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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