hcju/isabelleps huggingface dataset:
ext_inject: fixes bar1 :: "'a"   and bar2 :: "'b"   and bar3 :: "'c"   and bar21 :: "'b \<times> 'a"   and bar32 :: "'c \<times> 'b"   and bar31 :: "'c \<times> 'a"   and more :: "'d"   and bar1' :: "'a"   and bar2' :: "'b"   and bar3' :: "'c"   and bar21' :: "'b \<times> 'a"   and bar32' :: "'c \<times> 'b"   and bar31' :: "'c \<times> 'a"   and more' :: "'d" shows "(\<lparr>bar1 = bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr> = \<lparr>bar1 = bar1', bar2 = bar2', bar3 = bar3', bar21 = bar21', bar32 = bar32', bar31 = bar31', \<dots> = more'\<rparr>) = (bar1 = bar1' \<and> bar2 = bar2' \<and> bar3 = bar3' \<and> bar21 = bar21' \<and> bar32 = bar32' \<and> bar31 = bar31' \<and> more = more')"
shows_sep_paren_def: fixes s :: "char list \<Rightarrow> char list" shows "shows_sep_paren s = ('' ('' +#+ s \<circ> shows '')'')"
locally_path_connected: fixes S :: "'c set" shows "locally path_connected S = (\<forall>V x. openin (top_of_set S) V \<and> x \<in> V \<longrightarrow> (\<exists>U. openin (top_of_set S) U \<and> path_connected U \<and> x \<in> U \<and> U \<subseteq> V))"
minimalize_offending_overapprox_gives_back_an_offending_flow: fixes G :: "'a graph"   and ff :: "('a \<times> 'a) list"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph G"   and "is_offending_flows (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "set (local.minimalize_offending_overapprox ff [] G nP) \<in> set_offending_flows G nP"
sorted_splay: fixes t :: "'b tree"   and x :: "'b"   and l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "Sorted_Less.sorted (inorder t)"   and "splay x t = \<langle>l, a, r\<rangle>" shows "Sorted_Less.sorted (inorder l @ x # inorder r)"
finfun_Ball_def: fixes A :: "'a \<Rightarrow>f bool"   and P :: "'a \<Rightarrow> bool" shows "finfun_Ball A P = Ball {x. A $ x} P"
eqI: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<sigma>' :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation A B F G \<sigma>'"   and "\<And>a. partial_magma.ide A a \<Longrightarrow> \<sigma> a = \<sigma>' a" shows "\<sigma> = \<sigma>'"
essential_BI: fixes x :: "'a"   and B :: "'a set"   and G :: "('a, 'b) graph_scheme"   and S :: "'a set" assumes "x \<in> B" shows "essential G B S x"
rel_simps_89: shows "(0::'a) \<noteq> - (1::'a)"
transT: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> \<box>\<box>F"
ortho_def: fixes h1 :: "nat \<Rightarrow> nat option"   and h2 :: "nat \<Rightarrow> nat option" shows "h1 \<bottom> h2 = (dom h1 \<inter> dom h2 = {})"
star_of_inject: fixes x :: "'a"   and y :: "'a" shows "(star_of x = star_of y) = (x = y)"
path_connectedin_def: fixes X :: "'c topology"   and S :: "'c set" shows "path_connectedin X S \<equiv> S \<subseteq> topspace X \<and> path_connected_space (subtopology X S)"
prob_space_axioms: shows "prob_space (M \<Otimes>\<^sub>M N)"
enabled_iff_sconst: fixes x :: "'s"   and \<omega> :: "'s stream" assumes "set_pmf (K x) = {x}" shows "enabled x \<omega> = (\<omega> = sconst x)"
subset_entailed: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N2 :: "'a set"   and N1 :: "'a set" assumes "consequence_relation Bot entails"   and "N2 \<subseteq> N1" shows "entails N1 N2"
split_list_first: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys \<cdot> x # zs \<and> x \<notin> set ys"
remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "card A = Suc (card (A - {x}))"
subst_mgt_trans: fixes \<delta> :: "'d \<Rightarrow> ('b, 'c) Term.term"   and \<theta> :: "'d \<Rightarrow> ('b, 'a) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('b, 'e) Term.term" assumes "\<theta> \<preceq>\<^sub>\<circ> \<delta>"   and "\<delta> \<preceq>\<^sub>\<circ> \<sigma>" shows "\<theta> \<preceq>\<^sub>\<circ> \<sigma>"
split_div': fixes P :: "nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" shows "P (m div n) = (n = 0 \<and> P 0 \<or> (\<exists>q. (n * q \<le> m \<and> m < n * Suc q) \<and> P q))"
simps_2: fixes s1 :: "'s1"   and s2 :: "'s2"   and b :: "'c" shows "local.parallel_oracle (s1, s2) (Inr b) = map_spmf (map_prod Inr (Pair s1)) (right s2 b)"
topo_topology: fixes B :: "'b set set" shows "topology (topo B)"
seq'_not_inj_on_card_UNIV: fixes x :: "'a" shows "\<not> inj_on (seq' x) {0..card UNIV}"
valid_graph_def: fixes G :: "('e, 'f) graph" shows "valid_graph G \<equiv> fst ` edges G \<subseteq> nodes G \<and> snd ` snd ` edges G \<subseteq> nodes G"
real_sqrt_minus: fixes x :: "real" shows "sqrt (- x) = - sqrt x"
YFtorObj2: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "(YFtor C @@ Y) @@ X = Hom\<^bsub>C\<^esub> X Y"
arg_min_if_finite_1: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite S"   and "S \<noteq> {}" shows "arg_min_on f S \<in> S"
assms_2: shows "qAbs xs x X $= qAbs xs' x' X'"
Exit_source: fixes prog :: "cmd"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" assumes "valid_edge prog a"   and "sourcenode a = (_Exit_)" shows "False"
select_convs_1: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'d \<Rightarrow> 'c"   and Dom :: "'c"   and In :: "'d list"   and Out :: "'d list"   and more :: "'e" shows "Pol \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = Pol"
rank_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "rank A = row_rank A"
assms: shows "\<chi> \<noteq> \<chi>\<^sub>0"
prems: shows "subseq as' (as_proj (a # as) vs)"
exhaust: fixes y :: "tf" obtains x1 :: "t list" and x2 :: "t list" where "y = x1 _> x2"
sprod_mHom_closed: fixes N :: "('i, 'b, 'j) Module_scheme"   and a :: "'b"   and f :: "'a \<Rightarrow> 'i" assumes "R module N"   and "a \<in> carrier R"   and "f \<in> mHom R M N" shows "sprod_mHom R M N a f \<in> mHom R M N"
IH: fixes gpv :: "('e, 'b, 'd) gpv" shows "expectation_gpv' (right_gpv gpv) = expectation_gpv'' gpv"
stable_pair_onE: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH" obtains "- Rf (Contracts.CH Ch) (snd XD_XH) = fst XD_XH" and "- Rf (Contracts.CD_on Pd ds) (fst XD_XH) = snd XD_XH"
induct: fixes x1 :: "'node SDG_node set"   and x2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and x3 :: "'node list"   and x4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and x5 :: "'edge list"   and x6 :: "'node list"   and x7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and P :: "'node SDG_node set \<Rightarrow> ('edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind) \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> 'edge list \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> bool" assumes "x1,x2 \<turnstile> (x3,x4) =x5\<Rightarrow>\<^sub>\<tau> (x6,x7)"   and "\<And>ms s S f. length ms = length s \<Longrightarrow> P S f ms s [] ms s"   and "\<And>S f ms s a ms' s' as ms'' s''. \<lbrakk>S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s'); S,f \<turnstile> (ms',s') =as\<Rightarrow>\<^sub>\<tau> (ms'',s''); P S f ms' s' as ms'' s''\<rbrakk> \<Longrightarrow> P S f ms s (a # as) ms'' s''" shows "P x1 x2 x3 x4 x5 x6 x7"
homogeneousD: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and s :: "'c \<Rightarrow>\<^sub>0 nat"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p"   and "s \<in> keys p"   and "t \<in> keys p" shows "deg_pm s = deg_pm t"
select_convs_1: fixes GST :: "'g \<Rightarrow> ('h, 'i, 'j, 'k) com list \<times> 'i option \<times> 'k"   and HST :: "('j \<times> 'h) list"   and more :: "'l" shows "GST \<lparr>GST = GST, HST = HST, \<dots> = more\<rparr> = GST"
simpnum_numbound0: fixes t :: "Ferrack.num" assumes "numbound0 t" shows "numbound0 (simpnum t)"
eval_map: fixes f :: "'a \<Rightarrow> atomUni fmUni"   and L :: "'a list"   and x :: "real" shows "(\<exists>l\<in>set (map f L). evalUni l x) = (\<exists>l\<in>set L. evalUni (f l) x)"
silent_move_equal_length_1: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" shows "length ms = length s"
sams_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "SAMS A B C D E F" shows "A \<noteq> B \<and> B \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F"
exhaust: fixes y :: "'c net_state" obtains x11 :: "nat" and x12 :: "'c" and x13 :: "nat set" where "y = NodeS x11 x12 x13"   | x21 :: "'c net_state" and x22 :: "'c net_state" where "y = SubnetS x21 x22"
elims: fixes x :: "('f, 'b) Term.term"   and y :: "nat" assumes "weight x = y" obtains xa :: "'b" where "x = Var xa" and "y = w0"   | f :: "'f" and ts :: "('f, 'b) Term.term list" where "x = Fun f ts" and "y = (let n = length ts; scff = scf (f, n) in w (f, n) + sum_list (map2 (\<lambda>ti i. weight ti * scff i) ts [0..<n]))"
asymp_equiv_at_top_imp_at: fixes a :: "real" assumes "(\<lambda>x. f (a - inverse x)) \<sim>[at_top] (\<lambda>x. g (a - inverse x))"   and "(\<lambda>x. f (a + inverse x)) \<sim>[at_top] (\<lambda>x. g (a + inverse x))" shows "f \<sim>[at a] g"
chain: fixes a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat" assumes "a \<in> s"   and "b \<in> s" shows "a \<le> b \<or> b \<le> a"
one_neq_zero: shows "(1::'a) \<noteq> (0::'a)"
almost_all_eventually_provable_restrict_to_G: fixes \<phi> :: "'b ltl"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" assumes "\<PP>\<^sub>\<infinity> \<phi> \<G> w"   and "Only_G \<G>" shows "\<PP>\<^sub>\<infinity> \<phi> (\<G> \<inter> \<^bold>G \<phi>) w"
card_atLeastAtMost: fixes l :: "nat"   and u :: "nat" shows "card {l..u} = Suc u - l"
enumerate_in_set: fixes S :: "'c set"   and n :: "nat" assumes "infinite S" shows "wellorder_class.enumerate S n \<in> S"
productF_path_forward: fixes w :: "('d \<times> 'e) list"   and r1 :: "'f list"   and r2 :: "'g list"   and A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and p1 :: "'f"   and p2 :: "'g" assumes "length w = length r1"   and "length r1 = length r2"   and "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path A (w || r1) p1 \<and> path B (w || r2) p2 \<or> target (w || r1 || r2) (p1, p2) = FAIL \<and> 0 < length w \<and> path A (butlast (w || r1)) p1 \<and> path B (butlast (w || r2)) p2 \<and> succ A (last w) (target (butlast (w || r1)) p1) = {} \<and> succ B (last w) (target (butlast (w || r2)) p2) \<noteq> {}"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path AB (w || r1 || r2) (p1, p2)"
approx_floatarith_Some_is_float: fixes p :: "nat"   and fa :: "floatarith"   and XS :: "((real \<times> real pdevs) \<times> real) list"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "approx_floatarith p fa XS = Some ((a, b), ba)"   and "list_all (\<lambda>((a, b), c). is_float a \<and> is_float c) XS" shows "is_float a \<and> is_float ba"
ground_factorization_is_sound: fixes C :: "'a eclause"   and D :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set"   and \<theta> :: "('a \<times> 'a trm) list" assumes "finite (cl_ecl C)"   and "factorization C D \<sigma> k C'"   and "ground_clause (subst_cl (cl_ecl D) \<theta>)" shows "clause_entails_clause (subst_cl (subst_cl (cl_ecl C) \<sigma>) \<theta>) (subst_cl (cl_ecl D) \<theta>)"
delta: fixes a :: "'c"   and g :: "'c \<Rightarrow> 'b" shows "(\<Sum>b. if b = a then g b else (0::'b)) = g a"
commute: fixes a :: "'e"   and b :: "'e" shows "sup a b = sup b a"
hyps: fixes i :: "nat"   and j :: "nat" shows "(i < row_length M * row_length M2 \<and> j < length M * length M2) \<and> mat (row_length M) (length M) M \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (M \<otimes> M2) ! j ! i = M ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
ereal_le_minus_iff: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" shows "(x \<le> z - y) = ((y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = - \<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z))"
iT_not_empty_2: fixes n :: "nat" shows "[\<dots>n] \<noteq> {}"
eucl_of_list_inner_eq: fixes xs :: "real list"   and ys :: "real list" assumes "length xs = DIM('a)"   and "length ys = DIM('a)" shows "eucl_of_list xs \<bullet> eucl_of_list ys = inner_lv_rel xs ys"
ss_exps_Vals_None: fixes vs :: "val list" shows "ss_exps (map Val vs) = None"
always_eventually: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<box>P) \<sigma>" shows "(\<diamond>P) \<sigma>"
oppositeUniqueness: fixes literal1 :: "Literal"   and literal2 :: "Literal" shows "(opposite literal1 = opposite literal2) = (literal1 = literal2)"
oreachable_local': fixes \<sigma> :: "'e"   and p :: "'f"   and A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'g"   and \<sigma>' :: "'e"   and p' :: "'f" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "(\<sigma>', p') \<in> oreachable A S U"
config_n_fst_init_length: fixes s0 :: "'b list"   and qs :: "'b list" shows "\<forall>(uu_, x, uu_)\<in>set_pmf (config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs). length x = length s0"
assms_1: shows "a \<otimes> b \<in> Units G"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes " x1 \<turnstile> x2 : x3 "   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> e\<^sub>1 T\<^sub>1 T\<^sub>2 e\<^sub>2 y. \<lbrakk> \<Gamma> \<turnstile> e\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2 ; \<And>z. P z \<Gamma> e\<^sub>1 (T\<^sub>1\<rightarrow>T\<^sub>2);  \<Gamma> \<turnstile> e\<^sub>2 : T\<^sub>1 ; \<And>z. P z \<Gamma> e\<^sub>2 T\<^sub>1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App e\<^sub>1 e\<^sub>2) T\<^sub>2"   and "\<And>x \<Gamma> T\<^sub>1 t T\<^sub>2 y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>;  (x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2 ; \<And>z. P z ((x, T\<^sub>1) # \<Gamma>) t T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (Lam [x].t) (T\<^sub>1\<rightarrow>T\<^sub>2)"   and "\<And>\<Gamma> n y. valid \<Gamma> \<Longrightarrow> P y \<Gamma> (Const n) TBase"   and "\<And>\<Gamma> y. valid \<Gamma> \<Longrightarrow> P y \<Gamma> Unit TUnit" shows "P z x1 x2 x3"
rel_funI: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
b2: shows " P \<longrightarrow>\<^sub>1 P''"
well_com_hadamard_n: shows "well_com (hadamard_n n)"
index_eq_index_conv: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "x \<in> set xs \<or> y \<in> set xs" shows "(index xs x = index xs y) = (x = y)"
translate: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "ccw a (a + b) (a + c) = ccw (0::'a) b c"
distinct_2: fixes x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Assume x2 \<noteq> Skip"
limproc_Sync_F_annex1: fixes Y :: "nat \<Rightarrow> 'c process"   and a :: "'c event list"   and b :: "'c event set"   and A :: "'c set"   and S :: "'c process" assumes "chain Y"   and "\<forall>n. (a, b) \<in> F (Y n \<lbrakk>A\<rbrakk> S)"   and "\<exists>x. a \<notin> D (Y x \<lbrakk>A\<rbrakk> S)" shows "\<exists>t u X. (\<forall>x. (t, X) \<in> F (Y x)) \<and> (\<exists>Y. (u, Y) \<in> F S \<and> a setinterleaves ((t, u), insert tick (ev ` A)) \<and> b = (X \<union> Y) \<inter> insert tick (ev ` A) \<union> X \<inter> Y)"
domain_make_vert_int: fixes n :: "int" assumes "0 \<le> n" shows "domain_block (make_vert_block (nat n)) = n"
forget_subst_trm: fixes a :: "name"   and trm :: "trm"   and x :: "trm" assumes "atom a \<sharp> trm" shows "subst a x trm = trm"
elims_2: fixes x :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.rank_invar x" obtains "x = []"   | t :: "('a, 'b) BinomialHeapStruc.BinomialTree" where "x = [t__]"   | t :: "('a, 'b) BinomialHeapStruc.BinomialTree" and t' :: "('a, 'b) BinomialHeapStruc.BinomialTree" and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "x = t # t' # bq" and "BinomialHeapStruc.rank t < BinomialHeapStruc.rank t' \<and> BinomialHeapStruc.rank_invar (t' # bq)"
\<sigma>_4: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>"
size_build: fixes ps :: "(real, 'a) vec list"   and ks :: "'a list" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)" shows "KD_Tree.size_kdt (build ks ps) = length ps"
unit_div_mult_self: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "times (divide b a) a = b"
length_merge: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "length (Common.merge f xs ys) = length xs + length ys"
suminf_ennreal2: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "summable f" shows "(\<Sum>i. ennreal (f i)) = ennreal (\<Sum>i. f i)"
well_base\<^sub>h_mono_base: fixes base :: "nat"   and M :: "hmultiset"   and base' :: "nat" assumes "well_base\<^sub>h base M"   and "2 \<le> base"   and "base \<le> base'" shows "well_base\<^sub>h base' M"
3: shows "{x. poly p x = 0 \<and> 0 < poly q x} \<inter> {x. poly p x = 0 \<and> poly q x < 0} = {}"
merge_restrict_1: fixes I :: "'d set"   and J :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" shows "merge I J (restrict x I, y) = merge I J (x, y)"
prob_space_exponential_density: fixes l :: "real" assumes "0 < l" shows "prob_space (density lborel (\<lambda>x. ennreal (exponential_density l x)))"
real_sqrt_le_iff': fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "(sqrt x \<le> y) = (x \<le> y\<^sup>2)"
suffix_same_states: shows "\<forall>k>j. S t k = S (t[j - 1 := t ! j, j := t ! (j - 1)]) k"
at_within_eq_bot_lemma: fixes x :: "'c"   and b :: "'c"   and c :: "'c" shows "at x within {b..c} = (if x < b \<or> c < b then bot else at x within {b..c})"
a: shows "\<forall>y.  |x\<rangle>  |x\<rangle> y \<le>  |x\<rangle> y"
dghm_proj_components_3: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>HomDom\<rparr> = dg_prod I \<AA>"
NotS: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>\<phi> in v] = ([\<phi> in v] \<longrightarrow> (\<forall>\<psi>. [\<psi> in v]))"
outputSupportDerivative: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto> a[b] \<prec> P'" shows "supp P' \<subseteq> supp P"
in_set_drop_conv_nth: fixes x :: "'a"   and n :: "nat"   and l :: "'a list" shows "(x \<in> set (drop n l)) = (\<exists>i\<ge>n. i < length l \<and> x = l ! i)"
simps_1: fixes q :: "hf" shows "ST.nextl q [] = q"
sliceable_M: fixes S :: "'b list set"   and v :: "'b option list"   and i :: "nat"   and \<pi> :: "'a prefix" assumes "mem_restr S v" shows "((i, v) \<in> M (pslice S \<pi>)) = ((i, v) \<in> M \<pi>)"
mul_and_mul_eq_imp_mul: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset"   and P :: "'a multiset" assumes "trans r"   and "(M, N) \<in> mul r"   and "(N, P) \<in> mul_eq r" shows "(M, P) \<in> mul r"
nn_integral_null_set: fixes N :: "'b set"   and M :: "'b measure"   and u :: "'b \<Rightarrow> ennreal" assumes "N \<in> null_sets M" shows "\<integral>\<^sup>+ x. u x * indicator N x \<partial>M = 0"
simps_3: fixes xs :: "('b, 'c) Combinators list" shows "singleCombinators (DenyAll # xs) = singleCombinators xs"
keys_pp_of_exps: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "keys_pp (fst (rep_nat_term (Term_Order.of_exps a b i))) \<subseteq> {0, 1}"
subterm_inj_on_alt_def: fixes f :: "'a \<Rightarrow> ('b, 'c) Term.term"   and A :: "'a set" shows "subterm_inj_on f A = (inj_on f A \<and> (\<forall>s\<in>f ` A. \<forall>u\<in>f ` A. (\<exists>v. v \<sqsubseteq> s \<and> v \<sqsubseteq> u) \<longrightarrow> s = u))"
fixed_points_def: shows "H'act.fixed_points = {m \<in> rcosets P. carrier H' \<subseteq> H'act.stabilizer m}"
eqv_def: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" shows "eqv \<phi> \<chi> = cnj (imp \<phi> \<chi>) (imp \<chi> \<phi>)"
bind_advantage: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> ('grp \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "Schnorr_\<Sigma>_commit.abstract_com.bind_advantage \<eta> \<A> \<le> Schnorr_\<Sigma>_commit.rel_advantage \<eta> (Schnorr_\<Sigma>_commit.adversary \<eta> \<A>)"
that: assumes "path w p"   and "target w p = p" shows "thesis"
valid_Act: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act) formula" shows "P \<Turnstile> Formula.Act \<alpha> x = (\<exists>\<alpha>'' x' P'. Formula.Act \<alpha> x = Formula.Act \<alpha>'' x' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
contour_integrable_continuous_linepath: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (closed_segment a b) f" shows "f contour_integrable_on linepath a b"
Un_absorb1: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B" shows "A \<union> B = B"
first_root_nat: fixes n :: "nat" shows "nth_root_nat 1 n = n"
check_eqv_def: fixes idx :: "idx"   and n :: "nat"   and r :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and s :: "(presb, unit) aformula" shows "WS1S_Presb.check_eqv idx n r s = ((ws1s_wf idx r \<and> ws1s_lformula r) \<and> (presb_wf n s \<and> presb_lformula s) \<and> (case WS1S_Presb.closure idx n (WS1S_Formula.norm (WS1S_Formula.RESTRICT r), Presburger_Formula.norm (Presburger_Formula.RESTRICT s)) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False))"
STEPFMT_2: shows "ch = add_mset (uh # rh) ceh"
intros: fixes C :: "sequent"   and A :: "form"   and Ps :: "sequent list" assumes "C = ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>)"   and "A \<noteq> ff" shows "leftPrincipal (Ps, C) A"
fs_int_d_pos: fixes k :: "nat" assumes "k \<le> length fs" shows "0 < d fs k"
hmset_of_enat_0: shows "hmset_of_enat 0 = 0"
nth_restrict: fixes i :: "nat"   and A :: "nat set"   and v :: "'b option list" assumes "i \<in> A"   and "i < length v" shows "restrict A v ! i = v ! i"
epiE: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and g' :: "('a S.E.term, 'a) S.arr" assumes "S.epi f"   and "S.seq g f"   and "S.seq g' f"   and "g \<cdot>\<^sub>S f = g' \<cdot>\<^sub>S f" shows "g = g'"