Nuprl Lemma : fl_all_decomp
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[phi:Point(face_lattice(I+i))].
(phi = (∀i.phi) ∨ phi ∧ (i=0) ∨ phi ∧ (i=1) ∈ Point(face_lattice(I+i)))
Proof
Definitions occuring in Statement :
fl_all: (∀i.phi),
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
add-name: I+i,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[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],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
names: names(I),
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
lattice-join: a ∨ b,
record-select: r.x,
face_lattice: face_lattice(I),
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,
fset-constrained-ac-lub: lub(P;ac1;ac2),
fset-ac-lub: fset-ac-lub(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,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
lattice-meet: a ∧ b,
fset-constrained-ac-glb: glb(P;ac1;ac2),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
fl_all: (∀i.phi),
fl-all-hom: fl-all-hom(I;i),
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
fset-image: f"(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
lattice-1: 1,
top: Top,
union-deq: union-deq(A;B;a;b),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
sq_type: SQType(T),
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
face_lattice-induction,
add-name_wf,
equal_wf,
lattice-point_wf,
face_lattice_wf,
lattice-join_wf,
fl_all_wf,
lattice-meet_wf,
fl0_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
fl1_wf,
sq_stable__equal,
lattice-0_wf,
names_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
istype-nat,
fset_wf,
iff_weakening_equal,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-join-1,
bdd-distributive-lattice_wf,
lattice-1_wf,
true_wf,
squash_wf,
rec_select_update_lemma,
fl_all-1,
istype-universe,
fl_all-join,
face_lattice-point-subtype,
f-subset-add-name,
subtype_rel_self,
istype-void,
bdd-distributive-lattice-subtype-distributive-lattice,
distributive-lattice-distrib,
lattice_properties,
bdd-distributive-lattice-subtype-lattice,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
eqff_to_assert,
set_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
fl_all-meet,
fl_all-fl0,
lattice-meet-0,
lattice-join-0,
lattice-meet-idempotent,
FL-meet-0-1,
not-added-name,
fl_all-fl1,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
intEquality,
independent_isectElimination,
closedConclusion,
natural_numberEquality,
independent_functionElimination,
lambdaFormation_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
equalityIsType1,
independent_pairFormation,
instantiate,
productEquality,
cumulativity,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
productElimination,
baseClosed,
imageMemberEquality,
dependent_set_memberEquality,
universeEquality,
imageElimination,
lambdaEquality,
voidEquality,
voidElimination,
isect_memberEquality,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIsType4,
baseApply,
promote_hyp,
hyp_replacement
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. \mforall{}[phi:Point(face\_lattice(I+i))]. (phi = (\mforall{}i.phi) \mvee{} phi \mwedge{} (i=0) \mvee{} phi \mwedge{} (i=1))
Date html generated:
2019_11_04-PM-05_34_36
Last ObjectModification:
2018_11_08-AM-11_12_51
Theory : cubical!type!theory
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