Nuprl Lemma : str-to-nat-plus-property
∀[s:Atom List]. ∀[n:ℕ]. (str-to-nat-plus(s;n) = (str-to-nat(s) + (n * 10^||s||)) ∈ ℤ)
Proof
Definitions occuring in Statement :
str-to-nat: str-to-nat(s),
str-to-nat-plus: str-to-nat-plus(s;n),
exp: i^n,
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
atom: Atom,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
implies: P ⇒ Q,
str-to-nat: str-to-nat(s),
str-to-nat-plus: str-to-nat-plus(s;n),
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
guard: {T},
uiff: uiff(P;Q),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
nat_wf,
equal_wf,
str-to-nat-plus_wf,
str-to-nat_wf,
exp_wf2,
length_wf_nat,
list_wf,
list_ind_nil_lemma,
length_of_nil_lemma,
exp0_lemma,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
list_ind_cons_lemma,
length_of_cons_lemma,
str1-to-nat_wf,
add_nat_wf,
multiply_nat_wf,
false_wf,
le_wf,
decidable__le,
add-is-int-iff,
intformand_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
length_wf,
mul-swap,
add_functionality_wrt_eq,
non_neg_length,
exp_add,
iff_weakening_equal,
exp1,
squash_wf,
true_wf,
mul-distributes-right,
mul-associates,
add-associates,
mul-commutes,
zero-mul,
zero-add,
add-swap,
add-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
atomEquality,
sqequalRule,
lambdaEquality,
hypothesis,
intEquality,
hypothesisEquality,
applyEquality,
setElimination,
rename,
addEquality,
multiplyEquality,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
lambdaFormation,
axiomEquality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[s:Atom List]. \mforall{}[n:\mBbbN{}]. (str-to-nat-plus(s;n) = (str-to-nat(s) + (n * 10\^{}||s||)))
Date html generated:
2017_04_17-AM-09_18_09
Last ObjectModification:
2017_02_27-PM-05_22_19
Theory : decidable!equality
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