Nuprl Lemma : biject-int-nat
∃f:ℤ ⟶ ℕ. Bij(ℤ;ℕ;f)
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
nat: ℕ,
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
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 ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b)
Lemmas referenced :
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
subtract_wf,
itermAdd_wf,
itermSubtract_wf,
itermMinus_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_minus_lemma,
nat_wf,
int_subtype_base,
biject_wf,
nat_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
set_subtype_base,
div_rem_sum,
nequal_wf,
rem_bounds_1,
less_than_wf,
add-is-int-iff,
multiply-is-int-iff,
false_wf,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
cut,
natural_numberEquality,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
sqequalRule,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
multiplyEquality,
approximateComputation,
independent_functionElimination,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
addEquality,
minusEquality,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
applyLambdaEquality,
Error :equalityIsType2,
remainderEquality,
setElimination,
rename,
divideEquality,
imageMemberEquality,
pointwiseFunctionality,
imageElimination
Latex:
\mexists{}f:\mBbbZ{} {}\mrightarrow{} \mBbbN{}. Bij(\mBbbZ{};\mBbbN{};f)
Date html generated:
2019_06_20-PM-01_18_59
Last ObjectModification:
2018_10_06-PM-06_09_25
Theory : int_2
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