hcju/isabelleps huggingface dataset:
liberal_witness_four: fixes x :: "'c"   and y :: "'c"   and u :: "'c"   and v :: "'c"   and A :: "'c set"   and i :: "'d"   and j :: "'d"   and Is :: "'d set" assumes "hasw [x, y, u, v] A"   and "hasw [i, j] Is" obtains P :: "'d \<Rightarrow> ('c \<times> 'c) set" where "profile A Is P" and "x \<^bsub>(P i)\<^esub>\<prec> y" and "u \<^bsub>(P j)\<^esub>\<prec> v" and "\<forall>i\<in>Is. v \<^bsub>(P i)\<^esub>\<prec> x \<and> y \<^bsub>(P i)\<^esub>\<prec> u"
Says_Server_message_form: fixes A :: "agent"   and K' :: "nat"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and Ticket :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs"   and "evs \<in> bankerb_gets" shows "K' = shrK A \<and> K \<notin> range shrK \<and> Ticket = Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<and> Key K \<notin> used before Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) on evs \<and> Tk = CT before Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) on evs"
hyp: shows "Ipoly bs (p ^\<^sub>p Suc n div 2) = Ipoly bs p ^ (Suc n div 2)"
simps_1: fixes \<phi> :: "'fmla" shows "rawpsubst \<phi> [] = \<phi>"
dist_stereographic_infinite: fixes M1 :: "riemann_sphere"   and M2 :: "riemann_sphere"   and m2 :: "complex" assumes "stereographic M1 = \<infinity>\<^sub>h"   and "stereographic M2 = of_complex m2" shows "dist_riemann_sphere' M1 M2 = 2 / sqrt (1 + (cmod m2)\<^sup>2)"
sigma_finite_pairs: shows "\<exists>F. (\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and> (\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and> (\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (Pi\<^sub>M I M)"
ide_in_hom_2: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>f : f \<Rightarrow>\<^sub>C f\<guillemotright>"
i_def: shows "i = (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a))"
p_dist_sup: fixes uminus :: "'c \<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 x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (sup x y) = inf (uminus x) (uminus y)"
match_ex_entry: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and pc :: "nat"   and start :: "nat"   and "end" :: "nat"   and catch_type :: "char list"   and handler :: "nat \<times> nat" shows "matches_ex_entry P C pc (start, end, catch_type, handler) = (start \<le> pc \<and> pc < end \<and> P \<turnstile> C \<preceq>\<^sup>* catch_type)"
admissible_conj: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "ccpo.admissible lub ord P"   and "ccpo.admissible lub ord Q" shows "ccpo.admissible lub ord (\<lambda>x. P x \<and> Q x)"
next_def: fixes \<Delta>1 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" shows "next \<Delta>1 = (if \<Delta>1 = \<Delta>e then {} else if \<Delta>1 = last \<Delta>s then {\<Delta>1, \<Delta>e} else {\<Delta>1, SOME \<Delta>2. (\<Delta>1, \<Delta>2) \<in> allConsec \<Delta>s, \<Delta>e})"
bind_return: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and N :: "'b measure"   and x :: "'a" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "x \<in> space M" shows "return M x \<bind> f = f x"
varsL_substL: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and l :: "('c, 'd) lit" shows "varsL (substL \<pi> l) = (\<Union>x\<in>varsL l. vars (\<pi> x))"
is_inf_uniq: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and i :: "'a"   and i' :: "'a" assumes "dlat le"   and "dpo.is_inf le x y i"   and "dpo.is_inf le x y i'" shows "i = i'"
leq_funD: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<sqsubseteq> g" shows "f x \<sqsubseteq> g x"
lm071: fixes a :: "('a \<times> 'b set) set" shows "Domain (pseudoAllocation a) \<subseteq> Domain a"
that: fixes r :: "'a list" assumes "u = p \<cdot> r"   and "r \<le>p w"   and "r \<noteq> \<epsilon>" shows "thesis"
simps_3: fixes v :: "fresh_t" shows "invKey (sesK v) = sesK v"
cofinite_not_ed_ou: fixes X :: "nat cofinite set" shows "\<not> ed_ou X"
echelon_form_of_invertible: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'c) vec, 'b) vec" assumes "is_bezout_ext bezout" shows "\<exists>P. invertible P \<and> P ** A = echelon_form_of A bezout \<and> echelon_form (echelon_form_of A bezout)"
Foundation: fixes i :: "name"   and z :: "tm" assumes "atom i \<sharp> z" shows "{} \<turnstile> All2 i z (Neg (Disjoint (Var i) z)) IMP z EQ Zero"
Cons_6: assumes "snd b \<noteq> 0"   and "punit.lt (snd b) adds t"   and "\<not> (t - punit.lt (snd b)) \<oplus> fst b \<prec>\<^sub>t u" shows "thesis"
simps_4: fixes c :: "int"   and P :: "pol"   and i :: "nat"   and Q :: "pol" shows "Pc c \<langle>+\<rangle> PX P i Q = PX P i (Q \<langle>+\<rangle> Pc c)"
hiding_advantage_ind_cpa_def: fixes key_gen :: "('a \<times> 'b) spmf"   and commit :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and valid_msg :: "'c \<Rightarrow> bool"   and \<A> :: "('b \<Rightarrow> (('c \<times> 'c) \<times> 'f) spmf) \<times> ('d \<Rightarrow> 'f \<Rightarrow> bool spmf)" shows "abstract_commitment.hiding_advantage_ind_cpa key_gen commit valid_msg \<A> \<equiv> \<bar>spmf (abstract_commitment.hiding_game_ind_cpa key_gen commit valid_msg \<A>) True - 1 / 2\<bar>"
sorted_list_of_fset_simps_1: fixes S :: "'c fset" shows "set (sorted_list_of_fset S) = fset S"
multiset_prime_factorization_code: shows "prime_factorization = (\<lambda>n. mset (prime_factorization_nat n))"
is_lasso_prpl_def: fixes prpl :: "'Q list \<times> 'Q list" shows "is_lasso_prpl prpl \<equiv> is_lasso_prpl_pre prpl \<and> (\<forall>A\<in>F. set (snd prpl) \<inter> A \<noteq> {})"
r_gzero: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and design_rep_number :: "int" assumes "constant_rep_design point_set block_collection design_rep_number" shows "0 < design_rep_number"
higher_differentiable_on_compose: fixes T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and n :: "nat"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "higher_differentiable_on T f n"   and "higher_differentiable_on S g n"   and "g ` S \<subseteq> T"   and "open S"   and "open T" shows "higher_differentiable_on S (f \<circ> g) n"
full_imp_height: fixes n :: "nat"   and t :: "'b tree23" assumes "full n t" shows "height t = n"
atMost_iff: fixes i :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "(i \<in> ord.atMost less_eq k) = less_eq i k"
weakPar1Guarded: fixes \<Psi> :: "'b"   and R :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> : R \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* Q"   and "guarded Q" shows "\<Psi> : R \<parallel> Q \<rhd> P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P' \<parallel> Q"
power2_eq_square: fixes a :: "'b list" shows "CoWBasic.power2 a = a \<cdot> a"
intro: fixes cs :: "(String.literal \<times> String.literal) set" assumes "wf_subclass cs" shows "wf_subclass_loc cs"
wf_sig_imp_exe_wf_sig: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "alist_conds cto"   and "alist_conds tao"   and "exe_osig_conds sa"   and "wf_sig (map_of cto, map_of tao, translate_osig sa)" shows "(exe_wf_osig sa \<and> fst ` set (snd sa) = fst ` set tao \<and> (\<forall>type\<in>fst ` set (snd sa). \<forall>ars\<in>snd ` set (the (lookup (\<lambda>k. k = type) (snd sa))). the (lookup (\<lambda>k. k = type) tao) = length ars)) \<and> (\<forall>ty\<in>snd ` set cto. typ_ok_sig (map_of cto, map_of tao, translate_osig sa) ty)"
of_finite_Float_def: fixes x :: "Float.float" shows "of_finite_Float x = (if is_normal_Float TYPE(('a, 'b) IEEE.float) x then normal_of_Float x else if is_denormal_Float TYPE(('a, 'b) IEEE.float) x then denormal_of_Float x else 0)"
cc_n_minus_2: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F"   and "2 < card X" shows "pm_powerprod.aa X f F (card X - 2) + pm_powerprod.bb ord X F (card X - 2) \<le> (poly_deg f)\<^sup>2 + 2 * poly_deg f"
error_at_worst_lemma: fixes x :: "real"   and a :: "('a, 'b) IEEE.float" assumes "\<bar>x\<bar> < threshold TYPE(('a, 'b) IEEE.float)"   and "is_finite a" shows "\<bar>error TYPE(('a, 'b) IEEE.float) x\<bar> \<le> \<bar>valof a - x\<bar>"
simps: fixes uu :: "'b extN" shows "uu\<^sup>\<star> = extN.Top"
cases: fixes a :: "('sf, 'sc, 'sz) formula" assumes "FRadmit a" obtains \<theta>1 :: "('sf, 'sz) trm" and \<theta>2 :: "('sf, 'sz) trm" where "a = Geq \<theta>1 \<theta>2"   | p :: "'sz" and args :: "'sz \<Rightarrow> ('sf, 'sz) trm" where "a = $\<phi> p args"   | \<phi> :: "('sf, 'sc, 'sz) formula" where "a = ! \<phi>'__" and "FRadmit \<phi>'__"   | \<phi> :: "('sf, 'sc, 'sz) formula" and \<psi> :: "('sf, 'sc, 'sz) formula" where "a = (\<phi>''__ && \<psi>)" and "FRadmit \<phi>''__" and "FRadmit \<psi>"   | \<phi> :: "('sf, 'sc, 'sz) formula" and x :: "'sz" where "a = Exists x \<phi>'''__" and "FRadmit \<phi>'''__"   | \<alpha> :: "('sf, 'sc, 'sz) hp" and \<phi> :: "('sf, 'sc, 'sz) formula" where "a = (\<langle> \<alpha> \<rangle> \<phi>)" and "PRadmit \<alpha>" and "FRadmit \<phi>"
bigthetaI_tendsto: fixes c :: "real"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "c \<noteq> 0"   and "((\<lambda>x. f x / g x) \<longlongrightarrow> c) F" shows "f \<in> \<Theta>[F](g)"
closed_eq_obs_eq_rvs': fixes m :: "'node"   and S :: "'node SDG_node set"   and m' :: "'node" assumes "obs_intra m \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs_intra m' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "rv S (CFG_node m) = rv S (CFG_node m')"
ide_map\<^sub>0_obj: fixes a :: "'c" assumes "C.obj a" shows "D.ide (\<tau>\<^sub>0 a)"
semilat_def: shows "semilat = (\<lambda>(A, r, f). order r A \<and> closed A f \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z))"
elims_3: fixes x :: "'a list" assumes "\<not> sorted x" obtains x :: "'a" and ys :: "'a list" where "xa__ = x # ys" and "\<not> (Ball (set ys) ((\<le>) x) \<and> sorted ys)"
left_treeI: fixes u :: "'a"   and s :: "'a"   and t :: "'a" assumes "u \<in> V"   and "s \<in> set (u \<leadsto> t)" shows "u \<in> left_tree s t"
hyps_1: shows "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
eval_prod_lam: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and g :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>g : a \<otimes> b \<rightarrow> c\<guillemotright>" shows "eval b c \<cdot> (\<Lambda> a b c g \<otimes> b) = g"
diff_diff_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - b - c = a - (b + c)"
prim_invar2_ctd_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" shows "prim_invar2_ctd Q \<pi> \<equiv> let A = A Q \<pi>; S = S A in prim_invar1 A \<and> \<pi> r = None \<and> Q r = \<infinity> \<and> (\<forall>(u, v)\<in>edges rg \<inter> (- S) \<times> S. Q u \<noteq> \<infinity>) \<and> (\<forall>u. Q u \<noteq> \<infinity> \<longrightarrow> \<pi> u \<noteq> None) \<and> (\<forall>u v. \<pi> u = Some v \<longrightarrow> v \<in> S \<and> (u, v) \<in> edges rg) \<and> (\<forall>u v d. Q u = enat d \<and> \<pi> u = Some v \<longrightarrow> d = w {u, v} \<and> (\<forall>v'\<in>S. (u, v') \<in> edges rg \<longrightarrow> d \<le> w {u, v'}))"
deriv: fixes z :: "complex"   and t :: "real" assumes "z \<in> U"   and "t \<in> {a..x}" shows "((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)"
c_pair_sum_inj: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "c_pair x1 y1 = c_pair x2 y2" shows "x1 + y1 = x2 + y2"
flush_append_write: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and m :: "nat \<Rightarrow> nat" shows "flush (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W]) m = (flush sb m)(a := v)"
assms: fixes k :: "nat" shows "f \<bullet> ns k \<noteq> \<zero>"
simps_12: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x2 :: "'b"   and y2 :: "'c" shows "rel_sort_key R (Final x2) (Final y2) = R x2 y2"
crdt_ops_rem_spec: fixes xs :: "('a \<times> 'b) list"   and x :: "'a \<times> 'b"   and deps :: "'b \<Rightarrow> 'a set"   and ys :: "('a \<times> 'b) list"   and zs :: "('a \<times> 'b) list" assumes "spec_ops (xs @ [x]) deps"   and "crdt_ops (ys @ [x] @ zs) deps"   and "set (xs @ [x]) = set (ys @ [x] @ zs)" shows "crdt_ops (ys @ zs) deps"
simps: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b list"   and a3 :: "'b" shows "rtrancl_path r a1 a2 a3 = ((\<exists>x. a1 = x \<and> a2 = [] \<and> a3 = x) \<or> (\<exists>x y ys z. a1 = x \<and> a2 = y # ys \<and> a3 = z \<and> r x y \<and> rtrancl_path r y ys z))"
prime: fixes p :: "nat" assumes "prime p" shows "divisor_count p = 2"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('l \<Rightarrow> 'l) \<Rightarrow> ('c, 'a, 'b, 'd, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'd, 'g option) RElement_scheme"   and v :: "'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
distinct_83: fixes x4 :: "val"   and x71 :: "char list"   and x72 :: "expr" shows "Val x4 \<noteq> x71:=x72"
Pi_pmf_insert: fixes A :: "'b set"   and x :: "'b"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A"   and "x \<notin> A" shows "Pi_pmf (insert x A) dflt p = map_pmf (\<lambda>(y, f). f(x := y)) (pair_pmf (p x) (Pi_pmf A dflt p))"
infinite_cts_filtration_def: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure" shows "infinite_cts_filtration p M F \<equiv> infinite_coin_toss_space p M \<and> infinite_cts_filtration_axioms M F"
gcsrp_map_bij: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "bij_to (gcsrp_map G H) (set_rcs G H) (gcsrp_map G H ` set_rcs G H)"
LeftDerives1_append_prefix: fixes v :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and w :: "symbol list"   and u :: "symbol list" assumes "LeftDerives1 v i r w"   and "is_word u" shows "LeftDerives1 (u @ v) (i + length u) r (u @ w)"
diagonal_to_Smith_PQ_JNF_dim: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and S :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat m n"   and "diagonal_to_Smith_PQ_JNF A bezout = (P, S, Q)" shows "P \<in> carrier_mat m m \<and> S \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
rtrancl3p_Cons: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and bs :: "'b list"   and a' :: "'a" shows "rtrancl3p r a (b # bs) a' = (\<exists>a''. r a b a'' \<and> rtrancl3p r a'' bs a')"
dom_MkArr: fixes A :: "'a E.term"   and B :: "'a E.term"   and F :: "'a" assumes "arr (MkArr A B F)" shows "local.dom (MkArr A B F) = MkIde A"
max_less_iff_conj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.linorder less_eq less" shows "less (ord.max less_eq x y) z = (less x z \<and> less y z)"
list_all_filter: fixes p :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_all p (filter p xs)"
simps_2: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "witness' Y (r # rs) = Pool (witness Y r rs) (witness Y r rs)"
Digamma_plus1: fixes z :: "'a" assumes "z \<noteq> (0::'a)" shows "Digamma (z + (1::'a)) = Digamma z + (1::'a) / z"
tauBoundTrans: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi" assumes "\<tau>.P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'" shows "False"
in_remove1_mset_neq: fixes a :: "'a"   and b :: "'a"   and C :: "'a multiset" assumes "a \<noteq> b" shows "(a \<in># remove1_mset b C) = (a \<in># C)"
FAccStatic\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and h :: "heap"   and ls :: "val list"   and sh :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, ls, sh)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,Static:t in D" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h, ls, sh)\<rangle>"
arev_neq: fixes G :: "('a, 'b) pre_digraph"   and arev :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev a \<noteq> a"
in_hom_char: fixes f :: "'d"   and a :: "'d"   and b :: "'d" shows "D'.in_hom f a b = (D'.arr a \<and> D'.arr b \<and> D'.arr f \<and> \<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>)"
internal_int_poly_factorization_def: fixes f :: "int poly" shows "local.internal_int_poly_factorization f = (case square_free_factorization_int f of (a, gis) \<Rightarrow> (a, concat (map (\<lambda>(g, i). map (\<lambda>h. (h, i)) (local.main_int_poly_factorization g)) gis)))"
Union_addSubsetSet: fixes b :: "'b set"   and As :: "'b set set" shows "\<Union> (addSubsetSet b As) = b \<union> \<Union> As"
t_A_t: fixes n :: "nat" assumes "n < length rs" shows "t_A n = int (t (s_A n) (rs ! n) (as ! n))"
assms_1: fixes i :: "nat" assumes "i < card X" shows "wellorder_class.enumerate X i = wellorder_class.enumerate Y i"
pmaxsimpchain: fixes xs :: "'a set list" assumes "pmaxsimpchain xs" shows "maxsimpchain xs"
const_to_constant: fixes a :: "'a" assumes "a \<in> carrier R" shows "to_fun (monom P a 0) = constant_function (carrier R) a"
rev_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "rev (filter P xs) = filter P (rev xs)"
equivalence_comp_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "equivalence x" shows "x * y \<sqinter> x * z = x * (y \<sqinter> x * z)"
repbNodes_in_Nodes: shows "repb ` Nodes n ll \<subseteq> Nodes n ll"
StateParallel_hoare: fixes c :: "'d upred"   and P :: "'d hrel"   and d\<^sub>1 :: "'d upred"   and Q :: "'d hrel"   and d\<^sub>2 :: "'d upred"   and a :: "'e \<Longrightarrow> 'd"   and b :: "'f \<Longrightarrow> 'd" assumes "\<lbrace>c\<rbrace> P \<lbrace>d\<^sub>1\<rbrace>\<^sub>u"   and "\<lbrace>c\<rbrace> Q \<lbrace>d\<^sub>2\<rbrace>\<^sub>u"   and "a \<bowtie> b"   and "a \<natural> d\<^sub>1"   and "b \<natural> d\<^sub>2" shows "\<lbrace>c\<rbrace> P |a|b|\<^sub>\<sigma> Q \<lbrace>d\<^sub>1 \<and> d\<^sub>2\<rbrace>\<^sub>u"
\<I>_def: shows "\<I> = (\<lambda>x. SOME t. \<Gamma> (Var x) = \<Gamma> t \<and> public_ground_wf_term t)"
distinct_21: fixes x1 :: "char list"   and x121 :: "expr"   and x122 :: "expr" shows "new x1 \<noteq> x121;; x122"
C_sat: shows "\<not> falsifies\<^sub>c (f (Suc n)) C"
goal3_4: fixes ass :: "'a \<Rightarrow> bool" shows "fi ass = val_ifex i ass"
composite_simps_3: shows "trg (\<nu> \<star> \<pi> \<star> \<rho>) = trg \<nu>"
hyperpow_Suc_le_self2: fixes r :: "real star"   and n :: "nat star" assumes "0 \<le> r"   and "r < 1" shows "r pow (n + 1) \<le> r"
power_mono_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq zero b"   and "0 < n" shows "less_eq (power.power one times a n) (power.power one times b n) = less_eq a b"
sum_strict_mono_ex1_complex_lex: fixes A :: "'a set"   and f :: "'a \<Rightarrow> complex"   and g :: "'a \<Rightarrow> complex" assumes "finite A"   and "\<forall>x\<in>A. f x \<le>\<^sub>\<complex> g x"   and "\<exists>a\<in>A. f a <\<^sub>\<complex> g a" shows "sum f A <\<^sub>\<complex> sum g A"
of_rat_1: shows "of_rat 1 = (1::'b)"
f: fixes x :: "'a" assumes "x \<in> S" shows "Q x (f x)"
intros_2: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "((Skip, mem), Stop, mem) \<in> sifum_lang.eval\<^sub>w_simple eval\<^sub>A eval\<^sub>B"
G2_bigo: shows "(\<lambda>x. G2 x - (x * (ln x)\<^sup>2 - x * ln x)) \<in> O(ln)"
H_rcoset_reps: shows "Supgroup.is_rcoset_replist G H_rcoset_reps"
assms: fixes x :: "nat" shows "f x = card {y \<in> v0. y < x}"
dg_prod_2_Cod_app: fixes g :: "V"   and f :: "V" assumes "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>" shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr> \<lparr>g, f\<rparr>\<^sub>\<bullet> = [\<AA>\<lparr>Cod\<rparr>\<lparr>g\<rparr>, \<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr>]\<^sub>\<circ>"
f'_def: fixes v :: "'a" shows "f' v = complex_of_real (f v) + \<i> * complex_of_real (f (\<i> *\<^sub>C v))"
add_rel_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> R \<times> S"   and "(r', s') \<in> R \<times> S" shows "add_rel (r / s) (r' / s') = (r \<cdot> s' + r' \<cdot> s) / (s \<cdot> s')"
clausal_saturated_complete: fixes M :: "'a literal multiset set"   and N :: "'a literal multiset set" assumes "G.saturated M N"   and "\<forall>I. \<not> I \<TTurnstile>s N" shows "{#} \<in> N"