`∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹].  uiff(↑null(remove_leading(x.P[x];L));(∀x∈L.↑P[x]))`

Proof

Definitions occuring in Statement :  remove_leading: `remove_leading(a.P[a];L)` l_all: `(∀x∈L.P[x])` null: `null(as)` list: `T List` assert: `↑b` bool: `𝔹` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` top: `Top` prop: `ℙ` or: `P ∨ Q` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` not: `¬A` true: `True` false: `False` cons: `[a / b]` bfalse: `ff` guard: `{T}` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` less_than': `less_than'(a;b)` listp: `A List+` sq_type: `SQType(T)` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` less_than: `a < b` squash: `↓T`
Lemmas referenced :  remove_leading_property remove_leading_wf set_wf list_wf not_wf assert_wf null_wf3 subtype_rel_list top_wf hd_wf listp_properties list-cases length_of_nil_lemma null_nil_lemma product_subtype_list length_of_cons_lemma null_cons_lemma length_wf_nat nat_wf decidable__lt false_wf not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel equal_wf less_than_wf length_wf append-nil l_all_iff l_member_wf assert_elim subtype_base_sq bool_wf bool_subtype_base l_all_wf2 assert_witness select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf 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 intformless_wf int_formula_prop_less_lemma int_seg_wf true_wf append_wf nil_wf l_all_append cons_wf l_all_cons reduce_hd_cons_lemma subtype_rel_set
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_functionElimination productElimination cumulativity sqequalRule lambdaEquality applyEquality functionExtensionality functionEquality independent_isectElimination isect_memberEquality voidElimination voidEquality because_Cache unionElimination independent_functionElimination natural_numberEquality promote_hyp hypothesis_subsumption lambdaFormation setElimination rename addEquality independent_pairFormation intEquality minusEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality setEquality addLevel levelHypothesis instantiate hyp_replacement applyLambdaEquality dependent_pairFormation int_eqEquality computeAll imageElimination axiomEquality independent_pairEquality universeEquality

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

Date html generated: 2018_05_21-PM-06_44_10
Last ObjectModification: 2017_07_26-PM-04_54_52

Theory : general

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