Nuprl Lemma : free-dlwc-1-join-irreducible
∀T:Type. ∀eq:EqDecider(T). ∀Cs:T ⟶ fset(fset(T)). ∀x,y:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).
(x ∨ y = 1 ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
⇐⇒ (x = 1 ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])))
∨ (y = 1 ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[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-join: a ∨ b,
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
top: Top,
fset-constrained-ac-lub: lub(P;ac1;ac2),
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
squash: ↓T,
true: True
Lemmas referenced :
equal_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,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-1_wf,
bdd-distributive-lattice_wf,
or_wf,
fset_wf,
deq_wf,
free-dlwc-1,
free-dlwc-join,
free-dlwc-point,
member-fset-minimals,
deq-fset_wf,
f-proper-subset-dec_wf,
fset-union_wf,
empty-fset_wf,
member-fset-union,
squash_wf,
true_wf,
lattice-join-1,
bdd-distributive-lattice-subtype-bdd-lattice,
iff_weakening_equal,
lattice-1-join
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
hypothesis,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
setElimination,
rename,
functionEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
unionElimination,
inlFormation,
inrFormation,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}Cs:T {}\mrightarrow{} fset(fset(T)).
\mforall{}x,y:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x])).
(x \mvee{} y = 1 \mLeftarrow{}{}\mRightarrow{} (x = 1) \mvee{} (y = 1))
Date html generated:
2020_05_20-AM-08_48_50
Last ObjectModification:
2017_07_28-AM-09_15_23
Theory : lattices
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