Nuprl Lemma : cardinality-le-int_seg
∀[x,y:ℤ]. ∀[n:ℕ]. (y - x) ≤ n supposing |{x..y-}| ≤ n
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cardinality-le: |T| ≤ n,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
uiff: uiff(P;Q),
lelt: i ≤ j < k,
inject: Inj(A;B;f),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
subtype_rel: A ⊆r B,
squash: ↓T,
true: True
Lemmas referenced :
decidable__lt,
nat_properties,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
less_than'_wf,
cardinality-le_wf,
int_seg_wf,
nat_wf,
surject-inverse,
pigeon-hole,
le_wf,
add-member-int_seg1,
lelt_wf,
equal_wf,
int_seg_properties,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int_seg,
itermAdd_wf,
int_term_value_add_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
unionElimination,
isectElimination,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_pairEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
applyEquality,
functionExtensionality,
lambdaFormation,
promote_hyp,
instantiate,
cumulativity,
addEquality,
imageElimination,
universeEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[x,y:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. (y - x) \mleq{} n supposing |\{x..y\msupminus{}\}| \mleq{} n
Date html generated:
2017_04_17-AM-07_46_22
Last ObjectModification:
2017_02_27-PM-04_18_36
Theory : list_1
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