### Nuprl Lemma : monotone-bar-induction8-2

`∀Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ`
`  ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. Q[n + 1;s.m@n]) `` Q[n;s])) `` (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. ∀m:{n...}. Q[m;f])) `` ⇃(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` 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` uall: `∀[x:A]. B[x]` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` sq_stable: `SqStable(P)` squash: `↓T` int_upper: `{i...}` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_apply: `x[s]` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_lambda: `λ2x y.t[x; y]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` 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` cand: `A c∧ B` outl: `outl(x)` isl: `isl(x)`
Lemmas referenced :  strong-continuity-rel all_wf int_upper_wf nat_wf upper_subtype_nat sq_stable__le subtype_rel_function int_seg_wf int_seg_subtype_nat istype-false subtype_rel_self implies-quotient-true exists_wf unit_wf2 equal-wf-T-base subtype_rel_union set_subtype_base lelt_wf istype-int int_subtype_base assert_wf isl_wf le_wf int_seg_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma istype-void int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf quotient_wf true_wf equiv_rel_true nat_properties decidable__equal_int intformnot_wf intformeq_wf itermAdd_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le seq-add_wf basic_bar_induction decidable__assert seq-adjoin_wf ext2Baire_wf squash_wf bool_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 iff_imp_equal_bool btrue_wf bfalse_wf btrue_neq_bfalse decidable__lt equal-wf-base-T seq-adjoin-is-seq-add iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  rename cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin Error :lambdaEquality_alt,  isectElimination setElimination because_Cache hypothesis sqequalRule applyEquality functionExtensionality hypothesisEquality independent_isectElimination natural_numberEquality independent_functionElimination imageMemberEquality baseClosed imageElimination independent_pairFormation Error :universeIsType,  Error :functionIsType,  Error :inhabitedIsType,  functionEquality unionEquality productEquality productElimination baseApply closedConclusion intEquality Error :unionIsType,  Error :dependent_set_memberEquality_alt,  approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :productIsType,  Error :equalityIsType3,  instantiate universeEquality addEquality unionElimination equalityTransitivity equalitySymmetry hyp_replacement Error :functionExtensionality_alt,  equalityElimination Error :equalityIsType2,  promote_hyp cumulativity Error :equalityIsType1,  applyLambdaEquality Error :equalityIsType4,  Error :inrEquality_alt

Latex:
\mforall{}Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  Q[n  +  1;s.m@n])  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  Q[m;f]))
{}\mRightarrow{}  \00D9(Q[0;\mlambda{}x.\mbot{}]))

Date html generated: 2019_06_20-PM-02_56_49
Last ObjectModification: 2018_10_04-PM-11_40_03

Theory : continuity

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