hcju/isabelleps huggingface dataset:
oghoare_seq_valid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and R :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c\<^sub>1 R, A"   and "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> R c\<^sub>2 Q, A" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P Seq c\<^sub>1 c\<^sub>2 Q, A"
sup_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = Collect (sup_class.sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
path_defs_3: fixes g :: "real \<Rightarrow> 'b" shows "pathfinish g = g 1"
paste_def: fixes P :: "('b \<times> 'c) set"   and Q :: "('b \<times> 'c) set" shows "P +< Q = Outside' (Domain Q) P \<union> Q"
is_zero_row_def: fixes i :: "'b"   and A :: "(('c, 'd) vec, 'b) vec" shows "is_zero_row i A = is_zero_row_upt_k i (ncols A) A"
star_trans_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.left_near_kleene_algebra plus times one less_eq less star" shows "times (star x) (star x) = star x"
rep_Tree\<^sub>p_def: shows "rep_Tree\<^sub>p \<equiv> quot_type.rep Rep_Tree\<^sub>p"
RedTryFail: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and a :: "'addr"   and D :: "String.literal"   and P :: "'addr J_prog"   and C :: "String.literal"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and V :: "String.literal"   and e2 :: "'addr expr" assumes "typeof_addr (hp s) a = \<lfloor>Class_type D\<rfloor>"   and "\<not> P \<turnstile> D \<preceq>\<^sup>* C" shows "extTA,P,t \<turnstile> \<langle>try Throw a catch(C V) e2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
taut_impl_intro: fixes P :: "'a upred"   and Q :: "'a upred" assumes "\<Sigma> \<sharp> P"   and "`P` \<Longrightarrow> `Q`" shows "`P \<Rightarrow> Q`"
A: shows "s\<^sub>0 \<Turnstile> s"
is_median_median: fixes xs :: "'b list" shows "is_median (median xs) xs"
whisker_can_right_0: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u"   and "partial_magma.ide V f"   and "bicategorical_language.Src src t = bicategorical_language.Prim\<^sub>0 (trg f)" shows "H (bicategory.can V H \<a> \<i> src trg u t) f = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp u (bicategorical_language.Prim f)) (bicategorical_language.Hcomp t (bicategorical_language.Prim f))"
f0u: fixes x :: "real" assumes "x \<in> {0..u}" shows "f x = f 0"
check_eqv_sound: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "slow.check_eqvRE \<Sigma> n r s"   and "M2L_Equivalence_Checking.wf \<Sigma> n r"   and "M2L_Equivalence_Checking.wf \<Sigma> n s" shows "lang \<Sigma> n r = lang \<Sigma> n s"
unwinding_theorem_BSD: fixes ur :: "('s \<times> 's) set" assumes "lrf ur"   and "osc ur" shows "BSD \<V> Tr\<^bsub>(induceES SES)\<^esub>"
inject_2: fixes x2 :: "Secrets"   and y2 :: "Secrets" shows "(sKS x2 = sKS y2) = (x2 = y2)"
sel_2: fixes x1 :: "exesignature"   and x2 :: "Core.term list" shows "exeaxioms_of (ExeTheory x1 x2) = x2"
distinct_fst_imp_distinct: fixes l :: "('a \<times> 'b) list" assumes "distinct (map fst l)" shows "distinct l"
inject_2: fixes x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap"   and y21 :: "(Term_Class.term \<times> sterm) list"   and y22 :: "(name, value) fmap" shows "(Vabs x21 x22 = Vabs y21 y22) = (x21 = y21 \<and> x22 = y22)"
alpha_def_1: shows "f \<bullet> \<alpha> = \<zero>"
IH: fixes XSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) XSS" shows "\<forall>XS\<in>set XSS. \<forall>xs\<in>Joints XS. 0 \<le> interpret_floatarith fa xs"
subst_not_img_fixed: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<in> fv (t \<cdot> s)"   and "v \<notin> range_vars s" shows "v \<in> fv t"
oarrivemsg_simps_7: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd" shows "oarrivemsg P \<sigma> \<tau>\<^sub>n = True"
peek_and_def: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool" shows "(P \<and>. p) = (\<lambda>Y s Z. P Y s Z \<and> p s)"
a: shows "gdpr_Kripke \<turnstile> EF sgdpr"
weak_reduction_correspondence_simulation_impl_WOC: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"   and "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*"   and "weak_reduction_correspondence_simulation Rel (STCal Source Target)" shows "weakly_operational_corresponding (TRel\<^sup>*)"
emeasure_lebesgue_affine: fixes m :: "real"   and \<delta> :: "'c"   and S :: "'c set" shows "emeasure lebesgue ((\<lambda>x. m *\<^sub>R x + \<delta>) ` S) = ennreal (\<bar>m\<bar> ^ DIM('c)) * emeasure lebesgue S"
has_integral_refl_1: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "(f has_integral (0::'b)) (cbox a a)"
is_dghm_axioms: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"
confluence_implies_ChurchRosser: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "relation.is_confluent r" shows "relation.is_ChurchRosser r"
simps: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and a :: "'b" shows "multithreaded_base.deadlockedp final r s a = ((\<exists>t x. a = t \<and> thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> multithreaded_base.must_sync final r t x (shr s) \<and> wset s t = None \<and> (\<forall>xa. multithreaded_base.can_sync r t x (shr s) xa \<longrightarrow> (\<exists>lt\<in>xa. final_thread.must_wait final s t lt (Collect (sup (multithreaded_base.deadlockedp final r s) (final_thread.final_thread final s)))))) \<or> (\<exists>t x ln. a = t \<and> thr s t = \<lfloor>(x, ln)\<rfloor> \<and> final_thread.all_final_except final s {x. multithreaded_base.deadlockedp final r s x} \<and> waiting (wset s t)) \<or> (\<exists>t x l t' ln. a = t \<and> thr s t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> waiting (wset s t) \<and> 0 < ln $ l \<and> has_lock (locks s $ l) t' \<and> t' \<noteq> t \<and> (multithreaded_base.deadlockedp final r s t' \<or> final_thread.final_thread final s t')))"
append_store_ids_imply_messages_same: fixes i :: "'a"   and mo :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and r :: "'a"   and e2 :: "'b" assumes "i = mo"   and "xs prefix of j"   and "(i, Append i e1) \<in> set (node_deliver_messages xs)"   and "(r, Store e2 mo r) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
a: shows "\<turnstile> (\<Delta> @ \<Gamma>) ok"
assms_3: shows "\<V> (insert (1 - punit.monom_mult (1::'b) (monomial 1 None) (extend_indets f)) (extend_indets ` F)) = {}"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_backto_standard_pos m rs (l, r) = (wadjust_backto_standard_pos_B m rs (l, r) \<or> wadjust_backto_standard_pos_O m rs (l, r))"
wp3tl: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "wellformed_policy3 p" shows "wellformed_policy3 (tl p)"
tendsto_inf: fixes X :: "'a \<Rightarrow> 'b"   and x :: "'b"   and net :: "'a filter"   and Y :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "(X \<longlongrightarrow> x) net"   and "(Y \<longlongrightarrow> y) net" shows "((\<lambda>i. inf (X i) (Y i)) \<longlongrightarrow> inf x y) net"
cong_scalar2_poly: fixes a :: "'c poly"   and b :: "'c poly"   and m :: "'c poly"   and k :: "'c poly" assumes "[a = b] (mod m)" shows "[k * a = k * b] (mod m)"
is_red_subset: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_red A p"   and "A \<subseteq> B" shows "is_red B p"
not_in_verts_G: shows "w \<notin> verts I"
divides_pirreducible_condition: fixes K :: "'a list set"   and q :: "'a list list"   and p :: "'a list list" assumes "pirreducible\<^bsub>K [X]\<^esub> K q"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "p divides\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q" shows "p \<in> Units (K [X]\<^bsub>K [X]\<^esub>) \<or> p \<sim>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q"
in_set_conv_nth: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>i<|xs|. xs ! i = x)"
integrable_moments: fixes k :: "nat" assumes "k \<le> n" shows "integrable M (\<lambda>x. X x ^ k)"
satC_append: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and c1 :: "('fsym, 'psym) lit list"   and c2 :: "('fsym, 'psym) lit list" shows "satC \<xi> (c1 @ c2) = (satC \<xi> c1 \<or> satC \<xi> c2)"
obj_at_cong: fixes s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and r :: "'c"   and P :: "('a, 'b, 'c) object \<Rightarrow> bool"   and P' :: "('a, 'b, 'c) object \<Rightarrow> bool"   and r' :: "'c"   and s' :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state" assumes "\<And>obj. sys_heap s r = Some obj \<Longrightarrow> P obj = P' obj"   and "r = r'"   and "s = s'" shows "obj_at P r s = obj_at P' r' s'"
minus_ereal_def: fixes x :: "ereal"   and y :: "ereal" shows "x - y = x + - y"
subset: shows "unforwarded_non_volatile_reads (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (r # sb)) W \<subseteq> A"
143: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> - y = x"
remdups_mset_singleton: fixes a :: "'b" shows "remdups_mset {#a#} = {#a#}"
simps_9: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a" shows "(case Empty of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = f1"
propositions_lemma_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<lambda>\<^sup>0 \<phi> \<^bold>= \<phi> in v]"
meet_below1: fixes x :: "'c"   and z :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> z" shows "x \<sqinter> y \<sqsubseteq> z"
induct: fixes x1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and x2 :: "'d"   and x3 :: "'e \<Rightarrow> 'f"   and x4 :: "'d"   and x5 :: "'e \<Rightarrow> 'f"   and P :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint) \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow>* \<langle>x4, x5\<rangle>"   and "\<And>A l u. P A l u l u"   and "\<And>A l u l' u' l'' u''. \<lbrakk>A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>; A \<turnstile> \<langle>l', u'\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>; P A l' u' l'' u''\<rbrakk> \<Longrightarrow> P A l u l'' u''" shows "P x1 x2 x3 x4 x5"
tabulation_data_axioms: shows "tabulation_data (\<cdot>) (\<star>) \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1"
strip_3: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> P x" shows "\<forall>x\<in>A. P x"
Max_singleton: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.linorder less_eq less" shows "linorder.Max less_eq {x} = x"
ensures_weaken_R: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B' :: "'b set" assumes "F \<in> A ensures A'"   and "A' \<subseteq> B'" shows "F \<in> A ensures B'"
simps_19: fixes e\<^sub>1 :: "('f, 'g, 'h) exp"   and e\<^sub>2 :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (e\<^sub>1;; e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
br_set_rel_alt: fixes s' :: "'c set"   and s :: "'d set"   and \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "((s', s) \<in> \<langle>br \<alpha> I\<rangle>set_rel) = (s = \<alpha> ` s' \<and> (\<forall>x\<in>s'. I x))"
orthogonal_comp_def: fixes W :: "'b set" shows "W\<^sup>\<bottom> \<equiv> {x. \<forall>y\<in>W. orthogonal y x}"
out_degree_def: fixes G :: "('e, 'f) pre_digraph"   and v :: "'e" shows "out_degree G v \<equiv> card (out_arcs G v)"
EF_step_star: fixes x :: "'a"   and y :: "'a"   and f :: "'a set" assumes "x \<rightarrow>\<^sub>i* y"   and "y \<in> f" shows "x \<in> EF f"
stails_sdrop': fixes xs :: "[:'a:]" shows "stails\<cdot>xs = smap\<cdot>(\<Lambda> i. sdrop\<cdot>i\<cdot>xs)\<cdot>(supto\<cdot>0\<cdot>(slength\<cdot>xs))"
connected_closed: fixes s :: "'a set" shows "connected s = (\<nexists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
contour_integral_mirror: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g" shows "contour_integral (- g) f = contour_integral g (\<lambda>x. - f (- x))"
minusinf_uset0: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "islin p"   and "\<not> Ifm vs (x # bs) (minusinf p)"   and "Ifm vs (x # bs) p" shows "\<exists>(c, s)\<in>set (uset p). - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<le> x"
dcharacter_inv_character: shows "dcharacter n (inv_character \<chi>)"
finite_UNIV_literal_def: shows "finite_UNIV = Phantom(String.literal) False"
ereal_INF_uminus_eq: fixes f :: "'b \<Rightarrow> ereal"   and S :: "'b set" shows "(INF x\<in>S. - f x) = - Sup (f ` S)"
pubkey_unused: fixes s :: "(agent \<times> msg) set"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "PriKey A \<notin> used s" shows "PubKey A \<notin> parts (used s)"
sqrt_differentiable_on: fixes f :: "'b \<Rightarrow> real"   and S :: "'b set" assumes "f differentiable_on S"   and "0 \<notin> f ` S" shows "(\<lambda>x. sqrt (f x)) differentiable_on S"
RPb: shows "\<forall>v. (v \<^bold>\<in> RPb) = (\<exists>u. u \<^bold>\<in> Pb \<and> v = u \<triangleleft> a)"
abs_eq: fixes x :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" assumes "eq_onp hereditarily_fs x x" shows "Formula.Not (Abs_formula x) = Abs_formula (Not\<^sub>\<alpha> x)"
image_image: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
dg_digraph_dg_singleton: fixes j :: "V" assumes "j \<in>\<^sub>\<circ> Vset \<alpha>" shows "digraph \<alpha> (\<Prod>\<^sub>D\<^sub>Gj\<in>\<^sub>\<circ>ZFC_in_HOL.set {j}. \<CC>)"
cont_both2: fixes y :: "'b ttree" shows "cont ((\<otimes>\<otimes>) y)"
Says_Server_message_form: fixes B :: "agent"   and M :: "nat"   and X :: "msg"   and Nb :: "nat"   and K :: "nat"   and evs :: "event list" assumes "Says Server B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>X, Nonce Nb, Key K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> orb" shows "K \<notin> range shrK \<and> (\<exists>A Na. X = Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>)"
kbo_std_basis_axioms: shows "kbo_std_basis ground_heads_var (>\<^sub>s) arity_sym arity_var wt_sym \<epsilon> \<delta> extf"
ctxt_supteq: fixes s :: "('d, 'e) Term.term"   and C :: "('d, 'e) ctxt"   and t :: "('d, 'e) Term.term" assumes "s = C\<langle>t\<rangle>" shows "s \<unrhd> t"
in_carrier_atlasI: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and c :: "('c, 'd) chart"   and x :: "'c" assumes "c_manifold charts k"   and "c \<in> c_manifold.atlas charts k"   and "x \<in> domain c" shows "x \<in> manifold.carrier charts"
idiff_eq_conv: fixes x :: "enat"   and y :: "enat"   and z :: "enat" shows "(x - y = z) = (if y < x then x = y + z else if x \<noteq> \<infinity> then z = 0 else z = \<infinity>)"
simps_2: fixes n :: "nat" shows "lang n Full = lists ((set \<circ> \<sigma> \<Sigma>) n)"
nths_atLeastAtMost_prefix: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "m \<le> n" shows "nths xs {0..m} \<le> nths xs {0..n}"
IH: shows "frag_cmul a (sum b I) = (\<Sum>i\<in>I. frag_cmul a (b i))"
find_mu: shows "\<lbrace>\<lambda>s. l s = lambda\<rbrace> find_mu \<lbrace>\<lambda>s. l s = lambda \<and> m s = mu\<rbrace>"
elims_2: fixes x :: "('c, 'd) fmap"   and xa :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "sat_precond_as x xa" obtains s :: "('c, 'd) fmap" where "x = sb__" and "xa = []"   | s :: "('c, 'd) fmap" and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" where "x = s" and "xa = a # as" and "fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as"
elims_2: fixes x :: "'edge list"   and xa :: "'edge list" assumes "valid_path_aux x xa" obtains cs :: "'edge list" where "x = csb__" and "xa = []"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> valid_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> valid_path_aux [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> valid_path_aux cs' as | _ \<Rightarrow> valid_path_aux cs as"
distinct_1: fixes x1 :: "'d" shows "ln x1 \<noteq> \<star>"
indets_empty_iff_poly_deg_zero: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "(indets p = {}) = (poly_deg p = 0)"
T_ins_tree_simple_bound: fixes t :: "'b tree"   and ts :: "'b tree list" shows "T_ins_tree t ts \<le> length ts + 1"
x_split: fixes x :: "'a" shows "x \<sqinter> nc + x \<sqinter> 1\<^sub>\<pi> = x"
coprime_divisors: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a dvd c"   and "b dvd d"   and "algebraic_semidom_class.coprime c d" shows "algebraic_semidom_class.coprime a b"
chainFreshFresh: fixes x :: "name"   and xvec :: "name list"   and y :: "name"   and p :: "'a" assumes "x \<sharp> xvec"   and "y \<sharp> xvec" shows "xvec \<sharp>* ([(x, y)] \<bullet> p) = xvec \<sharp>* p"
T_bal_def: fixes n :: "nat"   and xs :: "'b list" shows "T_bal n xs = time (bal_tm n xs)"
disc_moebius_ocircline: fixes M :: "moebius"   and H :: "ocircline" shows "disc (moebius_ocircline M H) = moebius_pt M ` disc H"
carr: shows "gso ` {0..<i} \<subseteq> carrier_vec n"
is_lrun_def: fixes l :: "nat \<Rightarrow> 'l"   and s :: "nat \<Rightarrow> 'c" shows "C.is_lrun l s \<equiv> s 0 = cs\<^sub>0 \<and> (\<forall>i. clstep (l i) (s i) (s (Suc i)))"
access_cbody: fixes c :: "class" shows "access (cbody c) = access c"
rep_eq: fixes x :: "'a prefix"   and xa :: "'a set \<times> nat" shows "Rep_prefix (psnoc x xa) = (if (case Rep_prefix x of [] \<Rightarrow> 0 | a # list \<Rightarrow> snd (last (Rep_prefix x))) \<le> snd xa then Rep_prefix x @ [xa] else [])"
divide_divide_eq_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a (divide b c) = divide (times a c) b"
distr\<^sub>0_def: fixes ko :: "'a nat_term_order"   and xs :: "('a \<times> 'b) list" shows "distr\<^sub>0 ko xs = MP_oalist (oalist_of_list_ntm (xs, ko))"
eff_Done: fixes s :: "'state"   and i :: "nat" shows "eff Done s i = s"
abelian_groupE_5: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "abelian_group R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
opeW: shows "openin (top_of_set ({0..1} \<times> S)) W"
sccs_verts_conv_scc_of: shows "G.sccs_verts = G.scc_of ` verts G"
leg_simps_4: shows "C.dom \<rho>.leg1 = \<rho>.apex"
proj4_eq_flag: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>4 (the (rt dip)) = the (flag rt dip)"
ratio_large: shows "\<forall>k\<ge>t. (aa k / bb k) powr (1 / (1 + \<epsilon>)) + 1 \<le> (aa (k + 1) / bb (k + 1)) powr (1 / (1 + \<epsilon>))"
span_redundant: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> cspan S" shows "cspan (insert x S) = cspan S"
lunit_naturality: fixes f :: "'a" assumes "arr f" shows "\<l>[cod f] \<cdot> (\<I> \<otimes> f) = f \<cdot> \<l>[local.dom f]"
order_hom: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b"   and f :: "'b poly" assumes "map_poly_inj_idom_divide_hom hom" shows "order (hom x) (map_poly hom f) = order x f"
sorted_append: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (xs @ ys) = (linorder.sorted less_eq xs \<and> linorder.sorted less_eq ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. less_eq x y))"
distinct_7: fixes x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf" shows "Z \<noteq> Pr x51 x52 x53"
vsv_vinsert_match_appI: fixes a :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f)" shows "vinsert \<langle>a, b\<rangle> (VLambda A f)\<lparr>a\<rparr> = b"
weak_tricolour_inv_def: shows "weak_tricolour_inv = (\<lambda>s. \<forall>x xa. black x s \<and> (x points_to xa) s \<and> white xa s \<longrightarrow> (\<exists>x. (x grey_protects_white xa) s))"
sel_3: fixes x4 :: "nat" shows "ticket (REL x4) = x4"
klift_prop1: fixes f :: "'e \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'e set" shows "bd\<^sub>\<F> (bd\<^sub>\<F> f \<circ> g) = bd\<^sub>\<F> f \<circ> bd\<^sub>\<F> g"
ln11: shows "ln 11 \<in> {5 / 3..290 / 99}"
inv2: shows "inverse u2 \<cdot> inverse u1 \<cdot> (u1 \<cdot> u2) = \<one>"
dom_eqI': fixes a :: "'a"   and f :: "'a" assumes "a \<in> domains f" shows "a = local.dom f"
safe_distance_1r_def: shows "safe_distance_1r = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / a\<^sub>e / 2"
tsubst_matcher_imp_raw_match_term_unchanged: fixes t :: "Core.term"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ"   and u :: "Core.term" assumes "tsubst t \<rho> = u" shows "raw_match_term t u (\<lambda>(idx, S). Some (\<rho> idx S)) = Some (\<lambda>(idx, S). Some (\<rho> idx S))"
le_Suc_diff_le: fixes x :: "nat"   and n :: "nat" assumes "x \<le> Suc n" shows "x - Suc 0 \<le> n"
sin_diff_sin: fixes w :: "'a"   and z :: "'a" shows "sin w - sin z = (2::'a) * sin ((w - z) / (2::'a)) * cos ((w + z) / (2::'a))"
bind_return_pmf': fixes N :: "'b pmf" shows "N \<bind> return_pmf = N"
closed_INT: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'a set" assumes "\<forall>x\<in>A. closed (B x)" shows "closed (\<Inter> (B ` A))"