hcju/isabelleps huggingface dataset:
lm062: fixes P :: "('a \<times> 'b) set" shows "inj_on fst P = inj_on snd (P\<inverse>)"
assms_4: fixes z :: "'a" assumes "z \<in> S" shows "f z \<noteq> 0"
l_set_mode_get_child_nodes_axioms_def: fixes set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_child_nodes_axioms set_mode_locs get_child_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
divide_right_mono_ennreal: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a \<le> b" shows "a / c \<le> b / c"
exp_of_real: fixes x :: "real" shows "exp (\<i> * complex_of_real x) = complex_of_real (cos x) + \<i> * complex_of_real (sin x)"
checker_r2_correctness: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "checker_r2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> = (check_precond_r2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> \<and> \<not> safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> {0..})"
valid_regions_I_cong: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and I' :: "'a \<Rightarrow> Regions.intv" assumes "Regions.valid_region X k I r"   and "\<forall>x\<in>X. I x = I' x" shows "Regions.region X I r = Regions.region X I' r \<and> Regions.valid_region X k I' r"
path_independent_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "path_independent_on A f = (\<forall>B C. B \<subseteq> A \<and> C \<subseteq> A \<longrightarrow> f (B \<union> C) = f (B \<union> f C))"
ant_minf_in_Ainteg: shows "(0, - 1) \<in> Ainteg"
basic_identity\<^sub>3_def: shows "(\<^bold>=\<^sub>3) \<equiv> \<lambda>F G. \<^bold>\<forall>x y. (\<^bold>\<lambda>z. \<lparr>F,z\<^sup>P,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>=\<^sub>1 (\<^bold>\<lambda>z. \<lparr>G,z\<^sup>P,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>=\<^sub>1 (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>) \<^bold>=\<^sub>1 (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>)"
simps_8: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Cos f) env = compute_tm_by_comp prec ord I a (Cos (Var 0)) (approx_tm prec ord I a f env) (\<lambda>x. True)"
limA: shows "(sum f \<longlongrightarrow> limA) (finite_subsets_at_top A)"
join_restrict_table: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set"   and B :: "nat set"   and Y :: "'b option list set"   and b :: "bool"   and v :: "'b option list" assumes "table n A X"   and "table n B Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A" shows "(v \<in> join X b Y) = (wf_tuple n (A \<union> B) v \<and> restrict A v \<in> X \<and> join_cond b Y (restrict B v))"
CeqStart: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators list"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "x \<notin> dom (C (list2FWpolicy a))"   and "a \<noteq> []"   and "b \<noteq> []" shows "C (list2FWpolicy (b @ a)) x = C (list2FWpolicy b) x"
that: fixes x :: "'a" assumes "h \<Turnstile> P x" shows "thesis"
appL: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "s $ u \<rightarrow>\<^sub>\<beta> t $ u"
exhaust: fixes y :: "'c rule" obtains x1 :: "'c match_expr" and x2 :: "action" where "y = Rule x1 x2"
l_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<oplus> a = c \<oplus> b"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "a = b"
absorb_iff1: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (a \<sqinter> b = a)"
all_list_constr_def: fixes L :: "nat list list"   and n :: "nat" shows "all_list_constr L n \<equiv> \<forall>x. List.member L x \<longrightarrow> list_constr x n"
of_int_eq_numeral_power_cancel_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and y :: "int"   and x :: "num"   and n :: "nat" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus y = power.power one times (numeral.numeral one plus x) n) = (y = numeral x ^ n)"
peirce: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<cdot> z\<^sup>\<smile> = 0) = (y ; z \<cdot> x\<^sup>\<smile> = 0)"
lhs_step_If: fixes b :: "bool"   and t :: "'a"   and m :: "'a"   and e :: "'a" assumes "b \<Longrightarrow> t \<le> m"   and "\<not> b \<Longrightarrow> e \<le> m" shows "(if b then t else e) \<le> m"
assms_1: shows "\<forall>\<beta>'\<in>benv_in_ve ve. Q \<beta>'"
orthogonal_columns_fst_QR_decomposition: fixes i :: "'c"   and A :: "(('a, 'c) vec, 'b) vec"   and ia :: "'c" assumes "column i (fst (QR_decomposition A)) \<noteq> column ia (fst (QR_decomposition A))" shows "column i (fst (QR_decomposition A)) \<bullet> column ia (fst (QR_decomposition A)) = 0"
assms_2: shows "\<forall>n'>n. \<not> \<parallel>c\<parallel>\<^bsub>t n'\<^esub>"
univ_poly_is_domain: fixes K :: "'a set" assumes "subring K R" shows "domain (K [X])"
merge_thms_3: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and x' :: "'b" shows "hn_invalid R x x' \<or>\<^sub>A hn_ctxt R x x' \<Longrightarrow>\<^sub>t hn_invalid R x x'"
rep_eq: fixes x :: "'a regular"   and xa :: "'a regular" shows "Rep_regular (x \<squnion> xa) = Rep_regular x \<squnion> Rep_regular xa"
simps_3: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and e :: "trm" assumes "x \<sharp> \<theta>" shows "\<theta><Lam [x].e> = Lam [x].\<theta><e>"
leg0_commutes: shows "\<nu>'.cod.leg0 \<cdot> \<nu>'.chine = \<nu>'.dom.leg0"
ntrs_xchange_context_s: fixes s :: "'a list"   and c :: "'a list multiset"   and ee :: "('b, 'c) label list"   and s' :: "'a list"   and csp :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and cn :: "'a list multiset" assumes "((s, c), ee, s', csp + c) \<in> ntrs fg"   and "mon_c fg cn \<subseteq> mon_c fg c" shows "((s, cn), ee, s', csp + cn) \<in> ntrs fg"
repr_inv: fixes x :: "'a" shows "abst isom (repr isom x) = x"
timpl_closure'_timpls_trancl_supset': fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' c \<subseteq> timpl_closure' {(a, b). (a, b) \<in> c\<^sup>+ \<and> a \<noteq> b}"
take_butlast: fixes n :: "nat"   and xs :: "'a list" assumes "n < |xs|" shows "take n (butlast xs) = take n xs"
fps_lr_inverse_nth_0_1: fixes f :: "'a fps"   and x :: "'a" shows "fps_nth (fps_left_inverse f x) 0 = x"
refl_onD2: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "refl_on A r"   and "(x, y) \<in> r" shows "y \<in> A"
meet_equality: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and i :: "'a" assumes "dlat le"   and "dpo.is_inf le x y i" shows "dlat.meet le x y = i"
vfsequence_vcpower: fixes n :: "V"   and xs :: "V"   and A :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> n" shows "vfsequence xs"
auto_ll_on_open_congI: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x" shows "auto_ll_on_open g Y"
vimage_of_vempty: fixes r :: "V" shows "r `\<^sub>\<circ> 0 = 0"
b_2: shows "b \<in> span (insert a S - {b})"
closed_vimage_snd: fixes S :: "'b set" assumes "closed S" shows "closed (snd -` S)"
verify: fixes M :: "bit list"   and emBits :: "nat"   and EM :: "bit list" assumes "emsapss_encode M emBits \<noteq> []"   and "EM = emsapss_encode M emBits" shows "emsapss_decode M EM emBits = True"
interior_convex_hull_3_minimal: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<not> collinear {a, b, c}"   and "DIM('a) = 2" shows "interior (convex hull {a, b, c}) = {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and> x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
homeomorphic_trans: fixes S :: "'b set"   and T :: "'c set"   and U :: "'d set" assumes "S homeomorphic T"   and "T homeomorphic U" shows "S homeomorphic U"
F_def: shows "F \<equiv> \<lambda>W x. if x \<in> S then setdist {x} (S - W) / supp_sum (\<lambda>V. setdist {x} (S - V)) \<C> else 0"
set_weave: fixes xs :: "'a list"   and A :: "nat set"   and ys :: "'a list" assumes "length xs = card {a \<in> A. a < length xs + length ys}"   and "length ys = card {a \<in> - A. a < length xs + length ys}" shows "set (weave A xs ys) = set xs \<union> set ys"
assms_4: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "q \<in> pmdl A"   and "q \<noteq> 0" shows "is_red B q"
zip_append: fixes xs :: "'d list"   and us :: "'e list"   and ys :: "'d list"   and vs :: "'e list" assumes "length xs = length us" shows "xs @ ys || us @ vs = (xs || us) @ (ys || vs)"
simps_12: fixes b :: "nat"   and I :: "\<I>"   and \<phi> :: "formula" shows "fvi b (Prev I \<phi>) = fvi b \<phi>"
bitlen_def: fixes a :: "int" shows "bitlen a = int (floorlog 2 (nat a))"
entails_frame'': fixes P :: "assn"   and B :: "assn"   and F :: "assn"   and A :: "assn" assumes "\<not> (P \<Longrightarrow>\<^sub>A B * F)"   and "A \<Longrightarrow>\<^sub>A B" shows "\<not> (P \<Longrightarrow>\<^sub>A A * F)"
gfp_exp_least: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "\<And>P. \<lbrakk>P \<tturnstile> t P; unitary P\<rbrakk> \<Longrightarrow> P \<tturnstile> Q"   and "unitary Q" shows "gfp_exp t \<tturnstile> Q"
lead_coeff_simp: fixes p :: "'a list" assumes "p \<noteq> []" shows "local.coeff p (degree p) = lead_coeff p"
morph: fixes u :: "'a list"   and v :: "'a list" shows "f (u \<cdot> v) = f u \<cdot> f v"
f0: shows "AE x in lebesgue_on T. 0 \<le> f x"
index_simp: fixes u :: "Index \<Rightarrow> 'a"   and v :: "Index \<Rightarrow> 'a" shows "(u = v) = (u none = v none \<and> u some = v some)"
simps_1: fixes T1 :: "'a trm"   and T2 :: "'a trm" shows "gA (Eq T1 T2) = Eq (gT T1) (gT T2)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>x n. (\<And>xa. \<lbrakk>xa = (x + n div x) div 2; xa < x\<rbrakk> \<Longrightarrow> P xa n) \<Longrightarrow> P x n" shows "P a0 a1"
Q_hit_exists: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_y_neq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post" obtains Q_hit :: "'c list" and Q_hit_pre :: "'c list" and Q_hit_post :: "'c list" where "Q_hit \<in> ProofStepInduct_NonTrivial.Q G v0 v1 paths P_new sep_size" and "y \<in> set Q_hit" and "Q_hit = Q_hit_pre @ y # Q_hit_post"
set_finsert: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" obtains B :: "'b fset" where "A = finsert x B" and "x |\<notin>| B"
xa_P_init: shows "addrow_mat (dim_row H) (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 * P_init \<in> carrier_mat m m"
powers_82: fixes x :: "'a" shows "x ^ 4 = x * x * x * x"
distinct_75: fixes x6 :: "'a fm"   and x91 :: "nat"   and x92 :: "'a fm" shows "Neg x6 \<noteq> ExN x91 x92"
weak_ranking_unique: fixes As :: "'a set list" assumes "is_weak_ranking As"   and "of_weak_ranking As = of_weak_ranking xs" shows "As = weak_ranking (of_weak_ranking xs)"
map_of_map_apsnd: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "('a \<times> 'c) list" shows "map_of (map (apsnd f) xs) = map_option f \<circ> map_of xs"
nMRtoNone: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []"   and "applied_rule_rev Cp x p = \<bottom>" shows "Cp (list2FWpolicy p) x = \<bottom>"
IH: assumes "j'' \<le> j" shows "fwi m n k i j i' j' \<le> fwi m n k i j'' i' j'"
WbisT_coinduct: fixes \<phi> :: "('test, 'atom) com \<Rightarrow> ('test, 'atom) com \<Rightarrow> bool"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "\<phi> c d"   and "\<And>c d. \<phi> c d \<Longrightarrow> \<phi> d c"   and "\<And>c s d t c' s'. \<lbrakk>\<phi> c d; s \<approx> t; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> \<exists>d' t'. (d, t) \<rightarrow>*c (d', t') \<and> s' \<approx> t' \<and> (\<phi> c' d' \<or> c' \<approx>wT d')"   and "\<And>c s d t s'. \<lbrakk>\<phi> c d; s \<approx> t; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> \<exists>t'. (d, t) \<rightarrow>*t t' \<and> s' \<approx> t'" shows "c \<approx>wT d"
is_SDG_path_matched: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" obtains ns' :: "'b SDG_node list" where "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns' n'" and "set ns \<subseteq> set ns'"
Domain_extend_act: fixes h :: "'e \<times> 'f \<Rightarrow> 'd"   and act :: "('e \<times> 'e) set" shows "Domain (extend_act h act) = extend_set h (Domain act)"
distributed_measurable: fixes M :: "'a measure"   and N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal" assumes "distributed M N X f" shows "X \<in> M \<rightarrow>\<^sub>M N"
cut_less_le_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t = I \<down>\<le> t - {t}"
span_minimal: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set"   and T :: "'d set" assumes "module scale"   and "S \<subseteq> T"   and "module.subspace scale T" shows "module.span scale S \<subseteq> T"
exhaust: fixes y :: "ty" obtains "y = Ity"   | "y = Rty"
simps_2: fixes a :: "'f"   and S :: "'f list"   and iostep :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p (a # S) iostep \<sigma> = (case iostep a \<sigma> of None \<Rightarrow> None | Some (out, \<sigma>') \<Rightarrow> case mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p S iostep \<sigma>' of None \<Rightarrow> None | Some (outs, \<sigma>'') \<Rightarrow> Some (out # outs, \<sigma>''))"
prems_1: shows "R\<^sup>*\<^sup>* x b"
eval_def: fixes cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "eval cid t t' n \<gamma> \<equiv> (\<exists>i\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>) \<and> \<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (the_enat \<langle>cid #\<^bsub>enat n\<^esub>inf_llist t\<rangle>) \<or> (\<exists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>) \<and> \<not> (\<exists>i'\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i'\<^esub>) \<and> \<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (\<^bsub>cid\<^esub>\<down>\<^bsub>t\<^esub>n) \<or> (\<nexists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>) \<and> \<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) n"
is_bezout_def: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" shows "is_bezout bezout = (\<forall>a b. let (p, q, gcd_a_b) = bezout a b in p * a + q * b = gcd_a_b \<and> gcd_a_b dvd a \<and> gcd_a_b dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd gcd_a_b))"
is_pure_assn_basic_simps_1: shows "is_pure_assn false"
cong: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "linorder.merge_lists less_eq \<equiv> linorder.merge_lists less_eq"
cases: fixes valid_edge :: "'g \<Rightarrow> bool"   and src :: "'g \<Rightarrow> 'h"   and trg :: "'g \<Rightarrow> 'h"   and knd :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and E :: "'h"   and X :: "'h"   and a :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node" assumes "lift_valid_edge valid_edge src trg knd E X a" obtains aa :: "'g" and e :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node" where "a = e__" and "valid_edge aa" and "src aa \<noteq> E \<or> trg aa \<noteq> X" and "e__ = (Node (src aa), knd aa, Node (trg aa))"   | e :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node" where "a = ea__" and "ea__ = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E)"   | e :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node" where "a = eb__" and "eb__ = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit)"   | e :: "'h LDCFG_node \<times> ('i, 'j, 'k, 'l) edge_kind \<times> 'h LDCFG_node" where "a = e" and "e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit)"
simple_function_compose2: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "simple_function M f"   and "simple_function M g" shows "simple_function M (\<lambda>x. h (f x) (g x))"
valid_SDG_CFG_node: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "valid_node (parent_node n)"
\<beta>: shows "\<guillemotleft>w'.the_\<nu> \<cdot> local.inv w.the_\<nu> : s\<^sub>1 \<star> w \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright>"
assms_1: shows "f \<le> - - (- h \<sqinter> g)"
append_one_prefix: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys"   and "length xs < length ys" shows "prefix (xs @ [ys ! length xs]) ys"
Some: shows "applied_rule_rev C x p = \<lfloor>y\<rfloor>"
butlast_append: fixes xs :: "'b list"   and ys :: "'b list" shows "butlast (xs \<cdot> ys) = (if ys = \<epsilon> then butlast xs else xs \<cdot> butlast ys)"
intros_11: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and n :: "nat"   and s' :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> =n\<Rightarrow> t"
span_eq: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set"   and T :: "'e set" assumes "module scale" shows "(module.span scale S = module.span scale T) = (S \<subseteq> module.span scale T \<and> T \<subseteq> module.span scale S)"
propagate_pointstamps_eq: fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc" assumes "c_work c loc \<noteq> {#}\<^sub>z" shows "c_pts c = c_pts (SOME c'. next_propagate c c')"
w_add: fixes b1 :: "'a \<Rightarrow> real"   and b2 :: "'a \<Rightarrow> real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> w x \<in> UNIV"   and "b1 \<in> diff_fun_space"   and "b2 \<in> diff_fun_space" shows "w (b1 + b2) = w b1 + w b2"
embedding_map_graph: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "embedding_map X (prod_topology X Y) (\<lambda>x. (x, f x)) = continuous_map X Y f"
bound_scheme_inst_type: fixes t :: "typ"   and B :: "nat \<Rightarrow> type_scheme"   and sch :: "type_scheme" assumes "mk_scheme t = bound_scheme_inst B sch" shows "\<exists>S. \<forall>x\<in>bound_tv sch. B x = mk_scheme (S x)"
sub_sup_closed: fixes x :: "'b"   and y :: "'b" shows "! (d x) \<squnion> ! (d y) = ! (d (! (d (! (d x) \<squnion> ! (d y)))))"
summable_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'c" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'c)" shows "summable f"
perron_frobenius_spectral_type_based: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and d :: "nat" assumes "non_neg_mat A"   and "\<forall>x. poly (charpoly A) x = 0 \<longrightarrow> x \<le> 1"   and "\<forall>x. cmod x = 1 \<longrightarrow> order x (map_poly complex_of_real (charpoly A)) \<le> d" shows "\<exists>c1 c2. \<forall>k a. a \<in> elements_mat_h (matpow A k) \<longrightarrow> \<bar>a\<bar> \<le> c1 + c2 * real k ^ (d - 1)"
precompact_locally_finite_open_coverE: fixes charts :: "('c, 'd) chart set" obtains W :: "nat \<Rightarrow> 'c set" where "manifold.carrier charts = \<Union> (range W)" and "\<And>i. open (W i)" and "\<And>i. compact (closure (W i))" and "\<And>i. closure (W i) \<subseteq> manifold.carrier charts" and "locally_finite_on (manifold.carrier charts) UNIV W"
minus_cont: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> bcontfun"   and "g \<in> bcontfun" shows "(\<lambda>x. f x - g x) \<in> bcontfun"
herbrand_evalt'_2: fixes e :: "nat \<Rightarrow> 'a hterm"   and hts :: "'a hterm list" shows "evalts e HApp (terms_of_hterms hts) = hts"
internal_iface_name_to_set: fixes i :: "char list"   and p_iface :: "char list" shows "internal_iface_name_match i p_iface = (p_iface \<in> internal_iface_name_to_set i)"
succE: fixes x :: "hf"   and y :: "hf" assumes "x \<^bold>\<in> succ y" obtains "x = y"   | "x \<^bold>\<in> y"
fundamental_theorem_of_calculus_strong: fixes S :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "finite S"   and "a \<le> b"   and "\<And>x. x \<in> {a..b} - S \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "continuous_on {a..b} f" shows "(f' has_integral f b - f a) {a..b}"
Uint64_signed_def: fixes i :: "integer" shows "Uint64_signed i = (if i < - 9223372036854775808 \<or> 9223372036854775808 \<le> i then undefined Uint64 i else Uint64 i)"
set_sel_2: fixes a :: "'a llist"   and x :: "'a" assumes "\<not> lnull a"   and "x \<in> lset (ltl a)" shows "x \<in> lset a"
Throw: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b" shows "\<Gamma>\<turnstile>Throw \<down> Normal s"
nn_integral_cong: fixes M :: "'c measure"   and u :: "'c \<Rightarrow> ennreal"   and v :: "'c \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
val_minus_eq: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "v (-\<^sub>a x) = v x"
that: fixes d :: "'a extended list list" assumes "mapM (concurrent s l u) (substr s l u) = Inr d \<and> set (concat d) = InString ` I ` {x. is_concurrent pm sm x}" shows "thesis"