Nuprl Lemma : select-remove-first

`∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)}  ⟶ 𝔹]. ∀[i:ℕ||remove-first(P;L)||].`
`  (remove-first(P;L)[i] ~ L[i] supposing ∀j:ℕi + 1. (¬↑(P L[j]))`
`  ∧ remove-first(P;L)[i] ~ L[i + 1] supposing ∃j:ℕi + 1. (↑(P L[j])))`

Proof

Definitions occuring in Statement :  remove-first: `remove-first(P;L)` l_member: `(x ∈ l)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type` sqequal: `s ~ 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` or: `P ∨ Q` remove-first: `remove-first(P;L)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` decidable: `Dec(P)` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` subtract: `n - m` l_all: `(∀x∈L.P[x])`
Lemmas referenced :  length-remove-first-le 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 less_than_wf list-cases list_ind_nil_lemma length_of_nil_lemma stuck-spread istype-base int_seg_properties int_seg_wf le_wf l_member_wf nil_wf bool_wf product_subtype_list colength-cons-not-zero colength_wf_list istype-false subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le list_ind_cons_lemma length_of_cons_lemma cons_member cons_wf subtype_rel_sets eqtt_to_assert eqff_to_assert bool_subtype_base bool_cases_sqequal assert-bnot nat_wf length_wf remove-first_wf list_wf list-subtype select_wf assert_wf not_wf lelt_wf decidable__lt false_wf select_cons_tl_sq int_seg_cases int_seg_subtype add-is-int-iff length_wf_nat subtract-add-cancel select_member add-member-int_seg2 add-subtract-cancel select-cons-tl all_wf length-remove-first subtract-is-int-iff exists_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis Error :lambdaFormation_alt,  setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  productElimination independent_pairEquality axiomSqEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  unionElimination baseClosed Error :functionIsType,  Error :setIsType,  promote_hyp hypothesis_subsumption Error :equalityIsType1,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination Error :equalityIsType4,  baseApply closedConclusion applyEquality intEquality Error :inlFormation_alt,  Error :functionExtensionality_alt,  Error :inrFormation_alt,  equalityElimination Error :equalityIsType3,  cumulativity universeEquality Error :productIsType,  setEquality functionExtensionality addEquality lambdaFormation voidEquality isect_memberEquality lambdaEquality dependent_pairFormation dependent_set_memberEquality inlFormation pointwiseFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].  \mforall{}[i:\mBbbN{}||remove-first(P;L)||].
(remove-first(P;L)[i]  \msim{}  L[i]  supposing  \mforall{}j:\mBbbN{}i  +  1.  (\mneg{}\muparrow{}(P  L[j]))
\mwedge{}  remove-first(P;L)[i]  \msim{}  L[i  +  1]  supposing  \mexists{}j:\mBbbN{}i  +  1.  (\muparrow{}(P  L[j])))

Date html generated: 2019_06_20-PM-01_42_54
Last ObjectModification: 2018_10_15-PM-05_48_05

Theory : list_1

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