### Nuprl Lemma : monotone-bar-induction-strict2

`∀B,Q:n:ℕ ⟶ {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)}  ⟶ ℙ.`
`  ((∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .`
`      (B[n;s] `` (∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) `` B[n + 1;s.m@n]))))`
`  `` (∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .  (B[n;s] `` ⇃(Q[n;s])))`
`  `` (∀n:ℕ. ∀s:{s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)} .`
`        ((∀m:ℕ. (strictly-increasing-seq(n + 1;s.m@n) `` ⇃(Q[n + 1;s.m@n]))) `` ⇃(Q[n;s])))`
`  `` (∀alpha:StrictInc. ∃m:ℕ. B[m;alpha])`
`  `` ⇃(Q[0;λx.⊥]))`

Proof

Definitions occuring in Statement :  strict-inc: `StrictInc` quotient: `x,y:A//B[x; y]` strictly-increasing-seq: `strictly-increasing-seq(n;s)` 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]` implies: `P `` Q` true: `True` set: `{x:A| B[x]} ` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` 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` 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_lambda: `λ2x y.t[x; y]` strictly-increasing-seq: `strictly-increasing-seq(n;s)` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` squash: `↓T` seq-add: `s.x@n` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` guard: `{T}` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` true: `True` quotient: `x,y:A//B[x; y]` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` strict-inc: `StrictInc`
Lemmas referenced :  all_wf strict-inc_wf exists_wf nat_wf strict-inc-subtype int_seg_wf strictly-increasing-seq_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 true_wf equiv_rel_true monotone-bar-induction2 decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf less_than_wf squash_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int int_seg_properties intformeq_wf int_formula_prop_eq_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int decidable__strictly-increasing-seq quotient-member-eq equal-wf-base it_wf top_wf subtype_rel_dep_function unit_wf2 make-strict_wf int_seg_subtype_nat false_wf subtype_rel_self make-strict-agrees iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality hypothesisEquality dependent_functionElimination setEquality functionEquality natural_numberEquality setElimination rename because_Cache dependent_set_memberEquality addEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll universeEquality cumulativity independent_functionElimination productElimination hyp_replacement equalitySymmetry imageElimination equalityTransitivity equalityElimination int_eqReduceTrueSq promote_hyp instantiate int_eqReduceFalseSq imageMemberEquality baseClosed pointwiseFunctionalityForEquality pertypeElimination productEquality applyLambdaEquality

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

Date html generated: 2017_04_20-AM-07_23_43
Last ObjectModification: 2017_02_27-PM-05_59_45

Theory : continuity

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