Nuprl Lemma : sublist_pair

`∀[T:Type]. ∀L:T List. ∀i,j:ℕ||L||.  [L[i]; L[j]] ⊆ L supposing i < j`

Proof

Definitions occuring in Statement :  sublist: `L1 ⊆ L2` select: `L[n]` length: `||as||` cons: `[a / b]` nil: `[]` list: `T List` int_seg: `{i..j-}` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` int_seg: `{i..j-}` sublist: `L1 ⊆ L2` exists: `∃x:A. B[x]` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than: `a < b` squash: `↓T` nat: `ℕ` less_than': `less_than'(a;b)` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` ge: `i ≥ j ` guard: `{T}` increasing: `increasing(f;k)` subtract: `n - m` 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` nequal: `a ≠ b ∈ T ` select: `L[n]` cons: `[a / b]` eq_int: `(i =z j)`
Lemmas referenced :  member-less_than length_of_cons_lemma length_of_nil_lemma ifthenelse_wf eq_int_wf int_seg_wf increasing_wf istype-void istype-le select_wf cons_wf int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt length_wf intformless_wf int_formula_prop_less_lemma nil_wf non_neg_length itermAdd_wf int_term_value_add_lemma istype-less_than length_wf_nat nat_properties list_wf istype-universe eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot neg_assert_of_eq_int decidable__equal_int int_subtype_base int_seg_subtype_special int_seg_cases
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation_alt cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis independent_isectElimination sqequalRule dependent_functionElimination Error :memTop,  dependent_pairFormation_alt lambdaEquality_alt productElimination imageElimination universeIsType natural_numberEquality productIsType dependent_set_memberEquality_alt independent_pairFormation voidElimination functionExtensionality applyEquality because_Cache functionIsType equalityIstype unionElimination approximateComputation independent_functionElimination int_eqEquality addEquality equalityTransitivity equalitySymmetry applyLambdaEquality inhabitedIsType instantiate universeEquality equalityElimination promote_hyp cumulativity intEquality hypothesis_subsumption

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}i,j:\mBbbN{}||L||.    [L[i];  L[j]]  \msubseteq{}  L  supposing  i  <  j

Date html generated: 2020_05_19-PM-09_42_10
Last ObjectModification: 2020_01_04-PM-08_26_15

Theory : list_1

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