Nuprl Lemma : str-to-nat-to-str
∀[n:ℕ]. (str-to-nat(nat-to-str(n)) = n ∈ ℤ)
Proof
Definitions occuring in Statement :
str-to-nat: str-to-nat(s),
nat-to-str: nat-to-str(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat-to-str: nat-to-str(n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
str-to-nat: str-to-nat(s),
str-to-nat-plus: str-to-nat-plus(s;n),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
str1-to-nat: str1-to-nat(a),
eq_atom: x =a y,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
list_ind_cons_lemma,
list_ind_nil_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nequal-le-implies,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
div_rem_sum,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rem_bounds_1,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
squash_wf,
iff_weakening_equal,
length_of_cons_lemma,
length_of_nil_lemma,
nat-to-str_wf,
divide_wf,
list_wf,
remainder_wf,
equal-wf-base-T,
list_subtype_base,
atom_subtype_base,
list_induction,
all_wf,
str-to-nat-plus_wf,
append_wf,
str1-to-nat_wf,
add_nat_wf,
str-to-nat_wf,
exp_wf2,
length_wf_nat,
length-append,
str-to-nat-plus-property,
add_functionality_wrt_eq,
length_wf,
add-associates,
mul-distributes-right,
mul-associates,
mul-commutes,
mul-swap,
zero-mul,
add-zero,
zero-add,
add-commutes,
exp_add,
exp1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
because_Cache,
productElimination,
unionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
addEquality,
addLevel,
baseClosed,
imageMemberEquality,
divideEquality,
imageElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
universeEquality,
remainderEquality,
productEquality,
atomEquality,
multiplyEquality,
functionEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[n:\mBbbN{}]. (str-to-nat(nat-to-str(n)) = n)
Date html generated:
2017_04_17-AM-09_18_20
Last ObjectModification:
2017_02_27-PM-05_23_10
Theory : decidable!equality
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