### Nuprl Lemma : min-increasing-sequence-prop1

`∀b:ℕ ⟶ ℕ. ∀n,x,k:ℕ.  ((min-increasing-sequence(b;n;x) = (inl k) ∈ (ℕ?)) `` (x ≤ (b k)))`

Proof

Definitions occuring in Statement :  min-increasing-sequence: `min-increasing-sequence(a;n;k)` nat: `ℕ` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` isl: `isl(x)` min-increasing-sequence: `min-increasing-sequence(a;n;k)` exposed-bfalse: `exposed-bfalse` 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)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf istype-less_than le_witness_for_triv unit_wf2 min-increasing-sequence_wf decidable__le intformnot_wf int_formula_prop_not_lemma istype-le union_subtype_base nat_wf set_subtype_base le_wf int_subtype_base unit_subtype_base subtract-1-ge-0 istype-nat btrue_neq_bfalse bfalse_wf btrue_wf primrec0_lemma primrec-unroll lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma le_int_wf assert_of_le_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  productElimination equalityTransitivity equalitySymmetry Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :equalityIstype,  Error :unionIsType,  because_Cache Error :dependent_set_memberEquality_alt,  unionElimination applyEquality intEquality baseApply closedConclusion baseClosed sqequalBase Error :functionIsType,  applyLambdaEquality Error :equalityIsType4,  Error :productIsType,  equalityElimination promote_hyp instantiate cumulativity

Latex:
\mforall{}b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mforall{}n,x,k:\mBbbN{}.    ((min-increasing-sequence(b;n;x)  =  (inl  k))  {}\mRightarrow{}  (x  \mleq{}  (b  k)))

Date html generated: 2019_06_20-PM-03_07_13
Last ObjectModification: 2018_12_06-PM-11_57_09

Theory : continuity

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