Nuprl Lemma : general-cantor-to-int-uniform-continuity-half-squashed
∀B:ℕ ⟶ ℕ+. ∀F:(i:ℕ ⟶ ℕB[i]) ⟶ ℤ. ⇃(∃n:ℕ. ∀f,g:i:ℕ ⟶ ℕB[i]. ((f = g ∈ (i:ℕn ⟶ ℕB[i])) ⇒ ((F f) = (F g) ∈ ℤ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
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,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
implies: P ⇒ Q,
nat_plus: ℕ+,
guard: {T},
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
listp: A List+,
sq_exists: ∃x:A [B[x]],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
less_than: a < b,
l_exists: (∃x∈L. P[x]),
istype: istype(T),
isl: isl(x),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
uiff: uiff(P;Q),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹
Lemmas referenced :
general-uniform-continuity-from-fan-ext,
int_seg_wf,
istype-nat,
nat_plus_wf,
istype-int,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
nat_plus_properties,
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,
decidable__lt,
nat_properties,
istype-false,
le_wf,
sq_exists_wf,
nat_wf,
trivial-quotient-true,
length_wf,
upto_wf,
map_wf,
length_upto,
map-length,
imax-list-nat,
int_seg_properties,
imax-list-ub,
select_wf,
nat_plus_subtype_nat,
select-upto,
top_wf,
subtype_rel_list,
select-map,
strong-continuity2-half-squash-surject-biject-ext,
countable-nsub-family,
biject-int-nat,
int_subtype_base,
set_subtype_base,
zero-le-nat,
subtype_rel_dep_function,
istype-assert,
unit_subtype_base,
union_subtype_base,
subtype_rel_self,
int_seg_subtype_nat,
subtype_rel_function,
bfalse_wf,
btrue_wf,
assert_wf,
isect_wf,
equal-wf-base-T,
all_wf,
unit_wf2,
exists_wf,
implies-quotient-true2,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eq_int_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
bool_wf,
assert_of_eq_int,
eqtt_to_assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
applyEquality,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
independent_functionElimination,
intEquality,
dependent_functionElimination,
functionIsType,
universeIsType,
setElimination,
rename,
productIsType,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
functionEquality,
dependent_set_memberFormation_alt,
functionExtensionality,
imageElimination,
productElimination,
productEquality,
sqequalBase,
isectIsType,
unionIsType,
spreadEquality,
inlEquality_alt,
unionEquality,
closedConclusion,
instantiate,
promote_hyp,
equalityElimination,
isect_memberFormation_alt,
cumulativity,
dependent_pairEquality_alt
Latex:
\mforall{}B:\mBbbN{} {}\mrightarrow{} \mBbbN{}\msupplus{}. \mforall{}F:(i:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[i]) {}\mrightarrow{} \mBbbZ{}. \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:i:\mBbbN{} {}\mrightarrow{} \mBbbN{}B[i]. ((f = g) {}\mRightarrow{} ((F f) = (F g))))
Date html generated:
2020_05_19-PM-10_04_56
Last ObjectModification:
2019_11_22-AM-10_51_12
Theory : continuity
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