Nuprl Lemma : combinations-n-intersecting
∀n,t:ℕ. ∀[A:Type]. (A ~ ℕ(n * t) + 1 ⇒ n-intersecting(A;Combination(((n - 1) * t) + 1;A);n))
Proof
Definitions occuring in Statement :
n-intersecting: n-intersecting(A;T;n),
combination: Combination(n;T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
n-intersecting: n-intersecting(A;T;n),
member: t ∈ T,
nat: ℕ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
le: A ≤ B,
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
combination: Combination(n;T),
sq_stable: SqStable(P),
cand: A c∧ B,
respects-equality: respects-equality(S;T),
uiff: uiff(P;Q),
nat_plus: ℕ+,
l_all: (∀x∈L.P[x])
Lemmas referenced :
istype-int,
length_wf_nat,
combination_wf,
subtract_wf,
set_subtype_base,
le_wf,
int_subtype_base,
list_wf,
equipollent_wf,
int_seg_wf,
istype-universe,
istype-nat,
mul_bounds_1a,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equipollent-decidable-equal,
decidable__equal_int_seg,
ge_wf,
istype-less_than,
le_witness_for_triv,
list-cases,
length_of_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
spread_cons_lemma,
decidable__equal_int,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__le,
length_of_cons_lemma,
equipollent-zero,
l_exists_wf_nil,
not_wf,
l_member_wf,
int_seg_properties,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-nsub,
l_exists_wf,
cons_wf,
equipollent-partition,
decidable__l_exists_better-extract,
decidable__not,
decidable__l_member,
sq_stable__le,
length_wf,
non_neg_length,
equipollent-length,
equipollent-subtract2,
equipollent-subtype,
no_repeats_wf,
subtype_rel_sets,
respects-equality-set,
subtype-respects-equality,
sq_stable__l_member,
l_exists_cons,
multiply-is-int-iff,
false_wf,
add-is-int-iff,
equipollent-non-zero,
l_all_wf2,
decidable__l_all-better-extract,
not-l_exists,
select_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
equalityIstype,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
because_Cache,
closedConclusion,
natural_numberEquality,
applyEquality,
sqequalRule,
intEquality,
lambdaEquality_alt,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
inhabitedIsType,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
intWeakElimination,
productElimination,
equalityTransitivity,
functionIsTypeImplies,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
baseApply,
baseClosed,
setEquality,
setIsType,
imageMemberEquality,
cumulativity,
productIsType,
functionIsType,
pointwiseFunctionality
Latex:
\mforall{}n,t:\mBbbN{}. \mforall{}[A:Type]. (A \msim{} \mBbbN{}(n * t) + 1 {}\mRightarrow{} n-intersecting(A;Combination(((n - 1) * t) + 1;A);n))
Date html generated:
2019_10_15-AM-11_24_51
Last ObjectModification:
2018_11_30-AM-09_53_14
Theory : general
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