Nuprl Lemma : face-lattice-hom-unique
∀T:Type. ∀eq:EqDecider(T). ∀L:BoundedDistributiveLattice. ∀eqL:EqDecider(Point(L)). ∀f0,f1:T ⟶ Point(L).
∀[g,h:Hom(face-lattice(T;eq);L)].
g = h ∈ Hom(face-lattice(T;eq);L)
supposing (∀x:T. (g (x=0) ∧ g (x=1) = 0 ∈ Point(L)))
∧ (∀x:T. ((g (x=0)) = (h (x=0)) ∈ Point(L)))
∧ (∀x:T. ((g (x=1)) = (h (x=1)) ∈ Point(L)))
Proof
Definitions occuring in Statement :
face-lattice1: (x=1),
face-lattice0: (x=0),
face-lattice: face-lattice(T;eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-0: 0,
lattice-meet: a ∧ b,
lattice-point: Point(l),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
guard: {T},
implies: P ⇒ Q,
face-lattice: face-lattice(T;eq),
fl-deq: fl-deq(T;eq),
bdd-lattice: BoundedLattice,
cand: A c∧ B,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
compose: f o g
Lemmas referenced :
uall_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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
lattice-1_wf,
all_wf,
face-lattice0_wf,
face-lattice1_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
deq_wf,
fl-point,
subtype_rel_weakening,
fset_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
fset-member_wf,
deq-fset_wf,
not_wf,
set_wf,
ext-eq_inversion,
free-dlwc-basis,
face-lattice-constraints_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
deq-implies,
fl-deq_wf,
lattice-hom-fset-join,
subtype_rel_transitivity,
bdd-lattice_wf,
fset-image_wf,
lattice-fset-meet_wf,
free-dlwc-inc_wf,
free-dist-lattice-with-constraints_wf,
lattice-fset-join_wf,
squash_wf,
decidable_wf,
decidable-equal-deq,
true_wf,
iff_weakening_equal,
fset-image-compose,
lattice-hom-fset-meet,
face-lattice0-is-inc,
face-lattice1-is-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
independent_isectElimination,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
axiomEquality,
functionEquality,
setEquality,
unionEquality,
inlEquality,
inrEquality,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
independent_pairFormation,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
unionElimination
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:BoundedDistributiveLattice. \mforall{}eqL:EqDecider(Point(L)).
\mforall{}f0,f1:T {}\mrightarrow{} Point(L).
\mforall{}[g,h:Hom(face-lattice(T;eq);L)].
g = h
supposing (\mforall{}x:T. (g (x=0) \mwedge{} g (x=1) = 0))
\mwedge{} (\mforall{}x:T. ((g (x=0)) = (h (x=0))))
\mwedge{} (\mforall{}x:T. ((g (x=1)) = (h (x=1))))
Date html generated:
2017_10_05-AM-00_41_14
Last ObjectModification:
2017_07_28-AM-09_16_25
Theory : lattices
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