Nuprl Lemma : face-lattice-hom-is-id
∀I:fset(ℕ)
∀[h:Hom(face_lattice(I);face_lattice(I))]
h = (λx.x) ∈ Hom(face_lattice(I);face_lattice(I))
supposing ∀x:names(I). (((h (x=0)) = (x=0) ∈ Point(face_lattice(I))) ∧ ((h (x=1)) = (x=1) ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
names: names(I),
bounded-lattice-hom: Hom(l1;l2),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fl1: (x=1),
face_lattice: face_lattice(I),
fl0: (x=0),
and: P ∧ Q,
cand: A c∧ B,
lattice-0: 0,
record-select: r.x,
face-lattice: face-lattice(T;eq),
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: ⋅,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
bounded-lattice-hom: Hom(l1;l2),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
lattice-hom: Hom(l1;l2),
guard: {T},
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
face-lattice-hom-unique,
names_wf,
names-deq_wf,
face-lattice_wf,
face_lattice-deq_wf,
face-lattice0_wf,
face-lattice1_wf,
lattice-0_wf,
bdd-distributive-lattice_wf,
lattice-1_wf,
lattice-meet_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-point_wf,
equal_wf,
lattice-join_wf,
all_wf,
face_lattice_wf,
fl0_wf,
fl1_wf,
bounded-lattice-hom_wf,
fset_wf,
nat_wf,
fl-meet-0-1,
iff_weakening_equal,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
setElimination,
rename,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality,
productElimination,
instantiate,
productEquality,
cumulativity,
universeEquality,
independent_isectElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}I:fset(\mBbbN{})
\mforall{}[h:Hom(face\_lattice(I);face\_lattice(I))]
h = (\mlambda{}x.x) supposing \mforall{}x:names(I). (((h (x=0)) = (x=0)) \mwedge{} ((h (x=1)) = (x=1)))
Date html generated:
2017_10_05-AM-01_13_09
Last ObjectModification:
2017_07_28-AM-09_30_44
Theory : cubical!type!theory
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