Nuprl Lemma : coded-code-seq
∀k:ℕ. ∀s:ℕk ⟶ ℕ. (coded-seq(code-seq(k;s)) = <k, s> ∈ (k:ℕ × (ℕk ⟶ ℕ)))
Proof
Definitions occuring in Statement :
coded-seq: coded-seq(x),
code-seq: code-seq(k;s),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
code-seq: code-seq(k;s),
coded-seq: coded-seq(x),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
coded-pair: coded-pair(m),
subtract: n - m,
tsqrt: tsqrt(n),
isqrt: isqrt(x),
integer-sqrt-ext,
genrec-ap: genrec-ap,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
triangular-num: t(n),
eq_int: (i =z j),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
int_upper: {i...},
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
pi2: snd(t),
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
less_than: a < b,
label: ...$L... t
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
false_wf,
le_wf,
int_seg_properties,
int_seg_wf,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
lelt_wf,
nat_wf,
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,
nequal-le-implies,
zero-add,
code-pair_wf,
subtract_wf,
int_upper_properties,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
code-seq1_wf,
itermAdd_wf,
int_term_value_add_lemma,
coded-pair_wf,
add-associates,
add-swap,
add-commutes,
squash_wf,
true_wf,
coded-code-pair,
iff_weakening_equal,
set_subtype_base,
int_subtype_base,
decidable__lt,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
le-add-cancel,
less_than_wf,
coded-seq1_wf,
subtract-add-cancel,
coded-code-seq1,
integer-sqrt-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
functionExtensionality,
applyEquality,
functionEquality,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
hypothesis_subsumption,
addEquality,
applyLambdaEquality,
productEquality,
minusEquality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
imageMemberEquality,
baseClosed
Latex:
\mforall{}k:\mBbbN{}. \mforall{}s:\mBbbN{}k {}\mrightarrow{} \mBbbN{}. (coded-seq(code-seq(k;s)) = <k, s>)
Date html generated:
2019_06_20-PM-02_40_43
Last ObjectModification:
2019_06_12-PM-00_28_38
Theory : num_thy_1
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