Nuprl Lemma : assert-lattice-ble
∀[l:LatticeStructure]. ∀[eq:EqDecider(Point(l))]. ∀[a,b:Point(l)]. uiff(↑lattice-ble(l;eq;a;b);a ≤ b)
Proof
Definitions occuring in Statement :
lattice-ble: lattice-ble(l;eq;a;b),
lattice-le: a ≤ b,
lattice-point: Point(l),
lattice-structure: LatticeStructure,
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
lattice-le: a ≤ b,
lattice-ble: lattice-ble(l;eq;a;b),
member: t ∈ T,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
eqof: eqof(d),
assert: ↑b,
ifthenelse: if b then t else f fi ,
prop: ℙ,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A
Lemmas referenced :
lattice-meet_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
true_wf,
equal_wf,
lattice-point_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
false_wf,
assert_wf,
lattice-ble_wf,
lattice-le_wf,
deq_wf,
lattice-structure_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
independent_pairFormation,
isect_memberFormation,
natural_numberEquality,
axiomEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[l:LatticeStructure]. \mforall{}[eq:EqDecider(Point(l))]. \mforall{}[a,b:Point(l)].
uiff(\muparrow{}lattice-ble(l;eq;a;b);a \mleq{} b)
Date html generated:
2017_10_05-AM-00_33_23
Last ObjectModification:
2017_07_28-AM-09_13_45
Theory : lattices
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