Nuprl Lemma : free-DeMorgan-algebra-hom-unique
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[dm:DeMorganAlgebra]. ∀[eq2:EqDecider(Point(dm))].
∀f:T ⟶ Point(dm)
∀[g,h:dma-hom(free-DeMorgan-algebra(T;eq);dm)].
g = h ∈ dma-hom(free-DeMorgan-algebra(T;eq);dm) supposing ∀i:T. ((g <i>) = (h <i>) ∈ Point(dm))
Proof
Definitions occuring in Statement :
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dma-hom: dma-hom(dma1;dma2),
DeMorgan-algebra: DeMorganAlgebra,
dminc: <i>,
lattice-point: Point(l),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
so_apply: x[s],
guard: {T},
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
dminc: <i>,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
dmopp: <1-i>,
top: Top,
lattice-meet: a ∧ b,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-join: a ∨ b,
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
fset-union: x ⋃ y,
l-union: as ⋃ bs,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
free-dist-lattice-hom-unique2,
union-deq_wf,
DeMorgan-algebra-subtype,
all_wf,
equal_wf,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
dminc_wf,
dma-hom_wf,
free-DeMorgan-algebra_wf,
deq_wf,
DeMorgan-algebra_wf,
free-dma-hom-is-lattice-hom,
squash_wf,
true_wf,
trivial-equal,
subtype_rel_self,
iff_weakening_equal,
free-dma-neg,
dm-neg-inc,
free-dma-point-subtype,
dma-neg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
cumulativity,
lambdaEquality,
instantiate,
productEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
functionEquality,
universeEquality,
hyp_replacement,
unionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
voidElimination,
voidEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[dm:DeMorganAlgebra]. \mforall{}[eq2:EqDecider(Point(dm))].
\mforall{}f:T {}\mrightarrow{} Point(dm)
\mforall{}[g,h:dma-hom(free-DeMorgan-algebra(T;eq);dm)]. g = h supposing \mforall{}i:T. ((g <i>) = (h <i>))
Date html generated:
2018_05_22-PM-09_55_43
Last ObjectModification:
2018_05_20-PM-10_14_03
Theory : lattices
Home
Index