`∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹].  (remove_leading(x.P[x];L) ∈ {L:T List| (¬↑null(L)) `` (¬↑P[hd(L)])} )`

Proof

Definitions occuring in Statement :  remove_leading: `remove_leading(a.P[a];L)` hd: `hd(l)` null: `null(as)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` not: `¬A` implies: `P `` Q` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
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` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` remove_leading: `remove_leading(a.P[a];L)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` bfalse: `ff` sq_stable: `SqStable(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` le: `A ≤ B` bool: `𝔹` unit: `Unit` bnot: `¬bb`
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 bool_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases 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_wf list_ind_nil_lemma nil_wf subtype_rel_list null_nil_lemma null_cons_lemma assert_wf hd_wf length_of_nil_lemma length_of_cons_lemma length_wf_nat sq_stable__le not_wf null_wf3 top_wf false_wf not-ge-2 condition-implies-le minus-add minus-one-mul add-swap minus-one-mul-top add-associates add-commutes add_functionality_wrt_le add-zero le-add-cancel2 list_ind_cons_lemma eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot reduce_hd_cons_lemma length_cons_ge_one cons_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry functionEquality cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality functionExtensionality imageMemberEquality minusEquality equalityElimination

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].
(remove\_leading(x.P[x];L)  \mmember{}  \{L:T  List|  (\mneg{}\muparrow{}null(L))  {}\mRightarrow{}  (\mneg{}\muparrow{}P[hd(L)])\}  )

Date html generated: 2018_05_21-PM-06_43_01
Last ObjectModification: 2017_07_26-PM-04_54_36

Theory : general

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