### Nuprl Lemma : monotone-bar-induction5

`∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.`
`  ((∀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. ⇃(∃n:ℕ. (B[n;alpha] ∧ (∀m:{n...}. (B[m;alpha] `` B[m + 1;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` and: `P ∧ Q` true: `True` 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` and: `P ∧ Q` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` sq_stable: `SqStable(P)` squash: `↓T` int_upper: `{i...}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x y.t[x; y]` guard: `{T}` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` assert: `↑b` bnot: `¬bb` sq_type: `SQType(T)` bfalse: `ff` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` ext2Baire: `ext2Baire(n;f;d)` cand: `A c∧ B` outl: `outl(x)` subtract: `n - m` nequal: `a ≠ b ∈ T ` less_than: `a < b` seq-append: `seq-append(n;m;s1;s2)` seq-add: `s.x@n` seq-adjoin: `s++t` quotient: `x,y:A//B[x; y]` isl: `isl(x)`
Lemmas referenced :  strong-continuity-rel subtype_rel_function nat_wf int_seg_wf int_seg_subtype_nat istype-false subtype_rel_self all_wf int_upper_wf upper_subtype_nat sq_stable__le int_upper_properties nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void 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 istype-le istype-int_upper prop-truncation-quot int_seg_properties intformless_wf int_formula_prop_less_lemma istype-nat quotient_wf exists_wf true_wf equiv_rel_true seq-add_wf isl_wf assert_wf int_subtype_base lelt_wf set_subtype_base subtype_rel_union le_wf int_formula_prop_eq_lemma intformeq_wf decidable__equal_int unit_wf2 seq-adjoin_wf decidable__assert basic_bar_induction ext2Baire_wf less_than_wf iff_weakening_uiff assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal eqff_to_assert assert_of_lt_int eqtt_to_assert lt_int_wf bool_wf squash_wf btrue_neq_bfalse bfalse_wf btrue_wf iff_imp_equal_bool equal_wf decidable__lt zero-add zero-mul add-mul-special add-associates add-commutes minus-one-mul int_term_value_subtract_lemma itermSubtract_wf subtract_wf primrec-wf2 int_seg_subtype add-zero subtract-add-cancel neg_assert_of_eq_int istype-top assert_of_eq_int eq_int_wf member_wf quotient-member-eq union_subtype_base unit_subtype_base istype-assert istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  rename cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin Error :lambdaEquality_alt,  productEquality applyEquality hypothesisEquality isectElimination hypothesis because_Cache natural_numberEquality setElimination independent_isectElimination sqequalRule independent_pairFormation functionEquality independent_functionElimination imageMemberEquality baseClosed imageElimination Error :dependent_set_memberEquality_alt,  addEquality unionElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  productElimination Error :functionIsType,  Error :inhabitedIsType,  Error :productIsType,  universeEquality intEquality closedConclusion baseApply Error :equalityIsType3,  equalitySymmetry equalityTransitivity functionExtensionality instantiate Error :unionIsType,  cumulativity promote_hyp Error :equalityIsType1,  equalityElimination Error :functionExtensionality_alt,  hyp_replacement applyLambdaEquality Error :inrEquality_alt,  Error :inlEquality_alt,  unionEquality Error :equalityIsType4,  Error :setIsType,  int_eqReduceFalseSq axiomSqEquality Error :isect_memberFormation_alt,  lessCases int_eqReduceTrueSq pertypeElimination pointwiseFunctionality Error :equalityIstype,  sqequalBase

Latex:
\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{}  \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{}n:\mBbbN{}.  (B[n;alpha]  \mwedge{}  (\mforall{}m:\{n...\}.  (B[m;alpha]  {}\mRightarrow{}  B[m  +  1;alpha])))))
{}\mRightarrow{}  \00D9(Q[0;\mlambda{}x.\mbot{}]))

Date html generated: 2019_06_20-PM-02_56_12
Last ObjectModification: 2018_12_06-PM-11_35_34

Theory : continuity

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