hcju/isabelleps huggingface dataset:
eq_vecI: fixes w :: "'d Matrix.vec"   and v :: "'d Matrix.vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $v i = w $v i"   and "dim_vec v = dim_vec w" shows "v = w"
Inf_empty: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
l3_inv1_def: shows "l3_inv1 \<equiv> {s. parts (ik s) \<inter> range LtK \<subseteq> ik s}"
wf_update_until_auxlist: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and pos :: "bool"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and ne :: "nat"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and args :: "Monitor.args" assumes "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> auxlist ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length auxlist) \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length auxlist) \<psi>) rel2"   and "fv \<phi> \<subseteq> fv \<psi>"   and "args_ivl args = I"   and "args_n args = n"   and "args_pos args = pos" shows "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> (update_until args rel1 rel2 (\<tau> \<sigma> (ne + length auxlist)) auxlist) ne"
total_well_order_extension: fixes p :: "('b \<times> 'b) set" assumes "wf p" shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"
cod_assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "D.cod \<a>\<^sub>D[a, b, c] = a \<otimes>\<^sub>D b \<otimes>\<^sub>D c"
e_or: shows "e0 \<noteq> e0' \<or> e1 \<noteq> e1'"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<exists>y\<in>carrier R. y \<oplus> x = \<zero>"
ipoly_0_0: fixes f :: "int poly" shows "(ipoly f (0::'a) = (0::'a)) = (poly f 0 = 0)"
prj1_dup: fixes a :: "'a" assumes "ide a" shows "\<l>[a] \<cdot> (\<t>[a] \<otimes> a) \<cdot> \<d>[a] = a"
P1: shows "B Out A' A \<and> B Out C' C \<and> E Out D' D \<and> E Out F' F \<and> A' B C' Cong3 D' E F'"
exec: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f"
scaleR_prod_def: fixes r :: "real"   and A :: "'c \<times> 'd" shows "r *\<^sub>R A = (r *\<^sub>R fst A, r *\<^sub>R snd A)"
assms_3: fixes x :: "('fun, 'var) Term.term" assumes "x \<in> set X" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m x"
exec_Suc: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and n :: "nat"   and c'' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" assumes "P \<turnstile> c \<rightarrow> c'"   and "P \<turnstile> c' \<rightarrow>^n c''" shows "P \<turnstile> c \<rightarrow>^(Suc n) c''"
prop_in_f_4_d: fixes v :: "i" shows "[\<^bold>\<not>Propositional A! in v]"
one_step_methodD: fixes m :: "'d set \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'd set \<Rightarrow> ('d set \<times> 'e) option nres"   and X0 :: "'d set"   and hl :: "real"   and hu :: "real"   and CX :: "'d set"   and z :: "('d set \<times> 'e) option nres" assumes "one_step_method m"   and "SPEC (case_option True (\<lambda>(res, err). nonempty res \<and> (\<forall>x0\<in>X0. \<forall>h\<in>{hl..hu}. x0 \<in> Csafe \<longrightarrow> 0 \<le> h \<longrightarrow> h \<in> existence_ivl0 x0 \<longrightarrow> (\<forall>h'\<in>{0..h}. flow0 x0 h' \<in> CX) \<longrightarrow> x0 + h *\<^sub>R ode x0 \<in> CX \<longrightarrow> flow0 x0 h \<in> res))) \<le> z" shows "m X0 hl hu CX \<le> z"
HStartBallot_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2a s"   and "HStartBallot s s' p" shows "Inv2a s'"
STAR_tan_zero: shows "(*f* tan) 0 = 0"
ex_less_of_nat_mult: fixes x :: "'c"   and y :: "'c" assumes "(0::'c) < x" shows "\<exists>n. y < of_nat n * x"
dbl_dec_simps_3: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one one = one"
closed_err_Union_lift2I: fixes AS :: "'b set set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err" assumes "\<forall>A\<in>AS. closed (err A) (lift2 f)"   and "AS \<noteq> {}"   and "\<forall>A\<in>AS. \<forall>B\<in>AS. A \<noteq> B \<longrightarrow> (\<forall>a\<in>A. \<forall>b\<in>B. a \<squnion>\<^bsub>f\<^esub> b = Err)" shows "closed (err (\<Union> AS)) (lift2 f)"
RInitInitFail: fixes P :: "(char list list \<times> char list exp) prog"   and e' :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and a :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and D :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e',s\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Error))"   and "P \<turnstile> \<langle>RI (D,throw a) ; Cs \<leftarrow> e,(h', l', sh'')\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>RI (C,e') ; D # Cs \<leftarrow> e,s\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<times> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat list \<times> nat" assumes "\<And>a x xs s. (\<And>xa. \<lbrakk>xa = s + a * x; cond1 b ys (x # xs) xa\<rbrakk> \<Longrightarrow> P a (Suc x) (xs, s)) \<Longrightarrow> P a x (xs, s)" shows "P a0 a1 a2"
next_def: fixes g :: "('d, 'c) generator" shows "list.next g = snd (generator g)"
cases: fixes x :: "'b list" obtains "x = []"   | xa :: "'b" and xs :: "'b list" where "x = xa # xs"
simps: fixes key :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "msort_key key xs = merge_all key (sequences key xs)"
take_len: fixes p :: "nat"   and w :: "'b list" assumes "p \<le> \<^bold>|w\<^bold>|" shows "\<^bold>|take p w\<^bold>| = p"
t_1: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<inter> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>))"
unique_connecting_path_unique: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto> w" shows "xs = v \<leadsto> w"
translation_exist: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "right_translations_of_group G composition unit"   and "f \<in> right_translations_of_group.translation G composition ` G" shows "\<exists>a\<in>G. f = right_translations_of_group.translation G composition a"
le_infI1: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and x :: "'c"   and b :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a x" shows "less_eq (inf a b) x"
that: fixes d :: "real" assumes "0 < d"   and "\<And>x'. \<lbrakk>x' \<in> S; dist x x' < d\<rbrakk> \<Longrightarrow> (\<forall>y\<in>f x. \<exists>y'. y' \<in> f x' \<and> dist y y' < e) \<and> (\<forall>y'\<in>f x'. \<exists>y. y \<in> f x \<and> dist y' y < e)" shows "thesis"
\<sigma>_eq_1_output_correct: fixes \<alpha>1 :: "nat"   and r :: "nat"   and u1 :: "nat"   and v1 :: "nat" shows "((\<^bold>g [^] \<alpha>1) [^] r \<otimes> \<^bold>g \<otimes> inv \<^bold>g) [^] u1 \<otimes> (\<^bold>g [^] \<alpha>1) [^] v1 \<otimes> x1 \<otimes> inv (((\<^bold>g [^] r) [^] u1 \<otimes> \<^bold>g [^] v1) [^] \<alpha>1) = x1"
continuous_map_fst: fixes X :: "'c topology"   and Y :: "'d topology" shows "continuous_map (prod_topology X Y) X fst"
Cones_SET_eq_\<iota>_img_cones: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and S :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and \<phi> :: "'ah \<times> 'ah \<Rightarrow> 'ah \<Rightarrow> 'ai"   and \<iota> :: "('ag \<Rightarrow> 'ah) \<Rightarrow> 'ai"   and a :: "'ah"   and \<Phi> :: "'ah \<Rightarrow> 'ai"   and a' :: "'ah" assumes "representation_of_cones_functor J C D S \<phi> \<iota> a \<Phi>"   and "partial_magma.ide C a'" shows "set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cones_functor.map J C D S \<iota> a') = \<iota> ` {\<chi>. Limit.cone J C D a' \<chi>}"
star_left_induct: fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z \<squnion> y \<sqinter> x \<le> x \<longrightarrow> top \<sqinter> z \<le> x"
bind_spmf_mono': fixes f :: "'c spmf"   and g :: "'c spmf"   and h :: "'c \<Rightarrow> 'd spmf"   and k :: "'c \<Rightarrow> 'd spmf" assumes "ord_spmf (=) f g"   and "\<And>x. ord_spmf (=) (h x) (k x)" shows "ord_spmf (=) (f \<bind> h) (g \<bind> k)"
hyps: fixes S :: "'a fm set" assumes "S \<union> {conjoin ps q} \<in> C" shows "S \<union> set ps \<union> {q} \<in> C"
Act\<^sub>\<alpha>_def: shows "Act\<^sub>\<alpha> \<equiv> (id ---> id ---> rep_Tree\<^sub>\<alpha> ---> abs_Tree\<^sub>\<alpha>) tAct"
opt_no_defection_def: fixes s :: "opt_v_state"   and round_votes :: "process \<Rightarrow> val option" shows "opt_no_defection s round_votes = (\<forall>a Q v. quorum_for Q v (last_vote s) \<and> a \<in> Q \<longrightarrow> round_votes a \<in> {None, Some v})"
lossless_S: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'a \<times> 'b" assumes "etp I domain range F F\<^sub>i\<^sub>n\<^sub>v" shows "y \<in> set_spmf I \<longrightarrow> lossless_spmf (ETP.etp.S range (fst y))"
tensor_mat_add2: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2"   and "m3 \<in> carrier_mat d2 d2" shows "tensor_mat m1 (m2 + m3) = tensor_mat m1 m2 + tensor_mat m1 m3"
Class_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and T :: "ty" assumes "subtype P (Class C) T" shows "\<exists>D. T = Class D"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> set xs"   and "y \<in> set ys"   and "z \<in> set zs"   and "P x y"   and "P y z" shows "P x z"
ground_heads_axioms_def: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set" shows "ground_heads_axioms arity_sym arity_var ground_heads_var \<equiv> (\<forall>f x. f \<in> ground_heads_var x \<longrightarrow> arity_var x \<le> arity_sym f) \<and> (\<forall>x. ground_heads_var x \<noteq> {})"
g_def: shows "g \<equiv> \<lambda>M. {t \<in> M. fv t = {}}"
B: shows "is_tangle_diagram xs \<and> is_tangle_diagram (basic (make_vert_block (nat (codomain_block y)))) \<longrightarrow> is_tangle_diagram (xs \<otimes> basic (make_vert_block (nat (codomain_block y))))"
assms_3: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "the (Mapping.lookup M f) < length slp"
simps_4: fixes P :: "'state"   and \<alpha> :: "'act"   and t :: "('b, 'pred, 'act) Tree" shows "valid_Tree P (tAct \<alpha> t) = (\<exists>\<alpha>'' t' P'. tAct \<alpha> t =\<^sub>\<alpha> tAct \<alpha>'' t' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> valid_Tree P' t')"
preserves_arr: fixes f :: "'a" assumes "C.arr f" shows "C.arr (src\<^sub>C f)"
monomial_inj: fixes s :: "'c"   and c :: "'d"   and t :: "'c"   and d :: "'d" assumes "monomial c s = monomial d t" shows "c = (0::'d) \<and> d = (0::'d) \<or> c = d \<and> s = t"
N_nonneg: fixes n :: "int" shows "0 \<le> N n"
surjective: fixes r :: "('a, 'b, 'c) TermEquivClT_scheme" shows "r = \<lparr>TDomain = TDomain r, TExprSet = TExprSet r, TCodomain = TCodomain r, \<dots> = TermEquivClT.more r\<rparr>"
wfp_on_mono: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "wfp_on Q B" shows "wfp_on P A"
interpret_floatarith_divide: fixes f :: "floatarith"   and g :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (f / g) xs = interpret_floatarith f xs / interpret_floatarith g xs"
step: fixes xs :: "'a llist" assumes "X xs"   and "\<not> lnull xs"   and "\<not> lnull (ltl xs)" shows "lhd xs \<le> lhd (ltl xs) \<and> (X (ltl xs) \<or> lsorted (ltl xs))"
filter_map_filter: fixes F :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "ord_class.filter F"   and "mono f"   and "\<forall>x y. f x \<le> y \<longrightarrow> (\<exists>z\<ge>x. y = f z)" shows "ord_class.filter (f ` F)"
connected_component_eq_empty: fixes S :: "'c set"   and x :: "'c" shows "(connected_component_set S x = {}) = (x \<notin> S)"
assms: fixes r :: "('f, 'g) aezfun" assumes "r \<in> FG" shows "smult' r (0::'a) = (0::'a)"
case_cong: fixes option :: "'a option"   and option' :: "'a option"   and f1 :: "'b"   and g1 :: "'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "option = option'"   and "option' = None \<Longrightarrow> f1 = g1"   and "\<And>x2. option' = Some x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case option of None \<Rightarrow> f1 | Some x2 \<Rightarrow> f2 x2) = (case option' of None \<Rightarrow> g1 | Some x2 \<Rightarrow> g2 x2)"
fresh_name'_fresh: fixes S :: "String.literal set" assumes "finite S" shows "fresh_name' S \<notin> S"
inj: shows "inj_on (\<lambda>y. {i. y = column i A}) (columns A)"
rm_vars_empty_1: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "rm_vars {} s = s"
scale_prod: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'c"   and a :: "'d"   and b :: "'e" assumes "module_prod s1 s2" shows "module_prod.scale s1 s2 x (a, b) = (s1 x a, s2 x b)"
max_diff_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "minus (ord.max less_eq x y) z = ord.max less_eq (minus x z) (minus y z)"
h': shows "h2 \<turnstile> set_child_nodes ptr (remove1 child children_h) \<rightarrow>\<^sub>h h'"
arc_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<exists>e. f e = top \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot)" shows "matrix_stone_relation_algebra.arc f"
preserve_rtranclp'': fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. R x y \<Longrightarrow> S (f x) (f y)"   and "\<And>y. S (f y) (g y)"   and "R\<^sup>*\<^sup>* x y" shows "S\<^sup>+\<^sup>+ (f x) (g y)"
in_set_conv_decomp: fixes x :: "'d"   and xs :: "'d list" shows "x mem xs = (\<exists>ys zs. xs = ys @ x # zs)"
obs2: shows " D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<mu>' (X \<tau>) - \<nu>' (X \<tau>)) on {t\<^sub>0--t}"
ideD_1: fixes a :: "'b" assumes "B.ide a" shows "B.arr a"
product_universe_implies_factor_universe: fixes x :: "i"   and y :: "i" assumes "O x y"   and "x \<otimes> y = u" shows "x = u"
smult_carrier_vec: fixes a :: "'b"   and v :: "'b vec"   and n :: "nat" shows "(a \<cdot>\<^sub>v v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
assms: shows "B \<inter>\<^sub>\<circ> A \<noteq> []\<^sub>\<circ>"
length_tp': fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list" assumes "ly = layout_of ap"   and "tp = tm_of ap" shows "length tp = 2 * sum_list (take (length ap) (layout_of ap))"
openin_imp_subset: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin (subtopology U S) T" shows "T \<subseteq> S"
pow_inductr: fixes z :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and i :: "nat" assumes "z \<union> y ; x \<subseteq> y" shows "z ; rel_d.power x i \<subseteq> y"
degree_map_poly_le: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly" shows "degree (map_poly f p) \<le> degree p"
greatest_unique: fixes x :: "'a upred"   and A :: "'a upred set"   and y :: "'a upred" assumes "greatest \<P> x A"   and "greatest \<P> y A" shows "x = y"
jnf_complexity_1_complex: fixes A :: "complex mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n n"   and "real_nonneg_mat A"   and "\<And>x. poly (char_poly A) x = 0 \<Longrightarrow> cmod x \<le> 1"   and "\<lbrakk>poly (char_poly A) 1 = 0; d + 1 < order 1 (char_poly A)\<rbrakk> \<Longrightarrow> dim_gen_eigenspace A 1 (d + 1) = order 1 (char_poly A)" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ d)"
hyps_3: shows "(h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)"
Sigma_mset_empty1: fixes B :: "'b \<Rightarrow> 'c multiset" shows "Sigma_mset {#} B = {#}"
map: fixes y :: "nat" shows "map_fm_binders (\<lambda>a x. lowerAtom (i + x) 1 a) F (y + x1) = map_fm_binders (\<lambda>a x. lowerAtom (i + x1 + x) 1 a) F y"
assms_3: fixes s :: "'a set" assumes "r \<in> s" shows "h s = g s"
zatmostonce: shows "count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) (cs ! i)) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) z \<le> 1"
indep_in_indep: fixes \<E> :: "'a set"   and X :: "'a set" assumes "indep_in \<E> X" shows "indep X"
hyps_4: shows "finite {M. set_mset M \<subseteq> insert x A \<and> size M = k}"
isTable_Cons: fixes E :: "'a \<Rightarrow> 'b"   and vs :: "'a list"   and a :: "'a"   and b :: "'b"   and ps :: "('a \<times> 'b) list" assumes "isTable E vs ((a, b) # ps)" shows "isTable E vs ps"
intAf: shows "integrable M (\<lambda>x. indicat_real A x * f x)"
assms_1: shows "\<Psi> \<turnstile> M \<leftrightarrow> K"
language_state_for_inputs_nonempty: fixes xs :: "'e list"   and M :: "('e, 'f, 'g) FSM"   and q :: "'g" assumes "set xs \<subseteq> inputs M"   and "completely_specified M"   and "q \<in> nodes M" shows "LS\<^sub>i\<^sub>n M q {xs} \<noteq> {}"
append_in_lists_conv: fixes xs :: "'d list"   and ys :: "'d list"   and A :: "'d set" shows "(xs @ ys \<in> lists A) = (xs \<in> lists A \<and> ys \<in> lists A)"
blinding_of_on_axioms: shows "blinding_of_on A rha boa"
fv_Upd: fixes v :: "Vars.var" shows "fv (Upd v) = fv v"
all_ops_def: fixes rects :: "'a rectangle list" shows "all_ops rects = sort (ins_ops rects @ del_ops rects)"
farkas_minkowsky_weyl_theorem_1_full_dim: fixes X :: "'a vec set"   and Bnd :: "'a" assumes "X \<subseteq> carrier_vec n"   and "finite X"   and "local.span X = carrier_vec n" shows "\<exists>nr A. A \<in> carrier_mat nr n \<and> cone X = polyhedral_cone A \<and> (X \<subseteq> \<int>\<^sub>v \<inter> Bounded_vec Bnd \<longrightarrow> A \<in> \<int>\<^sub>m \<inter> Bounded_mat (det_bound (n - 1) Bnd))"
nth_Abs_fls_ex_nat_lower_bound: fixes f :: "int \<Rightarrow> 'a"   and n :: "int" assumes "\<exists>m. \<forall>n>m. f (- int n) = (0::'a)" shows "fls_nth (Abs_fls f) n = f n"
diff_cancel: fixes a :: "'i" shows "a - a = (0::'i)"
simps_13: shows "(extN.Top \<le> extN.Top) = True"
F1_def: fixes X' :: "'a set" shows "F1 X' = - Rf CH X'"
sample_parametric: fixes R :: "'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "bi_unique R" shows "rel_fun (rel_pmf R) (rel_fun (rel_fun R (=)) (=)) sample sample"
added_files_Deliver_Store_diff_collapse: fixes e :: "'b"   and e' :: "'b"   and i :: "'c"   and mo :: "'c"   and j :: "'c" assumes "e \<noteq> e'" shows "added_files [Deliver (i, Store e mo j)] e' = []"
eval_in_carrier: fixes p :: "'a list"   and x :: "'a" assumes "set p \<subseteq> carrier R"   and "x \<in> carrier R" shows "R.eval p x \<in> carrier R"
simps_3: fixes c :: "'c"   and cs :: "'c list"   and t :: "'d"   and ts :: "'d list"   and u :: "'c \<Rightarrow> 'd" shows "set_clocks (c # cs) (t # ts) u = set_clocks cs ts (u(c := t))"
in_dom: fixes i :: "'a" assumes "i \<in> agents" shows "Order_Predicates.preorder_on alts (R i)"
stop_oracle_def: shows "stop_oracle = (\<lambda>c_o. (\<lambda>(idgs, s) x. case idgs of Inl x \<Rightarrow> return_spmf (None, idgs, s) | Inr xa \<Rightarrow> map_spmf (\<lambda>(y, s). (Some y, idgs, s)) (oracle c_o s x)) \<oplus>\<^sub>O\<^sup>S (\<lambda>(idgs, s) guess. return_spmf (case idgs of Inl x \<Rightarrow> (None, idgs, s) | Inr 0 \<Rightarrow> (None, Inl (guess, s), s) | Inr (Suc i) \<Rightarrow> (Some (), Inr i, s))))"
linear_subspace_image: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and S :: "('b, 'c) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "vec.subspace S" shows "vec.subspace (f ` S)"
M_def: shows "M \<equiv> \<lambda>E d. subterms_tuples E \<union> subterms_img d"
dflt_option_assn_def: fixes dflt :: "'d"   and A :: "'b \<Rightarrow> 'd \<Rightarrow> assn" shows "dflt_option_assn dflt A \<equiv> pure (dflt_option_rel_aux dflt O \<langle>the_pure A\<rangle>option_rel)"
tuple_since_None_None: fixes as :: "'a option list" assumes "Mapping.lookup tuple_since as = None" shows "Mapping.lookup tuple_since' as = None"
vsv_vimage_vinsert: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r `\<^sub>\<circ> vinsert a A = vinsert (r\<lparr>a\<rparr>) (r `\<^sub>\<circ> A)"
root_bound_1: fixes p :: "int poly"   and B :: "rat"   and x :: "real" assumes "root_bound p = B"   and "0 < degree p"   and "poly (real_of_int_poly p) x = 0" shows "norm x \<le> real_of_rat B"
induct: fixes P1 :: "pat \<Rightarrow> bool"   and P3 :: "(char list \<times> pat) list \<Rightarrow> bool"   and P2 :: "char list \<times> pat \<Rightarrow> bool"   and char_list_pat_prod_list :: "(char list \<times> pat) list" assumes "\<And>x. P1 (PVar x)"   and "\<And>x. P3 x \<Longrightarrow> P1 (PRcd x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P3 char_list_pat_prod_list"
distinct_2: fixes x21 :: "dB"   and x22 :: "dB"   and x1 :: "nat" shows "x21 \<degree> x22 \<noteq> Var x1"
inter_Set: fixes A :: "'b set"   and t :: "('b, unit) RBT.rbt" shows "A \<inter> Set t = rbt_filter (\<lambda>k. k \<in> A) t"
kinds_def: fixes xs :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list" shows "While_CFG.kinds xs \<equiv> map kind xs"
lhs_def: shows "lhs \<equiv> vector_space_ow (Collect (Domainp B))"
ih: fixes P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list" assumes "\<forall>a\<in>set P. case a of (E, k) \<Rightarrow> is_irreducible [C] E" shows "wstate_of_dstate (N, P, Q @ Q', n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, fst (reduce_all2 C Q') @ P, Q @ snd (reduce_all2 C Q'), n)"
intros: fixes i :: "nat"   and s :: "'b fset" assumes "i < size s" shows "list_of_fset s ! i |\<in>|\<^bsub>i\<^esub> s"