hcju/isabelleps huggingface dataset:
pdevs_val_msum_pdevs: fixes f :: "'b pdevs"   and n :: "nat"   and e :: "nat \<Rightarrow> real"   and g :: "'b pdevs" assumes "degree f \<le> n" shows "pdevs_val e (msum_pdevs n f g) = pdevs_val e f + pdevs_val (\<lambda>i. e (i + n)) g"
bind_mono: fixes B :: "('b \<Rightarrow> 'c option) \<Rightarrow> 'd option"   and C :: "'d \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'e option" assumes "monotone (flatf_ord None) option_ord B"   and "\<And>y. monotone (flatf_ord None) option_ord (C y)" shows "monotone (flatf_ord None) option_ord (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
L: assumes "B = add_mset s (B - {#s#})"   and "A = B - {#s#} + C" shows "P"
d_def: shows "d = Lcm (fpxs_root_order ` {f, g, h})"
abs_diff_triangle_ineq2: fixes a :: "'i"   and b :: "'i"   and c :: "'i" shows "\<bar>a - b\<bar> \<le> \<bar>a - c\<bar> + \<bar>c - b\<bar>"
bp_NE: fixes P :: "'a set set" assumes "bp P" shows "P \<noteq> {}"
times_singleton_eqpoll: fixes a :: "'a"   and A :: "'b set" shows "{a} \<times> A \<approx> A"
hom_0_iff: fixes x :: "int poly" shows "(Ring_Hom_Poly.of_int_poly x = 0) = (x = 0)"
assms_2: shows "AE x in M. proj (T x) = T2 (proj x)"
Key_unique_SesKey: fixes K :: "nat"   and SesKey :: "nat"   and B :: "agent"   and T :: "msg"   and Ticket :: "msg"   and evs :: "event list"   and K' :: "nat"   and B' :: "agent"   and T' :: "msg"   and Ticket' :: "msg" assumes "Crypt K \<lbrace>Key SesKey, Agent B, T, Ticket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt K' \<lbrace>Key SesKey, Agent B', T', Ticket'\<rbrace> \<in> parts (knows Spy evs)"   and "Key SesKey \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "K = K' \<and> B = B' \<and> T = T' \<and> Ticket = Ticket'"
errMOD_igAbsRenSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igVarIPresIGWls MOD"   and "igSubstIPresIGWls MOD"   and "igWlsDisj MOD"   and "igAbsRen MOD" shows "igAbsRenSTR (errMOD MOD)"
emeasure_restr_to_subalg: fixes M :: "'c measure"   and F :: "'c measure"   and A :: "'c set" assumes "subalgebra M F"   and "A \<in> sets F" shows "emeasure (restr_to_subalg M F) A = emeasure M A"
fps_mult_nth: fixes f :: "'b fps"   and g :: "'b fps"   and n :: "nat" shows "(f * g) $ n = (\<Sum>i = 0..n. f $ i * g $ (n - i))"
inject_7: fixes x8 :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and y8 :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "(DynCom x8 = DynCom y8) = (x8 = y8)"
ESem_Lam: fixes x :: "Vars.var"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> Lam [x]. e \<rbrakk>\<^bsub>\<rho>\<^esub> = (\<Lambda> x. x)\<cdot>(Fn\<cdot>(\<Lambda> v. \<lbrakk> e \<rbrakk>\<^bsub>\<rho>(x := v)\<^esub>))"
inverse_split: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier \<G>"   and "b \<in> carrier \<G>" shows "inv (a \<otimes> b) = inv a \<otimes> inv b"
homeomorphic_map_maps: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "homeomorphic_map X Y f = (\<exists>g. homeomorphic_maps X Y f g)"
listn_Cons_Suc: fixes l :: "'b"   and xs :: "'b list"   and n :: "nat"   and A :: "'b set" assumes "l # xs \<in> list n A" obtains n' :: "nat" where "n = Suc n'" and "l \<in> A" and "xs \<in> list n' A"
same_suffix_nil: fixes ys :: "'a list"   and xs :: "'a list" shows "suffix (ys @ xs) xs = (ys = [])"
that: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "homeomorphism (cbox a c) (cbox u w) f g"   and "f a = u"   and "f c = w"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = f1 x"   and "\<And>x. x \<in> cbox b c \<Longrightarrow> f x = f2 x" shows "thesis"
colossless_converterI: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter" assumes "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> colossless_gpv \<I>' (run_converter conv a)"   and "\<And>a b conv'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, conv') \<in> results_gpv \<I>' (run_converter conv a)\<rbrakk> \<Longrightarrow> colossless_converter \<I> \<I>' conv'" shows "colossless_converter \<I> \<I>' conv"
c3_2: fixes n :: "nat" assumes "1 \<le> n" shows "norm (h (real n)) \<le> c3 * norm (h (real (Suc n)))"
of_int_minus: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- z) = uminus (ring_1.of_int one plus zero minus z)"
Iod_Torder: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set" assumes "Torder D"   and "X \<subseteq> carrier D" shows "Torder (Iod D X)"
hol_cfc: shows "cnj \<circ> f \<circ> cnj holomorphic_on S \<inter> {z. Im z < 0}"
thetaZOParT_def: shows "thetaZOParT \<equiv> {(ParT cl, ParT dl) |cl dl. properL cl \<and> properL dl \<and> SbisL cl dl}"
Quotient3_prm: shows "Quotient3 (=p) abs_prm rep_prm"
lunit_char_2: fixes a :: "'a" assumes "ide a" shows "\<I> \<otimes> \<l>[a] = (\<iota> \<otimes> a) \<cdot> local.inv \<a>[\<I>, \<I>, a]"
OclSelf_at_pre_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e" shows "x@preH = OclSelf x H fst"
red_stat_is_stat_red: shows "reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
most_recent_write_recent: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and r :: "nat"   and w :: "nat"   and adal :: "'c \<times> addr_loc"   and w' :: "nat" assumes "P,E \<turnstile> r \<leadsto>mrw w"   and "adal \<in> action_loc P E r"   and "w' \<in> write_actions E"   and "adal \<in> action_loc P E w'" shows "E \<turnstile> w' \<le>a w \<or> E \<turnstile> r \<le>a w'"
bot_unique: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.order_bot bot less_eq less" shows "less_eq a bot = (a = bot)"
b: fixes i :: "'a" assumes "i \<in> A" shows "[b i = 1] (mod m i) \<and> [b i = 0] (mod prod m (A - {i}))"
eventually_elim2: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter"   and Q :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F"   and "\<And>i. \<lbrakk>P i; Q i\<rbrakk> \<Longrightarrow> R i" shows "eventually R F"
lhd_lSup_eq: fixes Y :: "'c llist set"   and xs :: "'c llist" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) Y"   and "xs \<in> Y"   and "\<not> lnull xs" shows "lhd (lSup Y) = lhd xs"
not_cong_3412: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Cong A B C D" shows "\<not> Cong C D A B"
sumx: shows "sum x {..p - Suc 0} = 1"
fn_norm_ge_zero: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real"   and f :: "'c \<Rightarrow> real" assumes "normed_vectorspace_with_fn_norm V norm"   and "Function_Norm.continuous V f norm" shows "0 \<le> \<Squnion>({0} \<union> {\<bar>f x\<bar> / norm x |x. x \<noteq> (0::'c) \<and> x \<in> V})"
not_in_support_flowD: fixes x :: "'a \<times> 'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "x \<notin> support_flow f" shows "f x = 0"
edges_def: fixes labeled_graph :: "('e, 'f) labeled_graph" shows "edges labeled_graph = (case labeled_graph of LG x1 x2 \<Rightarrow> x1)"
assms_2: fixes x :: "'a"   and \<pi> :: "nat \<Rightarrow> nat" assumes "x \<in> X"   and "\<pi> permutes A" shows "mpoly_map_vars \<pi> (f x) = f (g \<pi> x)"
bi_total_rel: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_total R1"   and "bi_total R2" shows "bi_total (rel_prod R1 R2)"
wordinterval_compress: fixes r :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_compress r) = wordinterval_to_set r"
subst_atm_list_def: fixes As :: "'a list"   and \<sigma> :: "'s" shows "As \<cdot>al \<sigma> = map (\<lambda>A. A \<cdot>a \<sigma>) As"
ts_def_1: shows "\<S>\<bullet>\<C> \<turnstile> vs : [] _> t1s"
simps_9: shows "push_forall (AllQ TrueF) = TrueF"
hyps_3: fixes disc_node :: "(_) node_ptr"   and disc_doc :: "(_) document_ptr" assumes "child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_node"   and "h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc"   and "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_doc |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_ancestors_di (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_doc)"
pac_idealI1: fixes p :: "int mpoly"   and A :: "int mpoly set" assumes "p \<in> A" shows "p \<in> pac_ideal A"
ub_imageD: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and u :: "'a"   and x :: "'b" assumes "class.po below"   and "po.is_ub below (f ` S) u"   and "x \<in> S" shows "below (f x) u"
gR_galois_imp22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>R y"   and "z \<preceq>\<^sub>R rquot y x" shows "D x z"
n_kat_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y = !x \<cdot> y \<cdot> !z) = (!x \<cdot> y \<le> y \<cdot> !z)"
deep: shows "3 \<le> length [10, 10, 10]"
\<b>_decreasing: fixes i :: "nat"   and j :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "i \<le> j" shows "\<b> ps j \<le> \<b> ps i"
assms_2: fixes j :: "'b" assumes "j \<in> J" shows "ideal (\<aa> j) R (+) (\<cdot>) \<zero> \<one>"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and a :: "'a" assumes "SATp \<Gamma> f a" obtains x :: "'a" where "a = x__" and "x__ \<in> A \<Gamma>"   | x :: "'a" where "a = x" and "weight \<Gamma> x \<le> d_IN f x"
analz_image_priEK: fixes evs :: "event list"   and C :: "agent"   and KK :: "nat set" assumes "evs \<in> set_pur" shows "(Key (priEK C) \<in> analz (Key ` KK \<union> knows Spy evs)) = (priEK C \<in> KK \<or> C \<in> bad)"
mult_strict_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less c d"   and "less zero b"   and "less_eq zero c" shows "less (times a c) (times b d)"
generate_pow: fixes a :: "'a" assumes "a \<in> carrier G" shows "generate G {a} = {a [^] k |k. k \<in> UNIV}"
snth_smap: fixes f :: "'b \<Rightarrow> 'a"   and s :: "'b stream"   and n :: "nat" shows "smap f s !! n = f (s !! n)"
Collect_permI: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'a \<Rightarrow> bool"   and pi :: "('b \<times> 'b) list" assumes "\<forall>x. P1 x = P2 x" shows "{pi \<bullet> x |x. P1 x} = {pi \<bullet> x |x. P2 x}"
solve_exec_ns_def: fixes s :: "('i \<times> 'a ns_constraint) list" shows "solve_exec_ns s \<equiv> case preprocess s of (t, as, trans_v, []) \<Rightarrow> case assert_all t as of Inl I \<Rightarrow> Inl I | Inr v \<Rightarrow> Inr (trans_v v) | (t, as, trans_v, i # x) \<Rightarrow> Inl [i]"
sup_least: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and z :: "'a" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L"   and "z \<in> carrier L"   and "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> z" shows "\<Squnion>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> z"
user_accessible_mod_user_reg: fixes new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and s :: "'b sparc_state"   and pa :: "word36" shows "user_accessible (s\<lparr>user_reg := new_user_reg\<rparr>) pa = user_accessible s pa"
ur: shows "\<exists>!x. root_cond (q, l, r) x"
set_mult_sub: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R" shows "set_mult R A B \<subseteq> carrier R"
argmin_takes_last: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "argmin f xs = xs ! Max {i. i < length xs \<and> (\<forall>x\<in>set xs. f (xs ! i) \<le> f x)}"
zero_lens_indep: fixes X :: "'e \<Longrightarrow> 'd" shows "0\<^sub>L \<bowtie> X"
less_list_empty: fixes xs :: "'b list" shows "([] < xs) = (xs \<noteq> [])"
reciprocal_zero: shows "reciprocal 0\<^sub>h = \<infinity>\<^sub>h"
se_star_sat_imp_sat: fixes c :: "('d, 'e) conf"   and ls :: "('d, 'e) label list"   and c' :: "('d, 'e) conf" assumes "se_star c ls c'"   and "Conf.sat c'" shows "Conf.sat c"
add_eq_conv_ex: fixes a :: "'c"   and M :: "'c multiset"   and b :: "'c"   and N :: "'c multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
fun_dual7: fixes f :: "'b set \<Rightarrow> 'c"   and g :: "'b set \<Rightarrow> 'c" shows "(f = \<partial> \<circ> g \<circ> (`) \<partial>) = (\<partial> \<circ> f \<circ> (`) \<partial> = g)"
simps_2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'g, 'i) new_thread_action"   and tas :: "('f, 'g, 'i) new_thread_action list" shows "redT_updTs ts (ta # tas) = redT_updTs (redT_updT ts ta) tas"
cases: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "('c, 'd) rose_tree\<^sub>m"   and a2 :: "('c, 'd) rose_tree\<^sub>m" assumes "blinding_of_tree h bo a1 a2" obtains t1 :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m" and t2 :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m" where "a1 = Tree\<^sub>m t1" and "a2 = Tree\<^sub>m t2" and "blinding_of_blindable (map_prod h (map (hash_tree h))) (rel_prod bo (list_all2 (blinding_of_tree h bo))) t1 t2"
\<Delta>_pos: shows "E = {} \<or> 0 < \<Delta>"
jumpNestingOkS_mono: fixes jmps' :: "jump set"   and c :: "stmt"   and jmps :: "jump set" assumes "jumpNestingOkS jmps' c"   and "jmps' \<subseteq> jmps" shows "jumpNestingOkS jmps c"
fund_sol_minimal': fixes z :: "nat \<times> nat" assumes "nontriv_solution z" shows "pell_valuation (case fund_sol of (x, y) \<Rightarrow> (int x, int y)) \<le> pell_valuation (case z of (x, y) \<Rightarrow> (int x, int y))"
fim: shows "f ` (cball a r \<inter> T) = cball a r \<inter> T"
"3_3": fixes x :: "'d" assumes "comp kx ky = Gt"   and "x = f ky (0::'b) vy"   and "oalist_inv_raw ((kx, vx) # xs)"   and "oalist_inv_raw ys"   and "map2_val_compat g"   and "map2_val_compat h" shows "fst ` set (map2_val_pair f g h ((kx, vx) # xs) ys) \<subseteq> fst ` set ((kx, vx) # xs) \<union> fst ` set ys"
eliminate_entries_gen_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and ad :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ad' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mul' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B' :: "'c mat"   and vs :: "nat \<Rightarrow> 'b"   and vs' :: "nat \<Rightarrow> 'c"   and i :: "nat"   and B :: "'b mat"   and j :: "nat" assumes "(R ===> R ===> R) ad ad'"   and "(R ===> R ===> R) mul mul'"   and "\<And>j. j < dim_row B' \<Longrightarrow> R (vs j) (vs' j)"   and "i < dim_row B'"   and "mat_rel R B B'" shows "mat_rel R (eliminate_entries_gen ad mul vs B i j) (eliminate_entries_gen ad' mul' vs' B' i j)"
Hermite_of_Least: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" assumes "\<not> is_zero_row i (Hermite_of A ass res bezout)"   and "ass_function ass"   and "res_function res"   and "is_bezout_ext bezout" shows "(LEAST n. Hermite_of A ass res bezout $ i $ n \<noteq> (0::'b)) = (LEAST n. echelon_form_of A bezout $ i $ n \<noteq> (0::'b))"
dfree_is_dsafe: fixes \<theta> :: "('d, 'e) trm" assumes "dfree \<theta>" shows "dsafe \<theta>"
proots_count_reciprocal: fixes P :: "'b poly"   and p :: "nat"   and A :: "'b set" assumes "degree P \<le> p"   and "P \<noteq> 0"   and "(0::'b) \<notin> A" shows "proots_count (reciprocal_poly p P) A = proots_count P {x. inverse x \<in> A}"
rel_fun_def: fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
mtx_invertible_def: fixes A :: "'a sq_mtx" shows "mtx_invertible A = (\<exists>A'. A' * A = 1 \<and> A * A' = 1)"
Redirect_Edge: shows "\<turnstile> Redirect_Edge  pre Color_Target"
fbrelationE: fixes r :: "V" assumes "fbrelation r" obtains "fpairs r = r"
m_defs_2: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis" shows "PL_Indis.mC_ZOC aval tval cval indis theta c d s t I0 P F \<equiv> PL_Indis.mC_ZOC_part c d s t I0 P F \<and> inj_on F P \<and> PL_Indis.mC_ZOC_wt aval tval cval c d s t I0 P F \<and> PL_Indis.mC_ZOC_eff_cont0 aval tval cval indis theta c d s t I0 F \<and> PL_Indis.mC_ZOC_eff_cont aval tval cval indis theta c d s t I0 P F"
IH: fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "distinct vs"   and "set vs \<inter> V' = {}"   and "integrable (state_measure (set vs) \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge (set vs) V' (\<sigma>, \<rho>)) e))"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> set vs \<union> V'" shows "extract_real (cexpr_sem \<rho> (integrate_vars \<Gamma> vs e)) = LINT \<sigma>|state_measure (set vs) \<Gamma>. extract_real (cexpr_sem (merge (set vs) V' (\<sigma>, \<rho>)) e)"
llist_le_trans: fixes r :: "'a llist"   and s :: "'a llist"   and t :: "'a llist" assumes "r \<le> s"   and "s \<le> t" shows "r \<le> t"
lastListInList: fixes list :: "'b list" assumes "list \<noteq> []" shows "last list \<in> set list"
bounded_by_mono: fixes a :: "real"   and P :: "'a \<Rightarrow> real"   and b :: "real" assumes "bounded_by a P"   and "a \<le> b" shows "bounded_by b P"
substEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "freshEnv zs z rho"   and "fresh zs z Y"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "freshEnv zs z (rho &[Y / y]_ys)"
exhaust: fixes y :: "'b option" obtains "y = None"   | x2 :: "'b" where "y = \<lfloor>x2\<rfloor>"
bisim: shows "strong_reduction_simulation Rel Cal \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Cal Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Cal P' \<and> (P', Q') \<in> Rel))"
abs_ish_def: fixes pats :: "Term_Class.term list"   and rhs :: "'a" shows "abs_ish pats rhs = (pats \<noteq> [] \<or> is_abs rhs)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' \<phi> \<psi>'"
face_distance_def: fixes F :: "'b set"   and D :: "'b set" shows "codomain.face_distance F D \<equiv> codomain.chamber_distance (codomain.closest_supchamber F D) D"
bordered_def: fixes w :: "'b list" shows "bordered w = (\<exists>b. b \<le>b w)"
appGetField: fixes F :: "char list"   and C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Getfield F C, P, pc, mxs, T\<^sub>r, s) = (\<exists>oT vT ST LT. s = (oT # ST, LT) \<and> P \<turnstile> C sees F,NonStatic:vT in C \<and> subtype P oT (Class C))"
E: assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c R,B" shows "S"
standard_identity_rules_4: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and l :: "'b Standard_Constant" assumes "standard' C G" shows "maintained (congruence_rule S_Idt l) G"
closed_except_self: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.closed_except frees t (frees t)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "category C"   and "span_in_category C (Dom \<mu>)"   and "span_in_category C (Cod \<mu>)"   and "arrow_of_spans_axioms C \<mu>" shows "arrow_of_spans C \<mu>"
winding_number_subpath_continuous: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "continuous_on {0..1} (\<lambda>x. winding_number (subpath 0 x \<gamma>) z)"
select_convs_5: fixes Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e" shows "Out \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = Out"
iwf_hyps_exist: fixes lc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and it :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form"   and "is" :: "nat list"   and i :: "nat"   and s :: "'subst"   and ants' :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" assumes "iwf lc it ent"   and "is \<in> it_paths it"   and "tree_at it is = HNode i s ants'"   and "fst ent |\<subseteq>| ass_forms" shows "subst s (freshen i anyP) \<in> hyps_along it is"
intros_7: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
negative_lx: assumes "- float_round_down prec (lb_pi prec) \<le> float_plus_down prec lx (- floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) * 2 * (if floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) < 0 then float_round_down prec (lb_pi prec) else float_round_up prec (ub_pi prec)))"   and "x - real_of_int k * (2 * pi) \<le> 0" shows "real_of_float (lb_cos prec (- float_plus_down prec lx (- floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) * 2 * (if floor_fl (float_divr prec (lx + float_round_down prec (lb_pi prec)) (2 * float_round_down prec (lb_pi prec))) < 0 then float_round_down prec (lb_pi prec) else float_round_up prec (ub_pi prec))))) \<le> cos x"