Nuprl Lemma : lattice-extend-is-hom-constrained
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))].
∀[f:T ⟶ Point(L)].
λac.lattice-extend-wc(L;eq;eqL;f;ac) ∈ Hom(free-dist-lattice-with-constraints(T;eq;x.Cs[x]);L)
supposing ∀x:T. ∀c:fset(T). (c ∈ Cs[x] ⇒ (/\(f"(c)) = 0 ∈ Point(L)))
Proof
Definitions occuring in Statement :
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
lattice-fset-meet: /\(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-0: 0,
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-0: 0,
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,
bfalse: ff,
btrue: tt,
empty-fset: {},
nil: [],
it: ⋅,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
order-preserving-map-is-bounded-lattice-hom,
free-dist-lattice-with-constraints_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-extend-wc_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-extend-wc-order-preserving,
lattice-le_wf,
lattice-extend-wc-meet,
lattice-extend-wc-join,
lattice-0_wf,
squash_wf,
true_wf,
lattice-extend-wc-1,
lattice-1_wf,
iff_weakening_equal,
all_wf,
fset_wf,
fset-member_wf,
deq-fset_wf,
lattice-fset-meet_wf,
decidable-equal-deq,
fset-image_wf,
deq_wf,
bdd-distributive-lattice_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
productElimination,
setElimination,
rename,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
axiomEquality,
functionEquality,
dependent_functionElimination,
isect_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))]. \mforall{}[L:BoundedDistributiveLattice].
\mforall{}[eqL:EqDecider(Point(L))]. \mforall{}[f:T {}\mrightarrow{} Point(L)].
\mlambda{}ac.lattice-extend-wc(L;eq;eqL;f;ac) \mmember{} Hom(free-dist-lattice-with-constraints(T;eq;x.Cs[x]);L)
supposing \mforall{}x:T. \mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} (/\mbackslash{}(f"(c)) = 0))
Date html generated:
2020_05_20-AM-08_50_25
Last ObjectModification:
2017_07_28-AM-09_15_44
Theory : lattices
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