### Nuprl Lemma : unsquashed-BIM-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_18
Last ObjectModification: 2016_11_08-PM-05_11_00

Theory : continuity

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