Nuprl Lemma : regular-consistency

`∀[x,y:ℝ]. ∀[n,m:ℕ+].  ((m * |(x n) - y n|) ≤ ((n * |(x m) - y m|) + (4 * n) + (4 * m)))`

Proof

Definitions occuring in Statement :  real: `ℝ` absval: `|i|` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` le: `A ≤ B` apply: `f a` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` nat_plus: `ℕ+` real: `ℝ` subtype_rel: `A ⊆r B` prop: `ℙ` nat: `ℕ` sq_stable: `SqStable(P)` squash: `↓T` true: `True` top: `Top` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` subtract: `n - m` regular-int-seq: `k-regular-seq(f)` sq_type: `SQType(T)`
Lemmas referenced :  less_than'_wf absval_wf subtract_wf nat_plus_wf real_wf sq_stable__le nat_wf nat_plus_properties decidable__le less_than_wf satisfiable-full-omega-tt intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf le_wf squash_wf true_wf add_functionality_wrt_eq absval_sym iff_weakening_equal le_functionality le_weakening add_functionality_wrt_le int-triangle-inequality minus-add minus-minus add-associates minus-one-mul add-commutes add-mul-special add-swap zero-mul zero-add add-zero subtype_base_sq int_subtype_base equal_wf absval_pos nat_plus_subtype_nat absval_mul mul-distributes mul-associates mul-swap mul-distributes-right
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality because_Cache extract_by_obid isectElimination addEquality multiplyEquality setElimination rename hypothesis applyEquality natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality voidElimination functionExtensionality independent_functionElimination imageMemberEquality baseClosed imageElimination minusEquality voidEquality dependent_set_memberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality computeAll universeEquality instantiate cumulativity

Latex:
\mforall{}[x,y:\mBbbR{}].  \mforall{}[n,m:\mBbbN{}\msupplus{}].    ((m  *  |(x  n)  -  y  n|)  \mleq{}  ((n  *  |(x  m)  -  y  m|)  +  (4  *  n)  +  (4  *  m)))

Date html generated: 2017_10_02-PM-07_13_47
Last ObjectModification: 2017_07_28-AM-07_20_03

Theory : reals

Home Index