hcju/isabelleps huggingface dataset:
is_sig_red_addsI: fixes f :: "'t \<Rightarrow>\<^sub>0 'b"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "f \<in> F"   and "t \<in> keys (rep_list p)"   and "rep_list f \<noteq> 0"   and "punit.lt (rep_list f) adds t"   and "ord_term_lin.is_le_rel sing_reg"   and "ordered_powerprod_lin.is_le_rel top_tail"   and "sing_reg (t \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)"   and "top_tail t (punit.lt (rep_list p))" shows "is_sig_red sing_reg top_tail F p"
mono: fixes c :: "'b"   and a :: "'b"   and d :: "'b"   and b :: "'b" assumes "c \<le> a"   and "d \<le> b" shows "max c d \<le> max a b"
subresultant_zero_ge: fixes f :: "'b poly"   and g :: "'b poly"   and J :: "nat"   and k :: "nat" assumes "degree f + (degree g - J) \<le> k"   and "degree f - J + (degree g - J) \<noteq> 0" shows "coeff (subresultant J f g) k = (0::'b)"
xor_bot: fixes x :: "'g" shows "x \<oplus> \<bottom> = x"
loxodromic_isometryD_1: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "isometry f"
points_index_gt0_impl_existance: fixes B :: "'b set multiset"   and ps :: "'b set" assumes "0 < B index ps" shows "\<exists>bl. bl \<in># B \<and> ps \<subseteq> bl"
low_eq: shows "mem\<^sub>1 =\<^sup>l mem\<^sub>2"
con_comp_rep: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" shows "Rep_process (P \<parallel> Q <p, q>) = (con_comp_failures P Q p q, con_comp_divergences P Q p q)"
hvzk_sim_inverse_rewrite: fixes h :: "'a"   and x1 :: "nat"   and x2 :: "nat"   and z1 :: "nat"   and c :: "nat"   and z2 :: "nat" assumes "h = \<^bold>g [^] x1 \<otimes> g' [^] x2" shows "\<^bold>g [^] ((z1 + order \<G> * c * x1 - c * x1) mod order \<G>) \<otimes> g' [^] ((z2 + order \<G> * c * x2 - c * x2) mod order \<G>) = \<^bold>g [^] z1 \<otimes> g' [^] z2 \<otimes> inv h [^] c"
parts_monotone: fixes c :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> parts G"   and "G \<subseteq> H" shows "c \<in> parts H"
csemiring_def: fixes R :: "('b, 'c) ring_scheme" shows "csemiring R \<equiv> semiring R \<and> Group.comm_monoid R"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c Heap" shows "for (x # xs) f = f x \<bind> (\<lambda>_. for xs f)"
Quotient3_rel_rep: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and a :: "'d"   and b :: "'d" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)"
complete_sets_sandwich: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "C \<in> sets M"   and "A \<subseteq> B"   and "B \<subseteq> C"   and "emeasure M A = emeasure M C"   and "emeasure M A < \<infinity>" shows "B \<in> sets M"
quicksort_by_rel_remove_acc: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sl :: "'a list"   and xs :: "'a list" shows "quicksort_by_rel R sl xs = quicksort_by_rel R [] xs @ sl"
from_nat_to_nat_id: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and x :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs (mod_type.to_nat Rep x) = x"
assoc: fixes a :: "'a upred"   and b :: "'a upred"   and c :: "'a upred" shows "((a \<and> b) \<and> c) = (a \<and> b \<and> c)"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_with_State_Idle_imp: fixes k :: "nat"   and s :: "nat \<Rightarrow> 'a"   and trans_fun :: "'b message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b message_af"   and c :: "'a"   and t0 :: "nat"   and t :: "nat"   and t1 :: "nat"   and localState :: "'a \<Rightarrow> 'c"   and output_fun :: "'a \<Rightarrow> 'd message_af" assumes "0 < k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "t0 = t * k"   and "t1 \<in> [0\<dots>,k - Suc 0] \<oplus> t0"   and "State_Idle localState output_fun trans_fun (localState (s t1))"   and "output_fun (s t1) \<noteq> NoMsg" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = output_fun (s t1)"
cpow_def: fixes r :: "('c \<times> 'c) set" shows "cpow r = |Pow (Field r)|"
Pair_equalI: fixes x :: "'b"   and p :: "'b \<times> 'c"   and y :: "'c" assumes "x \<equiv> fst p"   and "y \<equiv> snd p" shows "(x, y) \<equiv> p"
fps_deriv_linear: fixes a :: "'a"   and f :: "'a fps"   and b :: "'a"   and g :: "'a fps" shows "fps_deriv (fps_const a * f + fps_const b * g) = fps_const a * fps_deriv f + fps_const b * fps_deriv g"
set_subseqs_eq: fixes ys :: "'d list" shows "set (subseqs ys) = {xs. subseq xs ys}"
simps: fixes state :: "State"   and Vbl :: "nat set" shows "solve_loop state Vbl = (if getSATFlag state \<noteq> UNDEF then state else let state' = solve_loop_body state Vbl in solve_loop state' Vbl)"
distinct_9: fixes x2 :: "name"   and x3 :: "trm" shows "Var x2 \<noteq> suc x3"
\<rho>_l: fixes p :: "fm"   and i :: "int"   and bs :: "real list" assumes "iszlfm p (real_of_int i # bs)" shows "\<forall>(b, k)\<in>set (\<rho> p). 0 < k \<and> numbound0 b \<and> isint b (real_of_int i # bs)"
eq: shows "(\<i> * z * csqrt (1 - z\<^sup>2))\<^sup>2 = - z\<^sup>2 * (1 - z\<^sup>2)"
inf_sup_eq: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqinter> z = y \<sqinter> z"   and "x \<squnion> z = y \<squnion> z" shows "x = y"
all_squarefree: fixes q :: "rat poly" assumes "q \<in> set qs" shows "rsquarefree q"
minus_max_eq_min: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus (ord.max less_eq x y) = ord.min less_eq (uminus x) (uminus y)"
simps_1: fixes uu :: "'a"   and B :: "'a"   and uv :: "'b" shows "J_C.Cod (J_C.MkArr uu B uv) = B"
four_points_aligned_1: fixes t :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and s :: "real \<times> real" assumes "coll t p q"   and "coll t q r"   and "distinct5 t s p q r" shows "coll t r p"
prems: shows "F \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Arr\<rparr>"
basis: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
domIff: fixes a :: "'g"   and m :: "'g \<Rightarrow> 'h option" shows "(a \<in> dom m) = (m a \<noteq> None)"
permutation_inverse_works_2: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "p \<circ> inv' p = id"
state_var2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and new_state_var :: "sparc_state_var"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = s1\<lparr>state_var := new_state_var\<rparr> \<and> t2 = s2\<lparr>state_var := new_state_var\<rparr>" shows "low_equal t1 t2"
ex_zero: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "\<zero>\<^bsub>A\<^esub> \<in> carrier A"
sorted_poly_rel_vars_llist: fixes r :: "(char list list \<times> int) list"   and r' :: "int mpoly" assumes "(r, r') \<in> sorted_poly_rel O mset_poly_rel" shows "vars r' \<subseteq> \<phi> ` vars_llist r"
meta_allE: fixes P :: "'b \<Rightarrow> prop"   and x :: "'b" assumes "\<And>x. PROP P x" obtains "PROP P x"
assms_1: shows "length xs = card {a \<in> A. a < length xs + length ys}"
nv2L_Fax: fixes c' :: "(('fsym, 'a) efsym, ('psym, 'tp) epsym) lit list"   and l' :: "(('fsym, 'a) efsym, ('psym, 'tp) epsym) lit" assumes "c' \<in> Fax"   and "l' \<in> set c'" shows "nv2L l' = {}"
has_disc_negated_has_disc: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b match_expr" assumes "has_disc_negated disc neg m" shows "has_disc disc m"
cond_cons_def: fixes P :: "'a list \<Rightarrow> 'b \<Rightarrow> bool" shows "cond_cons P = (\<lambda>(ys, s). case ys of [] \<Rightarrow> True | a # list \<Rightarrow> P (a # list) s)"
ctr_transfer_1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_option R None None"
alw_cong: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and Q1 :: "'a stream \<Rightarrow> bool"   and Q2 :: "'a stream \<Rightarrow> bool" assumes "alw P \<omega>"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> = Q2 \<omega>" shows "alw Q1 \<omega> = alw Q2 \<omega>"
simulation1_\<tau>inf_step: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and tls1 :: "'h llist"   and s2 :: "'i" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "\<tau>trsys.\<tau>inf_step trsys1 \<tau>move1 s1 tls1"   and "bisim s1 s2" shows "\<exists>tls2. \<tau>trsys.\<tau>inf_step trsys2 \<tau>move2 s2 tls2 \<and> llist_all2 tlsim tls1 tls2"
res_e: shows "ord_resolve CAs DA AAs As \<sigma> E"
non_zero_neighbour: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z"   and "\<exists>\<^sub>F w in at z. f w \<noteq> 0" shows "\<forall>\<^sub>F w in at z. f w \<noteq> 0"
pi_it_to_it: fixes I :: "('d list \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'd list \<Rightarrow> 'd list" shows "proper_it (it_to_it I) (it_to_it I)"
inter_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(a \<in># A \<inter># B) = (a \<in># A \<and> a \<in># B)"
notunicastT: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>i\<^sub>p :: "'e \<Rightarrow> nat"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q), \<not>unicast (s\<^sub>i\<^sub>p \<xi>), \<xi>, q) \<in> seqp_sos \<Gamma>"
Inf_from_def: fixes N :: "('f \<times> 'l) set" shows "Inf_from N = {\<iota> \<in> Inf_FL. set (prems_of \<iota>) \<subseteq> N}"
that: fixes p' :: "('a \<times> 'a) list" assumes "path g u p' v"   and "simple p'"   and "set p' \<subseteq> set p" shows "thesis"
hyps_4: shows "\<Gamma> \<turnstile> s3 \<approx>\<^sub>L s2"
stake_cycle_eq: fixes u :: "'d list" assumes "u \<noteq> []" shows "stake (length u) (cycle u) = u"
IH: fixes n :: "int" shows "succs (ccomp c) n \<subseteq> {n..n + size (ccomp c)}"
f10: shows "\<ominus> (s \<otimes> fst x) \<in> carrier R"
assms_1: shows "\<forall>n w. prices Mkt x (Suc n) w = cls_val_process Mkt pf2 (Suc n) w"
simps_2: fixes b :: "'c" shows "local.plus_oracle (Inr b) = map_spmf (apfst Inr) (right s b)"
Collect_case_prod_mono: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "A \<le> B" shows "{(x, y). A x y} \<subseteq> {(x, y). B x y}"
cyclic_on_alldef: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set" shows "cyclic_on f S = (S \<noteq> {} \<and> (\<forall>s\<in>S. S = orbit f s))"
q: shows "q = sum_list (map2 (\<cdot>) (list_of_vec v) ps)"
simps_2: fixes A :: "'a mat"   and As :: "'a mat list" shows "diag_block_mat (A # As) = (let B = diag_block_mat As in four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B)"
derived_img: fixes K :: "'a set" assumes "K \<subseteq> carrier G" shows "derived H (h ` K) = h ` derived G K"
ID_cEnum: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "ID cEnum = Some (enum, enum_all, enum_ex)" shows "UNIV = set enum \<and> enum_all = Ball UNIV \<and> enum_ex = Bex UNIV"
h3: shows "\<forall>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<longrightarrow> list_constr (snd elem1 @ map ((+) (length qs1)) (snd elem2)) (length (qs1 @ qs2))"
monadic_nfoldli_def: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool nres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "monadic_nfoldli l c f s \<equiv> REC\<^sub>T (\<lambda>D (l, s). case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> c s \<bind> (\<lambda>b. if b then f x s \<bind> (\<lambda>s'. D (ls, s')) else RETURN s)) (l, s)"
wt_elim_cases_3: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and i :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In2 (e.[i])\<Colon>T" obtains Ta :: "ty" where "T = Inl Ta" and "E,dt\<Turnstile>e\<Colon>-Ta.[]" and "E,dt\<Turnstile>i\<Colon>-PrimT Integer"
semInt_prWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom" assumes "wlsSEM SEM" shows "prWls (semInt SEM) SEM"
A93_NotSource_level1: shows "isNotDSource level1 sA93"
chain_length_pos: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R xs" shows "0 < llength xs"
elims: fixes x :: "int \<Rightarrow> int \<Rightarrow> bool"   and xa :: "value option"   and xb :: "value option"   and y :: "trilean" assumes "MaybeBoolInt x xa xb = y" obtains f :: "int \<Rightarrow> int \<Rightarrow> bool" and a :: "int" and b :: "int" where "x = f" and "xa = Some (Num a)" and "xb = Some (Num b)" and "y = (if f a b then true else false)"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and uw_ :: "value option" where "x = uu___" and "xa = None" and "xb = uw___" and "y = invalid"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and va :: "String.literal" and uw_ :: "value option" where "x = uua___" and "xa = Some (Str va__)" and "xb = uw_" and "y = invalid"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and uv_ :: "value option" where "x = uub___" and "xa = uv___" and "xb = None" and "y = invalid"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and uv_ :: "value option" and va :: "String.literal" where "x = uu_" and "xa = uv_" and "xb = Some (Str va)" and "y = invalid"
length_greater_imp_not_empty: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "xs \<noteq> []"
lrr: shows "length rr = card {x. poly (real_of_int_poly p) x = 0}"
spp_rel_def: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and r :: "'t \<Rightarrow>\<^sub>0 'b" shows "spp_rel sp r = (r \<noteq> 0 \<and> r \<in> dgrad_sig_set' (length fs) dgrad \<and> lt r = fst sp \<and> rep_list r = snd sp)"
distinct_1: fixes x11 :: "'c \<times> 'd \<times> bool"   and x12 :: "'c option"   and x2 :: "'c" shows "Insert x11 x12 \<noteq> Delete x2"
assms_2: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
bounded_cball: fixes x :: "'b"   and e :: "real" shows "bounded (cball x e)"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "cod (R f) = R (cod f)"
simps_2: fixes uu :: "color"   and l :: "('a, 'c) rbt"   and x :: "'a"   and y :: "'c"   and r :: "('a, 'c) rbt"   and k :: "'a" shows "rbt_lookup (Branch uu l x y r) k = (if k < x then rbt_lookup l k else if x < k then rbt_lookup r k else Some y)"
find_zeros_from_vec_prop: fixes input_vec :: "rat vec" shows "\<forall>n<dim_vec input_vec. (input_vec $ n \<noteq> 0) = List.member (find_nonzeros_from_input_vec input_vec) n"
0: shows "is_subseq_of s (filtered_seq s (kth_res_equals k l))"
GM_def: shows "GM \<equiv> \<lparr>edge_rev = rev_G, edge_succ = perm_swap uw uv (perm_swap vw (rev_G uv) (fold perm_rem [rev_H uw, rev_H vw] (edge_succ HM)))\<rparr>"
fds_deriv_eq_imp_eq: fixes f :: "'a fds"   and g :: "'a fds" assumes "fds_deriv f = fds_deriv g"   and "fds_nth f (Suc 0) = fds_nth g (Suc 0)" shows "f = g"
PDG_BS_def: fixes S :: "'node set" shows "PDG_BS S \<equiv> {n'. \<exists>n. PDG.PDG_path sourcenode targetnode valid_edge Def Use standard_control_dependence n' n \<and> n \<in> S \<and> valid_node n}"
measurable_envelopeI_countable_cover: fixes A :: "'b set"   and C :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "A \<subseteq> \<Union> (range C)"   and "\<And>n. C n \<in> sets M"   and "\<And>n. emeasure M (C n) < \<infinity>" shows "\<exists>E\<subseteq>\<Union> (range C). measurable_envelope M A E"
hyps_2: shows "\<guillemotleft>\<guillemotleft>s\<guillemotright>\<guillemotright> = \<guillemotleft>s\<guillemotright>"
emb_Ball_lset_antimono: fixes Xs :: "'c set llist"   and Ys :: "'c set llist"   and x :: "'c" assumes "emb Xs Ys"   and "\<forall>Y\<in>lset Ys. x \<in> Y" shows "\<forall>X\<in>lset Xs. x \<in> X"
path_A: shows "path A (w || map fst tr) qA \<and> length w = length (map fst tr)"
resumption_ord_ABORT2: fixes r :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord r ABORT = (r = ABORT)"
C: shows "F (lim_proc (range (\<lambda>i. Y i [+] S))) \<subseteq> F (lim_proc (range Y) [+] S)"
abs_eq: fixes xa :: "'a \<Rightarrow> bool"   and x :: "'a \<Rightarrow> 'b" shows "plambda xa x = Abs_pfun (\<lambda>xb. if xa xb then Some (x xb) else None)"
multrow_mat: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and k :: "nat"   and a :: "'b" assumes "A \<in> carrier_mat n nc" shows "multrow k a A = multrow_mat n k a * A"
static_quantities_def: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "static_quantities p = (\<forall>asset\<in>support_set p. \<exists>c. p asset = (\<lambda>n w. c))"
takeWhile_eq_all_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(takeWhile P xs = xs) = (\<forall>x\<in>set xs. P x)"
poly_rel_one: shows "poly_rel (one_poly_i ff_ops) 1"
enum_def: fixes n :: "nat" shows "ListAux.enum n = {..<n}"
UP_ring_subring_mult: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "f \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "g \<otimes>\<^bsub>UP R\<^esub> f = g \<otimes>\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> f"
Abs: fixes t :: "dB" assumes "NF t" shows "NF (dB.Abs t)"
Red_F_model: fixes M :: "'f set"   and N :: "'f set" assumes "M \<Turnstile> N - Red_F N" shows "M \<Turnstile> N"
net_nhop_quality_increases: fixes n :: "net_tree" assumes "wf_net_tree n" shows "closed (pnet (\<lambda>i. paodv i \<langle>\<langle> qmsg) n) \<TTurnstile> netglobal (\<lambda>\<sigma>. \<forall>i dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip))"
6: shows "E ((\<box>a) \<^bold>\<cdot> a)"
collinear_subset: fixes T :: "'a set"   and S :: "'a set" assumes "collinear T"   and "S \<subseteq> T" shows "collinear S"
splitAt_is_sublist1R: fixes ram :: "'a"   and vs :: "'a list" assumes "ram \<in> set vs" shows "is_sublist (fst (splitAt ram vs) @ [ram]) vs"
deducts_eq_if_analyzed: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.analyzed arity public Ana M" shows "intruder_model.intruder_deduct arity public Ana M t = intruder_model.intruder_synth arity public M t"
simps_11: fixes F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "strip_guards F (Catch c\<^sub>1 c\<^sub>2) = Catch (strip_guards F c\<^sub>1) (strip_guards F c\<^sub>2)"
gcd_dvdI2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times b c" shows "dvd.dvd times (gcd a b) c"
exhaust: fixes y :: "'a conditional_action" obtains x1 :: "'a" where "y = Join x1"   | "y = Yield"
polyn_expr_tOp: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (n, f)"   and "pol_coeff S (m, g)" shows "\<exists>e. pol_coeff S (n + m, e) \<and> e (n + m) = f n \<cdot>\<^sub>r\<^bsub>S\<^esub> g m \<and> polyn_expr R X (n + m) (n + m, e) = polyn_expr R X n (n, f) \<cdot>\<^sub>r polyn_expr R X m (m, g)"
unit_vectors_is_genset: fixes n :: "nat" shows "LinearCombinations.module.span cpx_rng (module_cpx_vec n) (unit_vectors n) = carrier (module_cpx_vec n)"
is_tangle_diagram_weak_matrix_match2: fixes w2 :: "brick list"   and ws2 :: "wall"   and w1 :: "brick list" assumes "is_tangle_diagram (w2 * ws2)"   and "codomain_block w1 = 0" shows "weak_matrix_match2 (blockmat w1) (blockmat w2) (kauff_mat ws2)"
fail_env_def: fixes fail :: "'d" shows "fail_env fail = EnvT (\<lambda>r. fail)"
nprv_mono: fixes G :: "'fmla set"   and \<phi> :: "'fmla"   and F :: "'fmla set" assumes "nprv G \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "G \<subseteq> F"   and "\<phi> \<in> fmla" shows "nprv F \<phi>"
simps: fixes a1 :: "'a literal multiset set"   and a2 :: "'a literal multiset set" shows "(a1 \<triangleright> a2) = (\<exists>N M. a1 = M \<and> a2 = N \<and> N - M \<subseteq> concls_of (inferences_from M) \<and> M - N \<subseteq> Rf N)"
hyps_3: shows "map f w' \<in> map f w1 \<otimes>\<^bsub>\<alpha>\<^esub> map f w2'"
lincomb_list_carrier: fixes vs :: "rat vec list"   and c :: "nat \<Rightarrow> rat" assumes "set vs \<subseteq> Rn" shows "gs.lincomb_list c vs \<in> Rn"
fps_lr_inverse_gp_ring1_2: shows "fps_right_inverse (Abs_fps (\<lambda>n. 1::'b)) (1::'b) = Abs_fps (\<lambda>n. if n = 0 then 1::'b else if n = 1 then - (1::'b) else (0::'b))"
assms_1: shows "P = LCons v (LCons w P')"
abs_eq: fixes x :: "'a multiset \<times> 'a multiset" shows "mset_pos (abs_zmultiset x) = Abs_multiset (case x of (Mp, Mn) \<Rightarrow> count (Mp - Mn))"
Sigma_eq: shows "Sigma I \<FF>1 = Sigma I \<FF>2"
ccRestr_ccDelete: fixes x :: "Vars.var"   and G :: "CoCalls" shows "G G|` (- {x}) = cc_delete x G"
add_delete_edge: fixes G :: "'a graph"   and a :: "'a"   and b :: "'a" assumes "wf_graph G"   and "(a, b) \<in> edges G" shows "add_edge a b (delete_edge a b G) = G"
arr_iff_in_hom: fixes f :: "'a \<times> 'a" shows "CC.arr f = CC.in_hom f (CC.dom f) (CC.cod f)"
assms_1: shows "ccpo.admissible lSup lprefix P"
S_subset_reduced_imp_S_reduced: fixes T :: "'w set"   and ts :: "'w list" assumes "T \<in> Pow S"   and "reduced_word T ts" shows "S_reduced ts"
arr_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "arr f = (f \<noteq> Null \<and> Dom f \<in> {False, True} \<and> Cod f \<in> {False, True} \<and> Map f \<in> Collect (G.path (Dom f) (Cod f)))"
path_trans: fixes SES :: "('a, 'b) SES_rec"   and s1 :: "'a"   and l1 :: "'b list"   and s2 :: "'a"   and l2 :: "'b list"   and sn :: "'a" assumes "s1 l1\<Longrightarrow>\<^bsub>SES\<^esub> s2"   and "s2 l2\<Longrightarrow>\<^bsub>SES\<^esub> sn" shows "s1 (l1 @ l2)\<Longrightarrow>\<^bsub>SES\<^esub> sn"
Transset_def: fixes x :: "V" shows "Transset x \<equiv> \<forall>y\<in>elts x. y \<le> x"
discI_1: fixes llist :: "'c llist" assumes "llist = LNil" shows "lnull llist"