Nuprl Lemma : unsquashed-monotone-bar-induction3-false
¬(∀B,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. B[n;f]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀m:ℕ. (B[n;s] ⇒ B[n + 1;s.m@n]))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ Q[n;s]))
⇒ Q[0;λx.⊥]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-add: s.x@n,
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 :
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
so_lambda: λ2x y.t[x; y],
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s],
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
false: False,
implies: P ⇒ Q,
not: ¬A
Lemmas referenced :
int_formula_prop_less_lemma,
intformless_wf,
int_seg_properties,
equiv_rel_true,
true_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
exists_wf,
quotient_wf,
seq-add_wf,
le_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,
int_seg_wf,
nat_wf,
all_wf,
unsquashed-weak-continuity-false2,
unsquashed-BIM-implies-unsquashed-weak-continuity
Rules used in proof :
productElimination,
computeAll,
independent_pairFormation,
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
addEquality,
dependent_set_memberEquality,
functionExtensionality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
sqequalRule,
universeEquality,
hypothesisEquality,
cumulativity,
lambdaEquality,
applyEquality,
functionEquality,
isectElimination,
instantiate,
voidElimination,
hypothesis,
thin,
independent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mneg{}(\mforall{}B,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{}. B[n;f]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. \mforall{}m:\mBbbN{}. (B[n;s] {}\mRightarrow{} B[n + 1;s.m@n]))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} Q[n;s]))
{}\mRightarrow{} Q[0;\mlambda{}x.\mbot{}]))
Date html generated:
2016_11_09-AM-06_18_14
Last ObjectModification:
2016_11_08-PM-05_09_31
Theory : continuity
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