### Nuprl Lemma : find-ge-val_wf

`∀[T:Type]`
`  ∀[test:T ⟶ 𝔹]. ∀[n:ℤ]. ∀[f:{n...} ⟶ T].`
`    find-ge-val(test;f;n) ∈ v:T × {n':ℤ| (n ≤ n') ∧ (v = (f n') ∈ T) ∧ test v = tt}  `
`    supposing ∃m:{n...}. ∀k:{m...}. test (f k) = tt `
`  supposing value-type(T)`

Proof

Definitions occuring in Statement :  find-ge-val: `find-ge-val(test;f;n)` int_upper: `{i...}` value-type: `value-type(T)` btrue: `tt` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` product: `x:A × B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` find-ge-val: `find-ge-val(test;f;n)` int_upper: `{i...}` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` nat_plus: `ℕ+` spreadn: spread3 and: `P ∧ Q` subtype_rel: `A ⊆r B` guard: `{T}` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  find-xover-val_wf decidable__le full-omega-unsat intformnot_wf intformle_wf itermVar_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf istype-le decidable__lt intformless_wf itermConstant_wf int_formula_prop_less_lemma int_term_value_constant_lemma istype-less_than intformand_wf int_formula_prop_and_lemma bool_wf btrue_wf istype-int_upper upper_subtype_upper sq_stable__le value-type_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis Error :dependent_set_memberEquality_alt,  productElimination dependent_functionElimination unionElimination natural_numberEquality approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  Error :inhabitedIsType,  Error :lambdaFormation_alt,  spreadEquality Error :dependent_pairEquality_alt,  Error :setIsType,  Error :productIsType,  Error :equalityIstype,  applyEquality setElimination rename independent_pairFormation equalityTransitivity equalitySymmetry axiomEquality Error :functionIsType,  imageMemberEquality baseClosed imageElimination Error :isectIsTypeImplies,  instantiate universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}[test:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[n:\mBbbZ{}].  \mforall{}[f:\{n...\}  {}\mrightarrow{}  T].
find-ge-val(test;f;n)  \mmember{}  v:T  \mtimes{}  \{n':\mBbbZ{}|  (n  \mleq{}  n')  \mwedge{}  (v  =  (f  n'))  \mwedge{}  test  v  =  tt\}
supposing  \mexists{}m:\{n...\}.  \mforall{}k:\{m...\}.  test  (f  k)  =  tt
supposing  value-type(T)

Date html generated: 2019_06_20-PM-01_16_37
Last ObjectModification: 2019_01_09-PM-04_13_25

Theory : int_2

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