Nuprl Lemma : free-dlwc-1
∀[T:Type]
∀eq:EqDecider(T). ∀Cs:T ⟶ fset(fset(T)). ∀x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).
(x = 1 ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])) ⇐⇒ {} ∈ x)
Proof
Definitions occuring in Statement :
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
lattice-1: 1,
lattice-point: Point(l),
deq-fset: deq-fset(eq),
empty-fset: {},
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
lattice-1: 1,
record-select: r.x,
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,
btrue: tt,
fset-singleton: {x},
cons: [a / b],
empty-fset: {},
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
f-proper-subset: xs ⊆≠ ys,
f-subset: xs ⊆ ys,
squash: ↓T,
true: True
Lemmas referenced :
free-dlwc-point,
member-fset-singleton,
fset_wf,
deq-fset_wf,
empty-fset_wf,
fset-member_wf,
lattice-point_wf,
free-dist-lattice-with-constraints_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,
deq_wf,
fset-member_witness,
istype-universe,
assert-fset-antichain,
istype-assert,
fset-antichain_wf,
fset-all_wf,
fset-contains-none_wf,
fset-extensionality,
fset-singleton_wf,
fset-null_wf,
eqtt_to_assert,
assert-fset-null,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
mem_empty_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
Error :memTop,
hypothesis,
independent_pairFormation,
hypothesisEquality,
because_Cache,
productElimination,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
equalityIstype,
inhabitedIsType,
baseClosed,
sqequalBase,
universeIsType,
lambdaEquality_alt,
applyEquality,
instantiate,
productEquality,
cumulativity,
isectEquality,
functionIsType,
dependent_functionElimination,
independent_pairEquality,
independent_functionElimination,
functionIsTypeImplies,
axiomEquality,
universeEquality,
dependent_set_memberEquality_alt,
productIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
equalityTransitivity,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}Cs:T {}\mrightarrow{} fset(fset(T)).
\mforall{}x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).
(x = 1 \mLeftarrow{}{}\mRightarrow{} \{\} \mmember{} x)
Date html generated:
2020_05_20-AM-08_48_46
Last ObjectModification:
2020_02_04-PM-01_56_06
Theory : lattices
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