### Nuprl Lemma : first-success-is-inl

`∀[T:Type]. ∀[A:T ⟶ Type]. ∀[f:x:T ⟶ (A[x]?)]. ∀[L:T List]. ∀[j:ℕ||L||]. ∀[a:A[L[j]]].`
`  (first-success(f;L) = (inl <j, a>) ∈ (i:ℕ||L|| × A[L[i]]?)`
`  `⇐⇒` j < ||L|| ∧ ((f L[j]) = (inl a) ∈ (A[L[j]]?)) ∧ (∀x∈firstn(j;L).↑isr(f x)))`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` l_all: `(∀x∈L.P[x])` first-success: `first-success(f;L)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` assert: `↑b` isr: `isr(x)` less_than: `a < b` uall: `∀[x:A]. B[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` and: `P ∧ Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` pair: `<a, b>` product: `x:A × B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` 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` iff: `P `⇐⇒` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than: `a < b` squash: `↓T` guard: `{T}` rev_implies: `P `` Q` l_all: `(∀x∈L.P[x])` or: `P ∨ Q` first-success: `first-success(f;L)` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_lambda: so_lambda3 so_apply: `x[s1;s2;s3]` cons: `[a / b]` decidable: `Dec(P)` isr: `isr(x)` colength: `colength(L)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than': `less_than'(a;b)` subtype_rel: `A ⊆r B` firstn: `firstn(n;as)` list_ind: list_ind uiff: `uiff(P;Q)` pi2: `snd(t)` pi1: `fst(t)` cand: `A c∧ B` bfalse: `ff` btrue: `tt` ifthenelse: `if b then t else f fi ` assert: `↑b` outl: `outl(x)` isl: `isl(x)` true: `True` unit: `Unit` bool: `𝔹` bnot: `¬bb` istype: `istype(T)` subtract: `n - m`
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 member-less_than int_seg_properties intformeq_wf int_formula_prop_eq_lemma assert_witness list-cases length_of_nil_lemma stuck-spread istype-base list_ind_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le select_wf firstn_wf decidable__lt length_wf bfalse_wf btrue_wf subtract-1-ge-0 subtype_base_sq set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf length_of_cons_lemma list_ind_cons_lemma istype-nat list_wf unit_wf2 istype-universe satisfiable-full-omega-tt equal-wf-base-T less_than_wf equal_wf l_all_wf nil_wf assert_wf isr_wf l_member_wf int_seg_wf non_neg_length add-is-int-iff false_wf cons_wf istype-false lelt_wf first0 subtype_rel_list top_wf l_all_nil assert_of_bnot iff_weakening_uiff iff_transitivity eqff_to_assert assert_of_lt_int eqtt_to_assert bool_subtype_base bool_wf bool_cases lt_int_wf bnot_wf not_wf istype-assert l_all_cons outl_wf true_wf first-success_wf assert-bnot bool_cases_sqequal iff_weakening_equal select_cons_tl squash_wf subtype_rel_wf add-subtract-cancel select-cons-tl subtype_rel-equal add-member-int_seg2 equal_functionality_wrt_subtype_rel2 subtype_rel_self subtype_rel_union select-cons assert_of_le_int le_int_wf l_all_wf_nil btrue_neq_bfalse select-cons-hd isr-first-success
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut thin lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType productElimination independent_pairEquality imageElimination equalityTransitivity equalitySymmetry applyLambdaEquality axiomEquality functionIsTypeImplies inhabitedIsType isectIsTypeImplies unionElimination baseClosed promote_hyp hypothesis_subsumption equalityIstype because_Cache dependent_set_memberEquality_alt applyEquality instantiate baseApply closedConclusion intEquality sqequalBase functionIsType unionIsType universeEquality isect_memberFormation lambdaFormation dependent_pairFormation lambdaEquality isect_memberEquality voidEquality computeAll unionEquality productEquality inlEquality dependent_pairEquality functionExtensionality cumulativity setEquality productIsType addEquality inlEquality_alt dependent_pairEquality_alt pointwiseFunctionality setIsType Error :memTop,  hyp_replacement equalityIsType1 equalityIsType2 equalityElimination isectIsType imageMemberEquality equalityIsType3

Latex:
\mforall{}[T:Type].  \mforall{}[A:T  {}\mrightarrow{}  Type].  \mforall{}[f:x:T  {}\mrightarrow{}  (A[x]?)].  \mforall{}[L:T  List].  \mforall{}[j:\mBbbN{}||L||].  \mforall{}[a:A[L[j]]].
(first-success(f;L)  =  (inl  <j,  a>)  \mLeftarrow{}{}\mRightarrow{}  j  <  ||L||  \mwedge{}  ((f  L[j])  =  (inl  a))  \mwedge{}  (\mforall{}x\mmember{}firstn(j;L).\muparrow{}isr(f  x\000C)))

Date html generated: 2020_05_19-PM-09_41_49
Last ObjectModification: 2019_12_31-PM-07_21_56

Theory : list_1

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