Nuprl Lemma : strong-continuity2-no-inner-squash-bound
∀F:(ℕ ⟶ ℕ) ⟶ ℕ
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)
∀f:ℕ ⟶ ℕ
∃n:ℕ. (F f < n ∧ ((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ ((M m f) = (inl (F f)) ∈ (ℕ?))))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
isl: isl(x),
prop: ℙ,
so_lambda: λ2x.t[x],
outl: outl(x),
so_apply: x[s],
ge: i ≥ j ,
pi1: fst(t),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
guard: {T},
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bfalse: ff,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
cand: A c∧ B,
sq_type: SQType(T),
istype: istype(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y]
Lemmas referenced :
strong-continuity2-no-inner-squash,
nat_wf,
int_seg_wf,
unit_wf2,
subtype_rel_function,
int_seg_subtype_nat,
istype-void,
subtype_rel_self,
assert_wf,
btrue_wf,
bfalse_wf,
decidable__exists_int_seg,
less_than_wf,
decidable__and2,
decidable__assert,
decidable__lt,
nat_properties,
le_wf,
it_wf,
istype-false,
int_seg_subtype,
sq_stable__le,
le_weakening2,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
true_wf,
imax_wf,
add_nat_wf,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
decidable_wf,
exists_wf,
imax_ub,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
squash_wf,
istype-universe,
iff_weakening_equal,
isl_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert_elim,
btrue_neq_bfalse,
union_subtype_base,
set_subtype_base,
int_subtype_base,
unit_subtype_base,
subtype_rel_union,
quotient_wf,
all_wf,
equiv_rel_true,
quotient-member-eq,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
Error :functionIsType,
Error :universeIsType,
hypothesis,
Error :inhabitedIsType,
sqequalRule,
Error :productIsType,
isectElimination,
natural_numberEquality,
setElimination,
rename,
Error :unionIsType,
because_Cache,
Error :equalityIsType1,
applyEquality,
independent_isectElimination,
independent_pairFormation,
Error :inlEquality_alt,
Error :isectIsType,
unionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
instantiate,
Error :lambdaEquality_alt,
productEquality,
Error :isect_memberEquality_alt,
Error :dependent_pairFormation_alt,
functionExtensionality,
Error :dependent_set_memberEquality_alt,
Error :inrEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
int_eqEquality,
voidElimination,
addEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
functionEquality,
Error :inrFormation_alt,
universeEquality,
unionEquality,
cumulativity,
Error :inlFormation_alt,
intEquality,
pertypeElimination,
Error :equalityIsType4
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}
\mexists{}n:\mBbbN{}
(F f < n \mwedge{} ((M n f) = (inl (F f))) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} ((M m f) = (inl (F f)))))))
Date html generated:
2019_06_20-PM-02_53_46
Last ObjectModification:
2018_10_06-PM-11_55_25
Theory : continuity
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