hcju/isabelleps huggingface dataset:
prod_emb_subset_PiM: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and K :: "'b set"   and X :: "('b \<Rightarrow> 'c) set" shows "prod_emb I M K X \<subseteq> space (Pi\<^sub>M I M)"
arith_simps_10: fixes m :: "num" shows "m * num.One = m"
Red_I_subset_Liminf: fixes Ns :: "'f set llist"   and i :: "nat" assumes "chain (\<rhd>) Ns"   and "enat i < llength Ns" shows "Red_I (lnth Ns i) \<subseteq> Red_I (Liminf_llist Ns)"
u: shows "u \<equiv> getFr (x # map snd txs) (map fst txs) [\<phi>]"
sup_compl_top: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "x \<oplus> x\<^sup>c = \<tau>"
difibs: shows "(d\<inverse> \<union> f\<inverse> \<union> ov \<union> e \<union> f \<union> m \<union> b \<union> s\<inverse> \<union> s) O (b \<union> s \<union> m) \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse> \<union> d \<union> e \<union> s \<union> s\<inverse>"
compat_different_vars: fixes mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "fst (mems ! i) x = snd (mems ! i) x"   and "x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "mem\<^sub>1 x = mem\<^sub>2 x"
smallstepP_ps_prepend: fixes \<pi> :: "Syntax.term list"   and eP :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and eP' :: "Syntax.term"   and X :: "Syntax.term list" shows "\<lless>\<pi>, eP\<ggreater> P\<rightarrow> \<lless>\<pi>', eP'\<ggreater> = \<lless>X @ \<pi>, eP\<ggreater> P\<rightarrow> \<lless>X @ \<pi>', eP'\<ggreater>"
on_circline_opposite_ocircline_cmat: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" assumes "hermitean H \<and> H \<noteq> mat_zero"   and "z \<noteq> vec_zero" shows "on_circline_cmat_cvec (opposite_ocircline_cmat H) z = on_circline_cmat_cvec H z"
p1_not_equil: fixes x :: "real"   and y :: "real" assumes "p1 x y \<le> 0" shows "gx x y \<noteq> 0 \<or> gy x y \<noteq> 0"
is_valid_operator_sas_plus_then_2: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (precondition_of op). \<R>\<^sub>+ \<Psi> v \<noteq> {} \<and> a \<in> \<R>\<^sub>+ \<Psi> v"
pre_bfs2_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and succ :: "nat \<Rightarrow> nat list nres"   and src :: "nat"   and dst :: "nat" shows "Graph.pre_bfs2 c succ src dst \<equiv> Graph.init_state src \<bind> (\<lambda>s. while\<^sub>T (\<lambda>(f, PRED, C, N, d). f = False \<and> C \<noteq> {}) (\<lambda>(f, PRED, C, N, d). assert (C \<noteq> {}) \<bind> (\<lambda>_. IICF_Set.op_set_pick C \<bind> (\<lambda>v. let C = C - {v} in assert (v \<in> Graph.V c) \<bind> (\<lambda>_. succ v \<bind> (\<lambda>sl. Graph.inner_loop2 dst sl v PRED N \<bind> (\<lambda>(f, PRED, N). if f then return (f, PRED, C, N, d + 1) else assert (case (f, PRED, C, N, d) of (f, PRED, C, N, d) \<Rightarrow> \<not> f \<and> nf_invar' c src dst PRED C N d) \<bind> (\<lambda>_. if C = {} then let C = N; N = {}; d = d + 1 in return (f, PRED, C, N, d) else return (f, PRED, C, N, d)))))))) s \<bind> (\<lambda>(f, PRED, uu_, uu_, d). if f then return (Some (d, PRED)) else return None))"
IH: assumes "\<And>x. x \<in> F \<Longrightarrow> f x \<in>\<^sub>i g x" shows "sum f F \<in>\<^sub>i sum g F"
Bernstein_coeffs_01_x: fixes p :: "nat" assumes "p \<noteq> 0" shows "Bernstein_coeffs_01 p (monom 1 1) = map (\<lambda>i. real i / real p) [0..<p + 1]"
fin_cut_sameE: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "fin_cut_same x xs = ys" shows "\<exists>m. xs = ys @ replicate m x"
axioms_1: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "group G composition unit" shows "monoid G composition unit"
append_eq_append_conv2: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list"   and ts :: "'c list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
set_gb_red_aux: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "set (gb_red_aux bs ps) = trdsp (map fst bs) ` set ps - {0}"
rel_set_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
simps_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and z :: "Vars.var"   and body :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* (y, z)" shows "(Let \<Gamma> body)[y::=z] = Let \<Gamma>[y::h=z] body[y::=z]"
simps_2: fixes e :: "expr"   and es :: "expr list" shows "assignsEs (e # es) = assignsE e \<union> assignsEs es"
SUP_subset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
dghm_ObjMap_vdomain: shows "\<D>\<^sub>\<circ> (\<phi> i\<lparr>ObjMap\<rparr>) = \<CC>\<lparr>Obj\<rparr>"
merge_it_def: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and ys :: "'c list" shows "merge_it f xs ys = merge_it' f [] xs ys"
sup_ge2: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (y::'b) ((sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) y)"
IH: assumes "us \<in> lists \<langle>G\<rangle>" shows "Ref G us \<in> lists G\<^sub>+"
case_prodI2: fixes p :: "'e \<times> 'f"   and c :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
inject_2: fixes x3 :: "ty"   and x1 :: "name"   and x2 :: "btrm"   and y3 :: "ty"   and y1 :: "name"   and y2 :: "btrm" shows "(Bind x3 x1 x2 = Bind y3 y1 y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
interp_imp_Interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "C \<le> D"   and "D \<le> D'"   and "interp D \<Turnstile> C" shows "Interp D' \<Turnstile> C"
the_\<theta>_simps_1: shows "arr \<omega>.the_\<theta>"
single_diff: fixes k :: "'a"   and a :: "'b"   and b :: "'b" shows "monomial (a - b) k = monomial a k - monomial b k"
l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr assigned_nodes flatten_dom Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_tag_name get_tag_name_locs Shadow_DOM.get_root_node get_host get_host_locs find_slot Shadow_DOM.remove Shadow_DOM.append_child remove_shadow_root ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs Shadow_DOM.get_parent Shadow_DOM.to_tree_order"
is_public_def: fixes G :: "prog"   and qn :: "qtname" shows "is_public G qn = (case class G qn of None \<Rightarrow> case iface G qn of None \<Rightarrow> False | Some i \<Rightarrow> access i = Public | Some c \<Rightarrow> access c = Public)"
ProcSpec: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and p :: "'f" assumes "P \<subseteq> {s. \<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> return s t \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) Call p (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return c Q,A"
assms_5: shows "Marker \<notin> set (msgs c cid)"
"1_1": shows "(T, S) \<in> \<langle>A\<rangle>Rs1"
assms_8: shows "i (t + 3 + k + d) \<noteq> []"
join_def: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" shows "x \<squnion>\<^bsub>L\<^esub> y = \<Squnion>\<^bsub>L\<^esub>{x, y}"
assms: shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>count_space A < \<infinity>"
assms_3: assumes "\<gamma>' t n" shows "\<gamma>'' t n"
shC': shows "sh C' = \<lfloor>(sfs, i)\<rfloor>"
vNM_utilty_implies_continuity: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)" shows "continuous_vnm (\<P> outcomes) \<R>"
inf_sup_ord_4: fixes y :: "'b"   and x :: "'b" shows "y \<le> x \<sqinter> y"
bind_runit: fixes x :: "'c option" shows "Option.bind x Some = x"
urD_5: fixes y :: "real" shows "(real_of_1 (p, l, r) = y) = root_cond (p, l, r) y"
call_only_one_intra_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<exists>!a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<and> intra_kind (kind a')"
hidings_entailsD: fixes t :: "'a \<Rightarrow> 'b set"   and s :: "'a \<Rightarrow> 'c set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and k :: "'a"   and y :: "'c" assumes "t hidings s entails R"   and "x \<in> t k"   and "y \<in> s k" shows "R x y"
psubstEnv_substEnv_compose_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "goodEnv rho'"   and "good Y"   and "freshEnv ys y rho'" shows "rho &[Y / y]_ys &[rho'] = rho &[rho'] &[Y #[rho'] / y]_ys"
length_zip: fixes xs :: "'b list"   and ys :: "'c list" shows "length (zip xs ys) = min (length xs) (length ys)"
get_parent_def: shows "get_parent \<equiv> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent get_child_nodes"
B_equiv: fixes g :: "dB"   and f :: "dB"   and x :: "dB" shows "\<B> \<degree> g \<degree> f \<degree> x \<leftrightarrow> g \<degree> (f \<degree> x)"
poly_x_minus_y_as_comp: shows "poly_x_minus_y = (\<lambda>p. p \<circ>\<^sub>p x_y) \<circ> poly_lift"
abs_eq: fixes xa :: "'a"   and x :: "((real, 'a) vec, 'a) vec" shows "\<r>\<o>\<w> xa (to_mtx x) = row xa x"
a6: shows "\<alpha> sctni \<union> \<alpha> b = \<alpha> (uni sctni b)"
ccw_transitive_contr: fixes t :: "real \<times> real"   and s :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "ccw t s p"   and "ccw t s q"   and "ccw t s r"   and "ccw t p q"   and "ccw t q r"   and "ccw t r p"   and "distinct5 t s p q r" shows "False"
assms_2: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "(g has_derivative g' x) (at x within S)"
rep_eq: fixes x :: "'c" shows "(*\<^sub>v) (blinfun_mult_right x) = (*) x"
F_def: shows "F \<equiv> \<lambda>N x. if (\<exists>m\<le>N. x \<in> \<Union> (B m ` {..N})) \<and> (\<exists>n\<le>N. x \<in> B (m N x) n) then e (LEAST n. x \<in> B (m N x) n) else z"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (\<tau>\<^sub>0 a) = F.map\<^sub>0 a"
leofD: fixes m :: "'b nres"   and m' :: "'b nres" assumes "nofail m"   and "m \<le>\<^sub>n m'" shows "m \<le> m'"
l2_inv1_init: shows "init l2 \<subseteq> l2_inv1"
rel_distinct_18: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "\<not> rel_basic_type R Boolean OclVoid"
fin_A_n: fixes n :: "nat" shows "finite (A_n n)"
mrbst_join_Leaf_right: fixes t :: "'a tree" shows "mrbst_join t \<langle>\<rangle> = return_pmf t"
rel_star_contr: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'b) set" shows "X\<^sup>* O Y = (\<Union>n. rel_dioid.power X n O Y)"
dec_I_mk_I_satisfies_cong: fixes \<phi> :: "WS1S_Nameful.ws1s"   and bs1 :: "WS1S_Nameful.fo list"   and bs2 :: "WS1S_Nameful.so list"   and \<AA> :: "interp"   and I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset" assumes "set (fos \<phi>) \<subseteq> set bs1"   and "set (sos \<phi>) \<subseteq> set bs2"   and "\<AA> = mk_I I1 I2 bs1 bs2" shows "satisfies (dec_I1 \<AA> bs1) (dec_I2 \<AA> bs2) \<phi> = satisfies I1 I2 \<phi>"
group_eq_aux: fixes a :: "'b"   and b :: "'b" shows "a + (b - a) = b"
order_tendstoD_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and y :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "class.order_topology less_eq less open"   and "filterlim f (topological_space.nhds open y) F"   and "less y a" shows "\<forall>\<^sub>F x in F. less (f x) a"
subsumeseq_term_refl: fixes t :: "('a, 'b) Term.term" shows "t \<le>\<cdot> t"
\<Gamma>_wf_2: fixes x :: "'var" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<Gamma> (Var x))"
WT_gpv_OutD: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
assms: shows "xs \<noteq> []\<^sub>\<circ>"
da_elim_cases_42: fixes Env :: "env"   and B :: "lname set"   and v :: "Term.var"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (v:=e)\<guillemotright> A" obtains E :: "assigned" and vn :: "lname" where "v = LVar vn" and "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "nrm A = insert vn (nrm E)" and "brk A = brk E"   | V :: "assigned" where "\<forall>vn. v \<noteq> LVar vn" and "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> V" and "Env\<turnstile> nrm V \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
Array_widen: fixes P :: "'a prog"   and A :: "ty"   and T :: "ty" assumes "P \<turnstile> A\<lfloor>\<rceil> \<le> T" shows "(\<exists>B. T = B\<lfloor>\<rceil> \<and> P \<turnstile> A \<le> B) \<or> T = Class Object"
144: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- (x \<sqinter> y) \<sqinter> x \<squnion> z) \<sqinter> y = z \<sqinter> y"
nrows: fixes A :: "'b matrix"   and m :: "nat"   and n :: "nat" assumes "nrows A \<le> m" shows "Rep_matrix A m n = (0::'b)"
L_def: shows "L = lambda * (4 * 1 * deltaG TYPE('b) + 1 * 1 * C + 2 * K)"
init_fin_descend_thr_def: fixes ts :: "'c \<Rightarrow> (('f \<times> 'd) \<times> 'e \<Rightarrow>f nat) option" shows "init_fin_descend_thr ts = map_option (\<lambda>((s, x), ln). (x, ln)) \<circ> ts"
comb_spec2: fixes X :: "'b"   and b :: "'a"   and a :: "'a" assumes "Q X b" shows "comb a b \<le> SPEC (Q X)"
action_needed_asses_def: fixes a :: "('d, 'e) fmap \<times> 'f"   and s :: "('d, 'e) fmap" shows "action_needed_asses a s \<equiv> fmrestrict_set (action_needed_vars a s) s"
rho: fixes x :: "'var" assumes "x \<notin> domA h" shows "\<rho> x \<sqsubseteq> r x"
fmrestrict_fset_add_distrib: fixes A :: "'c fset"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmrestrict_fset A (m ++\<^sub>f n) = fmrestrict_fset A m ++\<^sub>f fmrestrict_fset A n"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT ($f x1a x2a)}. FVT (\<sigma> i))"
\<sigma>E: fixes i :: "nat"   and f :: "'f"   and ss :: "'a list" assumes "i \<in> set (\<sigma> (f, length ss))" shows "ss ! i \<in> set ss"
c2: shows "\<forall>\<psi>\<in>set ms. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[set ns]\<^sub>\<Sigma>\<^sub>1)"
const4_def: fixes x :: "'j" shows "const4 x \<equiv> \<lambda>_ _ _ _. x"
numer_cf5_pos: fixes x :: "real" assumes "- (7293 / 10 ^ 3) \<le> x" shows "0 < numer_cf5 x"
has_sort_Tv_refl: fixes oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and S :: "String.literal set"   and v :: "variable" assumes "wf_osig oss"   and "sort_ex (subclass oss) S" shows "has_sort oss (Tv v S) S"
induct: fixes x1 :: "'a list"   and x2 :: "'b list"   and r :: "('a \<times> 'b) set"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "(x1, x2) \<in> listrel r"   and "P [] []"   and "\<And>x y xs ys. \<lbrakk>(x, y) \<in> r; (xs, ys) \<in> listrel r; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
hom_0_iff: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "zero_hom_0 hom" shows "(hom x = (0::'b)) = (x = (0::'a))"
prodag_def: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" shows "a\<Pi>\<^bsub>I\<^esub> A = \<lparr>carrier = carr_prodag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A\<rparr>"
dimacs_to_var_def: fixes v :: "nat" shows "dimacs_to_var v \<equiv> if v < 1 + Suc (length ast\<delta>) * Suc h then Operator ((v - 1) mod Suc h) ((v - 1) div Suc h) else let k = v - 1 - Suc (length ast\<delta>) * Suc h in State (k mod Suc h) (k div Suc h)"
trms\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
ind: fixes j :: "name"   and i :: "name"   and A :: "fm" assumes "atom j \<sharp> (i, A)" shows "A(i::=Zero) IMP SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j)))) IMP SyntaxN.All i A \<in> induction_axioms"
hide_write0: fixes a :: "'a"   and B :: "'a set"   and P :: "'a process" assumes "a \<in> B" shows "(a \<rightarrow> P) \ B = P \ B"
UnI1_single: fixes a :: "'c"   and B :: "'c set" shows "a \<in> {a} \<union> B"
connected_simple_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "simple_path g" shows "connected (path_image g)"
freshSeqSubstName: fixes x :: "name"   and a :: "name"   and s :: "(name \<times> name) list" assumes "x \<noteq> a"   and "x \<sharp> s" shows "x \<noteq> seq_subst_name a s"
lspasl_a_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and h4 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0"   and "heap_sep_algebra.tern_rel plus sep_disj h3 h4 h1" shows "\<exists>h5. heap_sep_algebra.tern_rel plus sep_disj h3 h5 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h2 h4 h5 \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h3 h4 h1"
a: fixes y :: "real" assumes "y \<in> {l..u}" shows "a \<le> f' y"
weakTransition_def: shows "weakTransition \<equiv> transition \<union> {x. \<exists>P. x = (P, \<tau> \<prec> P)}"
2: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "t \<in> set M"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "comp_timpl_closure {t} (set TI') = timpl_closure_set {t} (set TI)"
auxifalse_assu: shows "\<forall>n\<ge>m. real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1))) < d (n + 1) * (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j)))"
exhaust: fixes y :: "'d seq_action" obtains x1 :: "'d" where "y = broadcast x1"   | x21 :: "nat set" and x22 :: "'d" where "y = groupcast x21 x22"   | x31 :: "nat" and x32 :: "'d" where "y = unicast x31 x32"   | x4 :: "nat" where "y = \<not>unicast x4"   | x5 :: "'d" where "y = send x5"   | x6 :: "nat" where "y = deliver x6"   | x7 :: "'d" where "y = receive x7"   | "y = \<tau>\<^sub>s"
update_since_def: fixes I :: "\<I>"   and pos :: "bool"   and rel1 :: "'a option list set"   and rel2 :: "'a option list set"   and nt :: "nat"   and aux :: "(nat \<times> 'a option list set) list" shows "update_since I pos rel1 rel2 nt aux = (let aux = case concat (map (\<lambda>(t, rel). if enat nt - enat t \<le> right I then [(t, join rel pos rel1)] else []) aux) of [] \<Rightarrow> [(nt, rel2)] | x # aux' \<Rightarrow> if fst x = nt then (fst x, snd x \<union> rel2) # aux' else (nt, rel2) # x # aux' in (foldr (\<union>) (concat (map (\<lambda>(t, rel). if left I \<le> nt - t then [rel] else []) aux)) {}, aux))"
swap_def: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "Fun.swap a b f = f(a := f b, b := f a)"
j_p_1: shows "(C, j) \<in># wP_of_wstate (lnth Sts (t_C + t_Cj))"
assms: shows "r\<inverse>\<^sub>\<circ> \<subseteq>\<^sub>\<circ> s"
PDG_BS_def: fixes S :: "'node set" shows "PDG_BS S \<equiv> {n'. \<exists>n. n' \<longrightarrow>\<^sub>d* n \<and> n \<in> S \<and> valid_node n}"
dis_rews_2: fixes x :: "A" shows "(is_A\<cdot>x = \<bottom>) = (x = \<bottom>)"
cexpr_type_Some_iff: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type" shows "(cexpr_type \<Gamma> e = Some t) = (\<Gamma> \<turnstile>\<^sub>c e : t)"
pm': shows "(case d of L \<Rightarrow> m \<otimes> LL | R \<Rightarrow> m \<otimes> UR) \<otimes> pm' = pm"
order_0I: fixes p :: "'b poly"   and a :: "'b" assumes "poly p a \<noteq> (0::'b)" shows "order a p = 0"
UP_a_comm: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
1: shows "{EqOnUC p p' u c |u c. (u, c) \<in> (UNIV - GH) \<times> UNIV} = (\<lambda>(u, c). EqOnUC p p' u c) ` ((UNIV - GH) \<times> UNIV)"
borel_measurable_times_ennreal: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
SUP_L_P_def: fixes pair :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and u :: "'d"   and i :: "'c" shows "DiagramTermination.SUP_L_P pair X u i = (SUP v\<in>{v. pair v i \<sqsubset> u}. X v i)"
continuous_map_iff_continuous: fixes S :: "'c set"   and g :: "'c \<Rightarrow> 'd" shows "continuous_map (top_of_set S) euclidean g = continuous_on S g"
Abs2: fixes x :: "'c"   and y :: "'d" assumes "T2 x y" shows "Abs2 x = y"
simps_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # ys) = (Ball (list.set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
Neg_SubstAtomicP_Fls: fixes y :: "tm"   and z :: "tm"   and v :: "tm"   and tm :: "tm"   and y' :: "tm" shows "{y EQ Q_Neg z, SubstAtomicP v tm y y'} \<turnstile> Fls"
ee_refl: fixes as :: "'a list" assumes "set as \<subseteq> carrier G" shows "essentially_equal G as as"