Nuprl Lemma : dM-hom-invariant
∀[I:fset(ℕ)]. ∀[J:{J:fset(ℕ)| I ⊆ J} ]. ∀[h:Hom(dM(J);dM(I))].
∀[x:Point(dM(I))]. ((h x) = x ∈ Point(dM(I)))
supposing ∀i:names(I). (((h <i>) = <i> ∈ Point(dM(I))) ∧ ((h <1-i>) = <1-i> ∈ Point(dM(I))))
Proof
Definitions occuring in Statement :
dM_opp: <1-x>,
dM_inc: <x>,
dM: dM(I),
names: names(I),
bounded-lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
deq: EqDecider(T),
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
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,
bool: 𝔹,
iff: P ⇐⇒ Q,
and: P ∧ Q,
assert: ↑b,
rev_implies: P ⇐ Q,
prop: ℙ,
true: True,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
nat: ℕ,
bdd-lattice: BoundedLattice,
bdd-distributive-lattice: BoundedDistributiveLattice,
top: Top,
union-deq: union-deq(A;B;a;b),
free-dml-deq: free-dml-deq(T;eq),
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,
dM-point-subtype,
sq_stable_from_decidable,
f-subset_wf,
nat_wf,
int-deq_wf,
decidable__f-subset,
dM_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
subtype_rel_transitivity,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
free-dml-deq_wf,
names_wf,
names-deq_wf,
equal_wf,
squash_wf,
true_wf,
dM-basis,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
all_wf,
dM_inc_wf,
names-subtype,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
dM_opp_wf,
bounded-lattice-hom_wf,
set_wf,
fset_wf,
lattice-fset-join_wf,
decidable_wf,
mk-DeMorgan-algebra-equal-bounded-lattice,
free-DeMorgan-lattice_wf,
dM-point,
fset-image_wf,
deq_wf,
deq-fset_wf,
union-deq_wf,
free-dl-point,
lattice-fset-meet_wf,
deq-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
applyEquality,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
hyp_replacement,
equalitySymmetry,
lambdaEquality,
equalityTransitivity,
universeEquality,
natural_numberEquality,
productEquality,
cumulativity,
isect_memberEquality,
axiomEquality,
intEquality,
voidElimination,
voidEquality,
lambdaFormation,
dependent_set_memberEquality,
productElimination,
functionEquality,
unionEquality,
unionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[J:\{J:fset(\mBbbN{})| I \msubseteq{} J\} ]. \mforall{}[h:Hom(dM(J);dM(I))].
\mforall{}[x:Point(dM(I))]. ((h x) = x) supposing \mforall{}i:names(I). (((h <i>) = <i>) \mwedge{} ((h ə-i>) = ə-i>))
Date html generated:
2018_05_23-AM-08_28_33
Last ObjectModification:
2018_05_20-PM-05_37_04
Theory : cubical!type!theory
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