### Nuprl Lemma : unsquashed-monotone-bar-induction3-false

¬(∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.
((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕQ[n 1;s.m@n])  Q[n;s]))
(∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕB[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: bottom: prop: so_apply: x[s1;s2] all: x:A. B[x] exists: x:A. B[x] not: ¬A implies:  Q true: True lambda: λx.A[x] function: x:A ⟶ B[x] add: m natural_number: \$n
Definitions unfolded in proof :  lelt: i ≤ j < k int_seg: {i..j-} guard: {T} so_lambda: λ2y.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: supposing a or: P ∨ Q decidable: Dec(P) all: x:A. B[x] ge: i ≥  so_apply: x[s1;s2] so_lambda: λ2x.t[x] nat: subtype_rel: A ⊆B uall: [x:A]. B[x] prop: member: t ∈ T false: False implies:  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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