hcju/isabelleps huggingface dataset:
simps: fixes a1 :: "(nat \<times> nat) multiset"   and a2 :: "(nat \<times> nat) multiset" shows "pw_leq a1 a2 = (a1 = {#} \<and> a2 = {#} \<or> (\<exists>x y X Y. a1 = {#x#} + X \<and> a2 = {#y#} + Y \<and> (x, y) \<in> pair_leq \<and> pw_leq X Y))"
min_lvar_not_in_bounds_def: fixes s :: "('a, 'b) state" shows "min_lvar_not_in_bounds s \<equiv> min_satisfying (\<lambda>x. \<not> in_bounds x \<langle>\<V> s\<rangle> (\<B> s)) (map lhs (\<T> s))"
f3: shows "\<forall>p ps rss nss m. map ((!) rss) (find_nonzeros_from_input_vec (solve_for_lhs p ps nss m)) = get_signs (reduce_system p (ps, m, nss, rss))"
uncaught_xcpt_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and xcp :: "nat"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)"   and f :: "val list \<times> val list \<times> char list \<times> char list \<times> nat"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "h xcp = \<lfloor>obj\<rfloor>"   and "P,\<Phi> |- (None, h, f # frs) [ok]" shows "P,\<Phi> |- find_handler P xcp h frs [ok]"
ASSERT_refine_left: fixes \<Phi> :: "bool"   and S :: "'a nres"   and R :: "('a \<times> 'b) set"   and S' :: "'b nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> S \<le> \<Down> R S'" shows "ASSERT \<Phi> \<bind> (\<lambda>_. S) \<le> \<Down> R S'"
assms_2: shows "w \<noteq> \<infinity>\<^sub>h"
f_shrink_defs_1: fixes xs :: "'b message_af list"   and k :: "nat" shows "xs \<div> k \<equiv> f_aggregate xs k last_message"
crename_credu: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and M' :: "trm" assumes "M[a\<turnstile>c>b] \<longrightarrow>\<^sub>c M'" shows "\<exists>M0. M0[a\<turnstile>c>b] = M' \<and> M \<longrightarrow>\<^sub>c M0"
simps: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a1 :: "'b set"   and a2 :: "'c" shows "fold_graph f z a1 a2 = (a1 = {} \<and> a2 = z \<or> (\<exists>x A y. a1 = insert x A \<and> a2 = f x y \<and> x \<notin> A \<and> fold_graph f z A y))"
S_def: shows "S \<equiv> \<lambda>c. snd (snd c)"
the_\<theta>_simps_2: shows "src \<omega>.the_\<theta> = src r\<^sub>0"
IDirProds_incl: fixes i :: "'d"   and I :: "'d set"   and S :: "'d \<Rightarrow> ('c \<Rightarrow> 'a) set" assumes "i \<in> I" shows "S i \<subseteq> IDirProds (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) S I"
differentiable_def: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "(f differentiable F) = (\<exists>D. (f has_derivative D) F)"
expansion_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B C. \<lbrakk>B \<subseteq> A; C \<subseteq> A\<rbrakk> \<Longrightarrow> f B \<inter> f C \<subseteq> f (B \<union> C)" shows "expansion_on A f"
cinner_gt_zero_iff: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(0 < cinner x x) = (x \<noteq> zero)"
ladder_last_j_def: fixes L :: "(nat \<times> nat \<times> nat) list" shows "ladder_last_j L = ladder_j L (length L - 1)"
s'_nonneg: fixes x :: "real" assumes "0 \<le> v\<^sub>o"   and "a\<^sub>o \<le> 0" shows "0 \<le> other.s' x"
reaches1_steps_append: fixes a :: "'a"   and b :: "'a"   and xs :: "'a list" assumes "a \<rightarrow>\<^sup>+ b"   and "steps xs"   and "hd xs = b" shows "\<exists>ys. steps (a # ys @ xs)"
LI_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant"   and I :: "'c set" shows "LI K v I = AMin (v ` I)"
weak_le_antisym: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "weak_partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x .=\<^bsub>L\<^esub> y"
ev_blockD: fixes n :: "nat"   and A :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "local.ev_block n A"   and "i < n"   and "j < n" shows "A $$ (i, i) = A $$ (j, j)"
assms_4: fixes ys :: "'varSort"   and y :: "'var"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "rho ys y = Some Y" shows "freshAbs ys y A \<or> fresh zs z Y"
isomorphic_objects_are_equal: fixes a :: "'a"   and b :: "'a" assumes "B.obj a"   and "B.obj b"   and "B.isomorphic a b" shows "a = b"
gamma4_is_compos_of_scal_and_g2: shows "gamma4 = (\<lambda>x. (g2 x, x)) \<circ> (\<lambda>x. a + (b - a) * x)"
branch_instr_low_equal_sub0: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "'c \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "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)))))" shows "low_equal t1 t2"
assms_4: shows "\<not> is_Prim\<^sub>0 t"
valid_W_invE_1: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name" assumes "r \<in> W (s p)"   and "valid_W_inv s" shows "marked r s"
f_def_1: shows "a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
cp_fun_inst: assumes "cp TYPE('a) TYPE('b) TYPE('c)"   and "cp TYPE('d) TYPE('b) TYPE('c)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)" shows "cp TYPE('a \<Rightarrow> 'd) TYPE('b) TYPE('c)"
equals_rsp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and xa :: "'c"   and xb :: "'c"   and ya :: "'c"   and yb :: "'c" assumes "Quotient3 R Abs Rep"   and "R xa xb"   and "R ya yb" shows "R xa ya = R xb yb"
fv\<^sub>e\<^sub>s\<^sub>t_cons: fixes x :: "('fun, 'var) extstrand_step"   and X :: "('fun, 'var) extstrand_step list" shows "fv\<^sub>e\<^sub>s\<^sub>t (x # X) = fv\<^sub>e\<^sub>s\<^sub>t [x] \<union> fv\<^sub>e\<^sub>s\<^sub>t X"
w2: shows "\<forall>n. r n \<in> M \<and> \<parallel>r n\<parallel>\<^sup>2 < d + 1 / (real n + 1)"
set_comp_red_monic_basis: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set (comp_red_monic_basis xs) = monic ` set (comp_red_basis xs)"
ffact_Suc_rev_nat: fixes n :: "nat"   and m :: "nat" shows "ffact (Suc n) m = (m - n) * ffact n m"
has_bochner_integral_reflect_real: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'c"   and i :: "'c" shows "has_bochner_integral (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) i = has_bochner_integral (lebesgue_on {a..b}) f i"
Second_def: shows "Second = {x. \<exists>b. x = Abs_OrdSum (1::'b, b)}"
abs_summable_on_reindex_bij_betw: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "bij_betw g A B" shows "((\<lambda>x. f (g x)) abs_summable_on A) = (f abs_summable_on B)"
polytope_imp_polyhedron: fixes S :: "'a set" assumes "polytope S" shows "polyhedron S"
induct: fixes P1 :: "'b ann_com option \<Rightarrow> 'b \<Rightarrow> 'b ann_com option \<Rightarrow> 'b \<Rightarrow> bool"   and P2 :: "'b com \<times> 'b \<Rightarrow> 'b com \<times> 'b \<Rightarrow> bool"   and x1 :: "'b ann_com option"   and x2 :: "'b"   and x3 :: "'b ann_com option"   and x4 :: "'b"   and x5 :: "'b com \<times> 'b"   and x6 :: "'b com \<times> 'b" assumes "\<And>r f s. P1 (Some (AnnBasic r f)) s None (f s)"   and "\<And>c0 s t c1. \<lbrakk>(Some c0, s) -1\<rightarrow> (None, t); P1 (Some c0) s None t\<rbrakk> \<Longrightarrow> P1 (Some (AnnSeq c0 c1)) s (Some c1) t"   and "\<And>c0 s c2 t c1. \<lbrakk>(Some c0, s) -1\<rightarrow> (Some c2, t); P1 (Some c0) s (Some c2) t\<rbrakk> \<Longrightarrow> P1 (Some (AnnSeq c0 c1)) s (Some (AnnSeq c2 c1)) t"   and "\<And>s b r c1 c2. s \<in> b \<Longrightarrow> P1 (Some (AnnCond1 r b c1 c2)) s (Some c1) s"   and "\<And>s b r c1 c2. s \<notin> b \<Longrightarrow> P1 (Some (AnnCond1 r b c1 c2)) s (Some c2) s"   and "\<And>s b r c. s \<in> b \<Longrightarrow> P1 (Some (AnnCond2 r b c)) s (Some c) s"   and "\<And>s b r c. s \<notin> b \<Longrightarrow> P1 (Some (AnnCond2 r b c)) s None s"   and "\<And>s b r i c. s \<notin> b \<Longrightarrow> P1 (Some (AnnWhile r b i c)) s None s"   and "\<And>s b r i c. s \<in> b \<Longrightarrow> P1 (Some (AnnWhile r b i c)) s (Some (AnnSeq c (AnnWhile i b i c))) s"   and "\<And>s b c t r. \<lbrakk>s \<in> b; atom_com c; ((c, s), Parallel [], t) \<in> (transition \<inter> {(x, y). P2 x y})\<^sup>*\<rbrakk> \<Longrightarrow> P1 (Some (AnnAwait r b c)) s None t"   and "\<And>i Ts c q s r t. \<lbrakk>i < length Ts; Ts ! i = (Some c, q); (Some c, s) -1\<rightarrow> (r, t); P1 (Some c) s r t\<rbrakk> \<Longrightarrow> P2 (Parallel Ts, s) (Parallel (Ts[i := (r, q)]), t)"   and "\<And>f s. P2 (Basic f, s) (Parallel [], f s)"   and "\<And>Ts c s. All_None Ts \<Longrightarrow> P2 (Seq (Parallel Ts) c, s) (c, s)"   and "\<And>c0 s c2 t c1. \<lbrakk>(c0, s) -P1\<rightarrow> (c2, t); P2 (c0, s) (c2, t)\<rbrakk> \<Longrightarrow> P2 (Seq c0 c1, s) (Seq c2 c1, t)"   and "\<And>s b c1 c2. s \<in> b \<Longrightarrow> P2 (Cond b c1 c2, s) (c1, s)"   and "\<And>s b c1 c2. s \<notin> b \<Longrightarrow> P2 (Cond b c1 c2, s) (c2, s)"   and "\<And>s b i c. s \<notin> b \<Longrightarrow> P2 (While b i c, s) (Parallel [], s)"   and "\<And>s b i c. s \<in> b \<Longrightarrow> P2 (While b i c, s) (Seq c (While b i c), s)" shows "((x1, x2) -1\<rightarrow> (x3, x4) \<longrightarrow> P1 x1 x2 x3 x4) \<and> (x5 -P1\<rightarrow> x6 \<longrightarrow> P2 x5 x6)"
map_of_append: fixes xs :: "('e \<times> 'f) list"   and ys :: "('e \<times> 'f) list" shows "map_of (xs @ ys) = map_of ys ++ map_of xs"
bisimBangPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q"   and "guarded P"   and "guarded Q" shows "\<Psi> \<rhd> !P \<sim> !Q"
select_return_top_ct_eq: fixes cts :: "ct list"   and c1 :: "ct"   and c2 :: "ct"   and ctm :: "ct list"   and cm :: "t list" assumes "select_return_top cts c1 c2 = TopType ctm"   and "3 \<le> length cts"   and "c_types_agree (TopType ctm) cm" shows "\<exists>c' cm'. cm = cm' @ [c'] \<and> ct_suffix (take (length cts - 3) cts) (to_ct_list cm') \<and> ct_eq c1 (TSome c') \<and> ct_eq c2 (TSome c')"
def_hashmap_size_char_def: shows "def_hashmap_size = (\<lambda>_. 32)"
real_alg_2'_code: fixes ri :: "root_info"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "real_alg_2' ri p l r = (if degree p = 1 then Rational (Rat.Fract (- coeff p 0) (coeff p 1)) else case normalize_bounds_1 (case tighten_poly_bounds_for_x p 0 l r (sgn (ipoly p r)) of (l', r', sr') \<Rightarrow> (p, l', r')) of (p', l, r) \<Rightarrow> Irrational (root_info.number_root ri r) (p', l, r))"
invertible_mat_def: fixes A :: "'b mat" shows "invertible_mat A \<equiv> square_mat A \<and> (\<exists>B. inverts_mat A B \<and> inverts_mat B A)"
IH: assumes "\<exists>v\<in>wfrestrictedvars\<^sub>s\<^sub>t S. t \<cdot> \<I> \<sqsubseteq> \<I> v" shows "\<exists>S\<^sub>p\<^sub>r\<^sub>e S\<^sub>s\<^sub>u\<^sub>f. \<not> (\<exists>w\<in>wfrestrictedvars\<^sub>s\<^sub>t S\<^sub>p\<^sub>r\<^sub>e. t \<cdot> \<I> \<sqsubseteq> \<I> w) \<and> ((\<exists>t'. S = S\<^sub>p\<^sub>r\<^sub>e @ send\<langle>t'\<rangle>\<^sub>s\<^sub>t # S\<^sub>s\<^sub>u\<^sub>f \<and> t \<cdot> \<I> \<sqsubseteq> t' \<cdot> \<I>) \<or> (\<exists>t' t''. S = S\<^sub>p\<^sub>r\<^sub>e @ \<langle>assign: t' \<doteq> t''\<rangle>\<^sub>s\<^sub>t # S\<^sub>s\<^sub>u\<^sub>f \<and> t \<cdot> \<I> \<sqsubseteq> t' \<cdot> \<I>))"
Ifun_star_of: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "star_of f \<star> star_of x = star_of (f x)"
append_one_genPrefix: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs, ys) \<in> genPrefix r"   and "length xs < length ys"   and "refl r" shows "(xs @ [ys ! length xs], ys) \<in> genPrefix r"
assn_aci_5: fixes x :: "assn"   and y :: "assn" shows "x \<or>\<^sub>A y = y \<or>\<^sub>A x"
that: assumes "\<bar>Brouwer_degree2 p f\<bar> = 1"   and "\<bar>Brouwer_degree2 p g\<bar> = 1"   and "Brouwer_degree2 p g = Brouwer_degree2 p f" shows "thesis"
integrable_cong_AE: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integrable M f = integrable M g"
diagonal_mat_def: fixes A :: "'b mat" shows "diagonal_mat A \<equiv> \<forall>i<dim_row A. \<forall>j<dim_col A. i \<noteq> j \<longrightarrow> A $$ (i, j) = (0::'b)"
path_connected_component_set: fixes S :: "'a set" shows "path_connected S = (\<forall>x\<in>S. path_component_set S x = S)"
subset_image_iff: fixes B :: "'e set"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "(B \<subseteq> f ` A) = (\<exists>AA\<subseteq>A. B = f ` AA)"
l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs"
Mapping_inverse: fixes y :: "'a \<Rightarrow> 'b option" assumes "y \<in> UNIV" shows "mapping.rep (mapping.Mapping y) = y"
arctan_bounded: fixes y :: "real" shows "- (pi / 2) < arctan y \<and> arctan y < pi / 2"
new_spairs_sorted: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "sorted_wrt pair_ord (new_spairs bs p)"
uniform_size_incomp: fixes bl :: "'a set" assumes "\<k> \<le> des.\<v> - 1"   and "bl \<in># \<B>" shows "incomplete_block bl"
assms: shows "Deliver (i, Delete x) \<in> set (history j)"
to_nat_on_def: fixes S :: "'b set" shows "to_nat_on S = (SOME f. if finite S then bij_betw f S {..<card S} else bij_betw f S UNIV)"
c_def: shows "c = (\<lambda>n. (- 1) ^ Suc j * complex_of_real (bernoulli' n) * pochhammer (of_nat (Suc n)) j')"
gcd_swap: fixes m :: "nat"   and n :: "nat" shows "Arith2.gcd m n = Arith2.gcd n m"
prWlsAll_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'd \<Rightarrow> 'c) \<Rightarrow> ('varSort, 'c) sAbs"   and SEM :: "('e, 'f, 'g, 'sort, 'h, 'c, 'i) semDom_scheme" shows "prWlsAll g gA SEM \<equiv> prWls g SEM \<and> prWlsAbs gA SEM"
f_upwards: fixes s :: "(nat \<times> form) list list"   and n :: "nat"   and m :: "nat" assumes "f s n = []" shows "f s (n + m) = []"
maximal_execution_def: fixes e :: "('r, 'l, 'v) expr"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "maximal_execution e s s' \<equiv> execution e s s' \<and> (\<nexists>s''. s' \<leadsto> s'')"
smap_iterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "smap f (siterate f x) = siterate f (f x)"
code: fixes R :: "('f, 'g, 'h) Resumption.resumption set" shows "resumption_lub R = (if \<forall>r\<in>R. Resumption.resumption.is_Done r then Resumption.resumption.Done (flat_lub None (Resumption.resumption.result ` R)) else Resumption.resumption.Pause (THE out. out \<in> Resumption.resumption.output ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r})) (\<lambda>inp. resumption_lub ((\<lambda>c. c inp) ` Resumption.resumption.resume ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r}))))"
state_simple: fixes ct :: "'s \<Rightarrow> 's pmf"   and s :: "'s" shows "state (simple ct s) = s"
square_free_factorization_order_root_mem: fixes p :: "'c poly"   and c :: "'c"   and bs :: "('c poly \<times> nat) list"   and a :: "'c poly"   and i :: "nat"   and x :: "'c" assumes "square_free_factorization p (c, bs)"   and "p \<noteq> 0"   and "(a, i) \<in> set bs"   and "poly a x = (0::'c)" shows "order x p = Suc i"
r2f2r_inv: shows "r2fnd \<circ> f2rnd = id"
omega_induct: fixes n :: "V"   and P :: "V \<Rightarrow> bool" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "P []\<^sub>\<circ>"   and "\<And>n. \<lbrakk>n \<in>\<^sub>\<circ> \<omega>; P n\<rbrakk> \<Longrightarrow> P (ZFC_in_HOL.succ n)" shows "P n"
subseq_conv_nths: fixes xs :: "'c list"   and ys :: "'c list" shows "subseq xs ys = (\<exists>N. xs = nths ys N)"
r_neg: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> \<ominus>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>"
fsp_def: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('j, 'h, 'k) Module_scheme"   and N :: "('l, 'h, 'm) Module_scheme"   and f :: "'j \<Rightarrow> 'l"   and H :: "'j set"   and A :: "'j set"   and g :: "'j \<Rightarrow> 'l" shows "fsp R M N f H A g = (g \<in> mHom R (mdl M (fgs R M A)) N \<and> (\<forall>z\<in>A. f z = g z) \<and> A \<subseteq> H)"
g_fas_def: shows "g_fas \<equiv> [floatarith.Minus (floatarith.Add (floatarith.Add (floatarith.Minus (floatarith.Var 0)) (floatarith.Mult (floatarith.Mult (floatarith.Num (Float 8 0)) (floatarith.Inverse (floatarith.Num (Float 100 0)))) (floatarith.Var 1))) (floatarith.Mult (floatarith.Var 0 ^\<^sub>e 2) (floatarith.Var 1))), floatarith.Minus (floatarith.Add (floatarith.Add (floatarith.Mult (floatarith.Num (Float 6 0)) (floatarith.Inverse (floatarith.Num (Float 10 0)))) (floatarith.Minus (floatarith.Mult (floatarith.Mult (floatarith.Num (Float 8 0)) (floatarith.Inverse (floatarith.Num (Float 100 0)))) (floatarith.Var 1)))) (floatarith.Minus (floatarith.Mult (floatarith.Var 0 ^\<^sub>e 2) (floatarith.Var 1))))]"
sat_precond_exec_as_proj_eq_proj_exec: fixes s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set" assumes "sat_precond_as s as" shows "exec_plan (fmrestrict_set vs s) (as_proj as vs) = fmrestrict_set vs (exec_plan s as)"
INF_2: fixes z :: "'a" assumes "z \<in> X" shows "dist y x \<le> dist y z"
le_numeral_extra_4: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one one"
2: shows "|\<Union>us s. abs (us, s)| <o |UNIV|"
legendre_aux_posD_2: fixes x :: "real"   and p :: "nat" assumes "0 < legendre_aux x p" shows "real p \<le> x"
reindex_cong: fixes l :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "F g A = F h B"
oCons_1: assumes "xs \<noteq> []" shows "\<exists>a\<in>set xs. Missing_List.min_list xs = a"
simps_1: shows "T_splay_max \<langle>\<rangle> = 1"
simps: fixes i :: "nat"   and j :: "nat"   and l :: "'a list" shows "swap_list i j l = l[j := l ! i, i := l ! j]"
from_hma\<^sub>m_def: fixes a :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a = Matrix.mat CARD('d) CARD('c) (\<lambda>(i, j). a $h from_nat i $h from_nat j)"
simps_2: fixes n1 :: "nat"   and n2 :: "nat" shows "(R n1 < I n2) = False"
hyps_3: assumes "NF t" shows "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>l t"
that: assumes "h' = Ref.set r' (f (get h r')) h"   and "r = f (get h r')" shows "thesis"
list_all2_Cons1: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list"   and ys :: "'h list" shows "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
butlast_take: fixes n :: "nat"   and xs :: "'c list" assumes "n \<le> length xs" shows "butlast (take n xs) = take (n - 1) xs"
p_not_div_choose_lemma: fixes K :: "nat"   and p :: "nat"   and j :: "nat"   and k :: "nat" assumes "\<And>i. Suc i < K \<Longrightarrow> multiplicity p (Suc i) = multiplicity p (Suc (j + i))"   and "k < K"   and "prime p" shows "multiplicity p (j + k choose k) = 0"
pair_measure_eqI: fixes M1 :: "'c measure"   and M2 :: "'d measure"   and M :: "('c \<times> 'd) measure" assumes "sigma_finite_measure M1"   and "sigma_finite_measure M2"   and "sets (M1 \<Otimes>\<^sub>M M2) = sets M"   and "\<And>A B. \<lbrakk>A \<in> sets M1; B \<in> sets M2\<rbrakk> \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)" shows "M1 \<Otimes>\<^sub>M M2 = M"
simps_17: fixes x1 :: "'c" shows "vars_term (Var x1) = {x1}"
det_linear_rows_sum: fixes S :: "nat set"   and a :: "nat \<Rightarrow> nat \<Rightarrow> 'b Matrix.vec"   and n :: "nat" assumes "finite S"   and "a \<in> {0..<n} \<rightarrow> S \<rightarrow> carrier_vec n" shows "Determinant.det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('b) n (a i) S)) = (\<Sum>f | (\<forall>i\<in>{0..<n}. f i \<in> S) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i). Determinant.det (mat\<^sub>r n n (\<lambda>i. a i (f i))))"
acyclic_6_symmetric: fixes x :: "'a" assumes "acyclic_6 x" shows "symmetric x"
F: fixes n :: "nat" assumes "n < length cl" shows "inj_on (F n) (P n)"
AE_ball_countable: fixes X :: "'c set"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'d measure" assumes "countable X" shows "(AE x in M. \<forall>y\<in>X. P x y) = (\<forall>y\<in>X. AE x in M. P x y)"
rec_def: shows "rec \<equiv> linepath lb (Complex (Re ub) (Im lb)) +++ linepath (Complex (Re ub) (Im lb)) ub +++ linepath ub (Complex (Re lb) (Im ub)) +++ linepath (Complex (Re lb) (Im ub)) lb"
\<K>_\<S>_closed: fixes s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "s \<in> \<S>" shows "\<Union> (set_pmf ` \<K> s) \<subseteq> \<S>"
simps_2: fixes as :: "'b set"   and rest :: "'b set list" shows "is_partition_impl (as # rest) = is_partition_impl rest \<bind> (\<lambda>all. if as \<inter> all = {} then Some (all \<union> as) else None)"
char_inv': fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c" assumes "character G \<chi>"   and "a \<in> carrier G" shows "\<chi> (inv\<^bsub>G\<^esub> a) = inverse (\<chi> a)"
DiffD1: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A\<setminus>B" shows "c \<in> A"
eint_add_left_cancel: fixes a :: "eint"   and b :: "eint"   and c :: "eint" shows "(a + b = a + c) = (a = \<infinity> \<or> b = c)"