Nuprl Lemma : dM-homs-equal
∀[I:fset(ℕ)]. ∀[l:BoundedLattice].
∀eq:EqDecider(Point(l))
∀[h1,h2:Hom(dM(I);l)].
h1 = h2 ∈ (Point(dM(I)) ⟶ Point(l))
supposing ∀i:names(I). (((h1 <i>) = (h2 <i>) ∈ Point(l)) ∧ ((h1 <1-i>) = (h2 <1-i>) ∈ Point(l)))
Proof
Definitions occuring in Statement :
dM_opp: <1-x>,
dM_inc: <x>,
dM: dM(I),
names: names(I),
bounded-lattice-hom: Hom(l1;l2),
bdd-lattice: BoundedLattice,
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
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,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
prop: ℙ,
and: P ∧ Q,
bdd-lattice: BoundedLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
implies: P ⇒ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
free-dl-inc: free-dl-inc(x),
fset-singleton: {x},
cons: [a / b],
nil: [],
it: ⋅,
dM_inc: <x>,
dminc: <i>,
dM_opp: <1-x>,
dmopp: <1-i>
Lemmas referenced :
dM-hom-basis,
lattice-point_wf,
dM_wf,
subtype_rel_set,
lattice-structure_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
all_wf,
names_wf,
equal_wf,
bounded-lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
dM_inc_wf,
dM_opp_wf,
bounded-lattice-hom_wf,
DeMorgan-algebra-structure_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
deq_wf,
bdd-lattice_wf,
fset_wf,
nat_wf,
deq-implies,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-fset-join_wf,
decidable_wf,
dM-point,
fset-image_wf,
deq-fset_wf,
union-deq_wf,
names-deq_wf,
lattice-fset-meet_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality,
productEquality,
cumulativity,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
voidElimination,
voidEquality,
unionEquality,
unionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[l:BoundedLattice].
\mforall{}eq:EqDecider(Point(l))
\mforall{}[h1,h2:Hom(dM(I);l)].
h1 = h2 supposing \mforall{}i:names(I). (((h1 <i>) = (h2 <i>)) \mwedge{} ((h1 ə-i>) = (h2 ə-i>)))
Date html generated:
2017_10_05-AM-01_00_22
Last ObjectModification:
2017_07_28-AM-09_25_44
Theory : cubical!type!theory
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