Nuprl Lemma : gen-continuity-is-false
¬(∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ∀f:ℕ ⟶ ℕ. ((P f) ⇒ ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) ⇒ (P g)))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
so_apply: x[s],
exists: ∃x:A. B[x],
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
zero-seq: 0s,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
nat_wf,
exists_wf,
int_upper_wf,
equal-wf-T-base,
int_upper_subtype_nat,
zero-seq_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
quotient_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
true_wf,
equiv_rel_true,
squash-from-quotient,
equal-wf-base-T,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
int_seg_properties,
decidable__equal_int,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_upper_properties,
int_subtype_base,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
setElimination,
rename,
hypothesisEquality,
intEquality,
applyEquality,
functionExtensionality,
baseClosed,
because_Cache,
functionEquality,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
cumulativity,
universeEquality,
independent_pairFormation,
imageElimination,
productElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
impliesFunctionality
Latex:
\mneg{}(\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((P f) {}\mRightarrow{} \00D9(\mexists{}k:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} (P g)))))
Date html generated:
2017_09_29-PM-06_10_10
Last ObjectModification:
2017_07_11-PM-05_33_38
Theory : continuity
Home
Index