Nuprl Lemma : code-coded-seq
∀x:ℕ. (let k,s = coded-seq(x) in code-seq(k;s) = x ∈ ℤ)
Proof
Definitions occuring in Statement :
coded-seq: coded-seq(x),
code-seq: code-seq(k;s),
nat: ℕ,
all: ∀x:A. B[x],
spread: spread def,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
coded-seq: coded-seq(x),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
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 ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
ge: i ≥ j ,
int_upper: {i...},
code-seq: code-seq(k;s),
eq_int: (i =z j),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
label: ...$L... t
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
false_wf,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
nat_wf,
int_subtype_base,
coded-pair_wf,
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
add-subtract-cancel,
code-coded-seq1,
decidable__lt,
not-lt-2,
not-equal-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
decidable__equal_int,
coded-code-pair,
code-pair_wf,
squash_wf,
true_wf,
code-coded-pair,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
intEquality,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
productEquality,
addEquality,
hyp_replacement,
applyLambdaEquality,
applyEquality,
minusEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}x:\mBbbN{}. (let k,s = coded-seq(x) in code-seq(k;s) = x)
Date html generated:
2018_05_21-PM-07_56_19
Last ObjectModification:
2017_07_26-PM-05_33_55
Theory : general
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