Nuprl Lemma : equipollent-nat-union-nsub
∀k:ℕ+. ℕ ~ ℕ + ℕk
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
union: left + right,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
isl: isl(x),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
subtract: n - m
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
lelt_wf,
nat_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
subtract_wf,
nat_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
int_seg_wf,
biject_wf,
nat_plus_wf,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
btrue_wf,
and_wf,
isl_wf,
bfalse_wf,
btrue_neq_bfalse,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_subtype_nat,
false_wf,
add-associates,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add-mul-special,
zero-mul,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
independent_isectElimination,
because_Cache,
inrEquality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
inlEquality,
natural_numberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
unionEquality,
functionExtensionality,
applyEquality,
applyLambdaEquality,
addEquality,
minusEquality
Latex:
\mforall{}k:\mBbbN{}\msupplus{}. \mBbbN{} \msim{} \mBbbN{} + \mBbbN{}k
Date html generated:
2018_05_21-PM-07_57_36
Last ObjectModification:
2017_07_26-PM-05_35_10
Theory : general
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