Nuprl Lemma : code-coded-seq1
∀[k:ℕ+]. ∀[x:ℕ]. (code-seq1(k;λn.coded-seq1(k - 1;x;n)) = x ∈ ℤ)
Proof
Definitions occuring in Statement :
coded-seq1: coded-seq1(k;x;n),
code-seq1: code-seq1(k;s),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
code-seq1: code-seq1(k;s),
primrec: primrec(n;b;c),
subtract: n - m,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
coded-seq1: coded-seq1(k;x;n),
nat: ℕ,
all: ∀x:A. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
decidable: Dec(P),
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_wf,
nat_plus_properties,
add-subtract-cancel,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
coded-pair_wf,
uall_wf,
code-seq1_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
le_wf,
coded-seq1_wf,
subtract_wf,
int_seg_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtract-add-cancel,
decidable__lt,
lelt_wf,
int_seg_wf,
nat_plus_wf,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
primrec-unroll,
lt_int_wf,
assert_of_lt_int,
itermAdd_wf,
int_term_value_add_lemma,
less_than_wf,
decidable__equal_int,
int_subtype_base,
coded-code-pair,
code-pair_wf,
squash_wf,
true_wf,
code-coded-pair,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
extract_by_obid,
lambdaFormation,
isectElimination,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
because_Cache,
productEquality,
dependent_set_memberEquality,
addEquality,
applyEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
spreadEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbN{}]. (code-seq1(k;\mlambda{}n.coded-seq1(k - 1;x;n)) = x)
Date html generated:
2019_06_20-PM-02_40_06
Last ObjectModification:
2019_06_12-PM-00_28_09
Theory : num_thy_1
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