Nuprl Lemma : lattice-extend-1
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[f:T ⟶ Point(L)].
(lattice-extend(L;eq;eqL;f;1) = 1 ∈ Point(L))
Proof
Definitions occuring in Statement :
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
free-dist-lattice: free-dist-lattice(T; eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-1: 1,
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
lattice-1: 1,
free-dist-lattice: free-dist-lattice(T; eq),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
all: ∀x:A. B[x],
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
empty-fset: {},
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
lattice-fset-meet: /\(s),
fset-singleton: {x},
lattice-fset-join: \/(s),
squash: ↓T,
cand: A c∧ B,
bdd-lattice: BoundedLattice,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
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,
deq_wf,
bdd-distributive-lattice_wf,
rec_select_update_lemma,
fset-image-singleton,
list_accum_nil_lemma,
reduce_nil_lemma,
reduce_cons_lemma,
squash_wf,
true_wf,
lattice-join-1,
lattice-0_wf,
lattice-1_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
functionEquality,
cumulativity,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
setElimination,
rename,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:BoundedDistributiveLattice]. \mforall{}[eqL:EqDecider(Point(L))].
\mforall{}[f:T {}\mrightarrow{} Point(L)].
(lattice-extend(L;eq;eqL;f;1) = 1)
Date html generated:
2020_05_20-AM-08_45_35
Last ObjectModification:
2017_07_28-AM-09_14_30
Theory : lattices
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