hcju/isabelleps huggingface dataset:
constructive_security_sim_obsf_def: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and \<A> :: "('y + 'z, 'aa + 'ab) distinguisher_obsf"   and adv :: "real" shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv \<equiv> exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd> \<longrightarrow> advantage \<A> (obsf_resource (sim |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) (obsf_resource real_resource) \<le> adv"
ab_diff_conv_add_uminus': fixes a :: "'d"   and b :: "'d" shows "a - b = - b + a"
sc_Pow: shows "scylinder (space S) ` lists G \<subseteq> Pow (streams (space S))"
F_anal_valid: shows "\<forall>twoC\<in>two_chain_typeI. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"
True_True_eps_concI: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> (eps L)\<^sup>*" shows "(True # p, True # q) \<in> (eps (RegExp2NAe.conc L R))\<^sup>*"
scale_one: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "(1::'c) \<cdot> x = x"
rel_frontier_retract_of_punctured_affine_hull: fixes S :: "'b set"   and a :: "'b" assumes "bounded S"   and "convex S"   and "a \<in> rel_interior S" shows "rel_frontier S retract_of affine hull S - {a}"
simps_32: fixes x6 :: "ty"   and x5 :: "String.literal" shows "x6\<lfloor>\<rceil> \<noteq> Class x5"
run_pause_option: fixes pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and out :: "'c"   and c :: "'d \<Rightarrow> ('b, 'a) optionT" shows "run_option (pause_option pause out c) = pause out (\<lambda>i. run_option (c i))"
has_parametrized_subprob_density_cong: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and N' :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> N x = N' x" shows "has_parametrized_subprob_density M N R f = has_parametrized_subprob_density M N' R f"
hom: shows "gs.d k = rat_of_int (gs.Gramian_determinant fs k)"
rb_aux_inv_last_Nil: fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_dom args" shows "snd (snd (fst (rb_aux args))) = []"
lift_def: fixes h :: "'j \<Rightarrow> 'i"   and inp :: "'h \<Rightarrow> 'j option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
pdevs_val_independent_pdevs_strict: fixes xs :: "(real \<times> real) list"   and e :: "nat \<Rightarrow> real" assumes "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"   and "list_all (\<lambda>x. x \<noteq> 0) xs"   and "e \<in> UNIV \<rightarrow> {- 1<..<1}" shows "\<exists>e'. e' \<in> UNIV \<rightarrow> {- 1<..<1} \<and> pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list (independent_pdevs xs))"
less_INF_D: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and i :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less y (Inf (f ` A))"   and "i \<in> A" shows "less y (f i)"
min_product_iff: fixes x :: "'b fset"   and A :: "'b fset set"   and B :: "'b fset set" shows "(x \<in> A \<otimes>\<^sub>m B) = ((\<exists>a\<in>A. \<exists>b\<in>B. x = a |\<union>| b) \<and> (\<forall>a\<in>A. \<forall>b\<in>B. a |\<union>| b |\<subseteq>| x \<longrightarrow> a |\<union>| b = x))"
indep_varD: fixes Ma :: "'c measure"   and A :: "'b \<Rightarrow> 'c"   and Mb :: "'c measure"   and B :: "'b \<Rightarrow> 'c"   and Xa :: "'c set"   and Xb :: "'c set" assumes "X.indep_var Ma A Mb B"   and "Xa \<in> sets Ma"   and "Xb \<in> sets Mb" shows "X.prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space (distr M S X)) = X.prob (A -` Xa \<inter> space (distr M S X)) * X.prob (B -` Xb \<inter> space (distr M S X))"
e: assumes "p \<notin> Units G"   and "\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides a \<otimes> b \<longrightarrow> p divides a \<or> p divides b" shows "P"
Sigma_Un_distrib1: fixes I :: "'d set"   and J :: "'d set"   and C :: "'d \<Rightarrow> 'e set" shows "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
finite_Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "finite (A - insert a B) = finite (A - B)"
remove_def: shows "remove \<equiv> l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove get_child_nodes set_child_nodes get_parent get_owner_document get_disconnected_nodes set_disconnected_nodes"
empt: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and x :: "'c" assumes "DFS succs is_node invariant ins memb empt"   and "is_node x" shows "\<not> memb x empt"
is_binary_functor: shows "binary_functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>) (\<cdot>\<^sub>S) local.map"
learn_lim_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" shows "learn_lim \<psi> U s \<equiv> environment \<psi> U s \<and> (\<forall>f\<in>U. \<exists>i. \<psi> i = f \<and> (\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i))"
le_list_def: fixes s :: "'c list"   and t :: "'c list" shows "(s \<le> t) = (\<exists>r. s @ r = t)"
poly_mult_one_1: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "domain R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p" shows "ring.poly_mult R [\<one>\<^bsub>R\<^esub>] p = p"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
wf_bs_polyadd: fixes bs :: "'b list"   and p :: "poly"   and q :: "poly" shows "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
of_complex_add: fixes x :: "complex"   and y :: "complex" shows "of_complex (x + y) = of_complex x + of_complex y"
eval_monom_1: fixes \<alpha> :: "'c \<Rightarrow> 'b" shows "eval_monom \<alpha> 1 = (1::'b)"
swap: fixes n :: "nat"   and bs :: "'a list"   and m :: "nat"   and t :: "poly" assumes "n < length bs"   and "m < length bs" shows "\<lparr>swap n m t\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>t\<rparr>\<^sub>p\<^bsup>bs[n := bs ! m, m := bs ! n]\<^esup>"
imirror_cut_less: fixes I :: "nat set"   and t :: "nat" assumes "finite I" shows "imirror I \<down>< mirror_elem t I = imirror_bounds (I \<down>> t) (iMin I) (Max I)"
list_update_def: shows "list_update \<equiv> rec_list (\<lambda>i v. []) (\<lambda>x xs xsa i v. case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # xsa j v)"
dom: shows "dom h\<^sub>1 = dom h\<^sub>1'"
of_nat_monom: fixes n :: "nat" shows "of_nat n = Polynomial.monom (of_nat n) 0"
poly_linear_exp_linear_factors: fixes b :: "'b"   and n :: "nat"   and as :: "'b list" assumes "[:b, 1::'b:] ^ n dvd (\<Prod>a\<leftarrow>as. [:a, 1::'b:])" shows "n \<le> length (filter ((=) b) as)"
simps_2: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list"   and x :: "'d" shows "sinks I D u (xs @ [x]) = (if (u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I) then insert (D x) (sinks I D u xs) else sinks I D u xs)"
scaleR_le_0_iff: fixes a :: "real"   and b :: "'b" shows "(a *\<^sub>R b \<le> (0::'b)) = (0 < a \<and> b \<le> (0::'b) \<or> a < 0 \<and> (0::'b) \<le> b \<or> a = 0)"
dist_eq_tr_1: shows "TT \<noteq> \<bottom>"
dependent_explicit: fixes s :: "'d set" shows "dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *\<^sub>R v) = (0::'d) \<and> (\<exists>v\<in>t. u v \<noteq> 0))"
rtrancl3p_refl: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g" shows "rtrancl3p r a [] a"
rotate_3: assumes "round_robin_reschedule t0 (queue @ [t']) n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>" shows "\<exists>x ln n. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> 0 < ln $ n \<and> \<not> waiting (wset s t) \<and> may_acquire_all (locks s) t ln"
assoc'_in_hom_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[f, g, h] : local.dom f \<star> local.dom g \<star> local.dom h \<Rightarrow> (cod f \<star> cod g) \<star> cod h\<guillemotright>"
Class_def: shows "Class = (\<lambda>a\<in>S. {b \<in> S. (b, a) \<in> E(\<alpha>)})"
size_delete_if_isin: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "isin t x" shows "size t = Suc (size (delete x t))"
wf_while_rel2: fixes mp :: "nat" shows "wf (while_rel2 mp)"
iTILL_cut_le: fixes n :: "nat"   and t :: "nat" shows "[\<dots>n] \<down>\<le> t = (if t \<le> n then [\<dots>t] else [\<dots>n])"
subgame_\<omega>_subset: fixes G :: "('c, 'd) ParityGame_scheme"   and V' :: "'c set" assumes "ParityGame G" shows "\<omega>\<^bsub>ParityGame.subgame G V'\<^esub> ` V\<^bsub>ParityGame.subgame G V'\<^esub> \<subseteq> \<omega>\<^bsub>G\<^esub> ` V\<^bsub>G\<^esub>"
rr: shows "rr = \<langle>rrr, (rra, rrlv), rrl\<rangle>"
strong_formula_induct': fixes P :: "formula \<Rightarrow> bool"   and n :: "nat" assumes "\<forall>A. (\<forall>B. size B < size A \<longrightarrow> P B) \<longrightarrow> P A" shows "\<forall>A. size A = n \<longrightarrow> P A"
map_inv_map_inv: fixes f :: "'c \<Rightarrow> 'd option" assumes "inj_on f (dom f)" shows "map_inv (map_inv f) = f"
seqSubstEqvt: fixes p :: "(name \<times> name) list"   and P :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list" shows "p \<bullet> P[<\<sigma>>] = p \<bullet> P[<p \<bullet> \<sigma>>]"
ball_subset_cball: fixes x :: "'b"   and e :: "real" shows "ball x e \<subseteq> cball x e"
decompose_mod_p: fixes n :: "nat" shows "[\<Sum>i<n. y ^ (n - Suc i) * x ^ i = int n * x ^ (n - 1)] (mod int p)"
learn_cp_imp_total: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_cp \<psi> U s" shows "learn_total \<psi> U s"
oalist_inv_map_raw: fixes xs :: "('b \<times> 'c) list \<times> 'b nat_term_order"   and f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'd" assumes "ko_ntm.oalist_inv xs"   and "\<And>a b. key_compare (key_order_of_nat_term_order_inv (snd xs)) (fst (f a)) (fst (f b)) = key_compare (key_order_of_nat_term_order_inv (snd xs)) (fst a) (fst b)" shows "ko_ntm.oalist_inv (map_raw f xs)"
model_existence: fixes C :: "(nat, 'a) form set set"   and S :: "(nat, 'a) form set"   and p :: "(nat, 'a) form"   and e :: "nat \<Rightarrow> nat hterm" assumes "consistency C"   and "S \<in> C"   and "infinite (- \<Union> (params ` S))"   and "p \<in> S"   and "closed 0 p" shows "eval e HApp (\<lambda>a ts. Pred a (terms_of_hterms ts) \<in> Extend S (mk_finite_char (mk_alt_consistency (close C))) from_nat) p"
t_def: shows "t = (r - cmod (p - a)) / 2"
singleton_conv: fixes a :: "'k" shows "{x. x = a} = {a}"
simps: fixes x :: "'b" shows "theRight (Inr x) = x"
eq: shows "ind_in_set A ` ({0..<M} \<inter> A) = {0..<card ({0..<M} \<inter> A)}"
H2_UNION: fixes P :: "H2 \<Rightarrow> 'd set" shows "\<Union> (range P) = \<Union> (P ` {H1, H2})"
simps: fixes P :: "'a stream \<Rightarrow> bool"   and a1 :: "nat"   and a2 :: "'a stream" shows "ev_at P a1 a2 = ((\<exists>\<omega>. a1 = 0 \<and> a2 = \<omega> \<and> P \<omega>) \<or> (\<exists>\<omega> n. a1 = Suc n \<and> a2 = \<omega> \<and> \<not> P \<omega> \<and> ev_at P n (stl \<omega>)))"
D5: shows "\<lfloor>\<lambda>w. \<forall>x. x actualizedAt w \<longrightarrow> (\<forall>xa. ((actualizedAt) xa \<^bold>\<rightarrow> xa explains x \<^bold>\<rightarrow> \<^bold>\<box>(E! x \<^bold>\<rightarrow> E! xa)) w)\<rfloor>"
sup_isotone: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a" assumes "- p \<le> - q"   and "- r \<le> - s" shows "- p \<squnion> - r \<le> - q \<squnion> - s"
circline_type_mk_circline: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "(A, B, C, D) \<in> hermitean_nonzero" shows "circline_type (mk_circline A B C D) = sgn (Re (A * D - B * C))"
prod_induct3: fixes P :: "'c \<times> 'd \<times> 'e \<Rightarrow> bool"   and x :: "'c \<times> 'd \<times> 'e" assumes "\<And>a b c. P (a, b, c)" shows "P x"
set_child_nodes_get_disconnected_nodes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>Shadow_DOM.set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
quotient_of_number_2: shows "quotient_of 1 = (1, 1)"
cases: fixes x :: "opinl" obtains "x = OpAddNum"   | "x = OpMulNum"
l: assumes "set (map snd txs) \<subseteq> var"   and "set (r # map fst txs) \<subseteq> trm"   and "set [] \<subseteq> fmla" shows "length (getFrN (map snd txs) (r # map fst txs) [] (length txs)) = length txs"
Delay_L3: fixes t :: "nat"   and y :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and x :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and t2 :: "nat" assumes "\<forall>t1\<le>t. y t1 = []"   and "Delay y i1 d x i2"   and "t2 \<le> t + d" shows "x t2 = []"
sset_siterate: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "i.R (siterate f x) = {(f ^^ n) x |n. True}"
hyps: fixes t :: "symbol \<times> character list" assumes "charslength (p @ ts) = charslength p"   and "t \<in> set ts" shows "chars_of_token t = []"
basic_trans_rules_22: fixes x :: "'e"   and y :: "'e"   and z :: "'e" assumes "x < y"   and "y \<le> z" shows "x < z"
e_leq_c_c_complement_transpose_general: fixes e :: "'a"   and c :: "'a"   and g :: "'a" assumes "e = minarc (c * - c\<^sup>T \<sqinter> g)"   and "regular c" shows "e \<le> c * - c\<^sup>T"
cut_less_Max_less: fixes I :: "'a set"   and t :: "'a" assumes "finite (I \<down>< t)"   and "I \<down>< t \<noteq> {}" shows "Max (I \<down>< t) < t"
cert_def: fixes A :: "agent"   and Ka :: "nat"   and T :: "msg"   and signK :: "nat" shows "cert A Ka T signK = signCert signK \<lbrace>Agent A, Key Ka, T\<rbrace>"
it_paths_eq: fixes t :: "'a rose_tree" shows "it_pathsP t = (\<lambda>x. x \<in> it_paths t)"
FOREACH_oci_to_monadic: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d set"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>0 :: "'c" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S c f \<sigma>0 = monadic_FOREACH R \<Phi> S (\<lambda>\<sigma>. RETURN (c \<sigma>)) f \<sigma>0"
order_pos_iff: fixes p :: "'b poly"   and x :: "'b" assumes "p \<noteq> 0" shows "(0 < order x p) = (poly p x = (0::'b))"
bind_gpv_unfold: fixes r :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "r \<bind> f = GPV (the_gpv r \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> the_gpv (f x) | IO out c \<Rightarrow> return_spmf (IO out (\<lambda>input. c input \<bind> f))))"
intro: fixes M :: "('a, 'b) message set"   and s :: "('a, 'b option) insert_message list"   and a :: "'a extended \<Rightarrow> 'a sort_key extended" assumes "is_associated_string M s"   and "a_conditions (insert_messages M) a" shows "integrate_insert_commute M a s"
T_def: shows "T \<equiv> \<lambda>x y. x = Rep y"
labels_multiset: fixes \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<alpha> :: "'a"   and r :: "('a \<times> 'a) set" assumes "length (labels \<sigma>) \<le> 1"   and "set (labels \<sigma>) \<subseteq> {\<alpha>}" shows "(r|labels \<sigma>|, {#\<alpha>#}) \<in> mul_eq r"
connected_component_UNIV: fixes x :: "'b" shows "connected_component_set UNIV x = UNIV"
set_K_cfg: fixes cfg :: "'a cfg" shows "set_pmf (K_cfg cfg) = cont cfg ` set_pmf (action cfg)"
not_1: shows "\<not> (y < 0 \<or> n < 0)"
frechet_derivative_id: fixes a :: "'b" shows "frechet_derivative id (at a) = id"
gen_lossless_gpv_cases: fixes gpv :: "('a, 'out, 'in) gpv" assumes "local.gen_lossless_gpv gpv" obtains p :: "('a, 'out, ('a, 'out, 'in) rpv) generat spmf" where "gpv = GPV p" and "lossless_spmf p" and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.gen_lossless_gpv (c input)"
H_smooth_on: fixes k :: "enat" shows "k-smooth_on UNIV H"
lookup_fix: fixes F :: "('a \<Rightarrow> 'b) \<rightarrow> 'a \<Rightarrow> 'b"   and y :: "'a" shows "(fix\<cdot>F) y = (F\<cdot>(fix\<cdot>F)) y"
root_unity_decomp: fixes n :: "nat" assumes "n \<noteq> 0" shows "root_unity n = (\<Prod>i\<leftarrow>[0..<n]. [:- cis (real i * 2 * pi / real n), 1:])"
that: fixes l' :: "'a list" assumes "hd l' = l ! (sl + 1)"   and "last l' = l ! sl"   and "rel_chain l' r"   and "length l' = length l - 1"   and "\<forall>i. i + 1 < length l' \<longrightarrow> (\<exists>j. j + 1 < length l \<and> l' ! i = l ! j \<and> l' ! (i + 1) = l ! (j + 1))" shows "thesis"
cos_diff: shows "cos x - cos y = (x - y) * - sin z"
Inum_Add: fixes s :: "Reflection_Examples.num"   and t :: "Reflection_Examples.num"   and vs :: "nat list" shows "Inum (Add s t) vs = Inum s vs + Inum t vs"
pre_oldest_configuration_obtain: fixes x :: "nat"   and q :: "'b"   and n :: "nat" assumes "x \<in> pre_oldest_tokens q n - {Suc n}" obtains q' :: "'b" where "oldest_token q' n = Some x" and "q = \<delta> q' (w n)"
inorder_insert: fixes t :: "'t"   and x :: "'a" assumes "inv t \<and> sorted (inorder t)" shows "inorder (insert x t) = ins_list x (inorder t)"
accept_def: shows "accept\<^sub>M = Alm_all token_succeeds"
take_length: fixes xs :: "'b list" shows "take (length xs) xs = xs"
pred_prod_inject: fixes P1 :: "'b \<Rightarrow> bool"   and P2 :: "'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" shows "pred_prod P1 P2 (a, b) = (P1 a \<and> P2 b)"
dims1_def: shows "st2.dims1 = nths dims {0..<Suc k}"
pivot_fun_zero_row_iff: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and nr :: "nat"   and i :: "nat" assumes "pivot_fun A f nc"   and "A \<in> carrier_mat nr nc"   and "i < nr" shows "(f i = nc) = (row A i = 0\<^sub>v nc)"
len32: fixes x :: "32 itself" shows "len_of x = 32"
tnth_TCons: fixes x :: "'a"   and xs :: "('a, 'b) tllist"   and n :: "nat" shows "tnth (TCons x xs) n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> tnth xs n')"
composite_is_tabulation: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (f \<star> g \<star> h) TfHgh.\<rho>\<sigma>.tab (tab\<^sub>0 (g \<star> h) \<star> TfHgh.p\<^sub>0) (tab\<^sub>1 f \<star> TfHgh.p\<^sub>1)"
properI: fixes P :: "heap \<times> nat set \<Rightarrow> bool" assumes "\<And>as h. P (h, as) \<Longrightarrow> in_range (h, as)"   and "\<And>as h h'. \<lbrakk>P (h, as); relH as h h'; in_range (h', as)\<rbrakk> \<Longrightarrow> P (h', as)" shows "proper P"
simps_1: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list" shows "x # xs \<cong> {y # ys, z # zs, P} = (if P x xs then x = y \<and> xs \<cong> {ys, z # zs, P} else x = z \<and> xs \<cong> {y # ys, zs, P})"
maximal_existence_flow_2: fixes x :: "real \<Rightarrow> real"   and K :: "real set"   and t0 :: "real"   and x0 :: "real"   and t :: "real" assumes "(x solves_ode f) K X"   and "x t0 = x0"   and "is_interval K"   and "t0 \<in> K"   and "K \<subseteq> T"   and "t \<in> K" shows "flow t0 x0 t = x t"
elims_2: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'a list"   and resOf :: "'b \<Rightarrow> 'a"   and x :: "'b trm" assumes "Signature TYPE('a) wtFsym wtPsym"   and "Signature.wt wtFsym arOf resOf x" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "'b" and Tl :: "'b trm list" where "x = Fn f Tl" and "wtFsym f \<and> list_all (Signature.wt wtFsym arOf resOf) Tl \<and> arOf f = map (Signature.tpOf resOf) Tl"
insert_commute: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" shows "insert x (insert y A) = insert y (insert x A)"
is_expansion_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and G :: "'a"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "trimmed G"   and "is_expansion F basis"   and "is_expansion G basis" shows "is_expansion (divide F G) basis"
translation_property': fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "(red F)\<^sup>*\<^sup>* p 0" shows "is_red F (p + q) \<or> is_red F q"
\<I>_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>"
mset_diff_cancel1elem: fixes a :: "'a"   and B :: "'a multiset" assumes "a \<notin># B" shows "{#a#} - B = {#a#}"
noStuck_startD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "'e" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Normal t" shows "s \<noteq> Stuck"
gamma_plus': fixes i1 :: "int"   and a1 :: "'av"   and i2 :: "int"   and a2 :: "'av" assumes "i1 \<in> \<gamma> a1"   and "i2 \<in> \<gamma> a2" shows "i1 + i2 \<in> \<gamma> (plus' a1 a2)"
not_restricted_def: fixes v :: "'a"   and G :: "'a gexp list" shows "not_restricted v G = (length (filter (\<lambda>g. gexp_constrains g (V v)) G) = 0)"
lookup_a2_map': fixes tp' :: "nat"   and m :: "(event_data option list, nat + nat) mapping"   and xs :: "event_data option list"   and tstp :: "nat + nat" assumes "Mapping.lookup a2_map tp' = Some m"   and "Mapping.lookup m xs = Some tstp" shows "tstp_lt tstp (nt - (left I - 1)) tp \<and> isl tstp = (0 < left I)"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "proper_design point_set block_collection"   and "block_design_axioms block_collection u_block_size" shows "block_design point_set block_collection u_block_size"
intF: fixes f :: "'fsym"   and al :: "'univ list" assumes "wtFsym f"   and "list_all2 intT (arOf f) al" shows "intT (resOf f) (intF f al)"
neg: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "Modules.module_hom s1 s2 f" shows "f (- x) = - f x"