hcju/isabelleps huggingface dataset:
unique_factorization_m_def: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "poly_mod.unique_factorization_m m f cfs = (poly_mod.Mf m ` Collect (poly_mod.factorization_m m f) = {poly_mod.Mf m cfs})"
inject: fixes x :: "'a"   and ya :: "'a" shows "(DeleteMessage x = DeleteMessage ya) = (x = ya)"
absorb2: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a \<sqinter> b = b"
diagonal_to_Smith_PQ_JNF_mod_ring_with_types: fixes A :: "'c mat"   and S :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "A \<in> carrier_mat CARD('d mod_ring) CARD('e mod_ring)"   and "S \<in> carrier_mat CARD('d mod_ring) CARD('e mod_ring)"   and "P \<in> carrier_mat CARD('d mod_ring) CARD('d mod_ring)"   and "Q \<in> carrier_mat CARD('e mod_ring) CARD('e mod_ring)"   and "diagonal_to_Smith_PQ_JNF A bezout = (P, S, Q)"   and "isDiagonal_mat A"   and "is_bezout_ext bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S"
maxi_stable: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd"   and A :: "nat set"   and k :: "'d" assumes "xs \<noteq> []"   and "maxi xs key \<in> A" shows "filter (\<lambda>x. key x = k) (nths xs (A - {maxi xs key}) @ [xs ! maxi xs key]) = filter (\<lambda>x. key x = k) (nths xs A)"
const_res_def: fixes k :: "nat"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "const_res k s = (THE n. \<forall>m. const_res_subseq k s m k = n)"
H_cons: fixes p :: "'a"   and p' :: "'a"   and q' :: "'a"   and q :: "'a"   and x :: "'a" assumes "t p \<le> t p'"   and "t q' \<le> t q"   and "H p' x q'" shows "H p x q"
verit_prod_simplify_1: fixes a :: "'c" shows "(1::'c) * a = a"
proj_specific_def: fixes n :: "'a"   and t :: "('fun, 'var) Term.term"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and Secrets :: "('fun, 'var) Term.term set" shows "proj_specific n t \<A> Secrets \<equiv> t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n \<A>)) - (Secrets \<union> {m. {} \<turnstile>\<^sub>c m})"
0: fixes z :: "'b" shows "1 + norm z \<noteq> 0"
DF_all_tickfree_traces1: fixes A :: "'b set" shows "T (DF A) = {s. set s \<subseteq> ev ` A}"
ring_tOp_assoc: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<cdot>\<^sub>r\<^bsub>R\<^esub> z = x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<cdot>\<^sub>r\<^bsub>R\<^esub> z)"
poincare_distance_zero_x_axis: fixes x :: "complex_homo" assumes "x \<in> unit_disc"   and "x \<in> circline_set x_axis" shows "poincare_distance 0\<^sub>h x = (let x' = to_complex x in \<bar>ln (Re ((1 - x') / (1 + x')))\<bar>)"
fresh_prodD_2: fixes a :: "'a"   and x :: "'b"   and y :: "'c" assumes "a \<sharp> (x, y)" shows "a \<sharp> y"
mpoly_of_poly_pCons: fixes i :: "nat"   and c :: "'b"   and p :: "'b poly" shows "mpoly_of_poly i (pCons c p) = Const c + Var i * mpoly_of_poly i p"
preserves_reflects_arr: fixes f :: "'a" shows "V.arr (cod f) = V.arr f"
mem_sphere: fixes y :: "'a"   and x :: "'a"   and e :: "real" shows "(y \<in> sphere x e) = (dist x y = e)"
distinct_length_le: fixes ys :: "'a list"   and xs :: "'a list" assumes "distinct ys"   and "set ys = set xs" shows "length ys \<le> length xs"
x: shows "{x. root_cond (p, l, r) x} = {real_of_1 (p, l, r)}"
echelon_form_upt_k_condition1': fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c"   and j :: "'c" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A"   and "i < j" shows "is_zero_row_upt_k j k A"
trans: fixes b :: "complex"   and a :: "complex"   and c :: "complex" assumes "b \<le>\<^sub>\<complex> a"   and "c \<le>\<^sub>\<complex> b" shows "c \<le>\<^sub>\<complex> a"
prems_3: fixes k :: "nat" assumes "k < Suc n" shows "f k \<le>\<^sub>L g k"
dual_additive_antitone: fixes f :: "'a \<Rightarrow> 'a" assumes "dual_additive f" shows "isotone (uminus \<circ> f)"
last_message_hold_take: fixes n :: "nat"   and xs :: "'b message_af list" shows "last_message_hold xs \<down> n = last_message_hold (xs \<down> n)"
step: fixes stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option" assumes "\<tau>Exec_mover ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"   and "\<tau>exec_move ci P t e h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp'')"   and "Q stk' loc' pc' xcp'" shows "Q stk'' loc'' pc'' xcp''"
flow_eq_rev0: fixes y :: "'a"   and t :: "real" shows "rev.flow0 y t = flow0 y (- t)"
wpc_simpl: fixes t :: "'e itself"   and Rs :: "(('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph) set"   and S :: "nat \<Rightarrow> ('f, 'h) labeled_graph" assumes "Simple_WPC t Rs S" shows "WPC t Rs S"
space_measure_of: fixes A :: "'b set set"   and \<Omega> :: "'b set"   and \<mu> :: "'b set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
sparse_row_matrix_minus: fixes A :: "(nat \<times> (nat \<times> 'a) list) list" shows "sparse_row_matrix (minus_spmat A) = - sparse_row_matrix A"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<mu>"
t_Lam_elim: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> Lam [x].t : T"   and "x \<sharp> \<Gamma>" obtains T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" where "(x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2" and "T = T\<^sub>1\<rightarrow>T\<^sub>2"
Catch_decomp_star_Fault: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d"   and f :: "'f" assumes "\<Gamma> \<turnstile> (Catch p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>* (Skip, Fault f)" shows "\<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Fault f) \<or> (\<exists>s'. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Throw, Normal s') \<and> \<Gamma> \<turnstile> (p\<^sub>2, Normal s') \<rightarrow>\<^sup>* (Skip, Fault f))"
Sample_L_buffer: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and buffer :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "Sample_L req dt a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st buffer" shows "buffer t = inf_last_ti dt t"
d1: shows "(sterm_sem I \<theta>1 has_derivative frechet I \<theta>1 (fst \<nu>)) (at (fst \<nu>))"
Afix_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "Afix \<Gamma>\<cdot>ae = (\<mu> x. ABinds \<Gamma>\<cdot>x \<squnion> ae)"
g_bex_def: fixes m :: "'s"   and P :: "'k \<times> 'v \<Rightarrow> bool" shows "g_bex m P \<equiv> iteratei m Not (\<lambda>kv \<sigma>. P kv) False"
l2_when_disjoint: fixes l :: "nat"   and l' :: "nat"   and i :: "int"   and i' :: "int" assumes "l \<le> l'"   and "(i + 1) * 2 ^ (l' - l) < i' \<or> i' < (i - 1) * 2 ^ (l' - l)" shows "l2_\<phi> (l', i') (l, i) = 0"
theorem_sf: fixes A :: "fm" assumes "{} \<turnstile> A" shows "Sigma_fm A"
S_inductive_sup_mem: fixes X :: "'a set" assumes "S_inductive_set D"   and "Chain D X" shows "Algebra2.Sup D X \<in> carrier D"
39: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> (- - x \<squnion> y) = top \<squnion> y"
white_def: fixes r :: "'d"   and s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'd) Model.local_state" shows "white r s \<equiv> obj_at (\<lambda>obj. obj_mark obj \<noteq> sys_fM s) r s"
divide_strict_right_mono: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b"   and "less zero c" shows "less (divide a c) (divide b c)"
eucl_of_list_map_mult_right: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> real"   and r :: "real" assumes "length xs = DIM('b)" shows "eucl_of_list (map (\<lambda>x. f x * r) xs) = r *\<^sub>R eucl_of_list (map f xs)"
n2im1: shows "\<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 \<le> \<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2"
mem_adjoin: fixes H :: "'a set"   and x :: "'a"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)"   and "x \<in> H"   and "a \<in> carrier (G\<lparr>carrier := H\<rparr>)" shows "x \<in> adjoin (G\<lparr>carrier := H\<rparr>) H a"
Un_UNIV_left: fixes B :: "'a set" shows "UNIV \<union> B = UNIV"
prems_2: shows "odd (int (\<Prod>i\<in>#add_mset p ps. i))"
abs: fixes s :: "lam"   and t :: "lam"   and x :: "name" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "(Lam [x].s) \<rightarrow>\<^sub>\<beta> (Lam [x].t)"
nn_integral_0_iff_AE: fixes u :: "'h \<Rightarrow> ennreal"   and M :: "'h measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (AE x in M. u x = 0)"
naturality: fixes f :: "'a" assumes "B.arr f" shows "B.\<ll>'.map (B.cod f) \<cdot>\<^sub>B B.map f = B.L f \<cdot>\<^sub>B B.\<ll>'.map (B.dom f)"
2: shows "(\<lceil>c\<rceil>\<^sub>< \<Rightarrow> \<lceil>d\<^sub>1\<rceil>\<^sub>>) |a|b|\<^sub>\<sigma> (\<lceil>c\<rceil>\<^sub>< \<Rightarrow> \<lceil>d\<^sub>2\<rceil>\<^sub>>) \<sqsubseteq> P |a|b|\<^sub>\<sigma> Q"
under_Field: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "under r a \<subseteq> Field r"
atom_def: fixes a :: "'b" shows "atom a = ([True], \<lambda>b s. if s = [True] \<and> b = Some a then {[False]} else {}, \<lambda>s. s = [False])"
simps_2: fixes m :: "'node"   and x :: "nat" shows "valid_SDG_node (Formal_in (m, x)) = (\<exists>a Q r p fs ins outs. valid_edge a \<and> kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<and> targetnode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length ins)"
poly_Nil: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.semiring_0 plus zero times" shows "semiring_0.poly plus zero times [] x = zero"
False: shows "\<not> registers (mmu s2) CR AND 1 \<noteq> 0"
hash: fixes h :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "blinding_respects_hashes h bo" shows "bo \<le> vimage2p h h (=)"
TS_prefix_containment: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat"   and xs' :: "'e list" assumes "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "prefix xs' xs" shows "vs @ xs' \<in> TS M2 M1 \<Omega> V m i"
aux6: fixes x :: "'a"   and y :: "'a" shows "(x + y) \<cdot> - x = y \<cdot> - x"
simps_2: fixes I :: "('e, 'f, 'd) interp"   and x :: "'d"   and t :: "('e, 'd) trm" shows "prog_sem I (x := t) = {(\<nu>, \<omega>). \<omega> = repv \<nu> x (dterm_sem I t \<nu>)}"
rep_eq: fixes x :: "('b, 'c) pp" shows "keys_pp x = keys (pp.mapping_of x)"
lookup_is_in_vec: fixes "is" :: "nat list"   and A :: "'b tensor" assumes "is \<lhd> dims A" shows "lookup A is \<in> set (vec A)"
y_def: shows "y = rec_list (s j) (\<lambda>j _ Y. \<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i)"
sig_red_zero_mono: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and F' :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "sig_red_zero sing_reg F p"   and "F \<subseteq> F'" shows "sig_red_zero sing_reg F' p"
not_cong_3421: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Cong A B C D" shows "\<not> Cong C D B A"
case_2: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat" shows "(case Newpkt x21 x22 of Pkt x xa \<Rightarrow> f1 x xa | Newpkt x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
cartesian_product2: fixes D :: "(('b \<times> 'c) \<times> 'd) set"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "finite D"   and "{(a, b). \<exists>c. g a b c \<noteq> (0::'e)} \<times> {c. \<exists>a b. g a b c \<noteq> (0::'e)} \<subseteq> D" shows "(\<Sum>(a, b). Sum_any (g a b)) = (\<Sum>(a, b, c). g a b c)"
multiset_eq_iff: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(M = N) = (\<forall>a. count M a = count N a)"
divmod_numeral_simps_14: fixes n :: "num" shows "(num.One \<le> n) = True"
vertex_heuristic_def: fixes X :: "'a set"   and v :: "'a" shows "vertex_heuristic X v = (\<forall>u\<in>V - X. card (neighbors v - X) \<le> card (neighbors u - X))"
times_stream_def: fixes x :: "'a stream"   and y :: "'a stream" shows "x * y = pure (*) \<diamondop> x \<diamondop> y"
atLeast0_lessThan_Suc: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "prod g {0..<Suc n} = prod g {0..<n} * g n"
bind_runit: fixes x :: "'d option" shows "x \<bind> Some = x"
iMin_Int_ge2: fixes A :: "'b set"   and B :: "'b set" assumes "A \<inter> B \<noteq> {}" shows "iMin B \<le> iMin (A \<inter> B)"
321: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) \<sqinter> x = bot"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" obtains "a = a \<sqinter> b"
reduce_rules_def: fixes \<delta> :: "('d, 'e) ta_rule set"   and P :: "'d set" shows "reduce_rules \<delta> P \<equiv> \<delta> \<inter> {r. rule_states r \<subseteq> P}"
Ord_odiff_eq: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<le> \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>" shows "\<alpha> + odiff \<beta> \<alpha> = \<beta>"
cfi_rel_def: shows "cfi_rel \<equiv> br flow_of_cf (RGraph c s t)"
outside_mono: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "outside T \<subseteq> outside S"
less_eq_V_0_iff: fixes x :: "V" shows "(x \<subseteq>\<^sub>\<circ> []\<^sub>\<circ>) = (x = []\<^sub>\<circ>)"
addsI: fixes a :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and k :: "'a" assumes "a = plus b k" shows "plus.adds plus b a"
tarski_absolute_space_axioms_def: fixes B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "tarski_absolute_space_axioms B \<equiv> (\<forall>a b. B a b a \<longrightarrow> a = b) \<and> (\<forall>a b c p q. B a p c \<and> B b q c \<longrightarrow> (\<exists>x. B p x b \<and> B q x a)) \<and> (\<forall>X Y. (\<exists>a. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B a x y) \<longrightarrow> (\<exists>b. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B x b y))"
pos_def: shows "po = pos'(c := 0, d := 2)"
round_robin_step_invar_Some: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and n0 :: "nat"   and \<sigma> :: "'t list \<times> nat"   and t' :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and \<sigma>' :: "'t list \<times> nat" assumes "deterministic I"   and "round_robin_step n0 \<sigma> s t' = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "round_robin_invar \<sigma> (dom (thr s))"   and "s \<in> I" shows "round_robin_invar \<sigma>' (dom (thr s) \<union> {t. \<exists>x m. NewThread t x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>})"
distinct_snd_polychain_of: fixes xs :: "(real \<times> real) list"   and x0 :: "real \<times> real" assumes "list_all (\<lambda>x. x \<noteq> 0) xs"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "distinct (map snd (polychain_of x0 xs))"
U_def: shows "U = (\<lambda>d. {x \<in> space M. 1 - d \<le> upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - deltaf d l * real l < u n x - u (n - l) ((T ^^ l) x)}})"
Proc_CFG_SeqSecond: fixes c\<^sub>2 :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and n' :: "label"   and c\<^sub>1 :: "cmd" assumes "c\<^sub>2 \<turnstile> n -et\<rightarrow>\<^sub>p n'"   and "n \<noteq> Entry" shows "c\<^sub>1;; c\<^sub>2 \<turnstile> n \<oplus> #:c\<^sub>1 -et\<rightarrow>\<^sub>p n' \<oplus> #:c\<^sub>1"
image_diff_atLeastAtMost: fixes d :: "'c"   and a :: "'c"   and b :: "'c" shows "(-) d ` {a..b} = {d - b..d - a}"
r2f_def: fixes R :: "('b \<times> 'c) set" shows "\<F> R = bd\<^sup>-\<^sub>\<F> ((``) R)"
smc_is_arrD_3: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
tab_a1_map'_keys: shows "table n L (Mapping.keys a1_map')"
Max_Collect_less_nat: fixes k :: "nat" shows "Max {i. i < k} = (if k = 0 then Max {} else k - 1)"
assms_6: shows "is_a_intersec R A C' A' C"
hom_one: shows "eval R R id x \<one>\<^bsub>P\<^esub> = \<one>"
inext_nth_gr_Min_conv: fixes I :: "nat set"   and n :: "nat" assumes "I \<noteq> {}" shows "(iMin I < I \<rightarrow> n) = (0 < n \<and> (\<nexists>a. I = {a}))"
filt_simp: fixes b :: "nat"   and p :: "nat \<Rightarrow> bool"   and n :: "nat" shows "([b] = filter p [0..<n]) = (p b \<and> b < n \<and> (\<forall>y<n. p y \<longrightarrow> b = y))"
sccs_verts_disjoint: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> sccs_verts"   and "T \<in> sccs_verts"   and "S \<noteq> T" shows "S \<inter> T = {}"
eq_intro: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- (x \<squnion> - (y \<squnion> z)) = - (y \<squnion> - (x \<squnion> z))" shows "x = y"
monad_alt_writerT': assumes "monad_alt Monad_Overloading.return Monad_Overloading.bind alt" shows "monad_alt Monad_Overloading.return Monad_Overloading.bind alt"
measure_space: fixes M :: "'b measure" shows "measure_space (space M) (sets M) (emeasure M)"
oDeriv_fixed: fixes F :: "ordinal \<Rightarrow> ordinal"   and n :: "ordinal" assumes "normal F" shows "F (oDeriv F n) = oDeriv F n"
qFreshAbs_preserves_alphaAbs1: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs :: "'varSort"   and z :: "'var"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs A"   and "qFreshAbs zs z A"   and "A $= A'" shows "qFreshAbs zs z A'"
assms: assumes "x \<notin>\<^sub>\<circ> A" shows "x = y"
FL_valid_Act_strong: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and X :: "'e"   and P :: "'a"   and f :: "'d"   and \<alpha> :: "'c"   and x :: "('f, 'b, 'c, 'd) formula" assumes "effect_nominal_ts satisfies transition effect_apply L"   and "finite (supp X)" shows "effect_nominal_ts.FL_valid satisfies transition effect_apply P (Act f \<alpha> x) = (\<exists>\<alpha>' x' P'. Act f \<alpha> x = Act f \<alpha>' x' \<and> transition (effect_apply f P) \<langle>\<alpha>',P'\<rangle> \<and> effect_nominal_ts.FL_valid satisfies transition effect_apply P' x' \<and> bn \<alpha>' \<sharp>* X)"
checkDeadProcs_def: fixes g :: "'b gState_scheme" shows "checkDeadProcs g = (let (uu_, soDied, procs, dchans) = removeProcs (procs g) in if soDied then g\<lparr>procs := procs, gState.channels := cleanChans dchans (gState.channels g)\<rparr> else g)"
strict_boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<sqsubset> ord.min (\<sqsubseteq>) b c" obtains "a \<sqsubset> b" and "a \<sqsubset> c"
simps: fixes x :: "nat"   and n :: "nat" shows "token_run x n = run \<up>af\<^sub>G (Abs \<phi>) (suffix x w) (n - x)"
advantage\<^sub>1_def: fixes \<A> :: "nat \<Rightarrow> ('message \<times> 'signature, unit + 'message, 'vkey + 'signature) gpv"   and \<eta> :: "nat" shows "advantage\<^sub>1 \<A> \<eta> = spmf (suf_cma\<^sub>1 \<A> \<eta>) True"
vn: fixes x :: "nat \<Rightarrow>\<^sub>0 nat" shows "(if lookup x x' = 0 then 1 else Var x' ^ lookup x x') = (if lookup x x' = 0 then 1 else MPoly_Type.monom (monomial (Suc 0) x') 1) + (if lookup x x' = 0 then 0 else 1) * qr (lookup x x') * ((MPoly_Type.monom (monomial (Suc 0) x') 1)\<^sup>2 - MPoly_Type.monom (monomial (Suc 0) x') 1)"
cart_basis_def: shows "cart_basis = {axis i (1::'a) |i. i \<in> UNIV}"
deg: shows "degree (primitive_part_fract p) \<noteq> 0"
naturally_isomorphic_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" shows "naturally_isomorphic A B F G = (\<exists>\<tau>. natural_isomorphism A B F G \<tau>)"
right_minus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus a) = zero"
aezfun_setspan_el_decomp_aezdeltafun: fixes c :: "('a, 'b) aezfun"   and A :: "'b set" assumes "c \<in> aezfun_setspan A" shows "\<exists>ras. set (map snd ras) \<subseteq> A \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
countable_card_of_nat: fixes A :: "'a set" shows "countable A = (|A| \<le>o |UNIV|)"
IH_1: fixes inputs :: "real vec list"   and j :: "nat" assumes "valid_net' m1"   and "map dim_vec inputs = input_sizes m1"   and "j < output_size' m1" shows "evaluate_net m1 inputs $ j = (\<Sum>is | is \<lhd> input_sizes m1. (\<Prod>k<length inputs. inputs ! k $ (is ! k)) * lookup (tensors_from_net m1 $ j) is)"
ope: shows "openin (top_of_set (affine hull S)) S"
int_xor_extra_simps_1: fixes w :: "int" shows "w XOR 0 = w"
assms_4: shows "is_WAIT (tts (s (Suc i)) t)"
asymp_equiv_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g"   and "g \<sim>[F] h" shows "f \<sim>[F] h"
Try\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and C :: "char list"   and i :: "nat"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>try e\<^sub>1 catch(C i) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"
pfa_length_def: shows "pfa_length \<equiv> arl_length"