Nuprl Lemma : free-dlwc-inc_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[x:T].
(free-dlwc-inc(eq;a.Cs[a];x) ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
Proof
Definitions occuring in Statement :
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
bfalse: ff,
assert: ↑b,
fset-antichain: fset-antichain(eq;ac),
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
true: True,
fset-all: fset-all(s;x.P[x]),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
false: False,
exists: ∃x:A. B[x]
Lemmas referenced :
free-dlwc-point,
fset-null_wf,
fset_wf,
fset-filter_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
assert_wf,
fset-singleton_wf,
eqtt_to_assert,
fset-antichain-singleton,
fset-antichain_wf,
fset-all_wf,
fset-contains-none_wf,
uiff_transitivity,
equal-wf-T-base,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
empty-fset_wf,
equal_wf,
deq_wf,
fset-all-iff,
deq-fset_wf,
member-fset-singleton,
assert-fset-contains-none,
fset-member_wf,
assert_witness,
assert-fset-null,
fset-filter-is-empty,
assert-deq-f-subset
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isect_memberFormation,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
functionEquality,
functionExtensionality,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
natural_numberEquality,
dependent_functionElimination,
axiomEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))]. \mforall{}[x:T].
(free-dlwc-inc(eq;a.Cs[a];x) \mmember{} Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
Date html generated:
2020_05_20-AM-08_48_41
Last ObjectModification:
2017_07_28-AM-09_15_17
Theory : lattices
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