hcju/isabelleps huggingface dataset:
bernoulli_conv_Stirling: fixes n :: "nat" shows "bernoulli n = (\<Sum>k\<le>n. (- 1) ^ k * fact k / real (k + 1) * real (Stirling n k))"
add_diff_cancel_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
cases: 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 a1 :: "char list"   and a2 :: "char list list"   and a3 :: "char list"   and a4 :: "ty list \<times> ty \<times> char list list \<times> expr"   and a5 :: "char list list" assumes "P \<turnstile> (a1,a2) selects a3 = a4 via a5" obtains C :: "char list" and M :: "char list" and mthd :: "ty list \<times> ty \<times> char list list \<times> expr" and Cs' :: "char list list" and Cs :: "char list list" where "a1 = C__" and "a2 = Cs__" and "a3 = M__" and "a4 = mthd__" and "a5 = Cs'__" and "P \<turnstile> C__ has least M__ = mthd__ via Cs'__"   | C :: "char list" and M :: "char list" and Cs :: "char list list" and mthd :: "ty list \<times> ty \<times> char list list \<times> expr" and Cs' :: "char list list" where "a1 = C" and "a2 = Cs" and "a3 = M" and "a4 = mthd" and "a5 = Cs'" and "\<forall>mthd Cs'. \<not> P \<turnstile> C has least M = mthd via Cs'" and "P \<turnstile> (C, Cs) has overrider M = mthd via Cs'"
comp_arr_ide: fixes a :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "ide a"   and "seq f a" shows "COMP f a = f"
of_class: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and c :: "String.literal"   and T :: "typ"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "const_type (sig \<Theta>) (const_of_class c) = Some (itselfT Core.aT \<rightarrow> propT)"   and "wf_type (sig \<Theta>) T"   and "has_sort (osig (sig \<Theta>)) T (insert c full_sort)" shows "\<Theta>,\<Gamma> \<turnstile> mk_of_class T c"
comp_GSMP_disjoint_def: fixes public :: "'a \<Rightarrow> bool"   and arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('a, 'b \<times> nat) Term.term list \<times> 'c"   and \<Gamma> :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('d, 'e) Term.term"   and A' :: "('a, 'b \<times> nat) Term.term list"   and B' :: "('a, 'b \<times> nat) Term.term list"   and A :: "('a, 'b \<times> nat) Term.term list"   and B :: "('a, 'b \<times> nat) Term.term list"   and C :: "('a, 'b \<times> nat) Term.term list" shows "comp_GSMP_disjoint public arity Ana \<Gamma> A' B' A B C \<equiv> let B\<delta> = B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set A))) in has_all_wt_instances_of \<Gamma> (set A') (set A) \<and> has_all_wt_instances_of \<Gamma> (set B') (set B\<delta>) \<and> finite_SMP_representation arity Ana \<Gamma> A \<and> finite_SMP_representation arity Ana \<Gamma> B\<delta> \<and> (\<forall>t\<in>set A. \<forall>s\<in>set B\<delta>. \<Gamma> t = \<Gamma> s \<and> mgu t s \<noteq> None \<longrightarrow> intruder_synth' public arity {} t \<and> intruder_synth' public arity {} s \<or> (\<exists>u\<in>set C. is_wt_instance_of_cond \<Gamma> t u) \<and> (\<exists>u\<in>set C. is_wt_instance_of_cond \<Gamma> s u))"
T_def: shows "T = (\<lambda>r. {x. infdist x S \<le> r})"
pdevs_apply_zero_pdevs: fixes i :: "nat" shows "pdevs_apply zero_pdevs i = (0::'b)"
sup_ge2: fixes y :: "'e multiset"   and x :: "'e multiset" shows "y \<subseteq># x \<union># y"
fixespointwise_cong: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "fun_eq_on f g A"   and "fixespointwise g A" shows "fixespointwise f A"
evalDdb_env_closed: fixes e :: "db"   and \<rho> :: "nat \<rightarrow> ValD"   and \<rho>' :: "nat \<rightarrow> ValD" assumes "closed e" shows "evalDdb e\<cdot>\<rho> = evalDdb e\<cdot>\<rho>'"
conf_def: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and v :: "'c val"   and T :: "ty" shows "heap_base.conf typeof_addr P h v T \<equiv> \<exists>T'. heap_base.typeof_h typeof_addr h v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
nth_tl: fixes n :: "nat"   and xs :: "'d list" assumes "n < length (tl xs)" shows "tl xs ! n = xs ! Suc n"
ranI: fixes m :: "'e \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and a :: "'e"   and b :: "'d" assumes "m a = \<lfloor>b\<rfloor>" shows "b \<in> ran m"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>x xs y ys. \<lbrakk>compare cmp x y = Greater \<Longrightarrow> P (x # xs) ys; compare cmp x y \<noteq> Greater \<Longrightarrow> P xs (y # ys)\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
Act_eqvt: fixes p :: "perm"   and f :: "'d"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c, 'd) formula" shows "p \<bullet> Act f \<alpha> x = Act (p \<bullet> f) (p \<bullet> \<alpha>) (p \<bullet> x)"
finfun_dom_def: fixes f :: "('a, 'b) finfun" shows "finfun_dom f = Abs_finfun (\<lambda>a. finfun_apply f a \<noteq> finfun_default f)"
prems_2: shows "R\<^sup>*\<^sup>* x c"
augment_flow_presv: fixes f' :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "Flow cf s t f'" shows "Flow c s t (augment f')"
IH: assumes "\<And>i. i < length vs \<Longrightarrow> (0::'a) \<le> us ! i \<and> us ! i \<le> vs ! i" shows "(0::'a) \<le> prod_list us \<and> prod_list us \<le> prod_list vs"
max_lin_independent_set_in_Span: fixes vs :: "'b list"   and us :: "'b list" assumes "set vs \<subseteq> V"   and "set us \<subseteq> RSpan vs"   and "R_lin_independent us" shows "length us \<le> length vs"
const_list_comb_neq_free: fixes name :: "name"   and args :: "'a list"   and name' :: "name" shows "name $$ args \<noteq> free name'"
is_none_bind: fixes f :: "'b option"   and g :: "'b \<Rightarrow> 'a option" shows "Option.is_none (f \<bind> g) = (Option.is_none f \<or> Option.is_none (g (the f)))"
g_rw: shows "mod_type_class.from_nat (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) = (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)"
setmonotone_limit: fixes f :: "'b set \<Rightarrow> 'b set" assumes "setmonotone f" shows "setmonotone (limit f)"
import_rel2_def: fixes R :: "('h \<times> 'g) set \<Rightarrow> ('j \<times> 'i) set \<Rightarrow> ('l \<times> 'k) set" shows "import_rel2 R \<equiv> \<lambda>A B c ci. \<up> (is_pure A \<and> is_pure B \<and> (ci, c) \<in> \<langle>the_pure A, the_pure B\<rangle>R)"
eucl_abs: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "abs x = comm_monoid_add.sum plus zero (\<lambda>i. scaleR \<bar>inner x i\<bar> i) Basis"
elims: fixes x :: "'b + 'a \<times> 'c"   and y :: "'a" assumes "|x|\<^sub>r = y" obtains r :: "'a" and uu_ :: "'c" where "x = Inr (r, uu_)" and "y = r"   | uv_ :: "'b" where "x = Inl uv_" and "y = undefined"
distinguishing_weak_formula_eqvt: fixes P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "\<not> P \<equiv>\<cdot> Q" shows "p \<bullet> distinguishing_weak_formula P Q = distinguishing_weak_formula (p \<bullet> P) (p \<bullet> Q)"
pinf_5: fixes t :: "'a" shows "\<exists>z. \<forall>x>z. (x < t) = False"
chain_tllist_terminal: fixes b :: "'g"   and A :: "('f, 'g) tllist set" assumes "Complete_Partial_Order.chain (tllist_ord b) A" shows "Complete_Partial_Order.chain (flat_ord b) {terminal xs |xs. xs \<in> A \<and> tfinite xs}"
domains_comp: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "D.domains (g \<cdot>\<^sub>D f) = D.domains f"
ct_suffix_to_ct_prefix: fixes xs :: "ct list"   and ys :: "ct list" shows "ct_suffix xs ys = ct_prefix (rev xs) (rev ys)"
subgroups_Inter_pair: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I (G\<lparr>carrier := S\<rparr>)"   and "subgroup J (G\<lparr>carrier := S\<rparr>)" shows "subgroup (I \<inter> J) (G\<lparr>carrier := S\<rparr>)"
c_never_blocked: fixes l :: "nat"   and s :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)" shows "C.can_step l s = (l < N)"
cases: fixes a1 :: "'action list"   and a2 :: "'action list" assumes "a1 =\<^sub>S a2" obtains a :: "'action" and b :: "'action" and u :: "'action list" and v :: "'action list" where "a1 = u @ [a] @ [b] @ v" and "a2 = u @ [b] @ [a] @ v" and "ind a b"
1: shows "(\<pi> m, \<pi>' l') \<in> pdg"
wf_flatten_INTERSECT: fixes X :: "'a atom rexp set"   and n :: "nat" assumes "finite X" shows "local.wf n (INTERSECT (sorted_list_of_set X)) = (\<forall>r\<in>X. local.wf n r)"
subcring_iff: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "subcring H R = cring (R\<lparr>carrier := H\<rparr>)"
keys_monom_mult: fixes c :: "'b"   and t :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'b" assumes "c \<noteq> (0::'b)" shows "keys (MPoly_Type_Class.punit.monom_mult c t p) = MPoly_Type_Class.punit.splus t ` keys p"
starfun_abs: fixes f :: "'b \<Rightarrow> 'a"   and N :: "'b star" shows "(*f* (\<lambda>n. \<bar>f n\<bar>)) N = \<bar>(*f* f) N\<bar>"
equivalent_objects_transitive: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "D.equivalent_objects a b"   and "D.equivalent_objects b c" shows "D.equivalent_objects a c"
remdups_enum_upto: fixes s :: "'b word"   and e :: "'b word" shows "remdups [s .e. e] = [s .e. e]"
NEC: fixes \<phi> :: "i \<Rightarrow> bool" assumes "\<lfloor>\<phi>\<rfloor>" shows "universal (\<lambda>w. (r) w \<^bold>\<rightarrow> \<phi>)"
ZFfunCompAssoc: fixes f :: "ZF"   and h :: "ZF"   and g :: "ZF" assumes "isZFfun f"   and "isZFfun h"   and "|cod|g = |dom|h"   and "isZFfun g"   and "|cod|f = |dom|g" shows "f |o| g |o| h = f |o| (g |o| h)"
cf_list_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" shows "cf_list R p = map p [0..<Suc (deg R p)]"
EnumLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and enm :: "('a, String.literal) phantom"   and lit :: "String.literal"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (EnumLiteral enm lit) : \<tau>" obtains lits :: "String.literal fset" where "\<tau> = (Enum enm)[1]" and "fmlookup literals enm = Some lits" and "lit |\<in>| lits"
cases: fixes a1 :: "'b"   and a2 :: "'b"   and R :: "('b \<times> 'b) set" assumes "(a1, a2) \<in> Omega_Algebra_Models.omega R" obtains x :: "'b" and y :: "'b" and z :: "'b" where "a1 = x" and "a2 = z" and "(x, y) \<in> R" and "(y, z) \<in> Omega_Algebra_Models.omega R"
vsubset_in_VsetI: fixes a :: "V"   and A :: "V"   and i :: "V" assumes "a \<subseteq>\<^sub>\<circ> A"   and "A \<in>\<^sub>\<circ> Vset i" shows "a \<in>\<^sub>\<circ> Vset i"
l_desc_def: fixes X :: "'s set"   and ct :: "'s \<Rightarrow> 's pmf"   and l :: "'s \<Rightarrow> 's"   and s :: "'s" shows "l_desc X ct l s = (s \<in> directed_towards S2 (SIGMA s:X. {l s}) \<and> v (simple ct s) \<le> v (simple ct (l s)) \<and> l s \<in> maximal (\<lambda>s. v (simple ct s)) (set_pmf (ct s)))"
mod_ex1: fixes a :: "nat" shows "(a mod 2 = Suc 0) = (\<exists>q. a = Suc (2 * q))"
oinvariant_oreachableI: fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<times> 'e \<Rightarrow> bool" assumes "\<And>\<sigma> s. (\<sigma>, s) \<in> oreachable A S U \<Longrightarrow> P (\<sigma>, s)" shows "A \<Turnstile> (S, U \<rightarrow>) P"
bij_betw_points_and_set: fixes a :: "'s" assumes "ide a" shows "bij_betw img (hom unity a) (local.set a)"
HPhase0Read_valueChosen2: fixes s :: "state"   and s' :: "state"   and qq :: "Proc"   and dd :: "Disk"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HPhase0Read s s' qq dd"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
step: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "(xs, ys) \<in> X" shows "lnull xs \<or> (\<exists>x xs' ys'. xs = LCons x xs' \<and> ys = LCons x ys' \<and> ((xs', ys') \<in> X \<or> xs' \<sqsubseteq> ys'))"
sets_PiF: fixes I :: "'c set set"   and M :: "'c \<Rightarrow> 'd measure" shows "sets (Pi\<^sub>F I M) = sigma_sets (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j)) {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
assms_2: shows "f1 ` S \<subseteq> T"
plus_sat_def: fixes x :: "'b sat"   and y :: "'b sat" shows "x + y = Abs_sat' (nat_of x + nat_of y)"
generalized_zipping_lemma: shows "\<forall>\<tau> lambda t1 t2. set \<tau> \<subseteq> E\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> set lambda \<subseteq> V\<^bsub>\<V>\<^esub> \<and> set t1 \<subseteq> E\<^bsub>ES1\<^esub> \<and> set t2 \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> @ t1 \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> @ t2 \<in> Tr\<^bsub>ES2\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES1\<^esub> = t1 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> t1 \<upharpoonleft> C\<^bsub>\<V>1\<^esub> = [] \<and> t2 \<upharpoonleft> C\<^bsub>\<V>2\<^esub> = [] \<longrightarrow> (\<exists>t. \<tau> @ t \<in> Tr\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> t \<upharpoonleft> V\<^bsub>\<V>\<^esub> = lambda \<and> t \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
C_Cinf: shows "C\<cdot>C\<^sup>\<infinity> = C\<^sup>\<infinity>"
root_newton_int_main_upper: fixes n :: "int"   and y :: "int"   and x :: "int"   and b :: "bool" assumes "n \<le> y ^ p"   and "0 \<le> y"   and "0 \<le> n"   and "root_newton_int_main y n = (x, b)" shows "n < (x + 1) ^ p"
dnf_and_def: fixes cnf1 :: "'b negation_type list list"   and cnf2 :: "'b negation_type list list" shows "dnf_and cnf1 cnf2 = concat (map (\<lambda>andlist1. map ((@) andlist1) cnf2) cnf1)"
finprod_infinite: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'a" assumes "infinite A" shows "finprod G f A = \<one>"
finite_Diff: fixes A :: "'a set"   and B :: "'a set" assumes "finite A" shows "finite (A - B)"
chainI: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'c" assumes "class.po below"   and "\<And>i. below (Y i) (Y (Suc i))" shows "po.chain below Y"
laplace_expansion_column: fixes A :: "'a mat"   and n :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "j < n" shows "Determinant.det A = (\<Sum>i<n. A $$ (i, j) * cofactor A i j)"
assms_3: shows "path_image \<gamma> \<subseteq> {w. b < a \<bullet> w}"
conjoined_in_consistency: fixes C :: "'b fm set set"   and S :: "'b fm set"   and ps :: "'b fm list"   and q :: "'b fm" assumes "consistency C"   and "subset_closed C"   and "S \<union> {conjoin ps q} \<in> C" shows "S \<union> set ps \<union> {q} \<in> C"
j_decisive: fixes A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and x :: "'a"   and y :: "'a"   and j :: "'b"   and u :: "'a"   and v :: "'a" assumes "has 3 A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "hasw [x, y] A"   and "semidecisive swf A Is {j} x y"   and "hasw [u, v] A" shows "decisive swf A Is {j} u v"
ivl_disj_int_two_4: fixes l :: "'b"   and m :: "'b"   and u :: "'b" shows "{l..m} \<inter> {m<..<u} = {}"
with_take_cols_from_matrix: shows "vec_space.rank (length (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) (take_cols_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))\<^sup>T (map snd (pivot_positions (gauss_jordan_single (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))\<^sup>T)))) = dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))"
elims_3: fixes x :: "bool"   and xa :: "v sem_env"   and xb :: "'b state"   and xc :: "top0 list"   and xd :: "'b state \<times> (v sem_env, v) result" assumes "\<not> evaluate_whole_prog x xa xb xc xd" obtains ck :: "bool" and env :: "v sem_env" and s1 :: "'b state" and tops :: "top0 list" and s2 :: "'b state" and res :: "(v sem_env, v) result" where "x = ck" and "xa = env" and "xb = s1" and "xc = tops" and "xd = (s2, res)" and "\<not> (if no_dup_mods tops (defined_mods s1) \<and> no_dup_top_types tops (defined_types s1) then evaluate_prog ck env s1 tops (s2, res) else s1 = s2 \<and> res = Rerr (Rabort Rtype_error))"
mat_vec_mult_index: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and v :: "'c list"   and i :: "nat"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m"   and "vec nc v"   and "i < nr" shows "matT_vec_multI ze pl ti (Matrix_Legacy.transpose nr m) v ! i = scalar_prodI ze pl ti (row m i) v"
hyps_correct: fixes n :: "'node"   and p1 :: "'outPort"   and p2 :: "'inPort" assumes "hyps n p1 = Some p2" shows "p1 |\<in>| outPorts n \<and> p2 |\<in>| inPorts n"
sup_fset_def: shows "(|\<union>|) \<equiv> (rep_fset ---> rep_fset ---> abs_fset) (@)"
pt_fresh_aux: fixes c :: "'b"   and a :: "'b"   and x :: "'a"   and b :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "c \<noteq> a"   and "a \<sharp> x"   and "c \<sharp> x" shows "c \<sharp> [(a, b)] \<bullet> x"
ev_alw_stl: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and x :: "'b stream" shows "ev (alw \<phi>) (stl x) = ev (alw \<phi>) x"
asymp_equiv_imp_eventually_le: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and c :: "real" assumes "f \<sim>[F] g"   and "1 < c" shows "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)"
2: shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G q'"
word_le_less_eq: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le> y) = (x = y \<or> x < y)"
simps: fixes a1 :: "'d \<Rightarrow> 'e"   and a2 :: "'d option"   and a3 :: "'d option"   and a4 :: "'e DBMEntry" shows "dbm_entry_val a1 a2 a3 a4 = ((\<exists>u r d. a1 = u \<and> a2 = Some r \<and> a3 = None \<and> a4 = Le d \<and> u r \<le> d) \<or> (\<exists>u c d. a1 = u \<and> a2 = None \<and> a3 = Some c \<and> a4 = Le d \<and> - u c \<le> d) \<or> (\<exists>u r d. a1 = u \<and> a2 = Some r \<and> a3 = None \<and> a4 = Lt d \<and> u r < d) \<or> (\<exists>u c d. a1 = u \<and> a2 = None \<and> a3 = Some c \<and> a4 = Lt d \<and> - u c < d) \<or> (\<exists>u r c d. a1 = u \<and> a2 = Some r \<and> a3 = Some c \<and> a4 = Le d \<and> u r - u c \<le> d) \<or> (\<exists>u r c d. a1 = u \<and> a2 = Some r \<and> a3 = Some c \<and> a4 = Lt d \<and> u r - u c < d) \<or> (\<exists>uu_ uv_ uw_. a1 = uu_ \<and> a2 = uv_ \<and> a3 = uw_ \<and> a4 = \<infinity>))"
scaleR_left_le_one_le: fixes x :: "'c"   and a :: "real" assumes "(0::'c) \<le> x"   and "a \<le> 1" shows "a *\<^sub>R x \<le> x"
laplace_integrand_integrable_on_Ici_iff: fixes M :: "real"   and c :: "real"   and f :: "real \<Rightarrow> complex"   and s :: "complex"   and a :: "real" assumes "exponential_order M c f"   and "c < Re s" shows "(laplace_integrand f s integrable_on {a..}) = (\<forall>k>a. laplace_integrand f s integrable_on {a..k})"
comp_null_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "C (partial_magma.null C) f = partial_magma.null C"
rep_eq: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and xa :: "('d, 'e) mapping_rbt" shows "RBT_Mapping2.fold x xa = RBT_Impl.fold x (impl_of xa)"
fair_through_make_step: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and X :: "nat \<Rightarrow> nat \<times> ('d \<times> nat \<times> nat) set"   and selector :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option" assumes "finite Rs"   and "set_of_graph_rules Rs"   and "graph (graph_of (X 0))"   and "\<forall>i. X (Suc i) = make_step selector (X i)"   and "valid_selector Rs selector" shows "fair_chain Rs (\<lambda>i. graph_of (X i))"
con_rews_3: fixes a :: "'b"   and b :: "'c" shows "(a :!: b = \<bottom>) = (a = \<bottom> \<or> b = \<bottom>)"
floorlog_ge_SucI: fixes b :: "nat"   and w :: "nat"   and x :: "nat" assumes "b ^ w \<le> x"   and "1 < b" shows "Suc w \<le> floorlog b x"
has_derivative_at_withinI: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "(f has_derivative f') (at x)" shows "(f has_derivative f') (at x within s)"
divisors_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_no_zero_divisors plus zero times"   and "times a b = zero" shows "a = zero \<or> b = zero"
parallel_converter2_comp2_out: fixes conv'' :: "('i, 'k, 'm, 'o) converter"   and conv :: "('j, 'l, 'q, 'r) converter"   and conv' :: "('q, 'r, 'n, 'p) converter" shows "conv'' |\<^sub>= conv \<odot> conv' = (1\<^sub>C |\<^sub>= conv) \<odot> (conv'' |\<^sub>= conv')"
neutral_const: fixes A :: "'b multiset" shows "(\<Sum>_\<in>#A. (0::'a)) = (0::'a)"
lexordp_char_char: shows "ord_class.lexordp = char.lexordp"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" shows "transfer (P)\<^sub>\<surd> s = s"
INT_iff: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "s2p (\<Inter> (\<P> B A)) b = (\<forall>x\<in>A. s2p (B x) b)"
measurable_case_nat_undefined: fixes A :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and t :: "pdf_type" shows "case_nat undefined \<in> state_measure A \<Gamma> \<rightarrow>\<^sub>M state_measure (Suc ` A) (case_nat t \<Gamma>)"
cong: fixes Pd :: "'a \<Rightarrow> ('b \<times> 'b) set" shows "Contracts.MaxR_f Pd \<equiv> Contracts.MaxR_f Pd"
bot_def: shows "bot = - (THE x. \<forall>y. x = y \<squnion> - y)"
iso_is_section: fixes f :: "'d" assumes "D.iso f" shows "D.section f"
wn_triangle1: fixes a :: "complex"   and b :: "complex"   and c :: "complex" assumes "0 \<in> interior (convex hull {a, b, c})" shows "\<not> (Im (a / b) \<le> 0 \<and> 0 \<le> Im (b / c))"
Diff_insert2: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A - Set.insert a B = A - {a} - B"
subset_smaller_list: fixes Z :: "'a set" shows "\<forall>a x. set (a # x) \<subseteq> Z \<longrightarrow> set x \<subseteq> Z"
hyps_free_def: fixes pth :: "(('d \<times> 'c) \<times> 'g \<times> 'h) list" shows "hyps_free pth = (\<forall>v\<^sub>1 p\<^sub>1 v\<^sub>2 p\<^sub>2. ((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> set pth \<longrightarrow> hyps (nodeOf v\<^sub>1) p\<^sub>1 = None)"