### Nuprl Lemma : quick-find_wf

`∀[n:ℕ+]. ∀[p:{n...} ⟶ 𝔹].  quick-find(p;n) ∈ {m:{n...}| ↑(p m)}  supposing ∃N:{n...}. ∀m:{N...}. (↑(p m))`

Proof

Definitions occuring in Statement :  quick-find: `quick-find(p;n)` int_upper: `{i...}` nat_plus: `ℕ+` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` exists: `∃x:A. B[x]` prop: `ℙ` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` int_upper: `{i...}` subtype_rel: `A ⊆r B` guard: `{T}` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` so_apply: `x[s]` nat: `ℕ` ge: `i ≥ j ` quick-find: `quick-find(p;n)` 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` has-value: `(a)↓`
Lemmas referenced :  exists_wf int_upper_wf all_wf assert_wf int_upper_subtype_int_upper int_upper_properties nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf bool_wf nat_plus_wf nat_properties itermConstant_wf intformless_wf int_term_value_constant_lemma int_formula_prop_less_lemma ge_wf less_than_wf le_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma nat_wf itermAdd_wf int_term_value_add_lemma eqtt_to_assert eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot value-type-has-value int-value-type itermMultiply_wf int_term_value_mul_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry extract_by_obid isectElimination setElimination rename because_Cache lambdaEquality applyEquality functionExtensionality hypothesisEquality independent_isectElimination applyLambdaEquality dependent_functionElimination unionElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionEquality lambdaFormation intWeakElimination independent_functionElimination addEquality dependent_set_memberEquality equalityElimination promote_hyp instantiate cumulativity callbyvalueReduce multiplyEquality

Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[p:\{n...\}  {}\mrightarrow{}  \mBbbB{}].    quick-find(p;n)  \mmember{}  \{m:\{n...\}|  \muparrow{}(p  m)\}    supposing  \mexists{}N:\{n...\}.  \mforall{}m:\{N...\}.  (\muparrow{}\000C(p  m))

Date html generated: 2017_10_01-AM-09_15_24
Last ObjectModification: 2017_07_26-PM-04_50_09

Theory : general

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