hcju/isabelleps huggingface dataset:
Kab_authentic: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> bankerberos" shows "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"
simps: fixes I :: "'c set"   and v :: "nat \<Rightarrow> 'd"   and BL :: "nat \<Rightarrow> 'd option"   and BU :: "nat \<Rightarrow> 'd option"   and IL :: "nat \<Rightarrow> 'c"   and IU :: "nat \<Rightarrow> 'c" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>b ((BL, BU), IL, IU) = ((\<forall>x c. BL x = Some c \<longrightarrow> IL x \<in> I \<longrightarrow> c \<le> v x) \<and> (\<forall>x c. BU x = Some c \<longrightarrow> IU x \<in> I \<longrightarrow> v x \<le> c))"
validCE_act: fixes n :: "nat"   and cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"   and "eval cid t t' n \<gamma>" shows "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (the_enat \<langle>cid #\<^bsub>enat n\<^esub>inf_llist t\<rangle>)"
no_illegal_updates_def: fixes t :: "'a transition_scheme"   and r :: "nat" shows "no_illegal_updates t r = (\<forall>u\<in>set (Updates t). fst u \<noteq> r)"
seq_CSP1: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is CSP1 healthy"   and "Q is CSP1 healthy" shows "P ;; Q is CSP1 healthy"
convex_interior_closure: fixes S :: "'a set" assumes "convex S" shows "interior (closure S) = interior S"
src_port_datatype_def: fixes x :: "int \<times> (DatatypeAddress \<times> int) \<times> (DatatypeAddress \<times> int) \<times> 'a" shows "src_port x \<equiv> (snd \<circ> fst \<circ> snd) x"
WT_converterD_1: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and q :: "'a" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g run_converter conv q \<surd>"
flowgraph_def: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" shows "flowgraph fg \<equiv> ((\<forall>u a v. (u, a, v) \<in> edges fg \<longrightarrow> proc_of fg u = proc_of fg v) \<and> (\<forall>p. proc_of fg (entry fg p) = p)) \<and> (\<forall>p. proc_of fg (return fg p) = p) \<and> (\<forall>p. initialproc fg p \<longrightarrow> mon fg p = {})"
list_diff_empty: fixes l :: "'c list" shows "the (list_diff l []) = l"
map_of_map_key_inverse_fun_eq: fixes xs :: "('b \<times> 'c) list"   and g :: "'d \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd"   and x :: "'d" assumes "\<And>x. x \<in> set xs \<Longrightarrow> g (f (fst x)) = fst x"   and "f (g x) = x" shows "map_of (map (\<lambda>(k, y). (f k, y)) xs) x = map_of xs (g x)"
le_Liminf_iff: fixes C :: "'a"   and F :: "'b filter"   and X :: "'b \<Rightarrow> 'a" shows "(C \<le> Liminf F X) = (\<forall>y<C. \<forall>\<^sub>F x in F. y < X x)"
simplex_in_fold_A_im_image: fixes x :: "'a set" assumes "x \<in> fold_A \<turnstile> A" shows "fold_A ` x = x"
tendsto_ident_at: fixes "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and s :: "'c set" assumes "class.topological_space open" shows "LIM x topological_space.at_within open a s. x :> topological_space.nhds open a"
game_sem_loop_fixpoint_mono: fixes X :: "(variable \<Rightarrow> real) set"   and I :: "interp"   and \<alpha> :: "game" shows "mono (\<lambda>Z. X \<union> game_sem I \<alpha> Z)"
signs_at_insert: fixes i :: "nat"   and qs :: "real poly list"   and p :: "real poly"   and x :: "real" assumes "i \<le> length qs" shows "insertAt i (squash (poly p x)) (signs_at qs x) = signs_at (insertAt i p qs) x"
majorities_intersect: fixes S :: "'b set"   and T :: "'b set" assumes "card UNIV < card S + card T" shows "S \<inter> T \<noteq> {}"
eq_image: shows "(\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"
soundness_diagonalize: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and P :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and S :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and Q :: "(('a, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" assumes "is_bezout_ext bezout"   and "local.diagonalize A bezout = (P, S, Q)" shows "invertible P \<and> invertible Q \<and> isDiagonal S \<and> S = P ** A ** Q"
Int_Union2: fixes B :: "'c set set"   and A :: "'c set" shows "\<Union> B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
df: fixes k :: "nat" assumes "k < n" shows "f k \<in> carrier_mat d d"
path_connectedin_path_components_of: fixes C :: "'c set"   and X :: "'c topology" assumes "C \<in> path_components_of X" shows "path_connectedin X C"
primorial'_primes_omega_le: fixes n :: "nat" assumes "0 < n" shows "primorial' (primes_omega n) \<le> n"
uminus_ow_def: fixes U :: "'a set"   and uminus :: "'a \<Rightarrow> 'a" shows "uminus_ow U uminus \<equiv> \<forall>a. a \<in> U \<longrightarrow> uminus a \<in> U"
m1'step: shows "(\<exists>p \<alpha>. \<langle>x:=\<^bsub>\<iota>\<^esub> e,m1'\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m2'\<rangle>) \<and> (\<forall>m''. (\<exists>p \<alpha>. \<langle>x:=\<^bsub>\<iota>\<^esub> e,m1'\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m''\<rangle>) \<longrightarrow> m'' = m2')"
map_comp: fixes g1 :: "'h \<Rightarrow> 'f"   and g2 :: "'i \<Rightarrow> 'g"   and f1 :: "'j \<Rightarrow> 'h"   and f2 :: "'k \<Rightarrow> 'i"   and v :: "('j, 'k) F" shows "F g1 g2 (F f1 f2 v) = F (g1 \<circ> f1) (g2 \<circ> f2) v"
bnds_sqrt': fixes x :: "float"   and prec :: "nat" shows "sqrt (real_of_float x) \<in> {real_of_float (lb_sqrt prec x)..real_of_float (ub_sqrt prec x)}"
allActBox: fixes F :: "'d \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and s :: "nat \<Rightarrow> 'b" shows "(\<box>[(\<forall>x. F x)]_v) s = (\<forall>x. (\<box>[F x]_v) s)"
vsubset_vintersection_rightD_1: fixes A :: "V"   and B :: "V"   and C :: "V" assumes "A \<subseteq>\<^sub>\<circ> B \<inter>\<^sub>\<circ> C" shows "A \<subseteq>\<^sub>\<circ> B"
hom_power: fixes x :: "'c poly"   and n :: "nat"   and a :: "'c" shows "poly (x ^ n) a = poly x a ^ n"
S_M_not_grounding_of_clss: fixes C :: "'a literal multiset" assumes "C \<notin> grounding_of_clss M" shows "S_M C = S C"
antisym: fixes b :: "'f"   and a :: "'f" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
assms_2: shows "\<Omega> \<inter> space N \<in> sets N"
not_cong_3412: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Cong A B C D" shows "\<not> Cong C D A B"
splitFace_edges_g': fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and g' :: "graph" assumes "pre_splitFace g ram1 ram2 f vs"   and "(f12, f21, g') = splitFace g ram1 ram2 f vs"   and "vs \<noteq> []" shows "\<E> g' = \<E> g \<union> Edges vs \<union> Edges (rev vs) \<union> {(ram2, last vs), (hd vs, ram1), (ram1, hd vs), (last vs, ram2)}"
map_sel_1: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e stream" shows "shd (smap f a) = f (shd a)"
NotL_inv': fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"
prems: shows "separateAtoms (At # L) = (a, b, c, d)"
nn_integral_spmf_def: fixes p :: "'c spmf" shows "nn_integral_spmf p = integral\<^sup>N (measure_spmf p)"
RES_leof_RES_iff: fixes X :: "'c set"   and Y :: "'c set" shows "(RES X \<le>\<^sub>n RES Y) = (X \<subseteq> Y)"
cond_assoc: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and c :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  Q \<triangleleft> c \<triangleright>  R) = (P \<triangleleft> b \<and> c \<triangleright>  (Q \<triangleleft> c \<triangleright>  R))"
distinct_nth_eq_iff: fixes xs :: "'b list"   and x :: "nat"   and y :: "nat" assumes "distinct xs"   and "x < length xs"   and "y < length xs" shows "(xs ! x = xs ! y) = (x = y)"
containFalsum: fixes \<Gamma> :: "'a form multiset"   and R :: "('a sequent list \<times> 'a sequent) set"   and C :: "'a form" assumes "ff \<in># \<Gamma>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* C, 0) \<in> derivable R*"
l4_16: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.FSC Bet Cong A B C D A' B' C' D'"   and "A \<noteq> B" shows "Cong C D C' D'"
ub2_sgn: shows "\<forall>x\<ge>ub2. sgn (poly p x) = sgn_pos_inf p"
upper_triangular_imp_lin_indpt_rows: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "upper_triangular A"   and "(0::'a) \<notin> set (diag_mat A)" shows "lin_indpt (set (rows A))"
callee': fixes x :: "'e"   and s :: "'b" assumes "x \<in> outs_\<I> \<I>" shows "results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"
isovar_greater_degree: fixes p :: "'b mpoly"   and var :: "nat" shows "\<forall>i>MPoly_Type.degree p var. isolate_variable_sparse p var i = 0"
t: fixes x :: "'a" assumes "x \<in> {a--b}" shows "{- t..t} \<subseteq> existence_ivl0 x"
impr_def: fixes a :: "'a G"   and b :: "'a G" shows "a r\<rightarrow> b \<equiv> Abs_G ((u - Rep_G a + Rep_G b) \<sqinter> u)"
lexord_eq_alt_def2: fixes xs :: "'a list"   and ys :: "'a list" shows "(RETURN \<circ>\<circ> lexord_eq) xs ys = REC\<^sub>T (\<lambda>f (xs, ys). case (xs, ys) of ([], x) \<Rightarrow> RETURN True | (xa # xs, []) \<Rightarrow> RETURN False | (xa # xs, y # ys) \<Rightarrow> if xa < y then RETURN True else if xa = y then f (xs, ys) else RETURN False) (xs, ys)"
pvalid_prefixlang: fixes p :: "(symbol \<times> character list) list"   and x :: "item"   and t :: "symbol" assumes "pvalid p x"   and "is_terminal t"   and "next_symbol x = Some t" shows "terminals p @ [t] \<in> \<L>\<^sub>P"
AE_count_space: fixes P :: "'d \<Rightarrow> bool"   and A :: "'d set" shows "almost_everywhere (count_space A) P = (\<forall>x\<in>A. P x)"
gauss_int_norm_normalize: fixes x :: "gauss_int" shows "gauss_int_norm (normalize x) = gauss_int_norm x"
bypass: fixes z :: "'a" assumes "z \<in> set p" shows "z \<notin> RF (TER g)"
AE_impI: fixes P :: "bool"   and Q :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "P \<Longrightarrow> almost_everywhere M Q" shows "AE x in M. P \<longrightarrow> Q x"
Dummy_eqvt: fixes \<pi> :: "perm"   and v :: "Vars.var" shows "\<pi> \<bullet> Dummy v = Dummy (\<pi> \<bullet> v)"
assms_2: shows "\<forall>\<tau>\<in>down (U s) t. x \<tau> \<in> x\<acute>=f & G on U S @ t\<^sub>0 s"
l11_49: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Cong B A B' A'"   and "Cong B C B' C'" shows "Cong A C A' C' \<and> (A \<noteq> C \<longrightarrow> Tarski_neutral_dimensionless.CongA Bet Cong B A C B' A' C' \<and> Tarski_neutral_dimensionless.CongA Bet Cong B C A B' C' A')"
splay_map_of_inorder: fixes x :: "'c"   and t :: "('c \<times> 'd) tree"   and l :: "('c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and r :: "('c \<times> 'd) tree" assumes "Splay_Map.splay x t = \<langle>l, a, r\<rangle>"   and "sorted1 (inorder t)" shows "AList_Upd_Del.map_of (inorder t) x = (if x = fst a then Some (snd a) else None)"
star_absorb: fixes x :: "'a"   and y :: "'a" assumes "x * y = bot" shows "x * y\<^sup>\<star> = x"
lowlink_eq_LowLink: fixes x :: "'a" assumes "x \<in> dom (discovered s)" shows "\<zeta> s x = Min (\<delta> s ` lowlink_set s x)"
A: fixes x :: "'a" assumes "x \<in> A" shows "B x \<lesssim> C x"
pmat_extension_def: fixes m :: "'a mat" shows "pmat_extension m = ptensor_mat m (1\<^sub>m d2)"
abs_split_split: fixes split_fun :: "('a btnode ref option \<times> 'a) array \<times> nat \<Rightarrow> 'a \<Rightarrow> nat Heap"   and n :: "nat"   and xs :: "('b \<times> 'c) list"   and p :: "'c" assumes "imp_split_smeq split_fun"   and "n < length xs"   and "\<forall>(uu_, s)\<in>set (take n xs). s < p"   and "case xs ! n of (x, s) \<Rightarrow> \<not> s < p" shows "abs_split xs p = (take n xs, drop n xs)"
adjacent_half_chamber_system_image_reverse: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>"   and "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "C \<sim> D" shows "(THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C) = D"
\<nu>_simps_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and \<omega> :: "'c"   and \<nu> :: "'c" assumes "uw\<theta>\<omega>\<nu> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<omega> \<nu>" shows "category.iso V \<nu>"
starrel_in_hypreal: fixes x :: "nat \<Rightarrow> 'b" shows "starrel `` {x} \<in> star"
init_def: fixes h :: "'grp"   and w :: "nat" shows "init h w = sample_uniform (order \<G>) \<bind> (\<lambda>r. return_spmf (r, \<^bold>g [^] r))"
leftderivations1_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.leftderivations1 \<NN> \<TT> \<RR> = {(u, v) |u v. CFG.leftderives1 \<NN> \<TT> \<RR> u v}"
ir_While': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com"   and Q\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> bval b s \<and> bval b' s') (c;; WHILE b DO c) (c';; WHILE b' DO c') Q\<^sub>1"   and "ir_hoare (\<lambda>s s'. P s s' \<and> \<not> bval b s \<and> \<not> bval b' s') SKIP SKIP Q\<^sub>2" shows "ir_hoare P (WHILE b DO c) (WHILE b' DO c') (\<lambda>s s'. Q\<^sub>1 s s' \<or> Q\<^sub>2 s s')"
module_hom_axioms: shows "module_hom (*\<^sub>R) (*\<^sub>R) g"
bounded_bilinear_scaleR: shows "bounded_bilinear (*\<^sub>R)"
intF_def: fixes f :: "'fsym"   and al :: "univ list" shows "intF f al \<equiv> if unprot (resOf f) \<or> protFw (resOf f) then eintF (Oldf f) (map2 ntsem (arOf f) al) else eintF (Tag (resOf f)) [eintF (Oldf f) (map2 ntsem (arOf f) al)]"
add: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "additive f" shows "f (x + y) = f x + f y"
assms_12: shows "sparse_row_matrix c1 \<le> c"
concat_take_Suc_empty: fixes n :: "nat"   and l :: "'a list list" assumes "n < length l"   and "l ! n = []" shows "concat (take (Suc n) l) = concat (take n l)"
le_trans_bot: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "le_trans (\<lambda>P s. 0) t"
cmod_eq_Im: fixes z :: "complex" assumes "Re z = 0" shows "cmod z = \<bar>Im z\<bar>"
L_lasthasxx_def: fixes x :: "'a"   and y :: "'a" shows "L_lasthasxx x y = Plus (seq [question (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y]) (seq [question (Atom y), Star (Times (Atom x) (Atom y)), Atom x, Atom x])"
Invariants_of_foTn: fixes f :: "'a \<Rightarrow> real"   and n :: "nat" assumes "integrable M f" shows "AE x in M. real_cond_exp M Invariants (f \<circ> T ^^ n) x = real_cond_exp M Invariants f x"
prfOf_def: fixes n :: "'b"   and \<phi> :: "'c" shows "prfOf n \<phi> \<equiv> bprv (PPf n \<langle>\<phi>\<rangle>)"
ratfps_Lcm: fixes A :: "'b ratfps set" assumes "A \<noteq> {}"   and "0 \<notin> A"   and "bdd_above (ratfps_subdegree ` A)" shows "Lcm A = ratfps_of_poly (monom (1::'b) (Sup (ratfps_subdegree ` A)))"
graph_rule_translation: fixes X :: "'a allegorical_term"   and Y :: "'a allegorical_term" shows "graph_rule (translation X, translation (A_Int X Y))"
poly_eq_insertion: fixes p :: "'b mpoly"   and v :: "nat"   and x :: "'b" assumes "vars p \<subseteq> {v}" shows "poly (mpoly_to_poly v p) x = insertion (\<lambda>v. x) p"
periodic_finite_ex: fixes d :: "int"   and P :: "int \<Rightarrow> bool" assumes "0 < d"   and "\<forall>x k. P x = P (x - k * d)" shows "(\<exists>x. P x) = (\<exists>j\<in>{1..d}. P j)"
assms_4: shows "snd a \<otimes> snd h [^] e = g' [^] z"
binomial_mono_left: fixes n :: "nat"   and N :: "nat"   and k :: "nat" assumes "n \<le> N" shows "n choose k \<le> N choose k"
Uniq_def: fixes P :: "'e \<Rightarrow> bool" shows "Uniq P \<equiv> \<forall>x y. P x \<longrightarrow> P y \<longrightarrow> y = x"
is_open_def: fixes T :: "'b set set"   and s :: "'b set" shows "(s open\<^bsub>T\<^esub>) = (s \<in> T)"
prems_1: shows "\<forall>a. r $ a \<noteq> None \<longrightarrow> r $ a = r' $ a"
successively_mono: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and Q :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "successively P xs"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y" shows "successively Q xs"
has_subprob_densityD_3: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "M = density N f"
Cons_4: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
simps_2: fixes M :: "'a"   and I :: "('a, 'b, 'c) input" shows "extractFrame (M\<lparr>I) = \<bottom>\<^sub>F"
right_adjoint_is_ide: fixes f :: "'a" assumes "is_right_adjoint f" shows "ide f"
subfrmlsn_id: fixes \<phi> :: "'b ltln" shows "\<phi> \<in> subfrmlsn \<phi>"
Cons_eq_map_D: fixes x :: "'c"   and xs :: "'c list"   and f :: "'d \<Rightarrow> 'c"   and ys :: "'d list" assumes "x # xs = map f ys" shows "\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs"
remdups_adj_distinct: fixes xs :: "'a list" assumes "distinct xs" shows "remdups_adj xs = xs"
q_def: shows "q = poly_subst (\<lambda>x. monomial (1::'a) (monomial 1 x) + monomial (a x) 0) p"
prob_space_distr: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d pmf"   and M' :: "'e measure" assumes "measure_pmf.random_variable M M' f" shows "prob_space (distr (measure_pmf M) M' f)"
assms_1: shows "PTA_Reachability.pairwise P xs"
has_limits_def: fixes uu :: "'c" shows "B.has_limits uu \<equiv> \<forall>J. category J \<longrightarrow> B.has_limits_of_shape J"
simps_2: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and ms :: "'c match_expr list"   and a :: "action"   and p :: "'d" shows "match_list \<gamma> (m # ms) a p = (if matches \<gamma> m a p then True else match_list \<gamma> ms a p)"
modes_respect_priv_ignores_old_vars: shows "ignores_old_vars modes_respect_priv"
sep_conj_sep_true: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "P h" shows "(P \<and>* (\<lambda>s. True)) h"
lin_indpt_list_length_eq_n_2: fixes n :: "nat"   and xs :: "'b vec list" assumes "cof_vec_space.lin_indpt_list n xs"   and "length xs = n" shows "vectorspace.basis class_ring (module_vec TYPE('b) n) (set xs)"
approximating_bigstep_fun_induct: fixes P :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> 'b rule list \<Rightarrow> state \<Rightarrow> bool"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list"   and s :: "state" assumes "\<And>\<gamma> p s. P \<gamma> p [] s"   and "\<And>\<gamma> p r rs X. P \<gamma> p (r # rs) (Decision X)"   and "\<And>\<gamma> p m a rs. \<lbrakk>\<not> matches \<gamma> m a p; P \<gamma> p rs Undecided\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided"   and "\<And>\<gamma> p m a rs. \<lbrakk>matches \<gamma> m a p; a = Log \<Longrightarrow> P \<gamma> p rs Undecided; a = Empty \<Longrightarrow> P \<gamma> p rs Undecided\<rbrakk> \<Longrightarrow> P \<gamma> p (Rule m a # rs) Undecided" shows "P \<gamma> p rs s"
implements_def: fixes pi :: "'a \<Rightarrow> 'b"   and Sa :: "('c, 'b) spec"   and Sc :: "('d, 'a) spec" shows "implements pi Sa Sc \<equiv> map pi ` obeh Sc \<subseteq> obeh Sa"
induced_subgraphE: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "induced_subgraph H G" obtains "subgraph H G" and "arcs H = {e \<in> arcs G. tail G e \<in> verts H \<and> head G e \<in> verts H}"
Labels_CondTrue: fixes c\<^sub>1 :: "cmd"   and l :: "nat"   and c :: "cmd"   and b :: "expr"   and c\<^sub>2 :: "cmd" assumes "labels c\<^sub>1 l c" shows "labels (if (b) c\<^sub>1 else c\<^sub>2) (l + 1) c"
vU: shows "insert (adjuster n w us + w) (set us) \<subseteq> carrier_vec n"
l_set_disconnected_nodes_get_child_nodes_def: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
order_trans: fixes x :: "'c list"   and y :: "'c list"   and z :: "'c list" assumes "subseq x y"   and "subseq y z" shows "subseq x z"
inf_chain_bad: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "inf_chain p f" shows "bad p (f i)"
measure_spmf_spmf_of_pmf: fixes p :: "'b pmf" shows "measure_spmf (spmf_of_pmf p) = measure_pmf p"
\<Sigma>_protocol_def: shows "\<Sigma>_AND.\<Sigma>_protocol \<equiv> \<Sigma>_AND.completeness \<and> \<Sigma>_AND.special_soundness \<and> \<Sigma>_AND.HVZK"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" obtains "a = a \<squnion> b"
Graph1: fixes M :: "node list"   and E :: "(nat \<times> nat) list" assumes "Roots \<subseteq> Blacks M"   and "Proper_Edges (M, E)"   and "\<forall>i<length E. \<not> BtoW (E ! i, M)" shows "Reach E \<subseteq> Blacks M"
C_1: shows "C < liminf (\<lambda>x. ereal (u x))"
induct: fixes P :: "'a vec list \<Rightarrow> ('a vec \<times> 'a) list \<Rightarrow> 'a list list \<Rightarrow> bool"   and a0 :: "'a vec list"   and a1 :: "('a vec \<times> 'a) list"   and a2 :: "'a list list" assumes "\<And>n_gs mus. P [] n_gs mus"   and "\<And>f fs n_gs mus. (\<And>x xa y xb. \<lbrakk>x = mus_adjuster f n_gs [] (0\<^sub>v n); (xa, y) = x; xb = y + f\<rbrakk> \<Longrightarrow> P fs ((xb, \<parallel>xb\<parallel>\<^sup>2) # n_gs) (xa # mus)) \<Longrightarrow> P (f # fs) n_gs mus" shows "P a0 a1 a2"