hcju/isabelleps huggingface dataset:
remove_vertex_path_from_to: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c"   and x :: "'c" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"   and "x \<in> V\<^bsub>G\<^esub>"   and "x \<notin> set xs" shows "v \<leadsto>xs\<leadsto>\<^bsub>Digraph.remove_vertex G x\<^esub> w"
intros: fixes 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 C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "D \<in> baseClasses Bs" shows "subcls1p P C D"
mult_minus1_right: fixes z :: "'a" shows "z * - (1::'a) = - z"
abs_eq: fixes x :: "'b poly" shows "- x = Abs_poly (\<lambda>n. - coeff x n)"
awalk_app_isoD: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and u :: "'g"   and p :: "'h list"   and v :: "'g" assumes "wf_digraph G"   and "pre_digraph.awalk (app_iso hom G) u p v"   and "pre_digraph.digraph_isomorphism G hom" shows "pre_digraph.awalk G (iso_verts (pre_digraph.inv_iso G hom) u) (map (iso_arcs (pre_digraph.inv_iso G hom)) p) (iso_verts (pre_digraph.inv_iso G hom) v)"
satisfies_bounded_nFAND: fixes \<AA> :: "'i"   and \<phi>s :: "('a, 'k) aformula list" shows "(\<AA> \<Turnstile>\<^sub>b nFAND \<phi>s) = (\<forall>\<phi>\<in>set \<phi>s. \<AA> \<Turnstile>\<^sub>b \<phi>)"
c: shows "(c1 \<inter>\<^sub>g c2) = Some c"
semilattice_set_axioms: shows "semilattice_set (\<sqinter>)"
preserves_reflects_arr: fixes f :: "'c" shows "D.arr (F f) = C.arr f"
finite_greaterThanAtMost: fixes l :: "nat"   and u :: "nat" shows "finite {l<..u}"
is_concrete_category: shows "concrete_category (Collect B.obj) Hom B.iso_class (\<lambda>_ _ _. Comp)"
P_new_hd_disjoint: fixes xs :: "'a list" assumes "xs \<in> paths" shows "hd (tl P_new) \<noteq> hd (tl xs)"
leg1_is_map: shows "is_left_adjoint (tab\<^sub>1 f)"
f_Exec_Stream_expand_aggregate_map_drop: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and k :: "nat"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c)) k ag \<up> n = f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<up> n \<odot> k) (f_Exec_Comp trans_fun (xs \<down> n \<odot> k) c))) k ag"
fa: shows "\<forall>x\<in>I. x \<in> s \<or> (\<exists>sl. sl \<noteq> [] \<and> tl sl \<noteq> [] \<and> (sl ! 0, sl ! (length sl - 1)) = ({x}, s) \<and> (\<forall>i<length sl - 1. \<turnstile>\<N>\<^bsub>(sl ! i, sl ! (i + 1))\<^esub>))"
cone_hull_empty_iff: fixes S :: "'a set" shows "(S = {}) = (cone hull S = {})"
chine_hcomp_trg_arr: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "chine_hcomp (trg \<mu>) \<mu> = \<langle>\<p>\<^sub>1[C.cod (Leg1 (Dom \<mu>)), Leg1 (Dom \<mu>)] \<lbrakk>C.cod (Leg1 (Dom \<mu>)), Leg1 (Cod \<mu>)\<rbrakk> Chn \<mu> \<cdot> \<p>\<^sub>0[C.cod (Leg1 (Dom \<mu>)), Leg1 (Dom \<mu>)]\<rangle>"
bot_least: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.order_bot bot less_eq less" shows "less_eq bot a"
image2_eqI: fixes b :: "'e"   and f :: "'f \<Rightarrow> 'e"   and x :: "'f"   and c :: "'g"   and g :: "'f \<Rightarrow> 'g"   and A :: "'f set" assumes "b = f x"   and "c = g x"   and "x \<in> A" shows "(b, c) \<in> BNF_Greatest_Fixpoint.image2 A f g"
permutation_inverse_compose: fixes p :: "'a \<Rightarrow> 'a"   and q :: "'a \<Rightarrow> 'a" assumes "permutation p"   and "permutation q" shows "inv' (p \<circ> q) = inv' q \<circ> inv' p"
invertible: shows "\<exists>g. g \<circ> f = id \<and> f \<circ> g = id"
simps_4: fixes x21 :: "nat"   and x22 :: "nat"   and x11 :: "nat"   and x12 :: "nat" shows "Eq x21 x22 \<noteq> Less x11 x12"
factorize_polys_undo_factorize_polys: fixes ps :: "rat poly list"   and ftrs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and x :: "real" assumes "factorize_polys ps = (ftrs, data)" shows "undo_factorize_polys data (sign_vec ftrs x) = sign_vec ps x"
linear: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "le_of_key_order ko x y \<or> le_of_key_order ko y x"
assms_2: fixes x :: "'b" assumes "x \<in> f ` {x. Domainp A x}" shows "B x x"
parts_insert2: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "parts (insert X (insert Y H)) = parts {X} \<union> parts {Y} \<union> parts H"
absolutely_continuousI_density: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "absolutely_continuous M (density M f)"
real_natfloor_nat: fixes x :: "real" assumes "x \<in> \<nat>" shows "real (nat \<lfloor>x\<rfloor>) = x"
is_sublist_at1: fixes as :: "'a list"   and bs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct (as @ bs)"   and "is_sublist [x, y] (as @ bs)"   and "x \<noteq> last as" shows "is_sublist [x, y] as \<or> is_sublist [x, y] bs"
IH: fixes ds :: "bool list"   and T :: "tree" assumes "branch p (delete ds T)" shows "branch p T \<or> p = ds"
assms: fixes i :: "'a" shows "P \<sqsubseteq> Q i"
emeasure_T_const: fixes s :: "'a cfg" shows "emeasure (MC.T s) (space (stream_space (count_space UNIV))) = 1"
eq2: shows "(Dec G u) \<cdot> (Dec G v) = [u \<cdot> v]"
sum_0_def: shows "sum_0 \<equiv> abel_semigroup_set.F (+) (0::'a)"
b_e_type_br_if_2: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Br_if i" shows "\<exists>ts_c ts''. ts = ts_c @ ts'' @ [T_i32] \<and> ts' = ts_c @ ts'' \<and> label \<C> ! i = ts''"
mset_pos_plus: fixes A :: "'a zmultiset"   and B :: "'a zmultiset" shows "mset_pos (A + B) = mset_pos A - mset_neg B + (mset_pos B - mset_neg A)"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1))))) \<and> t2 = snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2)))))"
\<R>_typed_step_no_await: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>',\<S>',P'"   and "tyenv_wellformed mds \<Gamma> \<S> P"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "pred P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"   and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>"   and "no_await c\<^sub>1" shows "\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>1, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
add_scalar_prod_distrib: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>3 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "(v\<^sub>1 + v\<^sub>2) \<bullet> v\<^sub>3 = v\<^sub>1 \<bullet> v\<^sub>3 + v\<^sub>2 \<bullet> v\<^sub>3"
LIM_zero_iff: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'c)) F = (f \<longlongrightarrow> l) F"
simps_6: fixes v :: "'c"   and vd :: "'c"   and ve :: "'c Tree"   and vf :: "'c Tree"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "siftDown (T v (T vd ve vf) (T va vb vc)) = (if val (T va vb vc) \<le> val (T vd ve vf) then if val (T vd ve vf) \<le> v then T v (T vd ve vf) (T va vb vc) else T (val (T vd ve vf)) (siftDown (set_val (T vd ve vf) v)) (T va vb vc) else if val (T va vb vc) \<le> v then T v (T vd ve vf) (T va vb vc) else T (val (T va vb vc)) (T vd ve vf) (siftDown (set_val (T va vb vc) v)))"
gcd_add2: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and m :: "'b"   and n :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd m (plus m n) = gcd m n"
body'_def: shows "body' \<equiv> \<mu> body'. \<Lambda> v. Fixrec.run\<cdot>(match_B\<cdot>v\<cdot>(\<Lambda> a. Fixrec.succeed\<cdot>(B\<cdot>A)))"
Rats_def: 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" assumes "class.field_char_0 divide inverse times one plus zero minus uminus" shows "field_char_0.Rats divide one plus zero minus = range (field_char_0.of_rat divide one plus zero minus)"
is_lubD2: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c"   and u :: "'c" assumes "class.po below"   and "po.is_lub below S x"   and "po.is_ub below S u" shows "below x u"
SN_def: fixes r :: "('a \<times> 'a) set" shows "SN r = (\<forall>x. SN_on r {x})"
ACI_comm: fixes r :: "'b rexp"   and s :: "'b rexp" shows "ACI (Plus r s) (Plus s r)"
all_positions_drop_length_take: fixes i :: "nat"   and w :: "'a list"   and x :: "'a list" assumes "i \<le> length w"   and "i \<le> length x"   and "\<forall>j<i. x ! j = w ! (length w + j - i)" shows "drop (length w - i) w = take i x"
that: fixes p :: "'a list"   and s :: "'a list" assumes "v = p \<cdot> u \<cdot> s" shows "thesis"
split_indicator_asm: fixes P :: "'a \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b" shows "P (indicator S x) = (\<not> (x \<in> S \<and> \<not> P (1::'a) \<or> x \<notin> S \<and> \<not> P (0::'a)))"
Comm1: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>e a<b> \<prec>\<^sub>e P'"   and "Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'" shows "P \<parallel> Q \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P' \<parallel> Q'"
mono_TokensAt: fixes k :: "nat" shows "mono (TokensAt k)"
arithmetic_simps_79: fixes a :: "'a" shows "a * (1::'a) = a"
empty_neutral_1: fixes x :: "'k multiset" shows "{#} + x = x"
process_rule_5_futures: fixes ys :: "'e list"   and P :: "'e process"   and xs :: "'e list"   and x :: "'e" assumes "ys \<in> future_divergences P xs" shows "ys @ [x] \<in> future_divergences P xs"
Ord_\<omega>_sum: fixes \<alpha>s :: "V list" assumes "list.set \<alpha>s \<subseteq> ON" shows "Ord (\<omega>_sum \<alpha>s)"
is_unifiers_mgu: fixes AAA :: "'a set set"   and \<sigma> :: "'s" assumes "finite AAA"   and "\<forall>AA\<in>AAA. finite AA"   and "mgu AAA = Some \<sigma>" shows "is_unifiers \<sigma> AAA"
eliminate_entries_convert: fixes J :: "nat"   and A :: "'c mat"   and I :: "nat"   and B :: "'c mat" assumes "J < dim_col A"   and "I < dim_row A"   and "dim_row B = dim_row A" shows "eliminate_entries (\<lambda>i. A $$ (i, J)) B I J = eliminate_entries_rec B I (map (\<lambda>i. (- A $$ (i, J), i)) (filter (\<lambda>i. i \<noteq> I) [0..<dim_row A]))"
comparison_principle_le_linear: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and \<phi> :: "real \<Rightarrow> real"   and \<phi>' :: "real \<Rightarrow> real" assumes "continuous_on {a..b} g"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> (\<phi> has_real_derivative \<phi>' t) (at t)"   and "\<phi> a \<le> 0"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> \<phi>' t \<le> g t *\<^sub>R \<phi> t" shows "\<forall>t\<in>{a..b}. \<phi> t \<le> 0"
assms: shows "optim_instr n (IOpUbx opubx) \<Sigma>1 = Ok (x', \<Sigma>2)"
Graph2: fixes T :: "nat"   and E :: "(nat \<times> nat) list"   and R :: "nat" assumes "T \<in> Reach E"   and "R < length E" shows "T \<in> Reach (E[R := (fst (E ! R), T)])"
compl_le_swap2: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and y :: "'d"   and x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq (uminus y) x" shows "less_eq (uminus x) y"
carrier_mult_of: fixes R :: "('a, 'b) ring_scheme" shows "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
join1_Some_restrict: fixes n :: "nat"   and A :: "nat set"   and x :: "'b option list"   and B :: "nat set"   and y :: "'b option list"   and z :: "'b option list" assumes "wf_tuple n A x"   and "wf_tuple n B y" shows "(join1 (x, y) = Some z) = (wf_tuple n (A \<union> B) z \<and> restrict A z = x \<and> restrict B z = y)"
singleton_eq_iff: fixes a :: "hf"   and b :: "hf" shows "(\<lbrace>a\<rbrace> = \<lbrace>b\<rbrace>) = (a = b)"
openin_Un: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin U S"   and "openin U T" shows "openin U (S \<union> T)"
simps: fixes a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> Arity \<times> Arity list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "a_consistent a1 a2 = (\<exists>ae \<Gamma> S a e as. a1 = (ae, a, as) \<and> a2 = (\<Gamma>, e, S) \<and> edom ae \<subseteq> domA \<Gamma> \<union> upds S \<and> Astack S \<sqsubseteq> a \<and> (ABinds \<Gamma>\<cdot>ae \<squnion> \<A>\<^bsub>a\<^esub> e \<squnion> AEstack as S) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae \<and> stack_consistent as S)"
aha2: shows "{xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) \<and> count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ((as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i))) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) xa \<le> 1} \<noteq> {}"
map_option_the: fixes x :: "'a option"   and f :: "'b \<Rightarrow> 'a"   and y :: "'b option" assumes "x = map_option f y"   and "x \<noteq> None" shows "the x = f (the y)"
orthogonal_matrix_id: shows "orthogonal_matrix (Finite_Cartesian_Product.mat (1::'a))"
poly_mapping_eq_iff: fixes a :: "'a \<Rightarrow>\<^sub>0 'b"   and b :: "'a \<Rightarrow>\<^sub>0 'b" shows "(a = b) = (lookup a = lookup b)"
cases: fixes x :: "bool list \<times> tree" obtains T :: "tree" where "x = ([], T)"   | d :: "bool" and ds :: "bool list" and T1 :: "tree" and T2 :: "tree" where "x = (d # ds, Branching T1 T2)"   | v :: "bool" and va :: "bool list" where "x = (v # va, Leaf)"
PPs_closed_image_lpp: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and X :: "'x set" assumes "F \<subseteq> P[X]" shows "lpp ` F \<subseteq> .[X]"
Mp_smult_2: fixes a :: "int"   and f :: "int poly" shows "Polynomial.smult a (Mp f) =m Polynomial.smult a f"
Inf_greatest: fixes A :: "('b \<times> 'c) set"   and z :: "'b \<times> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod z x" shows "less_eq_prod z (Inf_prod A)"
field_Units: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}"
scale_minus_right: fixes a :: "'c"   and x :: "'c" shows "a * - x = - (a * x)"
plus_adds_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus a b"   and "plus.adds plus c d" shows "plus.adds plus (plus a c) (plus b d)"
sum_shift_lb_Suc0_0_upt: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and f :: "nat \<Rightarrow> 'c"   and k :: "nat" assumes "class.comm_monoid_add plus zero"   and "f 0 = zero" shows "comm_monoid_add.sum plus zero f {Suc 0..<k} = comm_monoid_add.sum plus zero f {0..<k}"
simps_2: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'e option" shows "Some x \<bind> f = f x"
a2: shows "p - p' \<in> More_Modules.ideal polynomial_bool"
restrict_gpv_parametric: assumes "bi_unique (BNF_Def.Grp UNIV id)" shows "rel_fun (rel_\<I> (BNF_Def.Grp UNIV id) (=)) (rel_fun (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV id)) (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV id))) restrict_gpv restrict_gpv"
elims_3: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'c list" assumes "\<not> sorted_wrt x xa" obtains P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" and xb :: "'c" and ys :: "'c list" where "x = P" and "xa = xb # ys" and "\<not> (Ball (set ys) (P xb) \<and> sorted_wrt P ys)"
picard_lindeloef_autonomous_affine: fixes A :: "(('b, 'c) vec, 'c) vec"   and B :: "('b, 'c) vec"   and t\<^sub>0 :: "real" shows "picard_lindeloef (\<lambda>t s. A *v s + B) UNIV UNIV t\<^sub>0"
monoid_homomorphism_\<eta>_cong: fixes \<eta>' :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> R \<Longrightarrow> \<eta>' x = \<eta> x"   and "\<eta>' \<in> extensional R" shows "monoid_homomorphism \<eta>' R (\<cdot>) \<one> R' (\<cdot>') \<one>'"
subspace_image: fixes S :: "'b set" assumes "vs1.subspace S" shows "p.subspace ((\<lambda>x. (x, 0::'c)) ` S)"
LIST_FOREACH_def: fixes \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and tsl :: "'d list nres"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>0 :: "'c" shows "LIST_FOREACH \<Phi> tsl c f \<sigma>0 \<equiv> tsl \<bind> (\<lambda>xs. WHILE\<^sub>T\<^bsup>\<lambda>(it, \<sigma>). \<exists>xs'. xs = xs' @ it \<and> \<Phi> (set it) \<sigma>\<^esup> (FOREACH_cond c) (FOREACH_body f) (xs, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>))"
induces_meta_adjunction: shows "meta_adjunction C D G.F G G.\<phi> G.\<psi>"
nodup_all_lem: fixes x1 :: "'a"   and a :: "'a"   and l :: "'a list" assumes "nodup_all (x1 # a # l)" shows "insert x1 (insert a (set l)) - {x1} = insert a (set l)"
image_dual: shows "(`) uminus \<circ> (`) uminus = id"
that: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "distinct [a, b, c]"   and "{a, b, c} \<subseteq> A" shows "thesis"
martingale_def: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and X :: "'c \<Rightarrow> 'b \<Rightarrow> real" shows "martingale M F X = (Filtration.filtration M F \<and> (\<forall>t. integrable M (X t)) \<and> borel_adapt_stoch_proc F X \<and> (\<forall>t s. t \<le> s \<longrightarrow> AEeq M (real_cond_exp M (F t) (X s)) (X t)))"
H_4: shows "Reflected_Multivariate_Polynomial.degree (CN c n p) = Reflected_Multivariate_Polynomial.degree (CN c' n' p')"
IFind_5: fixes ass :: "'a \<Rightarrow> bool" shows "val_ifex (IF xv xb1 xb2) ass = val_ifex y ass"
\<Z>_Limit_\<alpha>\<omega>: shows "Limit (\<alpha> + \<omega> + \<omega>)"
wf_since_aux_def: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and aux :: "(nat \<times> 'a option list set) list"   and ne :: "nat" shows "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux ne = (sorted_wrt (\<lambda>x y. fst y < fst x) aux \<and> (\<forall>t X. (t, X) \<in> set aux \<longrightarrow> ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) (ne - 1) (Sincep pos \<phi> (point (\<tau> \<sigma> (ne - 1) - t)) \<psi>)) X) \<and> (\<forall>t. ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<longrightarrow> (\<exists>X. (t, X) \<in> set aux)))"
preserves_comp_def: fixes G :: "('a, 'b, 'a, 'b) functor" shows "preserves_comp G \<equiv> \<forall>f\<in>Ar. \<forall>g\<in>Ar. Cod f = Dom g \<longrightarrow> G \<^bsub>\<a>\<^esub> (g \<bullet> f) = G \<^bsub>\<a>\<^esub> g \<bullet>\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
wp1n_tl: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "p = DenyAll # tl p"
closure_closed_segment: fixes a :: "'a"   and b :: "'a" shows "closure (closed_segment a b) = closed_segment a b"
pow_sum: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q < deg R p" shows "deg R ((p \<oplus>\<^bsub>P\<^esub> q) [^]\<^bsub>P\<^esub> n) = deg R p * n"
cA_index: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and i :: "'a"   and j :: "'a" assumes "perron_frobenius A" shows "HMA_Connect.map_matrix complex_of_real A $h i $h j = complex_of_real (A $h i $h j)"
infty_zero_unbox_2: shows "Num (0::'a) = 0"
restr_upd: fixes r :: "'b \<Rightarrow> 'b ref"   and q :: "'b"   and t :: "'b ref"   and m :: "'b \<Rightarrow> bool" shows "(rel (r(q := t)) | m(q := True)) = (rel r | m(q := True))"
that: assumes "x \<in> closure s" shows "filterlim f l (at x within s)"
min_add_distrib_right: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus x (ord.min less_eq y z) = ord.min less_eq (plus x y) (plus x z)"
implies_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "f \<oslash> g = (\<lambda>e. f e \<leadsto> g e)"
unit_disc_fix_transitive: fixes u :: "complex_homo"   and u' :: "complex_homo" assumes "u \<in> unit_disc"   and "u' \<in> unit_disc" shows "\<exists>M. unit_disc_fix M \<and> moebius_pt M u = u'"
path_start_le_path_end': shows "\<gamma> 0 \<bullet> i \<le> \<gamma> 1 \<bullet> i"
neg_imp_zdiv_neg_iff: fixes b :: "int"   and a :: "int" assumes "b < 0" shows "(a div b < 0) = (0 < a)"
Mapping_keys_filterD: fixes k :: "'a"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) mapping" assumes "k \<in> Mapping.keys (Mapping.filter f m)" shows "\<exists>v. Mapping.lookup m k = Some v \<and> f k v"
predicate2D: fixes P :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'b"   and y :: "'d" assumes "P \<le> Q"   and "P x y" shows "Q x y"
rScope: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'" shows "Prop \<alpha> (\<lparr>\<nu>x\<rparr>P')"
simps_1: fixes P :: "'a \<Rightarrow> bool" shows "dropWhile P [] = []"