Nuprl Lemma : sub-free-dim-1
∀[K:CRng]. ∀[S,T:Type].
(∀s:T. ∀x:Point(sub-free-vs(K;S;T)). (↓∃k:|K|. (x = {<k, s>} ∈ Point(sub-free-vs(K;S;T))))) supposing
((∀x,y:T. (x = y ∈ T)) and
strong-subtype(T;S))
Proof
Definitions occuring in Statement :
sub-free-vs: sub-free-vs(K;S;T),
vs-point: Point(vs),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
pair: <a, b>,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|,
single-bag: {x}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
sub-free-vs: sub-free-vs(K;S;T),
vs-point: Point(vs),
sub-vs: (v:vs | P[v]),
mk-vs: mk-vs,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
fs-in-subtype: fs-in-subtype(K;S;T;f),
fs-predicate: fs-predicate(K;S;p.P[p];f),
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
crng: CRng,
rng: Rng,
prop: ℙ,
so_lambda: λ2x.t[x],
pi1: fst(t),
so_apply: x[s],
basic-formal-sum: basic-formal-sum(K;S),
cand: A c∧ B,
free-vs: free-vs(K;S),
formal-sum: formal-sum(K;S),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
respects-equality: respects-equality(S;T),
implies: P ⇒ Q,
strong-subtype: strong-subtype(A;B),
bfs-predicate: bfs-predicate(K;S;p.P[p];b),
pi2: snd(t),
subtype_rel: A ⊆r B,
guard: {T},
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
decidable: Dec(P),
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
rng_zero: 0,
empty-bag: {},
bfs-reduce: bfs-reduce(K;S;as;bs),
infix_ap: x f y,
bag: bag(T),
quotient: x,y:A//B[x; y],
zero-bfs: 0 * ss,
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
single-bag: {x},
formal-sum-add: x + y,
bag-append: as + bs,
append: as @ bs,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
uiff: uiff(P;Q),
monoid_p: IsMonoid(T;op;id),
true: True,
formal-sum-mul: k * x,
record-select: r.x,
record-update: r[x := v]
Lemmas referenced :
rec_select_update_lemma,
istype-void,
vs-point_wf,
sub-free-vs_wf,
strong-subtype_wf,
istype-universe,
crng_wf,
bag-summation_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
crng_all_properties,
rng_plus_comm2,
single-bag_wf,
respects-equality-quotient1,
basic-formal-sum_wf,
bfs-equiv_wf,
bfs-equiv-rel,
respects-equality-trivial,
bag_wf,
respects-equality-bag,
respects-equality-product,
subtype-respects-equality,
istype-base,
bag-member_wf,
pi2_wf,
bag_to_squash_list,
equal_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
list-subtype-bag,
nil_wf,
empty-bag_wf,
quotient-member-eq,
subtype_rel_self,
implies-bfs-equiv,
bag-append_wf,
formal-sum-mul_wf1,
bag-append-ident,
formal-sum-add_wf1,
zero-bfs_wf,
cons_wf,
bag-member-append,
bag-member-single,
bag-summation-append,
pi1_wf_top,
squash_wf,
true_wf,
bag-summation-single,
iff_weakening_equal,
formal-sum_wf,
respects-equality-list-bag,
subtype_quotient,
formal-sum-add_wf,
rng_sig_wf,
rng_one_wf,
empty_bag_append_lemma,
bag_map_single_lemma,
rng_times_wf,
rng_times_one,
bag-map_wf,
respects-equality-set,
free-vs_wf,
fs-in-subtype_wf,
basic-formal-sum-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
sqequalRule,
setElimination,
rename,
imageElimination,
productElimination,
imageMemberEquality,
hypothesisEquality,
baseClosed,
universeIsType,
isectElimination,
independent_isectElimination,
lambdaEquality_alt,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
because_Cache,
equalityIstype,
isectIsTypeImplies,
instantiate,
universeEquality,
dependent_pairFormation_alt,
productEquality,
productIsType,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
independent_functionElimination,
sqequalBase,
applyEquality,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
functionEquality,
intWeakElimination,
natural_numberEquality,
approximateComputation,
int_eqEquality,
axiomEquality,
unionElimination,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
baseApply,
closedConclusion,
intEquality,
voidEquality,
inlFormation_alt,
inrFormation_alt,
spreadEquality
Latex:
\mforall{}[K:CRng]. \mforall{}[S,T:Type].
(\mforall{}s:T. \mforall{}x:Point(sub-free-vs(K;S;T)). (\mdownarrow{}\mexists{}k:|K|. (x = \{<k, s>\}))) supposing
((\mforall{}x,y:T. (x = y)) and
strong-subtype(T;S))
Date html generated:
2019_10_31-AM-06_30_21
Last ObjectModification:
2019_08_19-PM-02_54_38
Theory : linear!algebra
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