### Nuprl Lemma : monotone-bar-induction6

`∀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]))))`
`  `` ⇃(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` so_apply: `x[s1;s2]` uimplies: `b supposing a` guard: `{T}` int_upper: `{i...}` sq_stable: `SqStable(P)` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` false: `False` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x y.t[x; y]` ge: `i ≥ j ` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` isl: `isl(x)` ext2Baire: `ext2Baire(n;f;d)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` true: `True` outl: `outl(x)` seq-adjoin: `s++t` seq-add: `s.x@n` seq-append: `seq-append(n;m;s1;s2)` less_than: `a < b` nequal: `a ≠ b ∈ T ` cand: `A c∧ B` quotient: `x,y:A//B[x; y]`
Lemmas referenced :  strong-continuity-rel int_upper_wf upper_subtype_nat sq_stable__le istype-nat prop-truncation-quot decidable__le full-omega-unsat intformnot_wf intformle_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf istype-le int_seg_properties intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_seg_wf quotient_wf nat_wf true_wf istype-int_upper equiv_rel_true nat_properties itermAdd_wf int_term_value_add_lemma seq-add_wf unit_wf2 int_seg_subtype_nat istype-false subtype_rel_function subtype_rel_self union_subtype_base set_subtype_base lelt_wf int_subtype_base unit_subtype_base istype-assert btrue_wf bfalse_wf subtype_rel_union basic_bar_induction assert_wf decidable__assert seq-adjoin_wf ext2Baire_wf squash_wf bool_wf isl_wf lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot iff_weakening_uiff less_than_wf istype-less_than le_wf btrue_neq_bfalse decidable__lt intformeq_wf int_formula_prop_eq_lemma decidable__equal_int eq_int_wf assert_of_eq_int istype-top neg_assert_of_eq_int quotient-member-eq member_wf 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 because_Cache hypothesis sqequalRule functionEquality isectElimination setElimination independent_isectElimination natural_numberEquality independent_functionElimination imageMemberEquality baseClosed imageElimination Error :functionIsType,  Error :inhabitedIsType,  Error :dependent_set_memberEquality_alt,  unionElimination approximateComputation Error :dependent_pairFormation_alt,  Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  productElimination int_eqEquality independent_pairFormation Error :productIsType,  addEquality universeEquality Error :unionIsType,  instantiate Error :equalityIstype,  intEquality baseApply closedConclusion sqequalBase equalitySymmetry equalityTransitivity hyp_replacement Error :functionExtensionality_alt,  equalityElimination promote_hyp cumulativity applyLambdaEquality functionExtensionality int_eqReduceTrueSq lessCases Error :isect_memberFormation_alt,  axiomSqEquality Error :isectIsTypeImplies,  int_eqReduceFalseSq pointwiseFunctionality pertypeElimination

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{}  \00D9(Q[0;\mlambda{}x.\mbot{}]))

Date html generated: 2019_06_20-PM-02_56_28
Last ObjectModification: 2018_12_06-PM-11_35_06

Theory : continuity

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