Nuprl Lemma : fb-to-cantor_wf
∀[b:ℕ ⟶ ℕ+]. ∀[f:n:ℕ ⟶ ℕb n]. ∀[k:ℕ]. (fb-to-cantor(b;f;k) ∈ 𝔹)
Proof
Definitions occuring in Statement :
fb-to-cantor: fb-to-cantor(b;f;n),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
fb-to-cantor: fb-to-cantor(b;f;n),
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
has-value: (a)↓,
int_seg: {i..j-},
nat_plus: ℕ+,
guard: {T},
lelt: i ≤ j < k,
sq_type: SQType(T),
subtract: n - m,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x])
Lemmas referenced :
mu_wf,
lt_int_wf,
sum_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
assert_of_lt_int,
sum_lower_bound,
assert_wf,
nat_wf,
value-type-has-value,
set-value-type,
int-value-type,
subtract_wf,
eq_int_wf,
equal_wf,
nat_plus_wf,
int_seg_properties,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
decidable__lt,
itermMultiply_wf,
int_term_value_mul_lemma,
less_than_functionality,
le_weakening,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
mu-property,
set_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
applyEquality,
functionExtensionality,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
dependent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
callbyvalueReduce,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality,
multiplyEquality,
applyLambdaEquality,
instantiate,
cumulativity
Latex:
\mforall{}[b:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}]. \mforall{}[f:n:\mBbbN{} {}\mrightarrow{} \mBbbN{}b n]. \mforall{}[k:\mBbbN{}]. (fb-to-cantor(b;f;k) \mmember{} \mBbbB{})
Date html generated:
2018_05_21-PM-07_58_02
Last ObjectModification:
2017_07_26-PM-05_35_30
Theory : general
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