### Nuprl Lemma : l_intersection-size

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b,c:T List].`
`  (no_repeats(T;a) `` no_repeats(T;b) `` a ⊆ c `` b ⊆ c `` ((||a|| + ||b||) ≤ (||c|| + ||(a ⋂ b)||)))`

Proof

Definitions occuring in Statement :  l_intersection: `(L1 ⋂ L2)` l_contains: `A ⊆ B` no_repeats: `no_repeats(T;l)` length: `||as||` list: `T List` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` le: `A ≤ B` implies: `P `` Q` add: `n + m` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` l_intersection: `(L1 ⋂ L2)` uimplies: `b supposing a` l_contains: `A ⊆ B` l_member: `(x ∈ l)` l_all: `(∀x∈L.P[x])` exists: `∃x:A. B[x]` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` lelt: `i ≤ j < k` and: `P ∧ Q` ge: `i ≥ j ` guard: `{T}` all: `∀x:A. B[x]` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` false: `False` nat: `ℕ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` less_than: `a < b` squash: `↓T` le: `A ≤ B` cand: `A c∧ B` pi1: `fst(t)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` inject: `Inj(A;B;f)` no_repeats: `no_repeats(T;l)` less_than': `less_than'(a;b)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` true: `True` label: `...\$L... t`
Lemmas referenced :  no_repeats_filter non_neg_length int_seg_properties length_wf decidable__le lelt_wf length_wf_nat nat_properties full-omega-unsat 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_not_lemma int_seg_wf select_wf decidable__lt int_formula_prop_less_lemma no_repeats_l_index-inj l_intersection_wf l_contains_wf no_repeats_wf le_witness_for_triv list_wf deq_wf less_than_wf nat_wf member-intersection select_member l_index_wf l_member_wf pigeon-hole add_nat_wf add-is-int-iff int_term_value_add_lemma int_formula_prop_eq_lemma false_wf le_wf lt_int_wf eqtt_to_assert assert_of_lt_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf subtract_wf intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_term_value_subtract_lemma int_formula_prop_wf itermAdd_wf deq-member_wf assert-deq-member add-member-int_seg1 intformeq_wf decidable__equal_int_seg int_seg_subtype_nat set_subtype_base int_subtype_base equal_wf squash_wf true_wf subtype_rel_self iff_weakening_equal not_wf decidable__equal_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache independent_isectElimination hypothesis sqequalRule promote_hyp productElimination Error :dependent_pairFormation_alt,  hypothesisEquality Error :functionExtensionality_alt,  Error :dependent_set_memberEquality_alt,  applyEquality independent_pairFormation natural_numberEquality setElimination rename dependent_functionElimination Error :universeIsType,  unionElimination equalityTransitivity equalitySymmetry applyLambdaEquality independent_functionElimination voidElimination approximateComputation Error :lambdaEquality_alt,  int_eqEquality Error :isect_memberEquality_alt,  Error :functionIsType,  Error :equalityIsType1,  imageElimination functionExtensionality cumulativity Error :inhabitedIsType,  universeEquality Error :productIsType,  addEquality pointwiseFunctionality baseApply closedConclusion baseClosed equalityElimination instantiate hyp_replacement Error :equalityIsType4,  intEquality imageMemberEquality productEquality Error :equalityIsType3,  Error :equalityIsType2

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[a,b,c:T  List].
(no\_repeats(T;a)
{}\mRightarrow{}  no\_repeats(T;b)
{}\mRightarrow{}  a  \msubseteq{}  c
{}\mRightarrow{}  b  \msubseteq{}  c
{}\mRightarrow{}  ((||a||  +  ||b||)  \mleq{}  (||c||  +  ||(a  \mcap{}  b)||)))

Date html generated: 2019_06_20-PM-01_58_00
Last ObjectModification: 2018_10_03-PM-10_28_04

Theory : decidable!equality

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