Nuprl Lemma : general-cantor-to-int-uniform-continuity
∀B:ℕ ⟶ ℕ+. ∀F:(k:ℕ ⟶ ℕB[k]) ⟶ ℤ. ∃n:ℕ. ∀f,g:k:ℕ ⟶ ℕB[k]. ((f = g ∈ (k:ℕn ⟶ ℕB[k])) ⇒ ((F f) = (F g) ∈ ℤ))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
gt: i > j,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
cand: A c∧ B,
guard: {T},
istype: istype(T),
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
le_witness,
int_formula_prop_eq_lemma,
intformeq_wf,
prop-truncation-implies,
mu-dec-property,
it_wf,
unit_wf2,
mu-dec_wf,
change-equality-type,
respects-equality-trivial,
not-gt-2,
respects-equality-function,
subtype_rel_transitivity,
istype-universe,
true_wf,
squash_wf,
decidable__equal_int,
decidable__not,
decidable__and2,
nat_plus_properties,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
not_wf,
decidable-exists-finite,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__equal_int_seg,
lelt_wf,
set_subtype_base,
le_weakening2,
int_seg_subtype,
decidable__all_int_seg,
nat_plus_subtype_nat,
nsub_finite,
finite-function,
decidable__lt,
le_wf,
int_subtype_base,
equal-wf-base,
istype-le,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
subtype_rel_dep_function,
istype-false,
int_seg_subtype_nat,
equal_wf,
all_wf,
nat_wf,
exists_wf,
implies-quotient-true,
istype-int,
nat_plus_wf,
int_seg_wf,
istype-nat,
general-cantor-to-int-uniform-continuity-half-squashed
Rules used in proof :
independent_pairEquality,
dependent_pairEquality_alt,
baseClosed,
imageMemberEquality,
universeEquality,
hyp_replacement,
functionExtensionality_alt,
cumulativity,
promote_hyp,
equalityElimination,
applyLambdaEquality,
equalityTransitivity,
inrFormation_alt,
functionExtensionality,
inlFormation_alt,
instantiate,
equalitySymmetry,
sqequalBase,
equalityIstype,
productIsType,
productEquality,
intEquality,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
imageElimination,
dependent_set_memberEquality_alt,
productElimination,
inhabitedIsType,
independent_pairFormation,
independent_isectElimination,
closedConclusion,
because_Cache,
functionEquality,
sqequalRule,
rename,
setElimination,
lambdaEquality_alt,
applyEquality,
natural_numberEquality,
isectElimination,
universeIsType,
functionIsType,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}B:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}. \mforall{}F:(k:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[k]) {}\mrightarrow{} \mBbbZ{}. \mexists{}n:\mBbbN{}. \mforall{}f,g:k:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[k]. ((f = g) {}\mRightarrow{} ((F f) = (F g)))
Date html generated:
2019_10_15-AM-10_26_13
Last ObjectModification:
2019_10_08-PM-05_31_18
Theory : continuity
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