Nuprl Lemma : equipollent-int_seg-shift
∀n:ℤ. ∀m:ℕ. {n..n + m-} ~ ℕm
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
subtract: n - m,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
istype-nat,
equipollent_same,
zero-mul,
add-mul-special,
add-swap,
add-zero,
minus-one-mul,
minus-zero,
add-associates,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermConstant_wf,
intformnot_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
istype-less_than,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
equipollent-int_seg,
subtract_wf,
lt_int_wf,
ifthenelse_wf,
int_seg_wf,
equipollent_functionality_wrt_equipollent
Rules used in proof :
universeIsType,
independent_pairFormation,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
approximateComputation,
voidElimination,
cumulativity,
instantiate,
promote_hyp,
equalityIstype,
dependent_pairFormation_alt,
equalitySymmetry,
equalityTransitivity,
sqequalRule,
independent_isectElimination,
equalityElimination,
unionElimination,
inhabitedIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination,
intEquality,
because_Cache,
natural_numberEquality,
hypothesis,
rename,
setElimination,
addEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbZ{}. \mforall{}m:\mBbbN{}. \{n..n + m\msupminus{}\} \msim{} \mBbbN{}m
Date html generated:
2019_10_15-AM-10_25_08
Last ObjectModification:
2019_10_01-PM-03_03_39
Theory : equipollence!!cardinality!
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