### Nuprl Lemma : monotone-bar-induction8-implies-3

`(∀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.⊥])))`
` (∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.`
`      ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` (∀m:ℕ. B[n + 1;s.m@n])))`
`      `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` ⇃(Q[n;s])))`
`      `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. ⇃(Q[n + 1;s.m@n])) `` ⇃(Q[n;s])))`
`      `` (∀alpha:ℕ ⟶ ℕ. ⇃(∃m:ℕ. B[m;alpha]))`
`      `` ⇃(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]` implies: `P `` Q` true: `True` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` so_apply: `x[s]` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` exists: `∃x:A. B[x]` so_lambda: `λ2x y.t[x; y]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` int_upper: `{i...}` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` squash: `↓T` label: `...\$L... t` sq_type: `SQType(T)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  nat_wf all_wf quotient_wf exists_wf subtype_rel_dep_function int_seg_wf int_seg_subtype_nat false_wf subtype_rel_self true_wf equiv_rel_true 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 int_upper_wf int_upper_subtype_nat int_seg_properties intformless_wf int_formula_prop_less_lemma implies-quotient-true int_upper_properties subtract_wf itermSubtract_wf int_term_value_subtract_lemma decidable__equal_int intformeq_wf int_formula_prop_eq_lemma equal_wf squash_wf subtype_base_sq int_subtype_base iff_weakening_equal add-zero set_wf less_than_wf primrec-wf2 seq-add-same
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination functionEquality introduction extract_by_obid isectElimination because_Cache sqequalRule lambdaEquality applyEquality functionExtensionality natural_numberEquality setElimination rename independent_isectElimination independent_pairFormation dependent_set_memberEquality addEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll cumulativity universeEquality instantiate productElimination imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed hyp_replacement

Latex:
(\mforall{}Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  \00D9(Q[n  +  1;s.m@n]))  {}\mRightarrow{}  \00D9(Q[n;s])))
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  \00D9(Q[m;f])))
{}\mRightarrow{}  \00D9(Q[0;\mlambda{}x.\mbot{}])))
{}\mRightarrow{}  (\mforall{}B,Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  (\mforall{}m:\mBbbN{}.  B[n  +  1;s.m@n])))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  \00D9(Q[n;s])))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  \00D9(Q[n  +  1;s.m@n]))  {}\mRightarrow{}  \00D9(Q[n;s])))
{}\mRightarrow{}  (\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}m:\mBbbN{}.  B[m;alpha]))
{}\mRightarrow{}  \00D9(Q[0;\mlambda{}x.\mbot{}])))

Date html generated: 2017_04_20-AM-07_22_02
Last ObjectModification: 2017_02_27-PM-05_57_28

Theory : continuity

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