hcju/isabelleps huggingface dataset:
eval_cases_12: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>throw e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains a :: "nat" where "e' = Throw a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>addr a,s'\<rangle>"   | "e' = THROW NullPointer" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>null,s'\<rangle>"   | e'a :: "char list exp" where "e' = throw e'a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
SN_trancl_imp_SN: fixes R :: "('b \<times> 'b) set" assumes "SN (R\<^sup>+)" shows "SN R"
surf_nz: fixes x :: "'a" assumes "x \<in> sphere (0::'a) 1 \<inter> affine hull S" shows "surf x \<noteq> (0::'a)"
Qg_unit: fixes N :: "'a set" assumes "G \<triangleright> N" shows "\<forall>x\<in>set_rcs G N. c_top G N N x = x"
the_inv_into_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'c"   and B :: "'b set" assumes "inj_on f A"   and "x \<in> f ` A"   and "A \<subseteq> B" shows "the_inv_into A f x \<in> B"
similar_matrix_refl: fixes A :: "(('a, 'b) vec, 'b) vec" shows "similar_matrix A A"
inf_K_enc_orb: fixes a :: "'a"   and b :: "'a" assumes "a \<le>\<^sub>\<K> b" shows "Orb a \<lless>\<^sub>\<K> Orb b"
output_vend_aux: fixes j :: "state stream"   and d :: "value" assumes "\<exists>p r t. j = make_full_observation drinks (Some 1) r p t" shows "alw (\<lambda>xs. label_eq ''vend'' xs \<and> output_eq [Some d] (stl xs) \<longrightarrow> check_exp (Ge (V (Rg 2)) (L (Num 100))) xs) j"
nonneg_primfun_list_iff: fixes fs :: "('a \<times> 'b) list" shows "nonneg_primfun_list fs = nonneg_list (map snd fs)"
assms: shows "b \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A"
ffUnion_least_rev: fixes A :: "'a fset fset"   and C :: "'a fset" assumes "ffUnion A |\<subseteq>| C" shows "fBall A (\<lambda>X. X |\<subseteq>| C)"
degree_map_poly_hom: fixes p :: "int poly" shows "Polynomial.degree (Ring_Hom_Poly.of_int_poly p) = Polynomial.degree p"
elims_2: fixes x :: "('d, 'e, 'f) FSM" assumes "well_formed x" obtains M :: "('d, 'e, 'f) FSM" where "x = M" and "finite_FSM M \<and> (\<forall>s1 x y. x \<notin> inputs M \<or> y \<notin> outputs M \<longrightarrow> succ M (x, y) s1 = {}) \<and> inputs M \<noteq> {} \<and> outputs M \<noteq> {}"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_mode type_wf set_mode set_mode_locs"   and "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_mode_get_disconnected_nodes_axioms set_mode_locs get_disconnected_nodes_locs" shows "l_set_mode_get_disconnected_nodes type_wf set_mode set_mode_locs get_disconnected_nodes get_disconnected_nodes_locs"
invar_def: fixes bq :: "('e, 'f) SkewBinomialTree list" shows "invar bq = (queue_invar bq \<and> rank_skew_invar bq)"
d_OUT_lift_SOURCE: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "d_OUT (lift f) SOURCE = value_flow \<Delta> f"
inj_linear_imp_inv_linear: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "Vector_Spaces.linear (*s) (*s) (inv f)"
find_witness_from_frontier: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c"   and c :: "('a, 'c) configuration"   and loc2 :: "'a" assumes "dataflow_topology summary results_in"   and "t \<in>\<^sub>A frontier (c_imp c loc2)"   and "dataflow_topology.inv_imps_work_sum summary results_in c" shows "\<exists>t' loc1 xs. graph.path summary loc1 loc2 xs \<and> t = results_in t' (foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b)) \<and> (t' \<in>\<^sub>A frontier (c_pts c loc1) \<or> zcount (c_work c loc1) t' < 0)"
sclose_subset_FV: fixes n :: "nat"   and s :: "char list"   and p :: "char list"   and t :: "sterm" shows "FV ({n \<leftarrow> [s,p]} t) \<subseteq> FV t"
r_2: shows "ball (x, y) r \<subseteq> S"
Inst: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and T :: "ref_ty"   and T' :: "ref_ty" assumes "E,dt\<Turnstile>e\<Colon>-RefT T"   and "is_acc_type (prg E) (pkg E) (RefT T')"   and "prg E\<turnstile>RefT T\<preceq>? RefT T'" shows "E,dt\<Turnstile>e InstOf T'\<Colon>-PrimT Boolean"
intros_2: fixes \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat" assumes "(\<pi>, m) \<in> IFC_def.scp prob"   and "IFC_def.is_cdi prob n \<pi> m" shows "(\<pi>, n) \<in> IFC_def.scp prob"
assms: shows "xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last xs = hd ys \<and> last xs \<noteq> (0::'a)"
LeastI2_wellorder: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
quasi_isometric_empty: fixes X :: "'b set"   and Y :: "'c set" assumes "X = {}"   and "quasi_isometric X Y" shows "Y = {}"
sets_stream_space_eq: fixes M :: "'b measure" shows "sets (stream_space M) = sets (SUP i. vimage_algebra (streams (space M)) (\<lambda>s. s !! i) M)"
cut_greater_Max_eq: fixes I :: "'b set"   and t :: "'b" assumes "finite I"   and "I \<down>> t \<noteq> {}" shows "Max (I \<down>> t) = Max I"
gcd_dvd1: fixes a :: "'d"   and b :: "'d" shows "gcd a b dvd a"
not_eq_order_implies_strict: fixes a :: "('c, 'd) Term.term"   and b :: "('c, 'd) Term.term" assumes "a \<noteq> b"   and "b \<unrhd> a" shows "b \<rhd> a"
cp_thm: fixes p :: "fm"   and d :: "int"   and bbs :: "bool list"   and bs :: "int list"   and i :: "int" assumes "iszlfm p"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d" shows "(\<exists>x. Ifm bbs (x # bs) p) = (\<exists>j\<in>{1..d}. Ifm bbs (j # bs) (minusinf p) \<or> (\<exists>b\<in>set (\<beta> p). Ifm bbs ((Inum (i # bs) b + j) # bs) p))"
open_segment_def: fixes a :: "'a"   and b :: "'a" shows "open_segment a b \<equiv> closed_segment a b - {a, b}"
card: fixes X :: "'a set" assumes "X \<subseteq> A" shows "card B * card X \<le> card A * card {y \<in> B. \<exists>x\<in>X. R x y}"
sigma_algebra_axioms_def: fixes M :: "'c set set" shows "sigma_algebra_axioms M \<equiv> \<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M"
group_byD: fixes b :: "'a"   and cs :: "'b set"   and f :: "'c \<Rightarrow> 'a \<times> 'b"   and as :: "'c set"   and a :: "'c"   and c :: "'b" assumes "(b, cs) \<in> group_by f as"   and "f a = (b, c)"   and "a \<in> as" shows "c \<in> cs"
inv_on_left_moving_in_middle_B_Bk_Oc: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and r :: "cell list"   and ires :: "cell list" assumes "inv_on_left_moving_norm (as, lm) (s, [], Oc # r) ires" shows "inv_on_left_moving_in_middle_B (as, lm) (s, [], Bk # Oc # r) ires"
fold_A_endo_X: shows "ChamberComplexEndomorphism X fold_A"
unique_nf_imp_confluence_on: fixes A :: "'a set" assumes "\<And>a b1 b2. \<lbrakk>a \<in> A; a \<rightarrow>\<^sup>* b1; a \<rightarrow>\<^sup>* b2; is_final b1; is_final b2\<rbrakk> \<Longrightarrow> b1 = b2"   and "wfp_on (\<rightarrow>)\<inverse>\<inverse> A"   and "dw_closed A" shows "is_confluent_on A"
intros_1: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and n :: "nat"   and n' :: "nat"   and h :: "'var" assumes "\<sigma> =\<^sub>L \<sigma>'"   and "cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'"   and "h \<in> local.reads (path \<sigma> n)"   and "(\<sigma>\<^bsup>n\<^esup>) h \<noteq> (\<sigma>'\<^bsup>n'\<^esup>) h"   and "\<forall>k<n. h \<notin> local.writes (path \<sigma> k)"   and "\<forall>k'<n'. h \<notin> local.writes (path \<sigma>' k')" shows "((\<sigma>, n), \<sigma>', n') \<in> cp"
l_get_owner_document_axioms: shows "l_get_owner_document Shadow_DOM.get_owner_document"
equal_alist_def: fixes xs :: "('a, 'b) alist"   and ys :: "('a, 'b) alist" shows "equal_class.equal xs ys \<equiv> impl_of xs = impl_of ys"
expand_all_subfrmls_propag: fixes n_ns :: "'a node \<times> 'a node set"   and B :: "'a ltlr set" assumes "all_subfrmls (fst n_ns) \<subseteq> B \<and> (\<forall>nd\<in>snd n_ns. all_subfrmls nd \<subseteq> B)" shows "expand n_ns \<le> SPEC (\<lambda>r. \<forall>nd\<in>snd r. all_subfrmls nd \<subseteq> B)"
Fvars_fmla_simps_3: fixes A :: "fmla"   and B :: "fmla" shows "Fvars (A AND B) = Fvars A \<union> Fvars B"
that: fixes u :: "'a" assumes "is_unit u"   and "b = u * a" shows "thesis"
sop_one_def: fixes R :: "('e, 'f) Ring_scheme"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and A :: "'g set" shows "sop_one R s A = (\<forall>x\<in>A. 1\<^sub>r\<^bsub>R\<^esub> \<^sub>s\<cdot> x = x)"
bnds_arctan: fixes l :: "float"   and u :: "float"   and prec :: "nat" shows "\<forall>x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real_of_float lx..real_of_float ux} \<longrightarrow> real_of_float l \<le> arctan x \<and> arctan x \<le> real_of_float u"
sgn_inverse: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "sgn (inverse a) = inverse (sgn a)"
mon_conv_real_def: fixes x :: "nat \<Rightarrow> real"   and y :: "real" shows "x\<up>y \<equiv> (\<forall>n. x n \<le> x (Suc n)) \<and> x \<longlonglongrightarrow> y"
simps: fixes x :: "'a extB"   and y :: "'a extB" shows "x * y = x \<sqinter> y"
m_o_anti_mono: fixes X :: "char list set"   and S2 :: "ivl Abs_State.st option"   and S1 :: "ivl Abs_State.st option" assumes "finite X"   and "domo S2 \<subseteq> X"   and "S1 \<sqsubseteq> S2" shows "m_o (m_st m_ivl) (2 * card X) S2 \<le> m_o (m_st m_ivl) (2 * card X) S1"
fold_cong: fixes a :: "'a"   and b :: "'a"   and xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" assumes "a = b"   and "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "fold f xs a = fold g ys b"
commit_def: fixes ck :: "nat \<times> nat"   and m :: "nat" shows "commit ck m = (let (a, b) = ck in return_spmf ((m + a) mod q, a, b))"
II: fixes it :: "'d set"   and \<sigma> :: "'c" assumes "it \<subseteq> dom m"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k v k' v'. k \<notin> it \<and> invar v \<and> m k = Some (\<alpha> v) \<and> k' \<in> it \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k, \<alpha> v) (k', \<alpha> v')" shows "P \<sigma>"
Sup_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. less_eq a b" shows "less_eq (Sup A) (Sup B)"
cod_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "cod f = (if arr f then MkIde (Cod f) else null)"
not_Some_eq: fixes x :: "'a option" shows "(\<forall>y. x \<noteq> Some y) = (x = \<bottom>)"
ideal_points_intersects_x_axis: fixes H :: "circline"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "is_poincare_line H"   and "ideal_points H = {i1, i2}"   and "H \<noteq> x_axis" shows "intersects_x_axis H = (Im (to_complex i1) * Im (to_complex i2) < 0)"
resCases: fixes x :: "name"   and P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Prop :: "ccs \<Rightarrow> bool" assumes "\<lparr>\<nu>x\<rparr>P \<longmapsto>\<alpha> \<prec> P'"   and "\<And>P'. \<lbrakk>P \<longmapsto>\<alpha> \<prec> P'; x \<sharp> \<alpha>\<rbrakk> \<Longrightarrow> Prop (\<lparr>\<nu>x\<rparr>P')" shows "Prop P'"
points_index_one_unique_block: fixes B :: "'b set multiset"   and ps :: "'b set" assumes "B index ps = 1" shows "\<exists>!bl. bl \<in># B \<and> ps \<subseteq> bl"
inject: fixes x :: "'d \<Rightarrow> 'e"   and ya :: "'d \<Rightarrow> 'e" shows "(EnvT x = EnvT ya) = (x = ya)"
ifex_sat_list_subset: fixes t :: "'a ifex"   and u :: "('a \<times> bool) list" assumes "ifex_sat_list t = Some u" shows "fst ` set u \<subseteq> ifex_var_set t"
IsCompFun_def: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" shows "IsCompFun F G = (dom G = \<Union> (States ` F) \<and> \<Union> (ran G) = F - {Root F G} \<and> RootEx F G \<and> OneAncestor F G \<and> NoCycles F G)"
sum_of_prideals4: fixes I :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R I"   and "\<forall>l\<le>n. f l \<in> carrier R"   and "f ` {i. i \<le> n} \<subseteq> I" shows "sum_pr_ideals R f n \<subseteq> I"
assms: shows "\<forall>r. \<not> evaluate_decs False mn env s ds r"
\<R>\<^sub>L_id: fixes ls :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" shows "\<R>\<^sub>L id id ls = ls"
T_LUB: fixes S :: "nat \<Rightarrow> 'c process" assumes "chain S" shows "T (lim_proc (range S)) = \<Inter> (T ` range S)"
a: shows "\<forall>a\<in>A. 0 \<le> f a"
eq3: shows "card (const_lt_component ` (fst ` set (gs @ ab gs bs hs data') - {0}) - {None}) = card (const_lt_component ` (fst ` set (gs @ bs) - {0}) - {None}) + count_const_lt_components (fst aux)"
finite_Roots: fixes p :: "int poly" assumes "p \<in> P" shows "finite (Roots p)"
semigroup_mult_ow: shows "class.semigroup_mult = semigroup_mult_ow UNIV"
one: fixes I :: "'c"   and a :: "'b"   and b :: "'b"   and J :: "'c"   and c :: "'b"   and d :: "'b" assumes "Q I a b"   and "Q J c d"   and "a = b \<and> b = c \<and> c \<noteq> d \<or> a = b \<and> b \<noteq> c \<and> c \<noteq> d \<and> a \<noteq> d" shows "P I J"
simps_5: fixes uu :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and C :: "acom" shows "strip ({uu} WHILE b DO C) = WHILE b DO strip C"
finsert_absorb: fixes a :: "'d"   and A :: "'d fset" assumes "a |\<in>| A" shows "finsert a A = A"
eq_Nil_null: fixes xs :: "'e list" shows "(xs = []) = List.null xs"
fps_const_mult_left: fixes c :: "'a"   and f :: "'a fps" shows "fps_const c * f = Abs_fps (\<lambda>n. c * fps_nth f n)"
measurable_value': fixes M :: "'d measure" shows "value \<in> restrict_space (tree_sigma M) (- {\<langle>\<rangle>}) \<rightarrow>\<^sub>M M"
K5_not_comb_planar: fixes G :: "('c, 'd) pre_digraph" assumes "K\<^bsub>5\<^esub> G" shows "\<not> comb_planar G"
ereal_Inf': fixes A :: "real set" assumes "bdd_below A"   and "A \<noteq> {}" shows "ereal (Inf A) = Inf (ereal ` A)"
less_LiminfD: fixes y :: "'b"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "y < Liminf F f" shows "\<forall>\<^sub>F x in F. y < f x"
case_inf2: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word"   and P :: "32 Word.word \<Rightarrow> 32 Word.word \<Rightarrow> bool" assumes "\<lbrakk>w1 = POS_INF; w2 = POS_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = POS_INF; w2 = NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = POS_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = NEG_INF; w2 = POS_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = NEG_INF; w2 = NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = NEG_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 = POS_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 = NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2" shows "P w1 w2"
a: shows "<t1,[]> \<mapsto>* <t2,[]>"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_mult" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vrat_mult). vrat_mult\<lparr>a\<rparr> = b"
smult_closed: fixes r :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "m \<in> V" shows "r \<cdot> m \<in> V"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int"   and a2 :: "int" assumes "\<And>x y z. \<lbrakk>\<not> x \<le> y \<Longrightarrow> P (x - 1) y z; \<not> x \<le> y \<Longrightarrow> P (y - 1) z x; \<not> x \<le> y \<Longrightarrow> P (z - 1) x y; \<not> x \<le> y \<Longrightarrow> P (tak (x - 1) y z) (tak (y - 1) z x) (tak (z - 1) x y)\<rbrakk> \<Longrightarrow> P x y z" shows "P a0 a1 a2"
assms: shows "graph_homomorphism (LG e\<^sub>1 v) G f"
assms_3: shows "D f \<mapsto> f' at t within T"
bisim1AAssThrow2: fixes P :: "'b prog"   and h :: "'heap"   and i :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr expr1"   and e :: "'addr expr1"   and v :: "'addr val" assumes "P,i,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>ad\<rfloor>)" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 a) + pc, \<lfloor>ad\<rfloor>)"
le_ivl_iff_subset: fixes iv1 :: "ivl"   and iv2 :: "ivl" shows "(iv1 \<le> iv2) = (\<gamma>_ivl iv1 \<subseteq> \<gamma>_ivl iv2)"
vv_def: shows "vv \<equiv> \<lambda>j i. if i = Suc j then 1 else 0"
simps_71: fixes x12 :: "qtname"   and x31 :: "jump"   and x32 :: "stmt" shows "Init x12 \<noteq> x31\<bullet> x32"
b_pos: fixes b :: "real" assumes "b \<in> set bs" shows "0 < b"
strip_step: fixes S :: "(char list \<Rightarrow> int) set"   and C :: "(char list \<Rightarrow> int) set acom" shows "strip (step S C) = strip C"
snd_last: fixes xs :: "('a \<times> 'b) list" assumes "xs \<noteq> []" shows "snd (last xs) = last (map snd xs)"
zero_le_mult_iff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
liftn_liftn0: fixes t :: "dB"   and k :: "nat"   and n :: "nat" assumes "\<forall>i\<in>frees t. k \<le> i" shows "liftn n t k = liftn n t 0"
prj_chine_assoc_3: shows "cods.Prj\<^sub>0\<^sub>0 \<cdot> cods.chine_assoc = cods.Prj\<^sub>0"
to_st_nil_inv: fixes A :: "('fun, 'var) extstrand_step list" assumes "to_st A = []" shows "A = []"
transitive_closure_trans_3: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
IHP_1: fixes l2 :: "'a list"   and w1 :: "'a list"   and w2 :: "'a list" assumes "l1' @ l2 \<in> w1 \<otimes> w2" shows "\<exists>w11 w12 w21 w22. w1 = w11 @ w12 \<and> w2 = w21 @ w22 \<and> l1' \<in> w11 \<otimes> w21 \<and> l2 \<in> w12 \<otimes> w22"
poly_monom_identity_mod_p: shows "monom 1 CARD('a) - monom 1 1 = (\<Prod>x\<in>UNIV. [:0, 1:] - [:x:])"
inters_def: fixes l :: "Lines"   and m :: "Lines" shows "inters l m \<equiv> {P. incid P l \<and> incid P m}"
AE_not_in: fixes N :: "'b set"   and M :: "'b measure" assumes "N \<in> null_sets M" shows "AE x in M. x \<notin> N"
simps_2: fixes uv :: "'b" shows "is_literal_formula (\<^bold>\<not> (Atom uv)) = True"
1: shows "NR\<^sub>N n u l = ennreal p * (of_nat l + NR\<^sub>N (n - 1) u l) + ennreal q * (of_nat u + (\<Sum>k<n - 1. NR\<^sub>N (k + 1) u l * ennreal (B (n - 1) k))) + ennreal (q ^ n) * NR\<^sub>N n u l"
d: shows "real_of_int (d div g) rdvd real_of_int (c div g) * t"
hom_div: fixes x :: "'a poly"   and y :: "'a poly" shows "map_poly hom (x div y) = map_poly hom x div map_poly hom y"
h9a: shows "n = 0 \<longrightarrow> MPoly_Type.degree (sum f {0..n}) x = n"
ereal_inj_on_uminus: fixes A :: "ereal set" shows "inj_on uminus A"
simps_41: fixes x61 :: "label"   and x62 :: "expr"   and x63 :: "stmt"   and x2 :: "expr" shows "x61\<bullet> While(x62) x63 \<noteq> Expr x2"
assms_2: shows "PP \<longrightarrow> P \<and> Q'"
starlike_imp_Borsukian: fixes S :: "'a set" assumes "starlike S" shows "Borsukian S"
sig_of_spair_alt_spp: fixes p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" shows "sig_of_pair p = sig_of_pair_spp (app_pair spp_of p)"
sum_is_submodule: fixes N1 :: "'a vec set"   and N2 :: "'a vec set" assumes "LinearCombinations.submodule class_ring N1 V"   and "LinearCombinations.submodule class_ring N2 V" shows "LinearCombinations.submodule class_ring (subspace_sum N1 N2) V"
S11: shows "loc_upb n x = (let y = c_assoc_value (pr_gr (loc_upb (c_snd (c_snd n)) x)) (c_pair (c_snd (c_snd n)) x) in c_pair (c_pair n x) (loc_upb (c_snd (c_snd n)) x + loc_upb (c_fst (c_snd n)) y) + 1)"
that: fixes cfsx' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cfx :: "('var \<Rightarrow> 'val option) \<times> 'ret" assumes "transfers (kinds as) (cfsx @ cf # cfs) = cfsx' @ cfx # cfs"   and "transfers (kinds as) (cfsx @ cf # cfs') = cfsx' @ cfx # cfs'"   and "length cfsx' = length (upd_cs cs as)" shows "thesis"
takeAllSubset: fixes list :: "'a list"   and P :: "'a set" shows "set (takeAll (\<lambda>x. x \<in> P) list) \<subseteq> P"
simps_1: fixes G :: "prog"   and dynC :: "qtname"   and static_membr :: "bool" shows "G,NullT \<turnstile> dynC valid_lookup_cls_for static_membr = False"
\<Z>_vfsequenceI: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<SS>"   and "\<R>\<^sub>\<circ> \<SS> \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "\<Z>_vfsequence \<alpha> \<SS>"
stdin_of_getLine: fixes world :: "\<^url>"   and inp :: "char list"   and stdin :: "char list list" assumes "stdin_of world = inp # stdin" shows "stdin_of (exec getLine world) = stdin \<and> eval getLine world = String.implode inp"