hcju/isabelleps huggingface dataset:
legendre_aux_set_eq: fixes p :: "nat"   and x :: "real" assumes "prime p"   and "1 \<le> x" shows "{m. 0 < m \<and> real (p ^ m) \<le> x} = {0<..nat \<lfloor>log (real p) x\<rfloor>}"
less_add_same_cancel1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus a b) = less zero b"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<pi> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "arrow_of_spans C \<mu>"   and "arrow_of_spans C \<nu>"   and "arrow_of_spans C \<pi>"   and "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>"   and "two_composable_arrows_of_spans C prj0 prj1 \<nu> \<pi>" shows "three_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi>"
un_subset1: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> i \<le> i \<squnion> j"
borel_measurable_integrable': fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and N :: "'e measure" assumes "integrable M f"   and "g \<in> N \<rightarrow>\<^sub>M M" shows "(\<lambda>x. f (g x)) \<in> borel_measurable N"
pnormal_last_nonzero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times"   and "semiring_0.pnormal zero p" shows "last p \<noteq> zero"
sup_assoc: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "sup_prod (sup_prod x y) z = sup_prod x (sup_prod y z)"
bounded_lattice_bot_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c" shows "class.bounded_lattice_bot inf less_eq less sup bot \<equiv> class.lattice inf less_eq less sup \<and> class.order_bot bot less_eq less"
walk_last_edge: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G (xs @ ys)"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "last xs \<rightarrow>\<^bsub>G\<^esub> hd ys"
min_dist_split_2: fixes u :: "nat"   and d1 :: "nat"   and w :: "nat"   and d2 :: "nat"   and v :: "nat" assumes "dist u d1 w"   and "dist w d2 v"   and "min_dist u v = d1 + d2" shows "min_dist w v = d2"
return_time_eqI: fixes P :: "'d set"   and t :: "real"   and x :: "'d" assumes "closed P"   and "0 < t"   and "t \<in> existence_ivl0 x"   and "flow0 x t \<in> P"   and "\<And>s. \<lbrakk>0 < s; s < t\<rbrakk> \<Longrightarrow> flow0 x s \<notin> P" shows "return_time P x = t"
leq_implies_path: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C \<preceq>\<^sup>* D"   and "is_class P C"   and "wf_prog wf_md P" shows "\<exists>Cs. P \<turnstile> Path C to D via Cs "
runit_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.ide C a" shows "tensor (runit a) unity = C (tensor a (elementary_monoidal_category.\<iota> unity lunit)) (assoc a unity unity)"
all_irr_GIrrRep_repset: assumes "of_nat (card G) \<noteq> (0::'b)" shows "\<forall>U\<in>GIrrRep_repset. IrrFinGroupRepresentation G (*) U"
distinct_88: fixes x101 :: "nat"   and x102 :: "msg"   and x6 :: "nat" shows "Crypt x101 x102 \<noteq> Passwd x6"
SA_all: fixes \<Gamma> :: "binding list"   and T\<^sub>1 :: "ty"   and S\<^sub>1 :: "ty"   and X :: "tyvrs"   and S\<^sub>2 :: "ty"   and T\<^sub>2 :: "ty" assumes "\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1"   and "(TVarB X T\<^sub>1 # \<Gamma>)\<turnstile>S\<^sub>2<:T\<^sub>2" shows "\<Gamma>\<turnstile>(\<forall>X<:S\<^sub>1. S\<^sub>2)<:(\<forall>X<:T\<^sub>1. T\<^sub>2)"
product_nn_integral_singleton: fixes f :: "'a \<Rightarrow> ennreal"   and i :: "'i" assumes "f \<in> borel_measurable (M i)" shows "\<integral>\<^sup>+ x. f (x i) \<partial>Pi\<^sub>M {i} M = integral\<^sup>N (M i) f"
sc': shows "h \<turnstile> get_scdom_component ptr' \<rightarrow>\<^sub>r sc'"
simps_5: fixes f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
iso_def: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "Group.iso G H = {h \<in> hom G H. bij_betw h (carrier G) (carrier H)}"
induct: fixes x1 :: "('b, 'c, 'd, 'e) Tree"   and x2 :: "('b, 'c, 'd, 'e) Tree"   and P :: "('b, 'c, 'd, 'e) Tree \<Rightarrow> ('b, 'c, 'd, 'e) Tree \<Rightarrow> bool" assumes "(x1, x2) \<in> Tree_wf"   and "\<And>t tset. t \<in> set_bset tset \<Longrightarrow> P t (tConj tset)"   and "\<And>t. P t (tNot t)"   and "\<And>t f \<alpha>. P t (tAct f \<alpha> t)" shows "P x1 x2"
dbl_dec_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one x = minus (plus x x) one"
sup_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "Err.sup f = lift2 (\<lambda>x y. OK (x \<squnion>\<^bsub>f\<^esub> y))"
map_pre_coF_def: shows "map_pre_coF \<equiv> \<lambda>f1 f2. BNF_Composition.id_bnf \<circ> map_coF0 f1 f2 \<circ> BNF_Composition.id_bnf"
UN_extend_simps_5: fixes A :: "'g set"   and B :: "'h \<Rightarrow> 'g set"   and C :: "'h set" shows "A \<inter> \<Union> (B ` C) = (\<Union>x\<in>C. A \<inter> B x)"
simps: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" shows "local.inter t1 t2 = (if t1 = \<langle>\<rangle> then \<langle>\<rangle> else if t2 = \<langle>\<rangle> then \<langle>\<rangle> else case t1 of \<langle>l1, (a, x), r1\<rangle> \<Rightarrow> let (l2, b, r2) = split t2 a; l' = local.inter l1 l2; r' = local.inter r1 r2 in if b then join l' a r' else join2 l' r')"
filterlim_sequentially_Suc: fixes f :: "nat \<Rightarrow> 'e"   and F :: "'e filter" shows "(LIM x sequentially. f (Suc x) :> F) = filterlim f F sequentially"
assms_3: fixes u :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "\<forall>n. u n \<in> S"   and "b \<in> T"   and "u \<longlonglongrightarrow> b" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"
vars_monom_keys: fixes a :: "'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> (0::'b)" shows "vars (monom m a) = keys m"
StarLR_Zero: fixes r :: "'a language" shows "StarLR Zero r = Zero"
rfb_comp_pres: fixes R :: "('d \<times> 'e) set"   and S :: "('e \<times> 'c) set" shows "fb\<^sub>\<R> (R ; S) = fb\<^sub>\<R> R \<circ> fb\<^sub>\<R> S"
smult_power: fixes a :: "'c"   and p :: "'c poly"   and n :: "nat" shows "smult a p ^ n = smult (a ^ n) (p ^ n)"
Sym: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>e Q" shows "Q \<equiv>\<^sub>e P"
lborel_eq: shows "lborel = distr (Pi\<^sub>M Basis (\<lambda>b. lborel)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
twoCisTypeI_8: shows "(\<lambda>y. twoC (1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
ifp_def: fixes configuration_subj_obj :: "'a \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "abstract_policy_derivation.ifp configuration_subj_obj a b \<equiv> abstract_policy_derivation.sp_spec_subj_subj configuration_subj_obj a b \<or> abstract_policy_derivation.sp_spec_subj_subj configuration_subj_obj b a \<or> (\<exists>c y. abstract_policy_derivation.sp_spec_subj_subj configuration_subj_obj a c \<and> abstract_policy_derivation.sp_spec_subj_obj configuration_subj_obj c (PAGE y) WRITE \<and> abstract_policy_derivation.sp_spec_subj_obj configuration_subj_obj b (PAGE y) READ) \<or> a = b"
empty_eff_exec_eq: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap" assumes "fmdom' (snd a) = {}" shows "state_succ s a = s"
inv_unique: fixes y :: "'c \<Rightarrow> 'a"   and x :: "'c \<Rightarrow> 'a"   and y' :: "'c \<Rightarrow> 'a" assumes "y \<otimes>\<^bsub>F\<^esub> x = \<one>\<^bsub>F\<^esub>"   and "x \<otimes>\<^bsub>F\<^esub> y' = \<one>\<^bsub>F\<^esub>"   and "x \<in> carrier F"   and "y \<in> carrier F"   and "y' \<in> carrier F" shows "y = y'"
assms: fixes u :: "nat \<Rightarrow> 'a" assumes "cauchy_in\<^sub>N N u"   and "\<forall>n. u n \<in> space\<^sub>N N" shows "\<exists>x\<in>space\<^sub>N N. tendsto_in\<^sub>N N u x"
A: fixes U :: "'b set"   and xs :: "'a list"   and xs' :: "'a list" shows "U \<subseteq> range D \<longrightarrow> U \<noteq> {} \<longrightarrow> xs @ ys \<in> traces P \<longrightarrow> (\<forall>u. u \<in> range D \<and> (\<forall>v\<in>U. (v, u) \<notin> I) \<longrightarrow> (xs, xs') \<in> R u) \<longrightarrow> (\<forall>u. u \<in> range D \<and> (\<forall>v\<in>sinks_aux I D U ys. (v, u) \<notin> I) \<longrightarrow> (xs @ ys, xs' @ ipurge_tr_aux I D U ys) \<in> R u)"
simps: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and r :: "'a \<Rightarrow>\<^sub>0 'b" shows "sig_trd_spp_body bs v (p, r) = (case find_sig_reducer bs v (punit.lt p) 0 of None \<Rightarrow> (punit.tail p, r + monomial (punit.lc p) (punit.lt p)) | Some i \<Rightarrow> let b = snd (bs ! i) in (punit.tail p - local.punit.monom_mult (punit.lc p / punit.lc b) (punit.lt p - punit.lt b) (punit.tail b), r))"
left_adjoint_determines_right_up_to_iso: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "adjoint_pair f g"   and "adjoint_pair f g'" shows "g \<cong> g'"
is_unused_elem_def: fixes dflt :: "'c"   and A :: "'d \<Rightarrow> 'c \<Rightarrow> assn" shows "is_unused_elem dflt A \<equiv> \<forall>x. A x dflt = false"
c: shows "((\<approx>) OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
seqE_fst: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and r :: "'b rule"   and rs :: "'b rule list"   and s :: "state"   and t :: "state" assumes "\<gamma>,p\<turnstile> \<langle>r # rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t" obtains t' :: "state" where "\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow>\<^sub>\<alpha> t'" and "\<gamma>,p\<turnstile> \<langle>rs, t'\<rangle> \<Rightarrow>\<^sub>\<alpha> t"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns'\<rightarrow>m"   and "distinct ns'"   and "set ns' \<subseteq> set ns"   and "length ns' \<le> length ns" shows "thesis"
abs_in_BasisE: fixes y :: "'c" assumes "\<bar>y\<bar> \<in> Basis" obtains i :: "'c" where "i__ \<in> Basis" and "y = i__"   | i :: "'c" where "i \<in> Basis" and "y = - i"
Diff_single_insert: fixes A :: "'k set"   and x :: "'k"   and B :: "'k set" assumes "A - {x} \<subseteq> B" shows "A \<subseteq> insert x B"
inverse_def: fixes C :: "('e, 'd, 'f) Category_scheme"   and f :: "'d" shows "Cinv\<^bsub>C\<^esub> f \<equiv> THE g. cinv\<^bsub>C\<^esub> f g"
map_is_Nil_conv: fixes f :: "'m \<Rightarrow> 'l"   and xs :: "'m list" shows "(map f xs = []) = (xs = [])"
i_set_i_set0_mem_conv: fixes I :: "nat set" shows "(I \<in> i_set) = (I \<in> i_set0 \<and> I \<noteq> {})"
bij_betw_inv_into_twice: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set" assumes "bij_betw f A A'" shows "\<forall>a\<in>A. inv_into A' (inv_into A f) a = f a"
gen_model_eq: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'd) KripkeStructure"   and w :: "'d"   and w' :: "'d"   and \<phi> :: "('b, 'c) Kform" assumes "kripke M"   and "kripke M'"   and "gen_model M w = gen_model M' w"   and "w' \<in> worlds (gen_model M' w)" shows "M, w' \<Turnstile> \<phi> = M', w' \<Turnstile> \<phi>"
distinct_5: fixes x2 :: "'e" shows "Fault x2 \<noteq> Stuck"
PsiInv: shows "\<turnstile> Psi \<longrightarrow> \<box>PsiInv"
ParCptnOne: fixes P :: "'b com option list"   and s :: "'b" shows "[(P, s)] \<in> par_cptn"
cobounded_outside: fixes S :: "'b set" assumes "bounded S" shows "bounded (- outside S)"
tsub_def: fixes k :: "int"   and l :: "int" shows "tsub k l = max 0 (k - l)"
rec_2: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
subtype_weaken': fixes \<Gamma> :: "binding list"   and P :: "POPLmarkRecord.type"   and Q :: "POPLmarkRecord.type"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile> P <: Q"   and "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "\<Delta> @ \<Gamma> \<turnstile> \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 P <: \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 Q"
id_fixespointwise: fixes A :: "'b set" shows "fixespointwise id A"
b_combined_def: shows "b_combined \<equiv> sup b1 b2"
T2: fixes u :: "'a"   and w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "is_left_adjoint u"   and "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : g \<star> w \<Rightarrow> g \<star> w'\<guillemotright>"   and "(r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w) = ((r \<star> \<theta>') \<cdot> \<a>[r, f, w'] \<cdot> (\<rho> \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = g \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (f \<star> \<gamma>)"
l_set_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_shadow_root set_shadow_root_locs \<equiv> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs"
observable_io_targets_split: fixes M :: "('d, 'e, 'f) FSM"   and q1 :: "'f"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and q3 :: "'f"   and q2 :: "'f" assumes "observable M"   and "io_targets M q1 (vs @ xs) = {q3}"   and "io_targets M q1 vs = {q2}" shows "io_targets M q2 xs = {q3}"
subseteq_guards_Throw: fixes c :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g Throw" shows "c = Throw"
shift_pow: fixes i :: "nat" shows "- (x * ((- 1) ^ i * a (Suc i) * x ^ i)) = (- 1) ^ Suc i * a (Suc i) * x ^ Suc i"
f3: shows "gets (\<lambda>s. ucast (get_CWP (cpu_reg_val PSR s))) = get_curr_win ()"
fmdomI: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f"   and y :: "'e" assumes "fmlookup m x = Some y" shows "x |\<in>| fmdom m"
cases: fixes a1 :: "('c, 'd) tm"   and a2 :: "('c, 'd) tm" assumes "sub a1 a2" obtains s :: "('c, 'd) tm" where "a1 = sa__" and "a2 = sa__"   | s :: "('c, 'd) tm" and t :: "('c, 'd) tm" and u :: "('c, 'd) tm" where "a1 = sb__" and "a2 = App u__ t__" and "sub sb__ t__"   | s :: "('c, 'd) tm" and t :: "('c, 'd) tm" and u :: "('c, 'd) tm" where "a1 = s" and "a2 = App t u" and "sub s t"
0: shows "map (int \<xi>) (TE.getTvars (arOf f)) = al"
assms_2: shows "\<Gamma> \<turnstile> e : t"
intros: fixes r :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set"   and seq :: "'b sequent" assumes "r \<in> R" shows "extendRule seq r \<in> R*"
arr_def: fixes f :: "'a" shows "H.arr f \<equiv> H.domains f \<noteq> {} \<or> H.codomains f \<noteq> {}"
simps: fixes ws :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set" shows "step (ws, N) = (let (r, s) = hd ws; succs = map (\<lambda>a. let r' = delta a r; s' = delta a s in ((r', s'), post r', post s')) (\<sigma> n); new = remdups' snd (filter (\<lambda>(uu_, rs). rs \<notin> N) succs) in (tl ws @ map fst new, set (map snd new) \<union> N))"
add_le_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "(c + a \<le> c + b) = (a \<le> b)"
assms_5: fixes l :: "complex" shows "\<not> (inverse \<circ> f) \<midarrow>z\<rightarrow> l"
intros_1: fixes x :: "'d"   and y :: "'d" shows "ignore_Inl (Inl x) (Inl y)"
gc_invar_gc_init: fixes Ns :: "('f \<times> 'l) set llist" assumes "\<not> lnull Ns"   and "active_subset (lhd Ns) = {}" shows "gc_invar Ns 0"
segment_def: fixes a :: "'a"   and b :: "'a" shows "segment a b \<equiv> {x. \<exists>ab. [[a x b]] \<and> x \<in> ab \<and> path ab a b}"
semiring_normalization_rules_19: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lx :: "'a"   and rx :: "'a"   and ry :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times lx (times rx ry) = times rx (times lx ry)"
Field_Restr_subset: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Field (Restr r A) \<subseteq> A"
size_mset_set_eq_card: fixes A :: "'c set" assumes "finite A" shows "size (mset_set A) = card A"
proto: shows "closed (pnet (\<lambda>i. ptoy i \<langle>\<langle> qmsg) n) \<TTurnstile> netglobal (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips n. no (\<sigma> i) \<le> no (\<sigma> (nhid (\<sigma> i))))"
triangle: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "MC.tensor a (MC.lunit b) \<cdot> MC.assoc a MC.unity b = MC.tensor (MC.runit a) b"
comm_monoid_axioms: shows "comm_monoid (+) (0::'b)"
fmempty_of_list: shows "fmempty = fmap_of_list []"
conga_trivial_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "C \<noteq> D" shows "A B A CongA C D C"
Cod_preserves_Diag: fixes t :: "'c local.term" assumes "Diag t" shows "Diag (Cod t)"
goodBinp_def: fixes binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" shows "goodBinp binp \<equiv> liftAll goodAbs binp \<and> |{i. binp i \<noteq> None}| <o |UNIV|"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a option"   and b :: "'b option" assumes "rel_option R a b" obtains "a = None" and "b = None"   | x :: "'a" and y :: "'b" where "a = pure x" and "b = pure y" and "R x y"
hausdorff: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.t2_space open"   and "x \<noteq> y" shows "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
cases: fixes x :: "ground_action \<times> ((predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option)" obtains pre :: "object atom formula" and add :: "object atom formula list" and del :: "object atom formula list" and f :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" where "x = (Ground_Action pre (Effect add del), f)"
full_elims_2: fixes n :: "nat"   and l :: "'a tree23"   and p :: "int \<times> 'a"   and r :: "'a tree23" assumes "full n (Branch2 l p r)" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na r"
that_2: assumes "xs \<le>\<^sub>v vs"   and "ys <\<^sub>v ws" shows "thesis"
volatile_Write\<^sub>s\<^sub>b: shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}"
submonoid_is_comm_monoid: fixes H :: "'a set" assumes "submonoid H G" shows "Group.comm_monoid (G\<lparr>carrier := H\<rparr>)"
AllN_8: shows "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) c = Cv"
simps_1: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" shows "redT_updTs ts [] = ts"
get_ancestors_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "h \<turnstile> get_ancestors ptr \<rightarrow>\<^sub>r ancestors" shows "ptr \<in> set ancestors"
arr_dom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "arr (local.dom f)"
abs_eq: fixes x :: "'d" shows "nebsingleton x = Abs_nebset {x}"
has_fps_expansion_def: fixes f :: "'b \<Rightarrow> 'b"   and F :: "'b fps" shows "f has_fps_expansion F = (0 < fps_conv_radius F \<and> (\<forall>\<^sub>F z in nhds (0::'b). eval_fps F z = f z))"
K_statefun_def: fixes c :: "'a"   and x :: "'b" shows "K_statefun c x \<equiv> c"
int_le_0_imp_le_1: fixes x :: "int" assumes "x \<le> 0" shows "x \<le> 1"
hd_eq_first: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "xs ! 0 = hd xs"
ge: shows "sum g (K - {i}) \<le> sum f (K - {i})"
assms_3: shows "\<forall>y\<in>S. norm (v - p) \<le> norm (v - y)"
case_2: fixes f1 :: "CTypeId \<Rightarrow> 'a"   and f2 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "Arraytype"   and x22 :: "nat" shows "(case new_array x21 x22 of new_instance x \<Rightarrow> f1 x | new_array x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
subo': shows "Subobjs P C (rev (C' # Cs'))"
content_reflect_poly: fixes f :: "'a poly" shows "content (reflect_poly f) = content f"
shift_vars_def: fixes vs :: "nat list" shows "shift_vars vs = 0 # map Suc vs"
base: fixes \<psi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<phi> :: "'a stream \<Rightarrow> bool" assumes "\<psi> xs" shows "(\<phi> until \<psi>) xs"