### Nuprl Lemma : l-first-when-none

`∀[T:Type]. ∀[f:T ⟶ 𝔹]. ∀[L:T List].  l-first(x.f[x];L) ~ inr (λx.Ax)  supposing (∀x∈L.¬↑f[x])`

Proof

Definitions occuring in Statement :  l-first: `l-first(x.f[x];L)` l_all: `(∀x∈L.P[x])` list: `T List` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` not: `¬A` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` inr: `inr x ` universe: `Type` sqequal: `s ~ t` axiom: `Ax`
Definitions unfolded in proof :  l-first: `l-first(x.f[x];L)` 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` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` exposed-it: `exposed-it` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` nat_plus: `ℕ+` true: `True` select: `L[n]` subtract: `n - m`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 less_than_wf l_all_wf not_wf assert_wf l_member_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_ind_nil_lemma l_all_wf_nil product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_ind_cons_lemma bool_wf eqtt_to_assert cons_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot list_wf length_of_cons_lemma false_wf add_nat_plus length_wf_nat nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff lelt_wf length_wf add-member-int_seg2 non_neg_length select-cons-tl int_seg_properties add-subtract-cancel int_seg_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity applyEquality functionExtensionality setEquality equalityTransitivity equalitySymmetry because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination functionEquality universeEquality imageMemberEquality pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:T  List].    l-first(x.f[x];L)  \msim{}  inr  (\mlambda{}x.Ax)    supposing  (\mforall{}x\mmember{}L.\mneg{}\muparrow{}f[x])

Date html generated: 2017_04_17-AM-07_24_54
Last ObjectModification: 2017_02_27-PM-04_03_48

Theory : list_1

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