### Nuprl Lemma : gen-bar-ind-implies-monotone

`(∀P:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ`
`   ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. P[n + 1;s.m@n]) `` P[n;s])) `` (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. ∀m:{n...}. P[m;f])) `` ⇃(P[0;λx.⊥])))`
` (∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ. ∀bar:∀s:ℕ ⟶ ℕ. ⇃(∃n:ℕ. B[n;s]). ∀mon:∀n:ℕ. ∀m:ℕn. ∀s:ℕn ⟶ ℕ.  (B[m;s] `` B[n;s]).`
`    ∀base:∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` Q[n;s]). ∀ind:∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. Q[n + 1;s.m@n]) `` Q[n;s]).`
`      ⇃(Q[0;seq-normalize(0;⊥)]))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` seq-normalize: `seq-normalize(n;s)` 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]` nat: `ℕ` 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]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` so_apply: `x[s]` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` so_lambda: `λ2x y.t[x; y]` int_upper: `{i...}` squash: `↓T` label: `...\$L... t` sq_type: `SQType(T)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  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 int_seg_subtype_nat false_wf subtype_rel_dep_function int_seg_subtype int_seg_properties intformless_wf int_formula_prop_less_lemma subtype_rel_self quotient_wf exists_wf true_wf equiv_rel_true int_upper_wf int_upper_subtype_nat implies-quotient-true subtract_wf int_upper_properties 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 decidable__lt lelt_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination functionEquality introduction extract_by_obid isectElimination sqequalRule lambdaEquality natural_numberEquality setElimination rename because_Cache applyEquality functionExtensionality dependent_set_memberEquality addEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll universeEquality productElimination cumulativity instantiate imageElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed hyp_replacement applyLambdaEquality

Latex:
(\mforall{}P:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  P[n  +  1;s.m@n])  {}\mRightarrow{}  P[n;s]))
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  P[m;f]))
{}\mRightarrow{}  \00D9(P[0;\mlambda{}x.\mbot{}])))
{}\mRightarrow{}  (\mforall{}B,Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.  \mforall{}bar:\mforall{}s:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  B[n;s]).  \mforall{}mon:\mforall{}n:\mBbbN{}.  \mforall{}m:\mBbbN{}n.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.
(B[m;s]  {}\mRightarrow{}  B[n;s]).
\mforall{}base:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  Q[n;s]).  \mforall{}ind:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.
((\mforall{}m:\mBbbN{}.  Q[n  +  1;s.m@n])  {}\mRightarrow{}  Q[n;s]).
\00D9(Q[0;seq-normalize(0;\mbot{})]))

Date html generated: 2017_04_20-AM-07_35_16
Last ObjectModification: 2017_02_27-PM-06_03_39

Theory : continuity

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