### Nuprl Lemma : bag-in-subtype

`∀[A,B:Type].  ∀[b:bag(B)]. b ∈ bag(A) supposing ∀x:B. (x ↓∈ b `` (x ∈ A)) supposing strong-subtype(A;B)`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag: `bag(T)` strong-subtype: `strong-subtype(A;B)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` sq_stable: `SqStable(P)` implies: `P `` Q` uiff: `uiff(P;Q)` and: `P ∧ Q` squash: `↓T` prop: `ℙ` all: `∀x:A. B[x]` respects-equality: `respects-equality(S;T)` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` nat: `ℕ` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` or: `P ∨ Q` subtype_rel: `A ⊆r B` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` bag-append: `as + bs` append: `as @ bs` list_ind: list_ind single-bag: `{x}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_or: `a ↓∨ b`
Lemmas referenced :  sq_stable__respects-equality strong-subtype-iff-respects-equality respects-equality-bag bag_wf strong-subtype_wf istype-universe bag-member_wf equal-wf bag_to_squash_list all_wf member_wf 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 list-cases list-subtype-bag nil_wf product_subtype_list colength-cons-not-zero colength_wf_list istype-le list_wf 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 le_wf cons_wf istype-nat bag-append_wf single-bag_wf bag-member-append bag-member-single
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis independent_functionElimination because_Cache productElimination independent_isectElimination sqequalRule imageMemberEquality baseClosed imageElimination universeIsType inhabitedIsType instantiate universeEquality functionIsType equalityIstype dependent_functionElimination promote_hyp equalitySymmetry hyp_replacement applyLambdaEquality lambdaEquality_alt functionEquality isectEquality cumulativity rename lambdaFormation_alt setElimination intWeakElimination natural_numberEquality approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation axiomEquality equalityTransitivity functionIsTypeImplies unionElimination voidEquality closedConclusion applyEquality hypothesis_subsumption dependent_set_memberEquality_alt baseApply intEquality sqequalBase inlFormation_alt inrFormation_alt

Latex:
\mforall{}[A,B:Type].
\mforall{}[b:bag(B)].  b  \mmember{}  bag(A)  supposing  \mforall{}x:B.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  (x  \mmember{}  A))  supposing  strong-subtype(A;B)

Date html generated: 2019_10_15-AM-11_01_23
Last ObjectModification: 2019_08_15-PM-03_41_06

Theory : bags

Home Index