Nuprl Lemma : extend-face-term-unique
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[a:Point(face_lattice(I))].
a = extend-face-term(I;phi;u) ∈ Point(face_lattice(I))
supposing a ≤ phi ∧ (∀[g:{f:I ⟶ I| (phi f) = 1} ]. ((a)<g> = u(g) ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
extend-face-term: extend-face-term(I;phi;u),
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],
member: t ∈ T,
subtype_rel: A ⊆r B,
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
cand: A c∧ B,
cubical-subset: I,psi,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
names-hom: I ⟶ J,
cat-arrow: cat-arrow(C),
pi2: snd(t),
cube-cat: CubeCat,
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X)
Lemmas referenced :
extend-face-term-uniqueness,
cubical-term_wf,
cubical-subset_wf,
subtype_rel_self,
I_cube_wf,
face-presheaf_wf,
small_cubical_set_subtype,
face-type_wf,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
fset_wf,
nat_wf,
extend-face-term_wf,
extend-face-term-le,
extend-face-term-property,
names-hom_wf,
name-morph-satisfies_wf,
lattice-le_wf,
fl-morph_wf,
cubical-term-at_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
inhabitedIsType,
universeIsType,
applyEquality,
sqequalRule,
instantiate,
because_Cache,
lambdaEquality_alt,
productEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_pairFormation,
setIsType,
productIsType,
isectIsType,
equalityIsType1,
setElimination,
rename
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:Point(face\_lattice(I))]. \mforall{}[u:\{I,phi \mvdash{} \_:\mBbbF{}\}]. \mforall{}[a:Point(face\_lattice(I))].
a = extend-face-term(I;phi;u) supposing a \mleq{} phi \mwedge{} (\mforall{}[g:\{f:I {}\mrightarrow{} I| (phi f) = 1\} ]. ((a)<g> = u(g)))
Date html generated:
2019_11_05-AM-10_33_14
Last ObjectModification:
2018_11_08-PM-06_03_48
Theory : cubical!type!theory
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