Nuprl Lemma : int2nat2int
∀[i:ℤ]. (nat2int(int2nat(i)) = i ∈ ℤ)
Proof
Definitions occuring in Statement :
nat2int: nat2int(m),
int2nat: int2nat(i),
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
int2nat: int2nat(i),
nat2int: nat2int(m),
member: t ∈ T,
has-value: (a)↓,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
bfalse: ff,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
remainder: n rem m
Lemmas referenced :
eq_int_wf,
remainder_wfa,
value-type-has-value,
int-value-type,
lt_int_wf,
istype-top,
istype-void,
subtract_wf,
subtype_base_sq,
int_subtype_base,
nequal_wf,
eqtt_to_assert,
assert_of_eq_int,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
divide-exact,
neg_assert_of_eq_int,
istype-int,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
equal_wf,
add_functionality_wrt_eq,
mul_com,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
int_nzero_wf,
add_com,
subtype_rel_self,
decidable__lt,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermMinus_wf,
itermSubtract_wf,
intformand_wf,
istype-le,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
rem_invariant,
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
false_wf,
multiply-is-int-iff,
add-is-int-iff,
int_term_value_mul_lemma,
itermMultiply_wf,
rem_bounds_1,
div_rem_sum,
mul-commutes,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
callbyvalueReduce,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
closedConclusion,
natural_numberEquality,
because_Cache,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
lessCases,
independent_pairFormation,
baseClosed,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
promote_hyp,
voidElimination,
addEquality,
multiplyEquality,
minusEquality,
Error :equalityIstype,
dependent_functionElimination,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
instantiate,
cumulativity,
sqequalBase,
Error :universeIsType,
productElimination,
int_eqReduceTrueSq,
imageElimination,
Error :dependent_pairFormation_alt,
int_eqReduceFalseSq,
approximateComputation,
Error :lambdaEquality_alt,
applyEquality,
universeEquality,
int_eqEquality,
rename,
setElimination,
applyLambdaEquality,
baseApply,
pointwiseFunctionality
Latex:
\mforall{}[i:\mBbbZ{}]. (nat2int(int2nat(i)) = i)
Date html generated:
2019_06_20-PM-02_52_15
Last ObjectModification:
2019_03_06-AM-10_52_13
Theory : continuity
Home
Index