hcju/isabelleps huggingface dataset:
d_def: fixes fs :: "int vec list"   and k :: "nat" shows "fs.d fs k = gs.Gramian_determinant fs k"
simulation2: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl2 :: "'tl2"   and s2' :: "'s2" assumes "s1 \<approx> s2"   and "s2 -2-tl2\<rightarrow> s2'"   and "\<not> \<tau>move2 s2 tl2 s2'" shows "\<exists>s1' s1'' tl1. s1 -\<tau>1\<rightarrow>* s1' \<and> s1' -1-tl1\<rightarrow> s1'' \<and> \<not> \<tau>move1 s1' tl1 s1'' \<and> s1'' \<approx> s2' \<and> tl1 \<sim> tl2"
mat_of_cols_carrier_1: fixes n :: "nat"   and vs :: "'a Matrix.vec list" shows "mat_of_cols n vs \<in> carrier_mat n (length vs)"
connected_inV_iff: fixes c :: "nat \<times> nat \<Rightarrow> 'c"   and u :: "nat"   and v :: "nat" assumes "Graph.connected c u v" shows "(v \<in> Graph.V c) = (u \<in> Graph.V c)"
mono_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "mono_on f A"   and "B \<subseteq> A" shows "mono_on f B"
all_lower: shows "\<forall>u. u \<in># mset_lit L \<longrightarrow> (u \<lhd> \<sigma>, u \<lhd> \<eta>) \<in> trm_ord \<or> u \<lhd> \<sigma> = u \<lhd> \<eta>"
th: fixes x :: "nat"   and y :: "nat" shows "\<not> (x < y \<and> (\<forall>m. y \<noteq> x + m))"
bigo_lesseq3: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(h)"   and "\<forall>x. (0::'b) \<le> g x"   and "\<forall>x. g x \<le> f x" shows "g \<in> O(h)"
igSwapIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igSwapIGOp MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
tail_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tail p \<equiv> punit.lower p (punit.lt p)"
measure_pmf_of_set: fixes A :: "'b set" assumes "A \<noteq> {}"   and "finite A" shows "measure_pmf (pmf_of_set A) = uniform_measure (count_space UNIV) A"
trivial_hom_iff: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "(h ` carrier G = {\<one>\<^bsub>H\<^esub>}) = (kernel G H h = carrier G)"
exact_seq_length1: fixes t :: "'b monoid list \<times> ('b \<Rightarrow> 'b) list" assumes "exact_seq t" shows "length (fst t) = Suc (length (snd t))"
substRes3: fixes a :: "name"   and P :: "pi"   and x :: "name" shows "<\<nu>a>P[x::=a] = <\<nu>x>([(x, a)] \<bullet> P)"
assoc_naturality: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>[cod \<mu>, cod \<nu>, cod \<tau>] \<cdot> ((\<mu> \<star> \<nu>) \<star> \<tau>) = (\<mu> \<star> \<nu> \<star> \<tau>) \<cdot> \<a>[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>]"
simps_16: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Power a 0) vs = True"
fun_eq_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "fun_eq_on f g A"   and "order.greater_eq A B" shows "fun_eq_on f g B"
cases: fixes x :: "'a \<times> 'a tree" obtains x :: "'a" and t :: "'a tree" where "xa__ = (x, t)"
length_tl: fixes xs :: "'a list" shows "|tl xs| = |xs| - 1"
ty_REL: shows "TYREL (\<langle>Id\<rangle>hs.rel)"
Gromov_completion_cases: fixes z :: "'c Gromov_completion" obtains x :: "'c" where "z = to_Gromov_completion x"   | "z \<in> Gromov_boundary"
tr1: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> (f $ Bv 0)) g"
subpath_trivial: fixes g :: "real \<Rightarrow> 'a" shows "subpath 0 1 g = g"
adjoint_eval: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat" assumes "i < dim_col A"   and "j < dim_row A" shows "adjoint A $$ (i, j) = conjugate (A $$ (j, i))"
c2p_mono_p2c_strict: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 \<le> n"   and "n < \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n'" shows "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n < n'"
simps_1: fixes cs :: "'edge list" shows "valid_path_rev_aux cs [] = True"
inj_on_inv_into: fixes B :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "B \<subseteq> f ` A" shows "inj_on (inv_into A f) B"
leg1_composite: shows "cods.\<nu>\<pi>.leg1 = cods.\<nu>.leg1 \<cdot> cods.\<nu>\<pi>.prj\<^sub>1"
enough_flushs_axioms: shows "enough_flushs ts\<^sub>s\<^sub>b"
gram_schmidt_fs_Rn_axioms: shows "gram_schmidt_fs_Rn n (map of_int_hom.vec_hom fs)"
less_diff_conv: fixes i :: "nat"   and j :: "nat"   and k :: "nat" shows "(i \<sqsubset> j - k) = (i + k \<sqsubset> j)"
ts_invD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and I :: "'d \<Rightarrow> 'c option"   and ts :: "'d \<Rightarrow> ('e \<times> 'g \<Rightarrow>f nat) option"   and m :: "'f"   and t :: "'d"   and x :: "'e"   and ln :: "'g \<Rightarrow>f nat" assumes "ts_inv P I ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m"
pi_half_neq_two: shows "pi / 2 \<noteq> 2"
nuc_comp_distr: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and y :: "'b" assumes "nucleus f" shows "f (f (\<Squnion> X) \<cdot> y) = f (\<Squnion>x\<in>X. f (x \<cdot> y))"
X: fixes t :: "nat" shows "random_variable (count_space UNIV) (X t)"
axioms_1: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
seq_top: fixes c :: "'a::refinement_lattice" shows "\<top> ; (c::'a::refinement_lattice) = \<top>"
ideD_1: fixes a :: "(bool, unit) arr" assumes "ide a" shows "arr a"
diff_fun_zero: shows "diff_fun k (charts_submanifold W) 0"
normal_restrict_supergroup: fixes S :: "'a set set"   and N :: "'a set set" assumes "subgroup S (G Mod N)"   and "N \<lhd> G Mod N"   and "N \<subseteq> S" shows "N \<lhd> (G Mod N)\<lparr>carrier := S\<rparr>"
Tensor_left_id: fixes xs :: "'a list list" shows "[[id]] \<otimes> xs = xs"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "a \<cdot> (b \<cdot> x) = a \<cdot> b \<cdot> x"
cases: fixes r :: "point4d" obtains xc :: "int" and yc :: "int" and zc :: "int" and wc :: "int" where "r = \<lparr>xc = xc, yc = yc, zc = zc, wc = wc\<rparr>"
wt_cont_eff_Par_3: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "i < brn (cl ! n)" shows "eff (Par cl) s (brnL cl n + i) = eff (cl ! n) s i"
Notes_SSL_imp_used: fixes B :: "agent"   and A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Notes B \<lbrace>Agent A, Agent B, X\<rbrace> \<in> set evs" shows "X \<in> used evs"
assms_2: shows "c \<notin> normalize_sort cs (insert c s)"
R_decomp_exists: obtains R_pre :: "'a list" and z :: "'a" and R_post :: "'a list" where "R = R_pre @ z # R_post" and "Q.hitting_paths z" and "\<And>z'. z' \<in> set R_pre \<Longrightarrow> \<not> Q.hitting_paths z'"
fps_plus_def: shows "(+) = (\<lambda>f g. Abs_fps (\<lambda>n. fps_nth f n + fps_nth g n))"
unat_ucast_eq_unat_and_mask: fixes w :: "'c word" shows "unat (UCAST('c \<rightarrow> 'b) w) = unat (w AND mask LENGTH('b))"
legal_subst_zpassign: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" assumes "legal_zpassign A"   and "wary_subst \<rho>" shows "legal_zpassign (subst_zpassign \<rho> A)"
assms_3: shows "src\<^sub>C \<mu> = trg\<^sub>C \<nu>"
Epigraph_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> ereal" shows "Epigraph S f = {xy. fst xy \<in> S \<and> f (fst xy) \<le> ereal (snd xy)}"
fUnionM_empty: fixes return :: "'d fset \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d fset \<Rightarrow> 'e) \<Rightarrow> 'e" assumes "monad_commute return bind"   and "monad_duplicate return bind" shows "ffold (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A |\<union>| B)))) (return {||}) {||} = return {||}"
rev\<alpha>_no_x: shows "\<forall>a\<in>set rev\<alpha>. a \<notin> X"
thetaSSeqD_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis" shows "PL_Indis.thetaSSeqD aval tval cval indis \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. proper c1 \<and> proper d1 \<and> proper c2 \<and> proper d2 \<and> PL_Indis.discr aval tval cval indis c2 \<and> PL_Indis.discr aval tval cval indis d2 \<and> (c1, d1) \<in> PL_Indis.Sbis aval tval cval indis}"
distinct_1: fixes x21 :: "'d tree"   and x22 :: "'d"   and x23 :: "'d tree" shows "\<langle>\<rangle> \<noteq> \<langle>x21, x22, x23\<rangle>"
simps_1: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message"   and v :: "('a, 'b option) insert_message list" assumes "integrate_insert_commute_insert M a s m" shows "integrate_insert_commute_insert.is_certified_associated_string' M a m (Inr v) = (set v = to_woot_character (M \<union> {Insert m}) ` insert_messages (M \<union> {Insert m}) \<and> sorted_wrt (<) (map a (ext_ids v)))"
simps_2: fixes n :: "nat"   and x :: "'a" shows "replicate (Suc n) x = x # replicate n x"
jumpF_polyR_mod: fixes q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jumpF_polyR q p x = jumpF_polyR (q mod p) p x"
domains_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "domains (g \<cdot> f) = domains f"
gen_mult: fixes x :: "'c"   and G :: "('c, 'd) monoid_scheme"   and gens :: "'c set"   and y :: "'c" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"   and "y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
f_analytically_valid: shows "analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i"
finite_UNIV_finite_4_def: shows "finite_UNIV = Phantom(Enum.finite_4) True"
has_derivative_subset: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'f"   and x :: "'e"   and s :: "'e set"   and t :: "'e set" assumes "(f has_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_derivative f') (at x within t)"
simps_2: fixes t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and ts :: "'a tree list" shows "huffman (t\<^sub>1 # t\<^sub>2 # ts) = huffman (insortTree (uniteTrees t\<^sub>1 t\<^sub>2) ts)"
distinct_31: shows "CharactersOp \<noteq> ToBooleanOp"
i_shrink_eq_Msg_iSince_conv: fixes k :: "nat"   and m :: "'a message_af"   and s :: "nat \<Rightarrow> 'a message_af"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg" shows "((s \<div> k) t = m) = (s t2 = NoMsg. t2 \<S> t1 [t * k\<dots>,k - Suc 0]. s t1 = m)"
group_left_neutral: fixes a :: "'ag" assumes "a \<in> U" shows "\<^bold>1\<^sub>o\<^sub>w \<^bold>*\<^sub>o\<^sub>w a = a"
dim_diag_block_mat_1: fixes As :: "'a mat list" shows "dim_row (diag_block_mat As) = sum_list (map dim_row As)"
step_rel_def: shows "step_rel = Restr {(I'', I). \<exists>l c h k. cond I I'' l c h k} valid_I"
suffix_state_prop: fixes P :: "'g \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g"   and i :: "nat" shows "\<lceil>P\<rceil> (\<sigma> |\<^sub>s i) = P (\<sigma> i)"
unit_disc_fix_preserve_distance_formula: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance_formula (to_complex (moebius_pt M u)) (to_complex (moebius_pt M v)) = poincare_distance_formula (to_complex u) (to_complex v)"
integral_mult_left: fixes c :: "'d"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'd" assumes "c \<noteq> (0::'d) \<Longrightarrow> integrable M f" shows "LINT x|M. f x * c = integral\<^sup>L M f * c"
simps: fixes a1 :: "'state"   and a2 :: "'state" shows "a1 \<Rightarrow>\<^sub>S a2 = ((\<exists>P. a1 = P \<and> a2 = P) \<or> (\<exists>P P' P''. a1 = P \<and> a2 = P'' \<and> P \<rightarrow>\<^sub>S \<langle>S_action.Act \<tau>,P'\<rangle> \<and> P' \<Rightarrow>\<^sub>S P''))"
vector_eq_ldot: fixes y :: "'b"   and z :: "'b" shows "(\<forall>x. x \<bullet> y = x \<bullet> z) = (y = z)"
phl_while: fixes p :: "'a"   and s :: "'a"   and w :: "'a"   and x :: "'a" assumes "p \<cdot> s \<le> s \<cdot> p \<cdot> s"   and "p \<cdot> w \<le> w \<cdot> p \<cdot> w"   and "\<lbrace>p \<cdot> s\<rbrace>x\<lbrace>p\<rbrace>" shows "\<lbrace>p\<rbrace>(s \<cdot> x)\<^sup>\<star> \<cdot> w\<lbrace>p \<cdot> w\<rbrace>"
var_dist_def: shows "var_dist = fold (\<lambda>i t. t \<degree> Var i)"
Span_subgroup_props_4: fixes Us :: "'a list"   and v :: "'a" assumes "set Us \<subseteq> carrier R"   and "v \<in> Span K Us" shows "\<ominus> v \<in> Span K Us"
distinct_remove1_removeAll: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "remove1 x xs = removeAll x xs"
dg_prod_is_arrI: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and f :: "V"   and a :: "V"   and b :: "V" assumes "pdigraph_base \<alpha> I \<AA>"   and "vsv f"   and "\<D>\<^sub>\<circ> f = I"   and "vsv a"   and "\<D>\<^sub>\<circ> a = I"   and "vsv b"   and "\<D>\<^sub>\<circ> b = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>" shows "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b"
subgraph_mk_symmetric: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G" shows "subgraph (with_proj (mk_symmetric H)) (with_proj (mk_symmetric G))"
finite_vimage_iff: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'b set" assumes "bij h" shows "finite (h -` F) = finite F"
vals_in_P_equiv: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "v' \<in> P\<^bsub> K v\<^esub>" shows "v_equiv K v v'"
php_link: fixes h1 :: "'b hp"   and h2 :: "'b hp" assumes "php h1"   and "php h2" shows "php (link h1 h2)"
weight_le_def: fixes s :: "('f, 'g) Term.term"   and t :: "('f, 'g) Term.term" shows "weight_le s t \<equiv> vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t) \<and> weight s \<le> weight t"
Finite_Graph_EI: assumes "finite E" shows "Finite_Graph c"
wf_simulate_simple: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and g :: "'b \<Rightarrow> 'a" assumes "wf r"   and "\<And>x y. (x, y) \<in> r' \<Longrightarrow> (g x, g y) \<in> r" shows "wf r'"
lincomb_diff_left: fixes rs :: "'r list"   and ss :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ss \<subseteq> R"   and "set ms \<subseteq> M"   and "length rs \<le> length ss" shows "map2 (-) rs ss \<bullet>\<cdot> ms = rs \<bullet>\<cdot> ms - take (length rs) ss \<bullet>\<cdot> ms"
2: shows "- v \<sqinter> x \<sqinter> - (v \<sqinter> y) \<sqinter> - (- v \<sqinter> z) = - v \<sqinter> x \<sqinter> - z"
sigma_sets_eq: fixes M :: "'c measure" shows "sigma_sets (space M) (sets M) = sets M"
\<phi>: fixes F :: "nat \<Rightarrow> 'a set" assumes "\<And>n. compact (F n)"   and "\<And>n. F n \<noteq> {}"   and "\<And>n. \<phi> (F n)"   and "\<And>n. F (Suc n) \<subseteq> F n" shows "\<phi> (\<Inter> (range F))"
per_init'_def: fixes n :: "nat" shows "per_init' n \<equiv> {(i, i) |i. i < n}"
k_1: fixes G1 :: "'a poly"   and G2 :: "'a poly" shows "??.Subresultant.F G1 G2 (Suc (??.Subresultant.k G1 G2)) = 0"
simps_5: fixes t :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list"   and r :: "('c, 'd) bintree"   and ys :: "('c, 'd) bintree option list" shows "meld (Some t # xs) (Some r # ys) = None # add (Some (merge t r)) (meld xs ys)"
isStaticLoc_simps_3: fixes T :: "Arraytype"   and a :: "nat" shows "isStaticLoc (arrLenLoc T a) = False"
enat_arith_defs_3: fixes m :: "enat"   and n :: "enat" shows "m + n = (case m of enat m \<Rightarrow> case n of enat n \<Rightarrow> enat (m + n) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
Suc_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "l = length xs + length ys"   and "length xs = card {a \<in> A. a < length xs + length ys}"   and "length ys = card {a \<in> - A. a < length xs + length ys}" shows "nths (weave A xs ys) A = xs \<and> nths (weave A xs ys) (- A) = ys"
landau_trans_2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> o[F](h)"
int2: shows "integrable \<chi> {w. a \<le> f w} M"
ab_left_minus: fixes a :: "'c" shows "- a + a = (0::'c)"
Iio_Int_singleton: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c"   and x :: "'c" assumes "class.order less_eq less" shows "ord.lessThan less k \<inter> {x} = (if less x k then {x} else {})"
Cast: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and T :: "ty"   and T' :: "ty" assumes "E,dt\<Turnstile>e\<Colon>-T"   and "is_acc_type (prg E) (pkg E) T'"   and "prg E\<turnstile>T\<preceq>? T'" shows "E,dt\<Turnstile>Cast T' e\<Colon>-T'"
maxsimpchainD_adj: fixes xs :: "'a set list" assumes "folding_g.gallery xs" shows "adjacentchain xs"
rejects_input: fixes l :: "String.literal"   and d' :: "nat \<Rightarrow>f value option"   and i :: "value list" assumes "l \<noteq> STR ''coin''"   and "l \<noteq> STR ''vend''" shows "\<not> recognises_execution drinks 1 d' [(l, i)]"
last_conv_nth: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "List.last xs = xs ! (length xs - 1)"
fls_deriv_shift: fixes n :: "int"   and f :: "'a fls" shows "fls_deriv (fls_shift n f) = of_int (- n) * fls_shift (n + 1) f + fls_shift n (fls_deriv f)"
sel: fixes gpv :: "('a, 'c, 'e) gpv" shows "the_gpv (right_gpv gpv) = map_spmf (map_generat id Inr right_rpv) (the_gpv gpv)"
mono_minus_ivl: fixes i1 :: "ivl"   and i1' :: "ivl"   and i2 :: "ivl"   and i2' :: "ivl" assumes "i1 \<sqsubseteq> i1'"   and "i2 \<sqsubseteq> i2'" shows "minus_ivl i1 i2 \<sqsubseteq> minus_ivl i1' i2'"
rScope: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'" shows "Prop (\<lparr>\<nu>x\<rparr>P')"
IH_1: fixes g :: "'a" assumes "left_nesting f1 \<noteq> left_nesting g"   and "is_const (fst (strip_comb f1))" shows "match f1 g = None"
scaleR_left_mono: fixes x :: "'a"   and y :: "'a"   and a :: "real" assumes "x \<le> y"   and "0 \<le> a" shows "a *\<^sub>R x \<le> a *\<^sub>R y"
R1: fixes x :: "'b"   and y :: "'b"   and yl :: "'b bintree"   and yr :: "'b bintree"   and ts :: "'b bintree list"   and X :: "'b t_type" assumes "\<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr> \<in> t_ins_set X"   and "x \<le> y" shows "\<lparr>t_type.folding = False, item = x, subtrees = yl # Branch y yl yr # ts\<rparr> \<in> t_ins_set X"
homotopic_paths_linear: fixes g :: "real \<Rightarrow> 'b"   and h :: "real \<Rightarrow> 'b"   and S :: "'b set" assumes "path g"   and "path h"   and "pathstart h = pathstart g"   and "pathfinish h = pathfinish g"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> S" shows "homotopic_paths S g h"
nex: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F" shows "|~ \<circle>F"
sublist_CONS1_E: fixes h :: "'a"   and l1 :: "'a list"   and l2 :: "'a list" assumes "subseq (h # l1) l2" shows "subseq l1 l2"
coboundedI: fixes A :: "'d DBMEntry set"   and a :: "'d DBMEntry" assumes "finite A"   and "a \<in> A" shows "a \<preceq> linordered_monoid.Max A"
delta_pos_neg_least_2: fixes rm :: "('i \<Rightarrow> 'd) admS"   and rp :: "('i \<Rightarrow> 'd) admS" assumes "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "delta_pos \<le> rp"
\<phi>_def: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "\<phi> c d = (\<exists>c1' c2'. c = if tst then c1' else c2' \<and> d = if tst then c1' else c2' \<and> c1' \<approx>wT c2' \<and> c1' \<approx>wT c1' \<and> c2' \<approx>wT c2')"
added_files_Deliver_Store_same_collapse: fixes i :: "'c"   and e :: "'b"   and mo :: "'c"   and j :: "'c" shows "added_files [Deliver (i, Store e mo j)] e = [j]"
designI: assumes "\<And>b. b \<in># \<B>\<^sup>D \<Longrightarrow> b \<noteq> {}"   and "\<B>\<^sup>D \<noteq> {#}"   and "bl \<noteq> {}" shows "design bl \<B>\<^sup>D"
MGFLen_helper: fixes z :: "bit list"   and l :: "nat" assumes "MGF z l \<noteq> []" shows "l \<le> 2 ^ 32 * length (sha1 z)"
gcd_0_left_nat: fixes x :: "nat" shows "gcd 0 x = x"
final_topology_def: fixes X :: "'d set"   and Y :: "'e \<Rightarrow> 'f topology"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" shows "final_topology X Y f = topology (\<lambda>U. U \<subseteq> X \<and> (\<forall>i. openin (Y i) (f i -` U \<inter> topspace (Y i))))"
function_one_eval: fixes R :: "('c, 'd) ring_scheme"   and a :: "'e"   and S :: "'e set" assumes "ring_functions R"   and "a \<in> S" shows "\<one>\<^bsub>function_ring S R\<^esub> a = \<one>\<^bsub>R\<^esub>"
valid_W_inv_mo_co_W: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and y :: "'g"   and p :: "'d process_name" assumes "valid_W_inv s"   and "marked y s"   and "ghost_honorary_grey (s p) = {y}"   and "p \<noteq> sys" shows "valid_W_inv (s(p := s p\<lparr>W := insert y (W (s p)), ghost_honorary_grey := {}\<rparr>))"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "llist_of (x # xs) = LCons x (llist_of xs)"
5: shows "bprv (imp (softSubst \<chi> \<langle>\<chi>\<rangle> xx) (subst \<phi> \<langle>\<psi>\<rangle> xx))"
sum_case_split: fixes P :: "'a \<Rightarrow> bool"   and f :: "V \<Rightarrow> 'a"   and g :: "V \<Rightarrow> 'a"   and a :: "V" shows "P (sum_case f g a) = ((\<forall>x. a = ZFC_Cardinals.Inl x \<longrightarrow> P (f x)) \<and> (\<forall>y. a = ZFC_Cardinals.Inr y \<longrightarrow> P (g y)) \<and> (\<not> is_sum a \<longrightarrow> P undefined))"
glb_bin: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.po below"   and "below x y" shows "po.glb below {x, y} = x"
simps: fixes a :: "'k set" shows "finite a = (a = {} \<or> (\<exists>A a. a = insert a A \<and> finite A))"
headR_def: fixes t :: "('a, 'b) FingerTree" shows "FingerTree.headR t = FingerTreeStruc.headR (Rep_FingerTree t)"
hoare\<^sub>3_def: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c \<Rightarrow> ('d \<times> 'c) option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> M \<lbrace>Q\<rbrace> \<equiv> \<forall>\<sigma>. P \<sigma> \<longrightarrow> (case M \<sigma> of None \<Rightarrow> False | Some (x, \<sigma>') \<Rightarrow> Q x \<sigma>')"
assms_2: fixes \<rho> :: "'a basic_type" assumes "\<tau> = \<rho>[1]"   and "\<rho> < \<sigma>" shows "P"
generalise_output_preserves_updates: fixes t :: "transition"   and r :: "nat"   and p :: "nat" shows "Updates (generalise_output t r p) = Updates t"
assms_2: shows "s2 \<in> syzygy_module B"
thm_noncont_propos_2: fixes v :: "i" shows "[Impossible (p\<^sub>0\<^sup>-) in v]"
df_cont: fixes i :: "'a" assumes "i \<in> Basis" shows "(f' i \<longlongrightarrow> f' i x) (at x within {x. \<forall>j\<in>Basis. x \<bullet> j \<in> X j})"
sqrt_sum_squares_le_sum: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "sqrt (x\<^sup>2 + y\<^sup>2) \<le> x + y"