Nuprl Lemma : fl-morph-id
∀I:fset(ℕ). (<1> = (λx.x) ∈ (Point(face_lattice(I)) ⟶ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
fl-morph: <f>,
face_lattice: face_lattice(I),
nh-id: 1,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
face_lattice: face_lattice(I),
nh-id: 1,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
cand: A c∧ B,
lattice-0: 0,
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),
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
fl-morph: <f>,
implies: P ⇒ Q,
fl0: (x=0),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fl1: (x=1)
Lemmas referenced :
fl-lift-unique,
names_wf,
names-deq_wf,
face-lattice_wf,
face_lattice-deq_wf,
dM-to-FL_wf,
dm-neg_wf,
dM_inc_wf,
subtype_rel-equal,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
free-DeMorgan-lattice_wf,
face_lattice_wf,
dM-to-FL-neg2,
lattice-0_wf,
bdd-distributive-lattice_wf,
lattice-1_wf,
bounded-lattice-hom_wf,
equal_functionality_wrt_subtype_rel2,
fset_wf,
nat_wf,
fl0_wf,
squash_wf,
true_wf,
neg-dM_inc,
dM-to-FL-opp,
iff_weakening_equal,
fl1_wf,
dM-to-FL-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
because_Cache,
sqequalRule,
applyEquality,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
dependent_functionElimination,
setElimination,
rename,
independent_pairFormation,
dependent_set_memberEquality,
productElimination,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
independent_functionElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}I:fset(\mBbbN{}). (ə> = (\mlambda{}x.x))
Date html generated:
2017_10_05-AM-01_13_19
Last ObjectModification:
2017_07_28-AM-09_30_52
Theory : cubical!type!theory
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