### 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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