Nuprl Lemma : int_seg-cardinality-le
∀i,j:ℤ. |{i..j-}| ≤ if i ≤z j then j - i else 0 fi
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
le_int: i ≤z j,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
cardinality-le: |T| ≤ n,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
prop: ℙ,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
surject: Surj(A;B;f)
Lemmas referenced :
le_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
equal_wf,
add-member-int_seg1,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermVar_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_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
int_seg_wf,
surject_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
intEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionExtensionality,
addEquality
Latex:
\mforall{}i,j:\mBbbZ{}. |\{i..j\msupminus{}\}| \mleq{} if i \mleq{}z j then j - i else 0 fi
Date html generated:
2017_04_17-AM-07_45_27
Last ObjectModification:
2017_02_27-PM-04_17_23
Theory : list_1
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