Nuprl Lemma : coded-code-seq1
∀[k:ℕ+]. ∀s:ℕk ⟶ ℕ. ∀[n:ℕk]. (coded-seq1(k - 1;code-seq1(k;s);n) = (s n) ∈ ℤ)
Proof
Definitions occuring in Statement :
coded-seq1: coded-seq1(k;x;n),
code-seq1: code-seq1(k;s),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
coded-seq1: coded-seq1(k;x;n),
subtract: n - m,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
nat_plus: ℕ+,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
so_apply: x[s],
code-seq1: code-seq1(k;s),
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
coded-pair: coded-pair(m),
tsqrt: tsqrt(n),
isqrt: isqrt(x),
integer-sqrt-ext,
genrec-ap: genrec-ap,
le_int: i ≤z j,
lt_int: i <z j,
triangular-num: t(n),
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
label: ...$L... t
Lemmas referenced :
int_seg_wf,
nat_wf,
nat_plus_properties,
all_wf,
uall_wf,
equal_wf,
coded-seq1_wf,
subtract_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
le_wf,
subtract-add-cancel,
decidable__lt,
lelt_wf,
code-seq1_wf,
nat_plus_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
primrec1_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_subtype,
false_wf,
int_seg_cases,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
add-subtract-cancel,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
primrec-unroll,
lt_int_wf,
assert_of_lt_int,
itermAdd_wf,
int_term_value_add_lemma,
less_than_wf,
squash_wf,
true_wf,
coded-code-pair,
subtype_rel_function,
not-le-2,
not-equal-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
subtype_rel_self,
nat_properties,
iff_weakening_equal,
integer-sqrt-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
functionEquality,
rename,
hypothesisEquality,
setElimination,
addEquality,
lambdaEquality,
because_Cache,
intEquality,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
functionExtensionality,
applyEquality,
axiomEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
hypothesis_subsumption,
equalityElimination,
promote_hyp,
imageElimination,
universeEquality,
spreadEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}s:\mBbbN{}k {}\mrightarrow{} \mBbbN{}. \mforall{}[n:\mBbbN{}k]. (coded-seq1(k - 1;code-seq1(k;s);n) = (s n))
Date html generated:
2019_06_20-PM-02_39_58
Last ObjectModification:
2019_06_12-PM-00_28_05
Theory : num_thy_1
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