Nuprl Lemma : fdl-1-join-irreducible
∀[X:Type]
∀x,y:Point(free-dl(X)). (x ∨ y = 1 ∈ Point(free-dl(X)) ⇐⇒ (x = 1 ∈ Point(free-dl(X))) ∨ (y = 1 ∈ Point(free-dl(X))))
Proof
Definitions occuring in Statement :
free-dl: free-dl(X),
lattice-1: 1,
lattice-join: a ∨ b,
lattice-point: Point(l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
not: ¬A,
lattice-point: Point(l),
record-select: r.x,
free-dl: free-dl(X),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
eq_atom: x =a y,
free-dl-type: free-dl-type(X),
quotient: x,y:A//B[x; y],
lattice-join: a ∨ b,
so_lambda: λ2x y.t[x; y],
free-dl-join: free-dl-join(as;bs),
append: as @ bs,
list_ind: list_ind,
so_apply: x[s1;s2],
fdl-is-1: fdl-is-1(x)
Lemmas referenced :
equal_wf,
lattice-point_wf,
free-dl_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-1_wf,
bdd-distributive-lattice_wf,
or_wf,
fdl-is-1_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
fdl-eq-1,
free-dl-type_wf,
not_wf,
assert_wf,
equal-wf-base,
list_wf,
dlattice-eq_wf,
subtype_quotient,
dlattice-eq-equiv,
bl-exists_wf,
append_wf,
isaxiom_wf_list,
l_member_wf,
assert-bl-exists,
l_exists_append,
l_exists_wf,
assert_witness,
lattice_properties,
bdd-distributive-lattice-subtype-lattice
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
because_Cache,
independent_isectElimination,
setElimination,
rename,
universeEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
inlFormation,
inrFormation,
pointwiseFunctionalityForEquality,
functionEquality,
hyp_replacement,
pertypeElimination,
setEquality,
addLevel,
levelHypothesis,
impliesFunctionality,
impliesLevelFunctionality,
applyLambdaEquality
Latex:
\mforall{}[X:Type]. \mforall{}x,y:Point(free-dl(X)). (x \mvee{} y = 1 \mLeftarrow{}{}\mRightarrow{} (x = 1) \mvee{} (y = 1))
Date html generated:
2020_05_20-AM-08_42_58
Last ObjectModification:
2018_05_20-PM-10_11_34
Theory : lattices
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