Nuprl Lemma : equipollent-int-nat
ℤ ~ ℕ
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
nat: ℕ,
int: ℤ
Definitions unfolded in proof :
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
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,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
subtype_rel: A ⊆r B,
surject: Surj(A;B;f),
ge: i ≥ j ,
true: True,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B
Lemmas referenced :
int_formula_prop_less_lemma,
intformless_wf,
false_wf,
multiply-is-int-iff,
add-is-int-iff,
less_than_wf,
rem_bounds_1,
nequal_wf,
div_rem_sum,
equal-wf-base-T,
neg_assert_of_eq_int,
assert_of_eq_int,
true_wf,
eq_int_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
nat_properties,
biject_wf,
int_subtype_base,
nat_wf,
equal-wf-base,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
itermMinus_wf,
itermSubtract_wf,
itermAdd_wf,
subtract_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_pairFormation,
lambdaEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
independent_isectElimination,
because_Cache,
dependent_set_memberEquality,
dependent_functionElimination,
multiplyEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
addEquality,
minusEquality,
equalityEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
remainderEquality,
setElimination,
rename,
addLevel,
divideEquality,
introduction,
imageMemberEquality,
pointwiseFunctionality,
imageElimination
Latex:
\mBbbZ{} \msim{} \mBbbN{}
Date html generated:
2016_05_15-PM-05_25_32
Last ObjectModification:
2016_01_16-PM-00_29_47
Theory : general
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