Nuprl Lemma : face_lattice_hom_subtype
∀[I,J:fset(ℕ)]. Hom(face_lattice(J);face_lattice(I)) ⊆r Hom(face_lattice(I);face_lattice(I)) supposing I ⊆ J
Proof
Definitions occuring in Statement :
face_lattice: face_lattice(I),
bounded-lattice-hom: Hom(l1;l2),
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
cand: A c∧ B,
prop: ℙ,
lattice-0: 0,
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,
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
nat: ℕ,
lattice-meet: a ∧ b,
union-deq: union-deq(A;B;a;b),
top: Top,
lattice-join: a ∨ b
Lemmas referenced :
subtype_rel_dep_function,
lattice-point_wf,
face_lattice_wf,
face_lattice-point-subtype,
subtype_rel_self,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
lattice-1_wf,
bounded-lattice-hom_wf,
f-subset_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
rec_select_update_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
productElimination,
hypothesisEquality,
applyEquality,
extract_by_obid,
isectElimination,
hypothesis,
because_Cache,
sqequalRule,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
productEquality,
functionExtensionality,
intEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
voidElimination,
voidEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})].
Hom(face\_lattice(J);face\_lattice(I)) \msubseteq{}r Hom(face\_lattice(I);face\_lattice(I)) supposing I \msubseteq{} J
Date html generated:
2017_10_05-AM-01_15_27
Last ObjectModification:
2017_07_28-AM-09_32_03
Theory : cubical!type!theory
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