hcju/isabelleps huggingface dataset:
rel_Grp: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "rel_pmf (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_pmf x \<subseteq> A} (map_pmf f)"
poly_eqI2: fixes p :: "'b poly"   and q :: "'b poly" assumes "degree p = degree q"   and "\<And>i. i \<le> degree p \<Longrightarrow> coeff p i = coeff q i" shows "p = q"
length_map: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "length (map f xs) = length xs"
word_rev_rev: fixes w :: "'b word" shows "word_reverse (word_reverse w) = w"
inv_stereographic_circline_def: shows "inv_stereographic_circline \<equiv> map_fun rep_circline abs_plane inv_stereographic_circline_clmat_pv"
g_2: shows "\<forall>x\<in>B. g x = f x"
block_size_t: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and index :: "int" assumes "t_design point_set block_collection u_block_size grouping index" shows "grouping \<le> u_block_size"
coeff_pow_X: fixes i :: "nat"   and n :: "nat" shows "coeff (X ^ i) n = (if i = n then 1::'b else (0::'b))"
that: assumes "Mapping.lookup m\<^sub>1 (snd b) = Some {n \<in> Mapping.keys (phis g). snd b \<in> set (the (Mapping.lookup (phis g) n))}" shows "thesis"
traces_snd: shows "traces (ts_process {[], [Some b]}) = {[], [Some b]}"
algebraic_def: fixes x :: "'a" shows "algebraic x = (\<exists>p. (\<forall>i. poly.coeff p i \<in> \<int>) \<and> p \<noteq> 0 \<and> poly p x = (0::'a))"
return_slice2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n'' :: "'b SDG_node"   and p :: "'f"   and n' :: "'b SDG_node"   and n :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None (Some (p, False)) False n'"   and "n' \<in> SDG.sum_SDG_slice2 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n" shows "n'' \<in> SDG.sum_SDG_slice2 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n"
elims: fixes x :: "'a arctic_delta"   and y :: "char list \<Rightarrow> char list" assumes "local.shows_arctic_delta x = y" obtains i :: "'a" where "x = Num_arc_delta i" and "y = shows i"   | "x = MinInfty_delta" and "y = shows ''-inf''"
adm: fixes f :: "'f"   and g :: "'f"   and n :: "nat" assumes "w (f, 1) = 0" shows "pr_weak (f, 1) (g, n)"
uminus_mult_mat_vec: fixes v :: "'a vec"   and A :: "'a mat" assumes "dim_vec v = dim_col A" shows "- A *\<^sub>v v = - (A *\<^sub>v v)"
T_merge_bound: fixes ts\<^sub>1 :: "'b tree list"   and ts\<^sub>2 :: "'b tree list" assumes "invar ts\<^sub>1"   and "invar ts\<^sub>2" shows "real (T_merge ts\<^sub>1 ts\<^sub>2) \<le> 4 * log 2 (real (size (mset_heap ts\<^sub>1)) + real (size (mset_heap ts\<^sub>2)) + 1) + 1"
irbb_def: shows "bb\<^sup>-\<^sub>\<R> = \<R> \<circ> bb\<^sup>-\<^sub>\<F>"
ladder_i_cut: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "index < length L" shows "ladder_i (ladder_cut L n) index = ladder_i L index"
sup_exists: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "\<exists>s. is_lub L s A"
sys_reg_mod_privilege: fixes s' :: "'b sparc_state"   and w :: "word32"   and r :: "sys_reg"   and s :: "'b sparc_state" assumes "s' = sys_reg_mod w r s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
undeft_equiv: fixes \<theta> :: "trm option" shows "(\<theta> \<noteq> undeft) = (\<exists>t. \<theta> = Aterm t)"
simps_1: fixes y :: "'a" shows "flat_orA (allow (allow y)) = allow y"
fmdom_fmmap_keys: fixes f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fmdom (fmmap_keys f m) = fmdom m"
normf: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> topspace (Euclidean_space n)" shows "(norm (f x))\<^sup>2 = (\<Sum>i<n. (x i)\<^sup>2)"
AD_aux: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "AllowPortFromTo a b po \<in> set p"   and "DenyAllFromTo c d \<in> set p"   and "allNetsDistinct p"   and "singleCombinators p"   and "a \<noteq> c \<or> b \<noteq> d" shows "dom (Cp (AllowPortFromTo a b po)) \<inter> dom (Cp (DenyAllFromTo c d)) = {}"
associated_sym: fixes a :: "'a list"   and b :: "'a list" assumes "a \<sim>\<^bsub>K [X]\<^esub> b" shows "b \<sim>\<^bsub>K [X]\<^esub> a"
fnorm_correct_1: fixes xs :: "'a list"   and e :: "fexpr" assumes "in_carrier xs"   and "nonzero xs (Cond e)" shows "feval xs e = peval xs (Num e) \<oslash> peval xs (Denom e)"
e''_def: shows "e'' = (SUP t\<in>ball z d. cmod t + (cmod t)\<^sup>2)"
consistent_splitLeaf: fixes t :: "'a tree"   and b :: "'a"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat" assumes "consistent t"   and "b \<notin> alphabet t" shows "consistent (splitLeaf t w\<^sub>a a w\<^sub>b b)"
disc_1: fixes x1 :: "'g" shows "is_TNil (TNil x1)"
stl_konigDtree: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> bool \<times> 'g) set"   and Ord :: "('h \<times> 'h) set"   and \<sigma> :: "'g \<times> 'f \<Rightarrow> 'h"   and s' :: "'e"   and t :: "('e \<times> 'd) tree"   and pred :: "'e \<times> 'i \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'i \<Rightarrow> bool"   and a :: "'i"   and a'' :: "'i"   and inv :: "'e \<times> 'i \<Rightarrow> bool" assumes "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>"   and "s' |\<in>| (fst \<circ> tree.root) |`| cont t"   and "pred (fst (tree.root t), a) (snd (tree.root t)) (s', a'')"   and "inv (s', a'')" shows "\<exists>t' a'. t' |\<in>| cont t \<and> pred (fst (tree.root t), a) (snd (tree.root t)) (fst (tree.root t'), a') \<and> inv (fst (tree.root t'), a') \<and> stl (Infinite_Soundness.konigDtree inv pred t a) = Infinite_Soundness.konigDtree inv pred t' a'"
sm: fixes x :: "'a" assumes "x \<noteq> (0::'a)"   and "norm x < s" shows "(\<lambda>n. a n * x ^ n) sums f x"
is_red_adds_iff: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "punit.is_red F p = (\<exists>f\<in>F. \<exists>v\<in>keys p. f \<noteq> 0 \<and> local.punit.adds_term (lpp f) v)"
Hilbert_fun_cone_aux: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and z :: "nat" assumes "h \<in> P[X]"   and "h \<noteq> 0"   and "U \<subseteq> X"   and "homogeneous h" shows "Hilbert_fun (cone (h, U)) z = card {t \<in> .[U]. deg_pm t + poly_deg h = z}"
assms: fixes i :: "nat" assumes "\<not> fresh_floatarith f i" shows "xs ! i = ys ! i"
wf_typeE2_1: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"
prefix_lemma: fixes x :: "'b llist"   and A :: "'b set"   and y :: "'b llist" assumes "x \<in> A\<^sup>\<omega>"   and "y \<in> A\<^sup>\<omega>"   and "\<And>s. \<lbrakk>s \<in> A\<^sup>\<star>; s \<le> x\<rbrakk> \<Longrightarrow> s \<le> y" shows "x = y"
inverse_less_iff_less: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less zero b" shows "less (inverse a) (inverse b) = less b a"
c: shows "\<Gamma>1 \<subseteq> \<Gamma>2"
Gamma_seriesI: fixes g :: "nat \<Rightarrow> 'a"   and z :: "'a" assumes "(\<lambda>n. g n / Gamma_series z n) \<longlonglongrightarrow> (1::'a)" shows "g \<longlonglongrightarrow> Gamma z"
intro: fixes m :: "nat" assumes "0 < m" shows "HBT m"
fmdrop_set_insert: fixes x :: "'c"   and S :: "'c set"   and m :: "('c, 'd) fmap" shows "fmdrop_set (insert x S) m = fmdrop x (fmdrop_set S m)"
strict_coboundedI1: fixes c :: "'a Word.word"   and a :: "'a Word.word"   and b :: "'a Word.word" assumes "c <s a" shows "c <s signed.max a b"
simps_2: fixes p :: "'a \<times> 'b"   and ps :: "('a \<times> 'b) list" shows "map_of (p # ps) = map_of ps(fst p \<mapsto> snd p)"
distinct_9: shows "a\<^sub>2 \<noteq> a\<^sub>4"
j_xs: shows "distinct (j # xs) \<and> len m j j xs < \<one> \<and> j \<in> set (a # as) \<and> set xs \<subseteq> set as"
pp_of_term_of_pair: fixes t :: "'a"   and k :: "'b" shows "pprod.pp_of_term (t, k) = t"
srev_strict: shows "srev\<cdot>\<bottom> = \<bottom>"
deriv_c2: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>2 c\<^sub>2 Q,A"
inv: fixes h :: "'a"   and H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "h \<in> H" shows "inv\<^bsub>G\<^esub> h \<in> generate G H"
not_Some_eq2: fixes v :: "('b \<times> 'c) option" shows "(\<forall>x y. v \<noteq> Some (x, y)) = (v = None)"
simps_4: shows "OUTfromV data4 = {stA2}"
propa: shows "propa f (step (hd w) (ss ! hd w)) ss (tl w) = (ssa, wa)"
cong_trans_poly: fixes a :: "'c poly"   and b :: "'c poly"   and m :: "'c poly"   and c :: "'c poly" assumes "[a = b] (mod m)"   and "[b = c] (mod m)" shows "[a = c] (mod m)"
fst_swap: fixes x :: "'f \<times> 'e" shows "fst (prod.swap x) = snd x"
find_sig_reducer_SomeD_aux_1: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and u :: "'t"   and t :: "'a"   and i :: "nat"   and j :: "nat" assumes "find_sig_reducer bs u t i = Some j" shows "i \<le> j"
reach_gunodes: fixes i :: "nat"   and A :: "('d, 'c) nba"   and w :: "'d stream" shows "{i} \<times> reach A w i \<subseteq> gunodes A w"
xPy: assumes "x \<^bsub>r\<^esub>\<prec> y" shows "thesis"
g1D: fixes x :: "real" assumes "x \<in> {0..1} - insert 1 ((*) 2 ` S)" shows "g1 differentiable at x"
le1221: fixes A :: "'p"   and B :: "'p" shows "A B Le B A"
obt1: shows "p = w1 @ [(u1, v1)] @ w2"
Quotient3_rep_reflp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and a :: "'c" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep a)"
NatTransComp_defs_2: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "\<eta>1 \<bullet>1 \<eta>2 = \<lparr>NTDom = NTDom \<eta>1, NTCod = NTCod \<eta>2, NatTransMap = \<lambda>X. \<eta>1 $$ X ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 $$ X)\<rparr>"
linear_order_strict_order: fixes x :: "'a" assumes "((top \<le> x \<and> x \<sqinter> x \<le> x) \<and> x \<sqinter> id x \<le> top) \<and> x \<squnion> id x = top" shows "(x \<sqinter> - top \<le> - top \<and> x \<sqinter> - top \<sqinter> (x \<sqinter> - top) \<le> x \<sqinter> - top) \<and> x \<sqinter> - top \<squnion> id (x \<sqinter> - top) = - top"
phiDefs_Entry_empty: fixes g :: "'g" shows "phiDefs g (Entry g) = {}"
replace_comp_val_process: fixes Mkt :: "('c, 'd) discrete_market"   and x :: "'d"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>n w. prices Mkt x n w = val_process Mkt pf2 n w"   and "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. val_process Mkt (qty_replace_comp pf1 x pf2) n w = val_process Mkt pf1 n w"
ngt: shows "normf f \<le> real (n - 1) * e"
\<Delta>_naturality: shows "(\<mu> \<star> tab\<^sub>0 (local.dom \<mu>)) \<cdot> \<mu>.r.tab = (cod \<mu> \<star> \<mu>.the_\<theta>) \<cdot> \<a>[cod \<mu>, tab\<^sub>0 (cod \<mu>), \<mu>.chine] \<cdot> (\<mu>.s.tab \<star> \<mu>.chine) \<cdot> \<mu>.the_\<nu>"
nullhomotopic_nonsurjective_sphere_map: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "continuous_map (nsphere p) (nsphere p) f"   and "f ` topspace (nsphere p) \<noteq> topspace (nsphere p)" obtains a :: "nat \<Rightarrow> real" where "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f (\<lambda>x. a)"
exec_wfp_in_states: fixes p :: "prog"   and \<sigma> :: "char list \<Rightarrow> nat option"   and \<sigma>' :: "char list \<Rightarrow> nat option" assumes "wfp p"   and "\<sigma> \<in> states p"   and "body p \<rhd> \<sigma> \<leadsto> Some \<sigma>'" shows "\<sigma>' \<in> states p"
partition_on_def: fixes A :: "'a set"   and P :: "'a set set" shows "partition_on A P = (\<Union> P = A \<and> disjoint P \<and> {} \<notin> P)"
Chains_def: fixes r :: "('b \<times> 'b) set" shows "Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
R1_14_interm2_def: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and choice :: "bool \<times> bool" shows "R1_14_interm2 msgs choice = (let (m00, m01, m10, m11) = msgs; (c0, c1) = choice in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. S1_OT12 (S0, S1) () \<bind> (\<lambda>a. S1_OT12 (S2, S3) () \<bind> (\<lambda>b. R1_OT12 (S4, S5) c1 \<bind> (\<lambda>c. return_spmf (msgs, (S0, S1, S2, S3, S4, S5), a, b, c)))))))))))"
acrelprime: shows "somegcd G a c \<sim> \<one>"
pathstart_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "pathstart (subpath u v g) = g u"
IH_2: fixes t' :: "'a tree" assumes "del x r = Some t'" shows "\<Phi> t' - \<Phi> r \<le> 6 * e * real (height r)"
index_mat_multrow_5: fixes mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b"   and A :: "'b mat" shows "dim_col (mat_multrow_gen mul k a A) = dim_col A"
currentD_SAT: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f" shows "(x \<in> SAT \<Gamma> f) = (x \<in> A \<Gamma> \<or> d_IN f x = weight \<Gamma> x)"
update_assignment_alt_def: fixes u :: "('c \<times> 'd) list"   and as :: "'c \<Rightarrow> 'd" shows "update_assignment_alt u as = (\<lambda>v. case map_of u v of None \<Rightarrow> as v | Some n \<Rightarrow> n)"
wf_sees_clinit1: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and a :: "'b class"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "wf_prog wf_md P"   and "class P C = \<lfloor>a\<rfloor>"   and "P \<turnstile> C sees clinit, b :  Ts\<rightarrow>T = m in D" shows "b = Static \<and> Ts = [] \<and> T = Void \<and> D = C"
subdiv_ends_G_rev_2: shows "head I (rev_I uv) = u"
ifFalse: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e1 :: "('r, 'l, 'v) expr"   and e2 :: "('r, 'l, 'v) expr" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV F)) e1 e2])" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [e2])))"
\<delta>: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "\<delta> i = n i - n (Suc i)"
modulo: shows "(R ===> R ===> R) (arith_ops_record.modulo ops) (mod)"
assms_5: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
path_component_of_eq_empty: fixes X :: "'a topology"   and x :: "'a" shows "(path_component_of_set X x = {}) = (x \<notin> topspace X)"
Bex_set_replicate: fixes n :: "nat"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>set (replicate n a). P x) = (P a \<and> n \<noteq> 0)"
that: fixes k :: "nat"   and z :: "'a list" assumes "k < l"   and "z <p w"   and "w \<^sup>@ k \<cdot> z = u" shows "thesis"
us: shows "us = getFrN (map snd txs) (t # map fst txs) [] (length txs)"
remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "sorted_list_of_set A = insort x (sorted_list_of_set (A - {x}))"
Ints_prod: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<int>" shows "prod f A \<in> \<int>"
FGModuleEnd_def: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" shows "FGModuleEnd G smult V T \<equiv> FGModuleHom G smult V smult T \<and> FGModuleEnd_axioms V T"
a_set_tag_name_def: fixes ptr :: "(_) element_ptr"   and tag :: "char list" shows "local.a_set_tag_name ptr tag = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr tag_name_update tag)"
domain_eq: shows "Domainp pcr_fin_string = (\<lambda>s. length s \<le> MAX_STR)"
le_infI2: fixes b :: "'a"   and x :: "'a"   and a :: "'a" assumes "b \<le> x" shows "a \<sqinter> b \<le> x"
ar_coset_subsetD: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c"   and x :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "x \<in> carrier R"
exp_cf4_def: shows "exp_cf4 \<equiv> \<lambda>x. numer_cf4 x / numer_cf4 (- x)"
set_disconnected_nodes_get_disconnected_nodes_l_set_disconnected_nodes_get_disconnected_nodes: shows "l_set_disconnected_nodes_get_disconnected_nodes ShadowRootClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
real_cond_exp_charact: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "\<And>A. A \<in> sets F \<Longrightarrow> set_lebesgue_integral M A f = set_lebesgue_integral M A g"   and "integrable M f"   and "integrable M g"   and "g \<in> borel_measurable F" shows "AE x in M. real_cond_exp M F f x = g x"
upset_set_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.upset_set less_eq X = {y. \<exists>x\<in>X. less_eq x y}"
ray_to_frontier: fixes S :: "'a set"   and a :: "'a"   and l :: "'a" assumes "bounded S"   and "a \<in> interior S"   and "l \<noteq> (0::'a)" obtains d :: "real" where "0 < d" and "a + d *\<^sub>R l \<in> frontier S" and "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> interior S"
scale_cancel_right: fixes a :: "complex"   and x :: "'d"   and b :: "complex" shows "(a *\<^sub>C x = b *\<^sub>C x) = (a = b \<or> x = (0::'d))"
prems: shows "[:- b, 1::'a:] * p \<noteq> 0"
Rf_model: fixes I :: "'a set"   and N :: "'a literal multiset set" assumes "I \<Turnstile>s N - Rf N" shows "I \<Turnstile>s N"
map_ide: fixes a :: "'c" assumes "C.ide a" shows "Cones.map a = S.mkIde (\<iota> ` D.cones a)"
distrib_4: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
B: shows "AE x in M. 0 < real_cond_exp M Invariants (\<lambda>x. f1 x + C x) x"
holf: shows "f holomorphic_on S \<inter> {z. 0 < Im z}"
2: shows "FvarsT \<chi> = FvarsT r - {x} \<union> (if x \<in> FvarsT r then FvarsT t else {})"
liftM_ctxt_type: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and T :: "Syntax.type"   and U :: "Syntax.type"   and k :: "nat"   and T1 :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T \<Leftarrow> U" shows "\<Gamma> , \<Delta>\<langle>k:T1\<rangle> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t liftM_ctxt E k : T \<Leftarrow> U"
whilei_def: fixes p :: "'a"   and i :: "'a"   and x :: "'a" shows "while p inv i do x od = while p do x od"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) A.map A.map A.map"
prems: shows "P \<turnstile> (C,Cs) selects M = mthd' via Cs''"
compatibleI_with_proj: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" shows "compatible (with_proj G) (with_proj H)"
distinct_zipI1: fixes xs :: "'b list"   and ys :: "'c list" assumes "distinct xs" shows "distinct (zip xs ys)"
2: shows "continuous_map (subtopology (powertop_real UNIV) (standard_simplex p)) X f"
nat_pow_0: fixes x :: "nat \<Rightarrow> int" shows "[0] \<cdot>\<^bsub>UP INTEG\<^esub> x = \<zero>\<^bsub>UP INTEG\<^esub>"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns'\<rightarrow>m'"   and "prefix ns' ns"   and "m' \<notin> set (butlast ns')" shows "thesis"
\<F>_af: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "\<F> (af \<phi> (prefix i w)) (suffix i w) = \<F> \<phi> (suffix i w)"
lift_opnet_sync_action: fixes U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "net_tree" assumes "\<And>\<xi>. U \<xi> \<xi>"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). castmsg (I \<sigma>) a)"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<and> (\<forall>d. a \<noteq> i:deliver(d)) \<longrightarrow> S (\<sigma> i) (\<sigma>' i))"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>d. a = i:deliver(d)) \<longrightarrow> U (\<sigma> i) (\<sigma>' i))" shows "opnet onp p \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p) \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (I \<sigma>) a \<and> (a \<noteq> \<tau> \<and> (\<forall>i d. a \<noteq> i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p. S (\<sigma> i) (\<sigma>' i))) \<and> (a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p \<longrightarrow> \<sigma>' i = \<sigma> i)))"
assms: shows "coreflexive (w * z\<^sup>T)"
nth_tl: fixes n :: "nat"   and xs :: "'a list" assumes "n < length (tl xs)" shows "tl xs ! n = xs ! Suc n"