### Nuprl Lemma : rminimum_wf

`∀[n,m:ℤ].  ∀[x:{n..m + 1-} ⟶ ℝ]. (rminimum(n;m;k.x[k]) ∈ ℝ) supposing n ≤ m`

Proof

Definitions occuring in Statement :  rminimum: `rminimum(n;m;k.x[k])` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` member: `t ∈ T` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  squash: `↓T` less_than: `a < b` le: `A ≤ B` lelt: `i ≤ j < k` int_seg: `{i..j-}` so_apply: `x[s]` prop: `ℙ` and: `P ∧ Q` top: `Top` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` implies: `P `` Q` not: `¬A` or: `P ∨ Q` decidable: `Dec(P)` all: `∀x:A. B[x]` nat: `ℕ` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` rminimum: `rminimum(n;m;k.x[k])`
Lemmas referenced :  int_seg_wf int_seg_properties rmin_wf istype-less_than int_term_value_add_lemma int_formula_prop_less_lemma itermAdd_wf intformless_wf decidable__lt istype-le int_formula_prop_wf int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma istype-void int_formula_prop_and_lemma istype-int itermVar_wf itermSubtract_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf full-omega-unsat decidable__le subtract_wf real_wf primrec_wf
Rules used in proof :  inhabitedIsType isectIsTypeImplies functionIsType equalitySymmetry equalityTransitivity axiomEquality imageElimination productElimination rename setElimination productIsType addEquality applyEquality universeIsType independent_pairFormation voidElimination isect_memberEquality_alt int_eqEquality lambdaEquality_alt dependent_pairFormation_alt independent_functionElimination approximateComputation independent_isectElimination unionElimination natural_numberEquality dependent_functionElimination hypothesisEquality dependent_set_memberEquality_alt hypothesis thin isectElimination sqequalHypSubstitution extract_by_obid cut introduction isect_memberFormation_alt computationStep sqequalTransitivity sqequalReflexivity sqequalRule sqequalSubstitution

Latex:
\mforall{}[n,m:\mBbbZ{}].    \mforall{}[x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].  (rminimum(n;m;k.x[k])  \mmember{}  \mBbbR{})  supposing  n  \mleq{}  m

Date html generated: 2019_11_06-PM-00_29_20
Last ObjectModification: 2019_11_05-AM-11_55_32

Theory : reals

Home Index