triple_lemma,
lattice-le_wf,
face_lattice_wf,
uall_wf,
names-hom_wf,
name-morph-satisfies_wf,
equal_wf,
lattice-point_wf,
fl-morph_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
cubical-term-at_wf,
cubical-subset_wf,
subtype_rel_self,
I_cube_wf,
face-presheaf_wf,
face-type_wf,
cubical-term_wf,
small_cubical_set_subtype,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
fset_wf,
nat_wf,
lattice-le-order,
bdd-distributive-lattice-subtype-lattice,
face_lattice-le,
lattice-1_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-1-le-iff,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-hom-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
productEquality,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
setEquality,
lambdaEquality,
setElimination,
rename,
instantiate,
cumulativity,
independent_isectElimination,
independent_pairFormation,
productElimination,
independent_functionElimination,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:Point(face\_lattice(I))]. \mforall{}[u:\{I,phi \mvdash{} \_:\mBbbF{}\}]. \mforall{}[a,b:Point(face\_lattice(I))].
a = b
supposing a \mleq{} phi
\mwedge{} b \mleq{} phi
\mwedge{} (\mforall{}[g:\{f:I {}\mrightarrow{} I| (phi f) = 1\} ]. ((a)<g> = u(g)))
\mwedge{} (\mforall{}[g:\{f:I {}\mrightarrow{} I| (phi f) = 1\} ]. ((b)<g> = u(g)))
Date html generated:
2018_05_23-AM-11_12_45
Last ObjectModification:
2018_05_20-PM-08_17_00
Theory : cubical!type!theory
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