Nuprl Lemma : unsquashed-BIM-implies-unsquashed-weak-continuity-old
(∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.
((∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. Q[n + 1;s.m@n]) ⇒ Q[n;s]))
⇒ (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. Q[n;f]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀m:ℕ. (B[n;s] ⇒ B[n + 1;s.m@n]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ Q[n;s]))
⇒ Q[0;λx.⊥]))
⇒ (∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((∀i:ℕn. ((a i) = (b i) ∈ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
bottom: ⊥,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
int_upper: {i...},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
seq-add: s.x@n,
rep-seq-from: rep-seq-from(s;n;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_wf,
all_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,
seq-add_wf,
quotient_wf,
exists_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
true_wf,
equiv_rel_true,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
rep-seq-from_wf,
int_upper_wf,
int_upper_properties,
int_upper_subtype_int_upper,
rep-seq-from-prop3,
squash_wf,
iff_weakening_equal,
strong-continuity2-implies-weak,
implies-quotient-true,
rep-seq-from-prop1,
decidable__lt,
lelt_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eq_int_wf,
assert_of_eq_int,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rep-seq-from-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
instantiate,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
lambdaEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
natural_numberEquality,
setElimination,
rename,
because_Cache,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
independent_functionElimination,
applyLambdaEquality,
hyp_replacement,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
int_eqReduceTrueSq,
promote_hyp,
int_eqReduceFalseSq
Latex:
(\mforall{}B,Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}.
((\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. ((\mforall{}m:\mBbbN{}. Q[n + 1;s.m@n]) {}\mRightarrow{} Q[n;s]))
{}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. Q[n;f]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. \mforall{}m:\mBbbN{}. (B[n;s] {}\mRightarrow{} B[n + 1;s.m@n]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} Q[n;s]))
{}\mRightarrow{} Q[0;\mlambda{}x.\mbot{}]))
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}i:\mBbbN{}n. ((a i) = (b i))) {}\mRightarrow{} ((F a) = (F b))))
Date html generated:
2017_04_20-AM-07_21_38
Last ObjectModification:
2017_02_27-PM-05_58_04
Theory : continuity
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