Nuprl Lemma : unsquashed-monotone-bar-induction8-false
¬(∀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. ∀m:{n...}. Q[m;f])) ⇒ Q[0;λx.⊥]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-add: s.x@n,
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
bottom: ⊥,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
true: True,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
false: False,
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],
top: Top,
and: P ∧ Q,
so_apply: x[s],
int_upper: {i...},
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,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b
Lemmas referenced :
unsquashed-weak-continuity-false2,
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,
int_upper_wf,
int_upper_subtype_nat,
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_properties,
int_upper_subtype_int_upper,
rep-seq-from-prop3,
squash_wf,
iff_weakening_equal,
strong-continuity2-implies-weak,
implies-quotient-true,
decidable__lt,
lelt_wf,
rep-seq-from-prop1,
rep-seq-from-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
independent_functionElimination,
thin,
functionEquality,
hypothesis,
voidElimination,
instantiate,
isectElimination,
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,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mneg{}(\mforall{}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{}. \mforall{}m:\{n...\}. Q[m;f]))
{}\mRightarrow{} Q[0;\mlambda{}x.\mbot{}]))
Date html generated:
2017_04_20-AM-07_21_25
Last ObjectModification:
2017_02_27-PM-05_57_33
Theory : continuity
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