Nuprl Lemma : int-seg-cardinality-le
∀x,y:ℤ. ∀n:ℕ. |{x..y-}| ≤ n supposing ((y - x) ≤ n) ∧ x < y
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
guard: {T},
cardinality-le: |T| ≤ n,
int_seg: {i..j-},
uiff: uiff(P;Q),
lelt: i ≤ j < k,
surject: Surj(A;B;f)
Lemmas referenced :
less_than'_wf,
subtract_wf,
member-less_than,
cardinality-le_functionality,
int_seg_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than_wf,
le_wf,
nat_wf,
add-member-int_seg1,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
lelt_wf,
surject_wf,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
productEquality,
because_Cache,
addEquality
Latex:
\mforall{}x,y:\mBbbZ{}. \mforall{}n:\mBbbN{}. |\{x..y\msupminus{}\}| \mleq{} n supposing ((y - x) \mleq{} n) \mwedge{} x < y
Date html generated:
2018_05_21-PM-00_39_40
Last ObjectModification:
2018_05_19-AM-06_45_06
Theory : list_1
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