### Nuprl Lemma : baf-bar-monotone

`∀R,T:ℕ ⟶ ℕ ⟶ ℙ. ∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .`
`  (baf-bar(n,m.R[n;m];n,m.T[n;m];n;s)`
`  `` (∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) `` baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;s.m@n))))`

Proof

Definitions occuring in Statement :  baf-bar: `baf-bar(n,m.R[n; m];n,m.T[n; m];l;a)` strictly-increasing-seq: `strictly-increasing-seq(n;s)` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` baf-bar: `baf-bar(n,m.R[n; m];n,m.T[n; m];l;a)` and: `P ∧ Q` exists: `∃x:A. B[x]` member: `t ∈ T` int_seg: `{i..j-}` lelt: `i ≤ j < k` uall: `∀[x:A]. B[x]` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` less_than: `a < b` guard: `{T}` so_apply: `x[s1;s2]` seq-add: `s.x@n` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` so_lambda: `λ2x y.t[x; y]`
Lemmas referenced :  nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf lelt_wf int_seg_properties eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf intformle_wf int_formula_prop_eq_lemma int_formula_prop_le_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int nat_wf decidable__equal_int le_wf decidable__le seq-add_wf exists_wf int_seg_wf strictly-increasing-seq_wf baf-bar_wf set_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution independent_pairFormation productElimination thin promote_hyp hypothesis dependent_pairFormation setElimination rename dependent_set_memberEquality hypothesisEquality cut introduction extract_by_obid isectElimination dependent_functionElimination addEquality natural_numberEquality unionElimination independent_isectElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule computeAll because_Cache addLevel hyp_replacement equalitySymmetry applyEquality equalityElimination equalityTransitivity int_eqReduceTrueSq instantiate cumulativity independent_functionElimination int_eqReduceFalseSq functionExtensionality levelHypothesis productEquality universeEquality functionEquality

Latex:
\mforall{}R,T:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  \mforall{}n:\mBbbN{}.  \mforall{}s:\{s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}|  strictly-increasing-seq(n;s)\}  .
(baf-bar(n,m.R[n;m];n,m.T[n;m];n;s)
{}\mRightarrow{}  (\mforall{}m:\mBbbN{}.  (strictly-increasing-seq(n  +  1;s.m@n)  {}\mRightarrow{}  baf-bar(n,m.R[n;m];n,m.T[n;m];n  +  1;s.m@n))))

Date html generated: 2017_04_20-AM-07_26_28
Last ObjectModification: 2017_02_27-PM-05_59_51

Theory : continuity

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