hcju/isabelleps huggingface dataset:
Complete_def: fixes k :: "nat"   and I :: "item set" shows "Complete k I = I \<union> {uu_. \<exists>x y. uu_ = inc_item x k \<and> x \<in> bin I (item_origin y) \<and> y \<in> bin I k \<and> is_complete y \<and> next_symbol x = Some (item_nonterminal y)}"
times_zhmultiset_def: shows "(*) \<equiv> map_fun zhmsetmset (map_fun zhmsetmset ZHMSet) (\<lambda>M N. zmset_of (hmsetmset (HMSet (mset_pos M) * HMSet (mset_pos N))) - zmset_of (hmsetmset (HMSet (mset_pos M) * HMSet (mset_neg N))) + zmset_of (hmsetmset (HMSet (mset_neg M) * HMSet (mset_neg N))) - zmset_of (hmsetmset (HMSet (mset_neg M) * HMSet (mset_pos N))))"
in_set_conv_nth: fixes x :: "'k"   and xs :: "'k list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
singleton_insert_inj_eq: fixes b :: "'k"   and a :: "'k"   and A :: "'k set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"
homotopic_with_imp_subset2: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and X :: "'c set"   and Y :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "homotopic_with_canon P X Y f g" shows "g ` X \<subseteq> Y"
upper_bounds_sub: fixes X :: "'a set" assumes "X \<subseteq> carrier D" shows "upper_bounds D X \<subseteq> carrier D"
canon_ang_3: fixes \<alpha> :: "real" shows "\<exists>x. \<alpha> - \<downharpoonright>\<alpha>\<downharpoonleft> = 2 * real_of_int x * pi"
add_diff_assoc2: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "b - a + c = b + c - a"
UINF_ind_closed: fixes H :: "'a upred \<Rightarrow> 'a upred"   and P :: "'b \<Rightarrow> 'a upred"   and Q :: "'b \<Rightarrow> 'a upred" assumes "Continuous H"   and "\<And>i. P i = true"   and "\<And>i. Q i is H" shows "UINF P Q is H"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "sum_0 g (insert x A) = (if x \<in> A then sum_0 g A else g x + sum_0 g A)"
sv01: assumes "n \<noteq> 0" shows "s\<^bsub>01\<^esub> (f \<triangleright> n) = eval g [f \<triangleright> n]"
\<alpha>_limit_set_invariant: fixes x :: "'a" shows "invariant (\<alpha>_limit_set x)"
direct_decomp_independent_some_basis: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> local.subspace s" shows "local.independent (\<Union> (some_basis ` set ss))"
lookup_mult: fixes f :: "'b \<Rightarrow>\<^sub>0 'a"   and g :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "lookup (f * g) k = (\<Sum>l. lookup f l * (\<Sum>q. lookup g q when k = l + q))"
1: fixes f :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "f \<in> F'"   and "t \<in> keys (q f)" shows "deg_pm t + poly_deg f = poly_deg p"
fold_equality: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z :: "'e"   and B :: "'e set"   and y :: "'e" assumes "comp_fun_commute_on f A"   and "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "Finite_Set.fold f z B = y"
a: shows "elem \<lbrakk>\<Psi> (l, u) i\<rbrakk> (max l l', min u u')"
\<Delta>'''_sel_2: fixes x :: "'v"   and y :: "'v" shows "capacity \<Delta>'''.\<Delta>''' (x, y) = (if x = sink \<Delta>''' \<or> y = source \<Delta>''' then 0 else capacity \<Delta>''' (x, y))"
rep_preserves_seq: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.seq g f" shows "Seq (\<F>C.rep g) (\<F>C.rep f)"
FOREACHc_def: shows "FOREACH\<^sub>C \<equiv> FOREACH\<^sub>C\<^bsup>\<lambda>_ _. True\<^esup>"
induct: fixes x1 :: "('c, 'd) tm"   and x2 :: "('c, 'd) tm"   and P :: "('c, 'd) tm \<Rightarrow> ('c, 'd) tm \<Rightarrow> bool" assumes "x1 \<unrhd>\<^sub>e\<^sub>m\<^sub>b x2"   and "\<And>t. P t t"   and "\<And>t u s. \<lbrakk>t \<unrhd>\<^sub>e\<^sub>m\<^sub>b u; P t u; u \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s\<rbrakk> \<Longrightarrow> P t s" shows "P x1 x2"
altinsertd: shows "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) d = D"
assms_2: shows "backward_simulation step2 step3 final2 final3 order2 match2"
eval_poly_append: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and mc1 :: "('d monom \<times> 'c) list"   and mc2 :: "('d monom \<times> 'c) list" shows "eval_poly \<alpha> (mc1 @ mc2) = eval_poly \<alpha> mc1 + eval_poly \<alpha> mc2"
map_filter_M2_pure: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> ('b, 'c, 'd option) prog"   and h :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_filter_M2 f xs) h"
healthy_nearly_healthy: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "nearly_healthy t"
zcount_diff: fixes M :: "'a zmultiset"   and N :: "'a zmultiset"   and a :: "'a" shows "zcount (M - N) a = zcount M a - zcount N a"
inputInduct: fixes P :: "pi"   and a :: "name"   and u :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'"   and "\<And>a x P u C. \<lbrakk>x \<sharp> C; x \<noteq> u; x \<noteq> a\<rbrakk> \<Longrightarrow> F C (a<x>.P) a u (P[x::=u])"   and "\<And>P a u P' b C. \<lbrakk>P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a u P'"   and "\<And>P a u P' b c C. \<lbrakk>P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'; \<And>C. F C P a u P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a u P'"   and "\<And>P a u P' Q C. \<lbrakk>P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a u P'"   and "\<And>Q a u Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e Q'; \<And>C. F C Q a u Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a u Q'"   and "\<And>P a u P' Q C. \<lbrakk>P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a u (P' \<parallel> Q)"   and "\<And>Q a u Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e Q'; \<And>C. F C Q a u Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a u (P \<parallel> Q')"   and "\<And>P a u P' x C. \<lbrakk>P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'; x \<noteq> a; x \<noteq> u; x \<sharp> C; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a u (<\<nu>x>P')"   and "\<And>P a u P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'; \<And>C. F C (P \<parallel> !P) a u P'\<rbrakk> \<Longrightarrow> F C (!P) a u P'" shows "F C P a u P'"
maximal_ends_on_deadend: fixes P :: "'a llist" assumes "maximal_path P"   and "lfinite P"   and "\<not> lnull P" shows "deadend (llast P)"
assms_4: fixes x :: "nat" assumes "x\<^sub>1 \<le> x" shows "t x < x"
b_trans: fixes s :: "'e"   and b :: "'e list"   and T :: "('e, 'f) TS_scheme"   and t :: "'e" assumes "s # b \<in> beh T"   and "(s, t) \<in> TS.trans T" shows "t # s # b \<in> beh T"
less_imp_neq: fixes x :: "'k multiset"   and y :: "'k multiset" assumes "x \<subset># y" shows "x \<noteq> y"
right_hom_is_subcategory: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "partial_magma.arr V \<mu>" shows "subcategory V (weak_composition.right V H \<mu>)"
stat_ref_comp: shows "statically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
subdivision_induct: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and P :: "('a, 'b) pre_digraph \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> bool" assumes "subdivision (G, rev_G) (H, rev_H)"   and "bidirected_digraph G rev_G \<Longrightarrow> P G rev_G"   and "\<And>I rev_I H rev_H u v w uv uw vw. \<lbrakk>subdivision (G, rev_G) (I, rev_I); P I rev_I; subdivision_step I rev_I H rev_H (u, v, w) (uv, uw, vw)\<rbrakk> \<Longrightarrow> P H rev_H" shows "P H rev_H"
INF_greatest: fixes A :: "'d set"   and u :: "'e \<times> 'f"   and f :: "'d \<Rightarrow> 'e \<times> 'f" assumes "\<And>i. i \<in> A \<Longrightarrow> less_eq_prod u (f i)" shows "less_eq_prod u (Inf_prod (f ` A))"
scaleC_Pair: fixes r :: "complex"   and a :: "'d"   and b :: "'e" shows "r *\<^sub>C (a, b) = (r *\<^sub>C a, r *\<^sub>C b)"
FadmitD: shows "Fadmit \<sigma> (foldr (||) \<Delta> FF)"
\<I>\<^sub>\<tau>_1: shows "subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)"
diff_eq_diff_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a - b = c - d" shows "(a = b) = (c = d)"
elims: fixes x :: "nat"   and xa :: "'v \<times> 'v"   and y :: "ennreal" assumes "h_minus_aux x xa = y" obtains e :: "'v \<times> 'v" where "x = 0" and "xa = e__" and "y = 0"   | j :: "nat" and e :: "'v \<times> 'v" where "x = Suc j" and "xa = e" and "y = (if e \<in> set (cycle_edges (enum_cycle j)) then h_minus_aux j e + Min {h' e' - h_minus_aux j e' |e'. e' \<in> set (cycle_edges (enum_cycle j))} else h_minus_aux j e)"
inf_top_left: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and x :: "'c" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf top x = x"
T: shows "T \<in> M \<rightarrow>\<^sub>M M'"
dist: fixes n :: "nat"   and fs :: "'b vec list" assumes "gram_schmidt_fs_lin_indpt n fs" shows "distinct fs"
holomorphic_pre_zeta: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and a :: "real" assumes "f holomorphic_on A"   and "0 < a" shows "(\<lambda>z. pre_zeta a (f z)) holomorphic_on A"
rep_eq: fixes x :: "'a multiset"   and xa :: "'a multiset" shows "count (x - xa) = (\<lambda>a. count x a - count xa a)"
Atoms_def: fixes R :: "'b rexp set" shows "Atoms R \<equiv> \<Union> (atoms ` R)"
simps: fixes splitf :: "'b list \<Rightarrow> 'b list \<times> 'b list"   and xs :: "'b list" shows "chop splitf xs = (if reducing splitf then let pp = splitf xs in if fst pp = [] then ([], xs) else let qq = chop splitf (snd pp) in (fst pp # fst qq, snd qq) else undefined)"
get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf: fixes h :: "(_) heap"   and object_ptr :: "(_) object_ptr" assumes "ObjectClass.type_wf h" shows "(object_ptr |\<in>| object_ptr_kinds h) = (get object_ptr h \<noteq> None)"
wf_no_infinite_down_chainE: fixes r :: "('g \<times> 'g) set"   and f :: "nat \<Rightarrow> 'g" assumes "wf r" obtains k :: "nat" where "(f (Suc k), f k) \<notin> r"
augment_rels_def: fixes S :: "nat set"   and R :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" shows "augment_rels S R \<R> = (\<lambda>a. if a \<in> R then case \<R> a of None \<Rightarrow> Some (a \<in> S) | Some s \<Rightarrow> Some (s \<and> a \<in> S) else \<R> a)"
t_stop_zero: assumes "t_stop = 0" shows "v = 0"
tendsto_at_iff_tendsto_nhds: fixes g :: "'c \<Rightarrow> 'd"   and l :: "'c" shows "g \<midarrow>l\<rightarrow> g l = (g \<longlongrightarrow> g l) (nhds l)"
config'_induct: fixes P :: "'state \<Rightarrow> bool"   and init :: "'state \<times> 'b"   and A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and rs :: "'request list" assumes "P (fst init)"   and "\<And>s q a. P s \<Longrightarrow> P (step s q a)" shows "P (fst (config' A init rs))"
pt_bij3: fixes x :: "'d"   and y :: "'d"   and pi :: "('e \<times> 'e) list" assumes "x = y" shows "pi \<bullet> x = pi \<bullet> y"
h5b: assumes "0 \<le> Av"   and "Bv \<le> 0"   and "Av\<^sup>2 < Bv\<^sup>2 * Cv" shows "Av < - Bv * sqrt Cv"
dirichlet_prod_def: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "dirichlet_prod f g = (\<lambda>n. \<Sum>d | d dvd n. f d * g (n div d))"
wt_expr_eq: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and t :: "expr"   and U :: "tys" shows "(E,dt\<Turnstile>In1l t\<Colon>U) = (\<exists>T. U = Inl T \<and> E,dt\<Turnstile>t\<Colon>-T)"
fst_image_times: fixes A :: "'c set"   and B :: "'d set" shows "fst ` (A \<times> B) = (if B = {} then {} else A)"
assms: shows "iobs \<in> envObs a ` set envInit"
in_nres_rel_iff: fixes a :: "'c nres"   and b :: "'d nres"   and R :: "('c \<times> 'd) set" shows "((a, b) \<in> \<langle>R\<rangle>nres_rel) = (a \<le> \<Down> R b)"
box_below: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.po below"   and "below a b"   and "below c a"   and "below b d" shows "below c d"
fps_poly_swap2_def: fixes p :: "'b poly fps" shows "fps_poly_swap2 p = Abs_poly (\<lambda>m. Abs_fps (\<lambda>n. coeff (p $ n) m))"
rep_eq: fixes x :: "'a boolean_subset" shows "Rep_boolean_subset (- x) = - Rep_boolean_subset x"
set_ConsD: fixes y :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "y \<in>. x # xs" shows "y = x \<or> y \<in>. xs"
prj_simps_10: shows "Maps.arr SPN_fgh.Prj\<^sub>1"
keys_monom_multE: fixes v :: "'t"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (monom_mult c t p)" obtains u :: "'t" where "u \<in> keys p" and "v = t \<oplus> u"
source_UNIV: fixes x :: "'d \<Longrightarrow> 'e" assumes "wb_lens x" shows "\<S>\<^bsub>x\<^esub> = UNIV"
par_comp_constr_typed: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "par_comp \<A> Sec"   and "\<I> \<Turnstile> \<langle>unlabel \<A>\<rangle>"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>)" shows "(\<forall>l. \<I> \<Turnstile> \<langle>proj_unl l \<A>\<rangle>) \<or> (\<exists>\<A>''. prefix \<A>'' \<A> \<and> strand_leaks\<^sub>l\<^sub>s\<^sub>t \<A>'' Sec \<I>)"
ih: assumes "inj_on f (set_mset M)" shows "count (image_mset f M) y \<le> count M (inv_into (set_mset M) f y)"
simps: fixes a1 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and a2 :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" shows "enabled a1 a2 = (\<exists>\<omega> s. a1 = s \<and> a2 = \<omega> \<and> enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> G s)"
no_vars: shows "\<forall>v. Var v \<notin> subterms\<^sub>s\<^sub>e\<^sub>t M"
elims: fixes x :: "32 Word.word"   and xa :: "32 Word.word"   and y :: "32 Word.word" assumes "wmax x xa = y" obtains w1 :: "32 Word.word" and w2 :: "32 Word.word" where "x = w1" and "xa = w2" and "y = (if w1 <s w2 then w2 else w1)"
less_irrefl: fixes x :: "'a list" shows "\<not> x <lex x"
inverse_Real: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X" shows "inverse (Real.Real X) = (if Real.vanishes X then 0 else Real.Real (\<lambda>n. inverse (X n)))"
simulation_reaches1: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "A.reaches1 a a'"   and "Post.equiv' a b" shows "\<exists>b'. B.reaches1 b b' \<and> Post.equiv' a' b'"
class_add_has_fields_rev: fixes C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog"   and D :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class_add P (C, cdec) \<turnstile> D has_fields FDTs"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> D has_fields FDTs"
GreenThm_typeI_typeII_divisible_region_2: fixes s :: "(real \<times> real) set"   and i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain_typeI :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and two_chain_typeII :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and one_chain_typeI :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain_typeII :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII"   and "only_vertical_division one_chain_typeI two_chain_typeI"   and "boundary_chain one_chain_typeI"   and "only_horizontal_division one_chain_typeII two_chain_typeII"   and "boundary_chain one_chain_typeII"   and "common_boundary_sudivision_exists one_chain_typeI one_chain_typeII" shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. F x \<bullet> j) i x - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeII"
assms: shows "bal (Node (ls @ (sub, sep) # rs) t)"
if_SE_split_asm: fixes \<sigma> :: "'b"   and P :: "'b \<Rightarrow> bool"   and B\<^sub>1 :: "'b \<Rightarrow> (bool \<times> 'b) option"   and B\<^sub>2 :: "'b \<Rightarrow> (bool \<times> 'b) option" shows "(\<sigma> \<Turnstile> if\<^sub>S\<^sub>E P B\<^sub>1 B\<^sub>2) = (P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>1) \<or> \<not> P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>2))"
new_element_no_child_nodes: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []"
abs_e1': shows "\<bar>e1'\<bar> \<le> snd (trunc_bound_eucl p (fst (trunc_bound_eucl p (alpha * fst X)) + zeta))"
foreach_impl_body_def: fixes u :: "'v" shows "foreach_impl_body u \<equiv> \<lambda>(v, d) (Qi, \<pi>i). if v = r then (Qi, \<pi>i) else case (Q_lookup Qi v, M_lookup \<pi>i v) of (None, None) \<Rightarrow> (Q_update v d Qi, M_update v u \<pi>i) | (None, Some xa) \<Rightarrow> (Qi, \<pi>i) | (Some d', x) \<Rightarrow> if d < d' then (Q_update v d Qi, M_update v u \<pi>i) else (Qi, \<pi>i)"
set_simps_2: fixes x21 :: "'d"   and x22 :: "'d list" shows "set (x21 # x22) = insert x21 (set x22)"
axioms_1: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "ring_of_sets \<Omega> M" shows "semiring_of_sets \<Omega> M"
chainTransitionAppend_1: fixes P :: "pi"   and P' :: "pi"   and Rs :: "residual" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<Longrightarrow>\<^sub>l\<^sup>^Rs" shows "P \<Longrightarrow>\<^sub>l\<^sup>^Rs"
Star_def: fixes r :: "'b language" shows "Star r = StarLR One r"
nat_diff_right_cancel_le1: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m - k \<le> n - k"   and "k < m" shows "m \<le> n"
that: fixes n :: "int" assumes "real_of_int n * (2 * pi) \<in>\<^sub>r ivl" shows "thesis"
cases: fixes r :: "run_t" obtains role :: "role_t" and owner :: "agent" and partner :: "agent" where "r = \<lparr>role = role, owner = owner, partner = partner\<rparr>"
U_def: shows "U = {x \<in> space M. \<not> P x}"
nextVertices_in_face: fixes v :: "nat"   and f :: "face"   and n :: "nat" assumes "v \<in> \<V> f" shows "f\<^bsup>n\<^esup> \<bullet> v \<in> \<V> f"
A: shows "u \<turnstile>\<^bsub>v,n\<^esub> FW M' n"
finprod_Suc: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> {..Suc n} \<rightarrow> carrier G" shows "finprod G f {..Suc n} = f (Suc n) \<otimes> finprod G f {..n}"
iprod0_if_coeffs0: fixes cs :: "'a list"   and xs :: "'a list" assumes "\<forall>c\<in>set cs. c = (0::'a)" shows "\<langle>cs,xs\<rangle> = (0::'a)"
simps_7: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 strictly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. #\<^sub>\<le> \<rho> K\<^sub>2 n \<le> #\<^sub>< \<rho> K\<^sub>1 n}"
simps_6: fixes x2 :: "('b, 'c, 'd) Tree"   and x1 :: "('b, 'c, 'd) Tree set['b]" shows "tNot x2 \<noteq> tConj x1"
elims_2: fixes x :: "'at set"   and xa :: "'at Literal" assumes "x \<Turnstile> xa" obtains I :: "'at set" and A :: "'at" where "x = Ia__" and "xa = Pos Aa__" and "Aa__ \<in> Ia__"   | I :: "'at set" and A :: "'at" where "x = I" and "xa = Neg A" and "A \<notin> I"
fullAbs: shows "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"
ennreal_INF_const_minus: fixes I :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "I \<noteq> {}" shows "(SUP x\<in>I. c - f x) = c - Inf (f ` I)"
Asm: fixes P :: "'e set"   and p :: "'f"   and Q :: "'e set"   and A :: "'e set"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and F :: "'g set" assumes "(P, p, Q, A) \<in> \<Theta>" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Call p Q,A"
integral_on_image_ubound_nonneg: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> real"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f (g x)"   and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S" shows "f integrable_on g ` S \<and> integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x))"
inducts: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "flow \<Delta> f" obtains "\<And>e. f e \<le> capacity \<Delta> e" and "\<And>x. \<lbrakk>x \<noteq> source \<Delta>; x \<noteq> sink \<Delta>\<rbrakk> \<Longrightarrow> KIR f x"
isomorphic_implies_ide_1: fixes f :: "'d"   and g :: "'d" assumes "f \<cong>\<^sub>D g" shows "D.ide f"
semiring_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.semiring plus times \<equiv> class.ab_semigroup_add plus \<and> class.semigroup_mult times \<and> class.semiring_axioms plus times"
af\<^sub>G_implies_af_eval\<^sub>G': fixes S :: "'a ltl set"   and \<G> :: "'a ltl set"   and \<phi> :: "'a ltl"   and w :: "'a set list" assumes "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> w)"   and "\<And>\<psi>. G \<psi> \<in> \<G> \<Longrightarrow> S \<Turnstile>\<^sub>P G \<psi>"   and "\<And>\<psi> i. \<lbrakk>G \<psi> \<in> \<G>; i < length w\<rbrakk> \<Longrightarrow> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<psi> (drop i w))" shows "S \<Turnstile>\<^sub>P af \<phi> w"
M_axioms_def: shows "M_axioms \<equiv> (\<forall>x y. P x y \<longrightarrow> P y x \<longrightarrow> x = y) \<and> (\<forall>x y. PP x y = (P x y \<and> x \<noteq> y))"
card_1_singleton_iff: fixes A :: "'d set" shows "(card A = Suc 0) = (\<exists>x. A = {x})"
n_s_narrow: fixes X :: "char list set"   and S1 :: "'av st"   and S2 :: "'av st" assumes "finite X"   and "fun S1 = fun S2 on - X"   and "S2 \<le> S1"   and "S1 \<triangle> S2 < S1" shows "n\<^sub>s (S1 \<triangle> S2) X < n\<^sub>s S1 X"
functor_axioms: shows "functor (\<cdot>\<^sub>J) (\<cdot>) Cod_f.map"
nat_of_integer_code_post_1: shows "nat_of_integer 0 = 0"
rx_pos: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "lt_vec 0 x" shows "0 < r x"
sorted_list_of_set_eq_Nil_iff: fixes A :: "'a set" assumes "finite A" shows "(sorted_list_of_set A = []) = (A = {})"
content_division_of: fixes K :: "'a set"   and \<D> :: "'a set set"   and S :: "'a set" assumes "K \<in> \<D>"   and "\<D> division_of S" shows "content K = (\<Prod>i\<in>Basis. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)"
a: shows "(x, T') # \<Gamma> \<turnstile> e : T"
progress_ge: fixes \<phi> :: "'a MFOTL.formula"   and i :: "nat"   and \<sigma> :: "(char list \<times> 'a list) trace" assumes "MFOTL.future_reach \<phi> \<noteq> \<infinity>" shows "\<exists>j. i \<le> Monitor.progress \<sigma> \<phi> j"
simps_11: shows "extractFrame'' (\<bottom>\<^sub>c) = \<bottom>\<^sub>F"
aw0: shows "(adjuster n w us + w) \<bullet>c (adjuster n w us + w) \<noteq> (0::'a)"
intra_path_slp: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'" shows "n -as\<rightarrow>\<^bsub>sl\<^esub>* n'"