Nuprl Lemma : equipollent-int_seg
∀n,m:ℤ. {n..m-} ~ ℕif n <z m then m - n else 0 fi
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
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,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
equipollent: A ~ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
decidable: Dec(P),
surject: Surj(A;B;f)
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
equipollent_interval,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
itermSubtract_wf,
itermConstant_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
lelt_wf,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
because_Cache,
independent_functionElimination,
voidElimination,
lambdaEquality,
setElimination,
rename,
natural_numberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
applyLambdaEquality,
dependent_set_memberEquality,
functionExtensionality,
applyEquality
Latex:
\mforall{}n,m:\mBbbZ{}. \{n..m\msupminus{}\} \msim{} \mBbbN{}if n <z m then m - n else 0 fi
Date html generated:
2017_04_17-AM-09_31_34
Last ObjectModification:
2017_02_27-PM-05_31_43
Theory : equipollence!!cardinality!
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