### Nuprl Lemma : monotone-bar-induction8

`∀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` so_apply: `x[s]` int_upper: `{i...}` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` 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` true: `True` cand: `A c∧ B` outl: `outl(x)` isl: `isl(x)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  strong-continuity-rel all_wf int_upper_wf quotient_wf int_upper_subtype_nat subtype_rel_dep_function nat_wf int_seg_wf int_seg_subtype_nat false_wf subtype_rel_self true_wf equiv_rel_true prop-truncation-quot le_wf int_seg_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf exists_wf nat_properties decidable__le intformnot_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_add_lemma seq-add_wf unit_wf2 equal_wf subtype_rel_union assert_wf isl_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 less_than_wf decidable__equal_int lelt_wf bfalse_wf and_wf btrue_wf btrue_neq_bfalse intformeq_wf int_formula_prop_eq_lemma decidable__lt equal-wf-base-T int_subtype_base equiv_rel_wf seq-adjoin-is-seq-add iff_weakening_equal implies-quotient-true
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation rename cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin lambdaEquality isectElimination setElimination because_Cache hypothesis sqequalRule applyEquality functionExtensionality hypothesisEquality natural_numberEquality independent_isectElimination independent_pairFormation dependent_set_memberEquality productElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination functionEquality addEquality unionElimination cumulativity universeEquality unionEquality productEquality inlEquality imageElimination imageMemberEquality baseClosed addLevel hyp_replacement equalitySymmetry equalityTransitivity equalityElimination promote_hyp instantiate levelHypothesis applyLambdaEquality baseApply closedConclusion inrEquality

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

Date html generated: 2017_04_20-AM-07_20_50
Last ObjectModification: 2017_02_27-PM-05_56_04

Theory : continuity

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