Nuprl Lemma : kripke's-schema-contradicts-squashed-continuity1-rel
(∀A:ℙ. ⇃(∃a:ℕ ⟶ ℕ. (A ⇐⇒ ∃n:ℕ. ((a n) = 1 ∈ ℤ)))) ⇒ (¬(∀A:(ℕ ⟶ ℕ) ⟶ (ℕ ⟶ ℕ) ⟶ ℙ. squashed-continuity1-rel(A)))
Proof
Definitions occuring in Statement :
squashed-continuity1-rel: squashed-continuity1-rel(A),
quotient: x,y:A//B[x; y],
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
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 :
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
true: True,
less_than: a < b,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
replace-seq-from: replace-seq-from(s;n;k),
cons-nat-seq: cons-nat-seq(n;a),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
shift-seq: shift-seq(c;a),
squash: ↓T,
guard: {T},
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
squashed-continuity1-rel: squashed-continuity1-rel(A),
nat: ℕ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q
Lemmas referenced :
less_than_anti-reflexive,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
iff_weakening_equal,
squash_wf,
int_subtype_base,
int_formula_prop_eq_lemma,
intformeq_wf,
neg_assert_of_eq_int,
int_formula_prop_less_lemma,
intformless_wf,
decidable__equal_int,
int_seg_properties,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
less_than_wf,
top_wf,
assert_of_lt_int,
lt_int_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
replace-seq-from_wf,
cons-nat-seq_wf,
le_wf,
shift-seq_wf,
subtype_rel_self,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
equal_wf,
implies-quotient-true,
false_wf,
squash-from-quotient,
equiv_rel_true,
true_wf,
equal-wf-T-base,
iff_wf,
exists_wf,
quotient_wf,
squashed-continuity1-rel_wf,
nat_wf,
all_wf
Rules used in proof :
applyLambdaEquality,
int_eqReduceFalseSq,
promote_hyp,
imageMemberEquality,
sqequalAxiom,
isect_memberFormation,
lessCases,
int_eqReduceTrueSq,
equalityElimination,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
addEquality,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
productElimination,
voidElimination,
imageElimination,
independent_pairFormation,
natural_numberEquality,
productEquality,
independent_functionElimination,
rename,
setElimination,
dependent_functionElimination,
baseClosed,
intEquality,
independent_isectElimination,
because_Cache,
hypothesisEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
sqequalRule,
universeEquality,
hypothesis,
cumulativity,
functionEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
(\mforall{}A:\mBbbP{}. \00D9(\mexists{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (A \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. ((a n) = 1))))
{}\mRightarrow{} (\mneg{}(\mforall{}A:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. squashed-continuity1-rel(A)))
Date html generated:
2017_04_20-AM-07_35_56
Last ObjectModification:
2017_04_07-PM-06_39_38
Theory : continuity
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