Nuprl Lemma : lattice-fset-join_functionality_wrt_subset2
∀[l:BoundedLattice]. ∀[eq:EqDecider(Point(l))]. ∀[s1,s2:fset(Point(l))]. \/(s1) ≤ \/(s2) supposing s1 ⊆ {0} ⋃ s2
Proof
Definitions occuring in Statement :
lattice-fset-join: \/(s),
bdd-lattice: BoundedLattice,
lattice-0: 0,
lattice-le: a ≤ b,
lattice-point: Point(l),
fset-singleton: {x},
fset-union: x ⋃ y,
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
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,
implies: P ⇒ Q,
all: ∀x:A. B[x],
lattice-le: a ≤ b,
prop: ℙ,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice-le_transitivity,
bdd-lattice-subtype-lattice,
lattice-fset-join_wf,
decidable-equal-deq,
f-subset_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
fset-union_wf,
fset-singleton_wf,
lattice-0_wf,
fset_wf,
deq_wf,
bdd-lattice_wf,
lattice-le_weakening,
equal_wf,
squash_wf,
true_wf,
lattice-fset-join-union,
iff_weakening_equal,
lattice-join-0,
lattice-join_wf,
lattice-fset-join-singleton,
lattice-fset-join_functionality_wrt_subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
independent_functionElimination,
lambdaFormation,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
axiomEquality,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
setElimination,
rename,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[eq:EqDecider(Point(l))]. \mforall{}[s1,s2:fset(Point(l))].
\mbackslash{}/(s1) \mleq{} \mbackslash{}/(s2) supposing s1 \msubseteq{} \{0\} \mcup{} s2
Date html generated:
2020_05_20-AM-08_44_00
Last ObjectModification:
2017_07_28-AM-09_13_58
Theory : lattices
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