Nuprl Lemma : unsquashed-continuity-false-troelstra
¬(∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((a = b ∈ (ℕn ⟶ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ)))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
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,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
pi1: fst(t),
phi-star: Phi*,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
cand: A c∧ B,
isl: isl(x),
outl: outl(x),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
gamma-neighbourhood: gamma-neighbourhood(beta;n0),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
assert: ↑b,
bfalse: ff,
bnot: ¬bb,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
finite-nat-seq: finite-nat-seq(),
mk-finite-nat-seq: f^(n),
pi2: snd(t),
ext-finite-nat-seq: ext-finite-nat-seq(f;x),
append-finite-nat-seq: f**g,
int_seg: {i..j-},
less_than: a < b,
ge: i ≥ j ,
lelt: i ≤ j < k
Lemmas referenced :
istype-nat,
zero-seq_wf,
int_seg_wf,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
gamma-neighbourhood-prop1,
finite-nat-seq_wf,
phi-star_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-assert,
gamma-neighbourhood_wf,
mk-finite-nat-seq_wf,
btrue_wf,
bfalse_wf,
assert_elim,
union_subtype_base,
unit_wf2,
set_subtype_base,
le_wf,
int_subtype_base,
unit_subtype_base,
btrue_neq_bfalse,
append-finite-nat-seq_wf,
assert_wf,
equal-wf-base,
subtype_base_sq,
init-seg-nat-seq_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
extend-seq1-all-dec,
decidable_wf,
not_wf,
true_wf,
equal_wf,
squash_wf,
iff_weakening_equal,
gamma-neighbourhood-prop2,
decidable__equal_int,
ext-finite-nat-seq_wf,
lt_int_wf,
assert_of_lt_int,
istype-top,
iff_weakening_uiff,
less_than_wf,
istype-less_than,
int_seg_properties,
nat_properties,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
itermAdd_wf,
int_term_value_add_lemma,
gamma-neighbourhood-prop3,
gamma-neighbourhood-prop4,
intformeq_wf,
int_formula_prop_eq_lemma,
gamma-neighbourhood-prop5,
gamma-neighbourhood-prop6,
eq-seg-nat-seq_wf,
assert-eq-seg-nat-seq,
Troelstra-lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
Error :functionIsType,
Error :inhabitedIsType,
introduction,
extract_by_obid,
because_Cache,
promote_hyp,
productElimination,
sqequalRule,
Error :productIsType,
Error :equalityIstype,
Error :universeIsType,
isectElimination,
natural_numberEquality,
setElimination,
rename,
applyEquality,
independent_isectElimination,
independent_pairFormation,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
functionExtensionality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
Error :isect_memberEquality_alt,
voidElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
sqequalBase,
applyLambdaEquality,
Error :unionIsType,
productEquality,
Error :functionExtensionality_alt,
instantiate,
cumulativity,
equalityElimination,
imageElimination,
universeEquality,
imageMemberEquality,
Error :equalityIsType4,
Error :equalityIsType1,
addEquality,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
int_eqEquality,
unionEquality
Latex:
\mneg{}(\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((a = b) {}\mRightarrow{} ((F a) = (F b))))
Date html generated:
2019_06_20-PM-03_05_14
Last ObjectModification:
2018_12_06-PM-11_57_48
Theory : continuity
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