Nuprl Lemma : face-lattice-le-1
∀T:Type. ∀eq:EqDecider(T). ∀x,y:Point(face-lattice(T;eq)). (x ≤ y ⇐⇒ fset-ac-le(union-deq(T;T;eq;eq);x;y))
Proof
Definitions occuring in Statement :
face-lattice: face-lattice(T;eq),
lattice-le: a ≤ b,
lattice-point: Point(l),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
face-lattice: face-lattice(T;eq),
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a
Lemmas referenced :
deq_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
face-lattice_wf,
lattice-point_wf,
face-lattice-constraints_wf,
union-deq_wf,
free-dlwc-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
cumulativity,
applyEquality,
instantiate,
productEquality,
universeEquality,
independent_isectElimination
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}x,y:Point(face-lattice(T;eq)).
(x \mleq{} y \mLeftarrow{}{}\mRightarrow{} fset-ac-le(union-deq(T;T;eq;eq);x;y))
Date html generated:
2016_05_18-AM-11_40_20
Last ObjectModification:
2016_01_19-PM-07_12_28
Theory : lattices
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