hcju/isabelleps huggingface dataset:
has_as_equalizer_def: fixes f0 :: "(bool, bool list) concrete_category.arr"   and f1 :: "(bool, bool list) concrete_category.arr"   and e :: "(bool, bool list) concrete_category.arr" shows "J.has_as_equalizer f0 f1 e \<equiv> J.par f0 f1 \<and> limit_cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>J) (parallel_pair_diagram.map (\<cdot>\<^sub>J) f0 f1) (J.dom e) (parallel_pair_diagram.mkCone (\<cdot>\<^sub>J) f0 e)"
line_line2: fixes a :: "'b"   and b :: "'b"   and c :: "real"   and x :: "real" shows "{a -- {a -- b}\<^bsub>c\<^esub>}\<^bsub>x\<^esub> = {a -- b}\<^bsub>c * x\<^esub>"
skl1_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Nr :: "msg"   and gny :: "msg" shows "skl1_step3 Ra A B Nr gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xni, xgnx} \<and> sklvl1.guessed_frame Ra xgny = Some gny \<and> sklvl1.guessed_frame Ra xnr = Some Nr \<and> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> (can_signal s A B \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress s Rb) {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgny = Some gny \<and> sklvl1.guessed_frame Rb xnr = Some Nr \<and> sklvl1.guessed_frame Rb xni = Some (NonceF (Ra $ ni)) \<and> sklvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))))) \<and> (Ra = test \<longrightarrow> Exp gny (NonceF (Ra $ nx)) \<notin> synth (analz (ik s))) \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) else signalsInit s, signalsInit2 := if can_signal s A B then addSignal (signalsInit2 s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit2 s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) else signalsResp s, signalsResp2 := if can_signal s A B then addSignal (signalsResp2 s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp2 s\<rparr>}"
assms_2: shows "configuration p (Suc n) \<noteq> {}"
CFG_edge_no_Def_equal: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and a :: "'f"   and V :: "'i"   and s :: "'h" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val"   and "valid_edge a"   and "V \<notin> Def (sourcenode a)"   and "pred (kind a) s" shows "state_val (transfer (kind a) s) V = state_val s V"
const_disj_def: fixes c\<^sub>1 :: "'b"   and c\<^sub>2 :: "'b" shows "const_disj c\<^sub>1 c\<^sub>2 \<equiv> transl_rule (A_Cmp (A_Lbl (S_Const c\<^sub>1)) (A_Lbl (S_Const c\<^sub>2)) \<sqsubseteq> \<bottom>)"
A\<^sub>_prism_def: shows "A\<^sub>\<triangle> = \<lparr>prism_match = fromA, prism_build = A\<rparr>"
perm_3: fixes pi :: "(name \<times> name) list"   and x1 :: "('d, 'e, 'f) psi" shows "pi \<bullet> RTau x1 = RTau (pi \<bullet> x1)"
star_unfold_left: fixes A :: "'b list set" shows "star A = A @@ star A \<union> {[]}"
fps_left_inverse_eq_fps_right_inverse: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a" assumes "x * fps_nth f 0 = (1::'a)"   and "fps_nth f 0 * y = (1::'a)" shows "fps_left_inverse f x = fps_right_inverse f y"
prime_elem_linear_poly: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)"   and "coprime a b" shows "prime_elem [:a, b:]"
in_cosets_aux: fixes p :: "nat"   and x :: "nat" assumes "euler_witness_context p"   and "euler_witness (int x) p"   and "x \<in> carrier (euler_witness_context.G p)" shows "euler_witness_context.H p #>\<^bsub>euler_witness_context.G p\<^esub> x \<in> rcosets\<^bsub>euler_witness_context.G p\<^esub> euler_witness_context.H p"
x_2: shows "t \<sqsubseteq> (\<theta>1 \<circ>\<^sub>s \<theta>2) x"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a * b * c = a * (b * c)"
gfp_upperbound: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and X :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq X (f X)" shows "less_eq X (complete_lattice.gfp Sup less_eq f)"
prems: fixes k :: "nat" assumes "k < Suc n" shows "f k \<in> carrier_mat d2 d2"
Measurement_System_eq_intro: fixes x :: "('c, 'd, 'e, 'f) Measurement_System_scheme"   and y :: "('c, 'd, 'e, 'f) Measurement_System_scheme" assumes "mag x = mag y"   and "Quantity.dim x = Quantity.dim y"   and "Measurement_System.more x = Measurement_System.more y" shows "x = y"
IP_2: shows "(\<sigma>h, \<sigma>') \<in> (R - UNIV \<times> \<Sigma>)\<^sup>*"
completely_multiplicative_function_def: fixes f :: "nat \<Rightarrow> 'a" shows "completely_multiplicative_function f \<equiv> f 0 = (0::'a) \<and> f (Suc 0) = (1::'a) \<and> (\<forall>a b. 1 < a \<longrightarrow> 1 < b \<longrightarrow> f (a * b) = f a * f b)"
is_TNil_tSup: fixes b :: "'g"   and A :: "('f, 'g) tllist set" shows "is_TNil (tSup b A) = (\<forall>x\<in>A. is_TNil x)"
l1_inv2I: fixes x :: "l1_state" assumes "\<And>KE A B Rb. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; progress x Rb = Some {xpkE, xsk}; guessed_frame Rb xpkE = Some KE; can_signal x A B\<rbrakk> \<Longrightarrow> 0 < l1_state.signals x (Running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>)" shows "x \<in> l1_inv2"
W_def: fixes c :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "W c = sum (wt (fst c) (snd c))"
acyclic_asymmetric: fixes w :: "'a" assumes "pd_kleene_allegory_class.acyclic w" shows "asymmetric w"
minus_top_ennreal: fixes x :: "ennreal" shows "x - \<top> = (if x = \<top> then \<top> else 0)"
connected_component_refl: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "connected_component S x x"
simps_15: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (Plus e1 e2) (Times v va) = True"
induct: fixes P :: "(nat \<Rightarrow> nat option) \<Rightarrow> char list list \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> bool"   and a0 :: "nat \<Rightarrow> nat option"   and a1 :: "char list list"   and a2 :: "nat \<Rightarrow> nat option" assumes "\<And>s s'. P s [] s'"   and "\<And>s \<pi> \<pi>s s'. P (execute \<pi> s) \<pi>s s' \<Longrightarrow> P s (\<pi> # \<pi>s) s'" shows "P a0 a1 a2"
unrest_conj: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred"   and Q :: "'b upred" assumes "x \<sharp> P"   and "x \<sharp> Q" shows "x \<sharp> P \<and> Q"
bounded_iff: fixes b :: "'d"   and c :: "'d"   and a :: "'d" shows "(sup b c \<le> a) = (b \<le> a \<and> c \<le> a)"
assms: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<P> \<down> x \<^bold>\<rightarrow> \<^bold>\<diamond>mexistsActB x) w)\<rfloor>"
simps: fixes e :: "'c \<Rightarrow> 'a"   and data :: "'c"   and subtrees :: "'c rose_tree list" shows "embed_source_tree e (Tree (data, subtrees)) = Tree\<^sub>m (Unblinded (e data, map (embed_source_tree e) subtrees))"
a1: shows "s' = mem_mod a1 a2 v s \<and> get_S (cpu_reg_val PSR s) = 0"
bound_upD: fixes R :: "('a, 'b) ring_scheme"   and f :: "nat \<Rightarrow> 'a" assumes "ring R"   and "f \<in> up R" shows "\<exists>n. bound \<zero>\<^bsub>R\<^esub> n f"
Block: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>0 :: "char list exp"   and h\<^sub>0 :: "heap"   and l\<^sub>0 :: "sfields"   and V :: "char list"   and sh\<^sub>0 :: "sheap"   and e\<^sub>1 :: "char list exp"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and T :: "ty" assumes "P \<turnstile> \<langle>e\<^sub>0,(h\<^sub>0, l\<^sub>0(V := None), sh\<^sub>0)\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>" shows "P \<turnstile> \<langle>{V:T; e\<^sub>0},(h\<^sub>0, l\<^sub>0, sh\<^sub>0)\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1(V := l\<^sub>0 V), sh\<^sub>1)\<rangle>"
simps_2: fixes x :: "'b"   and y :: "'b"   and p :: "'c" shows "destNet (AllowPortFromTo x y p) = y"
Singleton: fixes v :: "'a" assumes "v \<in> V\<^bsub>remove_vertex x\<^esub>" shows "H.walk [v]"
kLim: shows "(f \<circ> s \<circ> r) \<longlonglongrightarrow> k"
assms: shows "rtrancl3p r a (bs @ bs') a''"
lookup_trie_update_with: fixes ks :: "'e list"   and v :: "'d"   and f :: "'d \<Rightarrow> 'd"   and t :: "('e, 'd) trie"   and ks' :: "'e list" shows "lookup_trie (trie_update_with' ks t v f) ks' = (if ks = ks' then Some (f (case lookup_trie t ks of None \<Rightarrow> v | Some v' \<Rightarrow> v')) else lookup_trie t ks')"
dom_src: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.dom (src\<^sub>D \<mu>) = src\<^sub>D \<mu>"
assms_1: fixes t :: "'a set"   and x :: "'a" assumes "openin (top_of_set S) t"   and "x \<in> t" shows "openin (top_of_set S) (connected_component_set t x)"
L_gen_kill: fixes c :: "com"   and X :: "char list set" shows "L c X = gen c \<union> (X - kill c)"
mat_ge_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "class.non_strict_order less_eq"   and "ord.mat_ge less_eq A B"   and "ord.mat_ge less_eq B C"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "ord.mat_ge less_eq A C"
list_allI: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "list_all P xs"
assms_2: fixes P :: "complex mat"   and Q :: "complex mat" assumes "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "P \<le>\<^sub>L Q" shows "WS P \<le>\<^sub>L WS Q"
zone_set_def: fixes Z :: "('e \<Rightarrow> 'f) set"   and r :: "'e list" shows "Z\<^bsub>r \<rightarrow> 0\<^esub> = {[r\<rightarrow>(0::'f)]u |u. u \<in> Z}"
Subst_all_satisfies_invariant: fixes ES :: "('b list set \<times> 'b trm set) set"   and Y :: "'b list set"   and yrhs :: "'b trm set" assumes "invariant (ES \<union> {(Y, yrhs)})" shows "invariant (Subst_all ES Y (Arden Y yrhs))"
simps_3: fixes v :: "int \<times> int"   and vb :: "int \<times> int"   and vc :: "(int \<times> int) list" shows "closest_pair (v # vb # vc) = (let (_, x, y) = closest_pair_rec (mergesort fst (v # vb # vc)) in (x, y))"
AMax_def: fixes A :: "ant set" shows "AMax A = (THE m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m))"
distinct_orbit_list: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x \<in> orbit f x" shows "distinct (orbit_list f x)"
same_prefix_prefix: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
eqI: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b" 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>'"
exec_instr_xcpt_h: fixes ins :: "instr list"   and pc :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and stk :: "val list"   and vars :: "val list"   and Cl :: "char list"   and M :: "char list"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and xcp :: "nat"   and T :: "ty"   and mxs :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and C :: "char list"   and loc :: "val list" assumes "fst (exec_instr (ins ! pc) P h stk vars Cl M pc frs) = \<lfloor>xcp\<rfloor>"   and "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"   and "P,\<Phi> |- (None, h, (stk, loc, C, M, pc) # frs) [ok]" shows "\<exists>obj. h xcp = \<lfloor>obj\<rfloor>"
cv: shows "candidate_values (trm_rep v S) C2 Cl_C2 gr_Cl_C2 gr_rhs gr_L2 L2 \<eta> lhs rhs v S"
eq0: shows "Im (z + \<i> * csqrt (1 - z\<^sup>2)) = 0"
UW: shows "set (us @ ws) \<subseteq> carrier_vec n"
surjective_embedding_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "(embedding_map X Y f \<and> f ` topspace X = topspace Y) = homeomorphic_map X Y f"
igOpIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igOpIPresIGWls MOD \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igWls MOD (stOf delta) (igOp MOD delta inp binp)"
refl_on_def': fixes A :: "'e set"   and r :: "('e \<times> 'e) set" shows "refl_on A r = ((\<forall>(x, y)\<in>r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
Cd_idem: fixes d :: "'b"   and B :: "'a set" shows "Cd d (Cd d B) = Cd d B"
DAnMT: shows "dom (Cp DenyAll) \<noteq> {}"
square_free_m_def: fixes m :: "int"   and f :: "int poly" shows "poly_mod.square_free_m m f = (poly_mod.Mp m f \<noteq> poly_mod.Mp m 0 \<and> (\<forall>g. degree (poly_mod.Mp m g) \<noteq> 0 \<longrightarrow> \<not> poly_mod.dvdm m (g * g) f))"
prod_mset_prod_list: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and xs :: "'k list" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one (mset xs) = monoid_mult.prod_list one times xs"
simps: fixes xs :: "nat set"   and x :: "nat"   and y :: "nat" shows "fresh2 xs x y = (if x \<notin> xs \<or> infinite xs then x else if y \<notin> xs then y else fresh2 xs (x - 1) (y + 1))"
inner_gt_zero_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(0 < inner x x) = (x \<noteq> zero)"
line_integral_on_path_eq_line_integral_on_equiv_chain_2: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_path \<gamma> one_chain"   and "boundary_chain one_chain"   and "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"   and "finite basis" shows "line_integral_exists F basis \<gamma>"
assms_3: shows "y \<in>\<^sub>\<circ> S"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<zero>\<^bsub>P\<^esub> [^]\<^bsub>P\<^esub> n = \<zero>\<^bsub>P\<^esub>"
LIMSEQ_const_div: fixes f :: "nat \<Rightarrow> real"   and a :: "real"   and c :: "real" assumes "f \<longlonglongrightarrow> a"   and "c \<noteq> 0" shows "(\<lambda>x. f x / c) \<longlonglongrightarrow> a / c"
bij_lens_inv_right: fixes X :: "'c \<Longrightarrow> 'd" assumes "bij_lens X" shows "X ;\<^sub>L inv\<^sub>L X = \<Sigma>"
bound_typ_instance_BVar: fixes sch :: "type_scheme"   and n :: "nat" shows "sch \<le> BVar n"
Min_ex: assumes "finite (f ` {i..j})"   and "f ` {i..j} \<noteq> {}" shows "\<exists>m\<in>f ` {i..j}. \<forall>n\<in>f ` {i..j}. m \<le> n"
poly_mult: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "poly (p * q) x = poly p x * poly q x"
mta_bisim_def: fixes tta1 :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and tta2 :: "'t \<times> ('l, 't, 'x2, 'm2, 'w, 'o) thread_action" shows "tta1 \<sim>T tta2 \<equiv> fst tta1 = fst tta2 \<and> snd tta1 \<sim>m snd tta2"
x_axis_integral_measurable: shows "(\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) \<in> borel_measurable lborel"
finite_funs\<^sub>s\<^sub>t: fixes S :: "('a, 'b) strand_step list" shows "finite (funs\<^sub>s\<^sub>t S)"
sup_bot_right_2: fixes x :: "'b" shows "x \<squnion> \<bottom> = x"
thr_init_fin_list_state': fixes s :: "status"   and \<sigma> :: "('c, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "thr (init_fin_lift_state s \<sigma>) = map_option (\<lambda>(x, ln). ((s, x), ln)) \<circ> thr \<sigma>"
OppositeCategory_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "Op C \<equiv> \<lparr>Obj = obj\<^bsub>C\<^esub>, Mor = mor\<^bsub>C\<^esub>, Dom = Cod C, Cod = Dom C, Id = Category.Id C, Comp = \<lambda>f g. g ;;\<^bsub>C\<^esub> f, \<dots> = Category.more C\<rparr>"
infsetsum_Times: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'d set" assumes "f abs_summable_on A \<times> B" shows "infsetsum f (A \<times> B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B. f (x, y))"
lassocr_inverse: shows "rassocl\<^sub>C \<odot> lassocr\<^sub>C = 1\<^sub>C"
finprod_cong': fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finprod G f A = finprod G g B"
map_option_case: fixes f :: "'g \<Rightarrow> 'f"   and y :: "'g option" shows "map_option f y = (case y of None \<Rightarrow> None | \<lfloor>x\<rfloor> \<Rightarrow> \<lfloor>f x\<rfloor>)"
same_upto_inv_upto_jb: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "same_upto n j A B"   and "inv_upto n ??.Jordan_Normal_Form_Existence.jb A j" shows "inv_upto n ??.Jordan_Normal_Form_Existence.jb B j"
zet_ext_eq: fixes A :: "'a zet"   and B :: "'a zet" shows "(A = B) = (\<forall>x. zin x A = zin x B)"
inj_on_iff_surj: fixes A :: "'a set"   and A' :: "'b set" assumes "A \<noteq> {}" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> A') = (\<exists>g. g ` A' = A)"
Suc_Suc: fixes a :: "nat"   and b :: "nat" shows "{Suc a..<Suc b} = Suc ` {a..<b}"
ordinal_plus_times_omega: fixes x :: "ordinal" shows "x + x * \<omega> = x * \<omega>"
A: shows "((s, c), ee, s', c') \<in> ntrs fg"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_val e x = pdevs_val e' (summarize_pdevs p I d x)"   and "\<And>i. i < d \<Longrightarrow> e i = e' i"   and "e' \<in> UNIV \<rightarrow> {- 1..1}" shows "thesis"
eventually_imp_frequently: fixes P :: "'a \<Rightarrow> bool" assumes "frequently P F" shows "eventually P F"
cuts_incr_decr: fixes xs :: "int list" shows "\<forall>c\<in>set (cuts xs). incr c \<or> decr c"
sets_uniform_count_measure: fixes A :: "'b set" shows "sets (uniform_count_measure A) = Pow A"
countable_network_axioms: shows "countable_network (residual_network f)"
basis_in_max_indep_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and x :: "'b" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis_in indep \<E> X"   and "x \<in> \<E> - X" shows "\<not> indep_system.indep_in indep \<E> (insert x X)"
if3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e1 :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and e2 :: "exp0"   and e3 :: "exp0" assumes "evaluate ck env s e1 (s', Rerr err)" shows "evaluate ck env s (exp0.If e1 e2 e3) (s', Rerr err)"
wf_mult1: fixes r :: "('k \<times> 'k) set" assumes "wf r" shows "wf (mult1 r)"
F: fixes x :: "'c" shows "monotone (fun_ord orda) orda (\<lambda>f. U1 (F (C1 f)) x)"
intros_4: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
a: fixes b :: "'a set" assumes "b \<in># \<B>"   and "x \<in> b" shows "x \<notin> b\<^sup>c"
mon: shows "monic (\<Prod>e\<leftarrow>es. [:- e, 1::'a:])"
analz_Crypt_if: fixes K :: "key"   and X :: "msg"   and H :: "msg set" shows "analz (insert (Crypt K X) H) = (if Key (invKey K) \<in> analz H then insert (Crypt K X) (analz (insert X H)) else insert (Crypt K X) (analz H))"
Ord_succ_mono: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Ord \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "ZFC_in_HOL.succ \<alpha> \<in>\<^sub>\<circ> ZFC_in_HOL.succ \<beta>"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_mult" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_mult" and "vrat_mult\<lparr>a\<rparr> = b"
simps_3: fixes A :: "hfm"   and B :: "hfm" shows "hfm_to_fm (A HOR B) = hfm_to_fm A OR hfm_to_fm B"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'a"   and llist :: "'b llist" shows "(case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = (if lnull llist then f1 else f2 (lhd llist) (ltl llist))"
converse_tranclp_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r y b \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>+\<^sup>+ z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
rank_of_eq_card_basis_in: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and B :: "'b set" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.basis_in indep \<E> B" shows "indep_system.lower_rank_of indep \<E> = card B"
line_eq_endpoint_iff_2: fixes a :: "'b"   and b :: "'b"   and i :: "real" shows "(a = {a -- b}\<^bsub>i\<^esub>) = (a = b \<or> i = 0)"
comp_simp: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "VVV.seq g f" shows "VVV.comp g f = VxVxV.comp g f"
proj_P2R: fixes \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "proj \<phi> x = P2R \<phi> `` {x}"
rGamma_complex_altdef: fixes z :: "complex" shows "rGamma z = (if z \<in> \<int>\<^sub>\<le>\<^sub>0 then 0 else exp (- ln_Gamma z))"
add_le_cancel_right: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset" shows "(a + c \<subseteq># b + c) = (a \<subseteq># b)"
Suc_double_not_eq_double: fixes m :: "nat"   and n :: "nat" shows "Suc (2 * m) \<noteq> 2 * n"
frangeI: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> r"
big_equal_iff_bigtheta: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(\<Omega>[F](f) = \<Omega>[F](g)) = (f \<in> \<Theta>[F](g))"
is_nan_float_sqrt: fixes x :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x" shows "IEEE.is_nan (float_sqrt x)"
assms_2: shows "net_ips s1 \<inter> net_ips s2 = {}"
eval_exprs_eq: fixes G :: "prog"   and s :: "state"   and t :: "expr list"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In3 t\<succ>\<rightarrow> (w, s') = (\<exists>vs. w = In3 vs \<and> G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<rightarrow> s')"
sort_ex_norm_sort: fixes s :: "String.literal set" assumes "sort_ex cs s" shows "sort_ex cs (normalize_sort cs s)"