hcju/isabelleps huggingface dataset:
simps: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b list"   and a3 :: "'b" shows "traces.trace step a1 a2 a3 = ((\<exists>final. a1 = final \<and> a2 = [] \<and> a3 = final) \<or> (\<exists>conf' T final conf. a1 = conf \<and> a2 = conf' # T \<and> a3 = final \<and> traces.trace step conf' T final \<and> step conf conf'))"
Confidentiality_A: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Key K \<notin> analz (knows Spy evs)"
imp_graph_skew_sym: fixes l\<^sub>1 :: "lit"   and l\<^sub>2 :: "lit"   and cnf :: "lit uprod set" assumes "(l\<^sub>1, l\<^sub>2) \<in> imp_graph cnf" shows "(negate l\<^sub>2, negate l\<^sub>1) \<in> imp_graph cnf"
h0: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly"   and i :: "nat" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"   and "i \<le> k" shows "subresultant_prs_locale.h \<delta> f i \<noteq> 0"
ide_char: fixes Arr :: "'a set"   and f :: "(bool, 'a list) concrete_category.arr" shows "partial_magma.ide (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) f = (f = quiver.Zero \<or> f = quiver.One)"
c_manifolds_axioms: shows "c_manifolds k charts1 charts2"
trunc_cfs: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "n < deg R p" shows "Cring_Poly.truncate R p n = p n"
orbit_cycle_non_elem: fixes a :: "'b"   and as :: "'b list" assumes "a \<notin> set as" shows "orbit \<langle>as\<rangle> a = {a}"
rewrite_MultiportPorts_one_nodisc_1: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and a :: "ipt_l4_ports negation_type" assumes "\<forall>a. \<not> disc (Src_Ports a)"   and "\<forall>a. \<not> disc (Dst_Ports a)" shows "\<not> has_disc disc (rewrite_MultiportPorts_one a)"
indices_eq: shows "{0..<m + n} = {..<n} \<union> (+) n ` {..<m}"
prems_3: fixes x :: "'a node \<times> 'a node set" assumes "((\<mu> U\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> \<eta> \<in> old (fst x) \<union> new (fst x) \<or> \<mu> \<in> old (fst x) \<union> new (fst x) \<and> \<mu> U\<^sub>r \<eta> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> U\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> \<eta> \<in> old nd \<or> \<mu> \<in> old nd \<and> \<mu> U\<^sub>r \<eta> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>n\<in>snd x. \<forall>nd\<in>snd x. \<mu> U\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming nd \<longrightarrow> \<eta> \<in> old n \<or> \<mu> \<in> old n \<and> \<mu> U\<^sub>r \<eta> \<in> old nd) \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (\<lambda>r. \<forall>n\<in>snd r. \<forall>nd\<in>snd r. \<mu> U\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming nd \<longrightarrow> \<eta> \<in> old n \<or> \<mu> \<in> old n \<and> \<mu> U\<^sub>r \<eta> \<in> old nd)"
epiE: fixes f :: "'s"   and g :: "'s"   and g' :: "'s" assumes "epi f"   and "seq g f"   and "seq g' f"   and "g \<cdot> f = g' \<cdot> f" shows "g = g'"
card_eq_const_sum: fixes A :: "'a set"   and k :: "real" assumes "finite A" shows "k * real (card A) = (\<Sum>x\<in>A. k)"
prod_ord_pair_ko_def: fixes ko :: "'a key_order" shows "prod_ord_pair_ko ko \<equiv> comparator.prod_ord_pair (key_compare ko)"
modify_tprog_fetch_odd: fixes st :: "nat"   and tp :: "(action \<times> nat) list" assumes "st \<le> length tp div 2"   and "0 < st" shows "modify_tprog tp ! Suc (Suc (4 * (st - Suc 0))) = action_map (fst (tp ! Suc (2 * (st - Suc 0))))"
induct: fixes P :: "'a list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "nat" assumes "\<And>n. P [] n"   and "\<And>x xs n. (\<And>nat. n = Suc nat \<Longrightarrow> P xs nat) \<Longrightarrow> P (x # xs) n" shows "P a0 a1"
mono_strict_invE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "f x < f y" obtains "less x y"
outside_in_components: fixes S :: "'a set" shows "(outside S \<in> components (- S)) = (connected (outside S) \<and> outside S \<noteq> {})"
equiv_Eps_preserves: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "equiv A r"   and "X \<in> A // r" shows "(SOME x. x \<in> X) \<in> A"
split_list_prop: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x"
times_left_mono: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "(0::'a) \<le> z"   and "y \<le> x" shows "y * z \<le> x * z"
encode2_replicate_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and i :: "nat"   and k :: "nat" assumes "grover_state_sig n f dims R K"   and "i < 2 ^ Suc k" shows "partial_state.encode2 (replicate (Suc k) 2) {0..<k} i = i div 2 ^ k"
assms_1: fixes a :: "nat"   and b :: "nat" assumes "a \<in> A"   and "b \<in> A" shows "gcd a b \<in> A"
quad_form_diagonal_iff: fixes k1 :: "complex"   and k2 :: "complex"   and z1 :: "complex" assumes "k1 \<noteq> 0"   and "is_real k1"   and "is_real k2"   and "Re k1 * Re k2 < 0" shows "(quad_form (z1, 1) (k1, 0, 0, k2) = 0) = (\<exists>\<phi>. z1 = rcis (sqrt (Re (- k2 / k1))) \<phi>)"
v_v': shows "\<forall>c\<in>X. v' (v c) = c"
one_FactGroup: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" shows "\<one>\<^bsub>G Mod N\<^esub> = N"
UP_map\<^sub>0_obj: fixes a :: "'a" assumes "B.obj a" shows "UP.map\<^sub>0 a = UP\<^sub>0 a"
independent_inj_on_image: fixes s :: "'b set" assumes "m1.independent s"   and "inj_on f (m1.span s)" shows "m2.independent (f ` s)"
step: fixes g :: "'c"   and x :: "'b"   and y :: "'a" assumes "\<And>x y. U g x = Some y \<Longrightarrow> P x y"   and "U (F g) x = Some y"   and "\<And>x. option_ord (U g x) (U f x)" shows "P x y"
lookup_simps_1: fixes e :: "('a, 'b, 'c) env" shows "lookup e [] = Some e"
FAddSA_ran: fixes G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and S :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "\<forall>T\<in>dom G. T \<noteq> S \<longrightarrow> the (G T) \<inter> the (G S) = {}"   and "S \<in> dom G"   and "States SA \<inter> dom G = {}" shows "ran (G [f+] (S, SA)) = insert {} (insert (insert SA (the (G S))) (ran G - {the (G S)}))"
l_zero: fixes a :: "'a" assumes "a \<in> carrier R" shows "\<zero> \<plusminus> a = a"
trace_callee_aux_append: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and p :: "'d spmf"   and xs :: "('e \<times> 'f) list"   and ys :: "('e \<times> 'f) list" shows "trace_callee_aux callee p (xs @ ys) = trace_callee_aux callee (trace_callee_aux callee p xs) ys"
simps_12: fixes f11 :: "'b"   and f12 :: "X \<Rightarrow> 'b \<Rightarrow> 'b"   and f13 :: "Y \<Rightarrow> 'a \<Rightarrow> 'b"   and f21 :: "X \<Rightarrow> 'b \<Rightarrow> 'a"   and f22 :: "Y \<Rightarrow> 'a \<Rightarrow> 'a"   and f23 :: "'a"   and x21 :: "X" shows "rec_Y f11 f12 f13 f21 f22 f23 (Y.D x21) = f21 x21 (rec_X f11 f12 f13 f21 f22 f23 x21)"
exec_instr_Checkcast: fixes C :: "char list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr (Checkcast C) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (if cast_ok P C h (hd stk) then (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh) else (\<lfloor>addr_of_sys_xcpt ClassCast\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh))"
mat_of_rows_list_def: fixes nc :: "nat"   and rs :: "'b list list" shows "mat_of_rows_list nc rs = mat (length rs) nc (\<lambda>(i, j). rs ! i ! j)"
vcg_conj: fixes coms :: "'a \<Rightarrow> ('b, 'c, 'd, 'e) com"   and p :: "'a"   and aft :: "'e \<Rightarrow> 'c set"   and I :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool"   and c :: "('b, 'c, 'd, 'e) com"   and Q :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool"   and R :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool" assumes "valid_syn coms p aft I c Q"   and "valid_syn coms p aft I c R" shows "valid_syn coms p aft I c (\<lambda>s. Q s \<and> R s)"
valid_W_inv_mo_co_mark: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and y :: "'g"   and p :: "'d process_name" assumes "valid_W_inv s"   and "white y s"   and "sys_mem_lock s = Some p"   and "filter is_mw_Mark (sys_mem_store_buffers p s) = []"   and "p \<noteq> sys" shows "valid_W_inv (s(p := s p\<lparr>ghost_honorary_grey := {y}\<rparr>, sys := s sys\<lparr>mem_store_buffers := (mem_store_buffers (s sys))(p := sys_mem_store_buffers p s @ [mw_Mark y (sys_fM s)])\<rparr>))"
tendsto_Re: fixes g :: "'a \<Rightarrow> complex"   and a :: "complex"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Re (g x)) \<longlongrightarrow> Re a) F"
lv_rel_def: shows "lv_rel = br eucl_of_list (\<lambda>xs. length xs = DIM('b))"
E_def: shows "E = {(515 / 10\<^sup>2, - (651 / 10 ^ 3))..(518 / 10\<^sup>2, - (647 / 10 ^ 3))}"
assms_3: shows "B \<equiv> \<Union>s\<in>A\<^sup>\<star>. {suff A s}"
many_strongly_connected_iff_4: fixes x :: "'a" shows "many_strongly_connected x = (x \<le> x\<^sup>T\<^sup>+)"
map_index'_append: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c list"   and b :: "'c list" shows "map_index' n f (a @ b) = map_index' n f a @ map_index' (n + length a) f b"
op_dghm_dghm_cn_comp: fixes \<GG> :: "V"   and \<FF> :: "V" shows "op_dghm (\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>) = op_dghm \<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> op_dghm \<FF>"
rrsmult_supp: fixes r :: "'r"   and f :: "'r \<Rightarrow> 'a" shows "supp (r \<currency> f) \<subseteq> R"
dghm_up_components_1: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" shows "dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr> = (\<lambda>a\<in>\<^sub>\<circ>\<CC>\<lparr>Obj\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<phi> i\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>)"
H_fcs_a_neigh: shows "rev.H.face_cycle_set rev.a_neigh = segment G.face_cycle_succ rev.a' a'"
list_of_oalist_ntm: fixes x :: "('a, 'b) oalist_ntm" shows "list_of_oalist_ntm x \<in> Collect ko_ntm.oalist_inv"
bij_UP: shows "bij_betw SC.UP UNIV SC.Univ"
simps_2: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr" shows "(ApplyL\<^sub>\<E> \<E> e1) [e] = Apply (\<E> [e]) e1"
mk_next_pow_dist: fixes i :: "nat"   and j :: "nat"   and \<phi> :: "'a ltln" shows "X\<^sub>n' (i + j) \<phi> = X\<^sub>n' i (X\<^sub>n' j \<phi>)"
compact_Int: fixes s :: "'b set"   and t :: "'b set" assumes "compact s"   and "compact t" shows "compact (s \<inter> t)"
erH_multTr1: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)"   and e :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S c"   and "pol_coeff S d"   and "pol_coeff S e"   and "fst e = fst c + fst d"   and "polyn_expr R X (fst c) c \<cdot>\<^sub>r polyn_expr R X (fst d) d = polyn_expr R X (fst c + fst d) e" shows "polyn_expr A Y (fst c) (cf_h h c) \<cdot>\<^sub>r\<^bsub>A\<^esub> polyn_expr A Y (fst d) (cf_h h d) = polyn_expr A Y (fst e) (cf_h h e)"
PDQ: shows "(P, D, Q) = diagonalize_2x2_JNF A"
the_\<nu>_simps_4: shows "local.dom \<omega>.the_\<nu> = r\<^sub>1"
finite_dimensional_vector_space_ow_def: fixes U\<^sub>V\<^sub>S :: "'c set"   and plus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero\<^sub>V\<^sub>S :: "'c"   and minus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale basis \<equiv> vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<and> finite_dimensional_vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis"
TER: shows "\<E> (TER f) - A \<Gamma> \<subseteq> TER h"
fx1: fixes t :: "'b"   and i :: "'a" assumes "t \<in> cbox a b"   and "i \<in> Basis" shows "continuous_on S (\<lambda>x. fx x t i)"
ode_to_fvo: fixes x :: "'d"   and I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE" assumes "x \<in> ODE_vars I ODE" shows "x \<in> FVO ODE"
cnf2bhv_suc: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "dynamic_component.lActive active c t \<le> n" shows "dynamic_component.cnf2bhv tCMP active c t (Suc n) = Suc (dynamic_component.cnf2bhv tCMP active c t n)"
SN_rel_ext_def: fixes P :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set"   and M :: "'a \<Rightarrow> bool" shows "SN_rel_ext P Pw R Rw M \<equiv> \<nexists>f t. (\<forall>i. (f i, f (Suc i)) \<in> SN_rel_ext_step P Pw R Rw (t i)) \<and> (\<forall>i. M (f i)) \<and> (\<exists>\<^sub>\<infinity>i. t i \<in> {top_s, top_ns}) \<and> (\<exists>\<^sub>\<infinity>i. t i \<in> {top_s, normal_s})"
Collect_conj_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "rel_ext (\<lambda>x. P x \<and> Q x) = rel_ext P \<inter> rel_ext Q"
exists_ge: fixes i :: "'a" shows "\<exists>k. i \<le> k"
add_ac_3: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b + (a + c) = a + (b + c)"
la: shows "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) signs"
cod_char: fixes f :: "'c" shows "C'.cod f = (if C'.arr f then C.cod f else C.null)"
icard_cartesian_product: fixes A :: "'a set"   and B :: "'b set" shows "icard (A \<times> B) = icard A * icard B"
select_index_union: fixes n :: "nat"   and i :: "nat" shows "{k |k. select_index n i k} \<union> {k |k. k < 2 ^ n \<and> \<not> select_index n i k} = {0..<2 ^ n}"
circular: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "dpo le"   and "le x y"   and "le y z"   and "le z x" shows "x = y \<and> y = z"
inf_continuous_mono: fixes F :: "'c \<Rightarrow> 'd" assumes "inf_continuous F" shows "mono F"
first_fresh_variable_not_in_lvars: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" shows "\<forall>var\<in>lvars (Tableau (preprocess' cs start)). var < FirstFreshVariable (preprocess' cs start)"
2: shows "\<exists>x y z. f b * b + f c * c = x * b + y * b + z * c"
simps_5: fixes I :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc ({I} WHILE b DO C) Q = (\<forall>s n. (I (Suc n) s \<longrightarrow> pre C (I n) s) \<and> (I (Suc n) s \<longrightarrow> bval b s) \<and> (I 0 s \<longrightarrow> \<not> bval b s \<and> Q s) \<and> vc C (I n))"
Abs_def: fixes z :: "ant" shows "Abs z = (if z < 0 then - z else z)"
A91_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. C \<noteq> sA92 \<and> C \<noteq> sA91 \<and> C \<noteq> sA81 \<longrightarrow> sA91 \<notin> Sources level1 C"
simps_3: fixes x1 :: "nat" shows "Entry \<noteq> Label x1"
nextMax_max_2: fixes x :: "nat set"   and v :: "nat" assumes "finite x"   and "v \<in> x" shows "v \<le> nextMax x"
not_exact_match_in_doubt_deny_approx_match: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and a :: "action"   and m :: "'c match_expr"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "a = action.Accept \<or> a = Reject \<or> a = action.Drop"   and "\<not> Semantics.matches \<gamma> m p" shows "(a = action.Drop \<or> a = Reject) \<and> Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p \<or> \<not> Matching_Ternary.matches (\<beta>, in_doubt_deny) m a p"
g_SOURCE: fixes x :: "'v" shows "g (SOURCE, \<langle>x\<rangle>) = (if x = b then \<alpha> else 0)"
iMOD_0_subset_conv: fixes r' :: "nat"   and r :: "nat"   and m :: "nat" shows "([ r', mod 0 ] \<subseteq> [ r, mod m ]) = (r' mod m = r mod m \<and> r \<le> r')"
RootedTree_def: fixes G :: "('c, 'd) Graph_scheme"   and root :: "'c" shows "RootedTree G root \<equiv> Tree G \<and> RootedTree_axioms G root"
filtration: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" assumes "filtrated_prob_space M F" shows "Filtration.filtration M F"
segment_Worder: fixes a :: "'a" shows "Worder (Iod D (segment D a))"
proper_def: fixes f :: "'b \<Rightarrow> ereal" shows "proper f = proper_on UNIV f"
subseqs_sub_mset: fixes ws :: "'a list"   and vs :: "'a list" assumes "ws \<in> set (subseqs vs)" shows "mset ws \<subseteq># mset vs"
table_wf_table: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set" shows "table n A X = wf_table n A (\<lambda>v. v \<in> X) X"
globalasimp: fixes P :: "(nat \<Rightarrow> 'd) \<times> 'e \<times> (nat \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "((nat \<Rightarrow> 'd) \<times> 'f) \<times> 'e \<times> (nat \<Rightarrow> 'd) \<times> 'f" shows "globala P s = P (fst (fst s), fst (snd s), fst (snd (snd s)))"
gbinomial_0_1: fixes a :: "'a" shows "a gchoose 0 = (1::'a)"
bit_eqI: fixes a :: "'a"   and b :: "'a" assumes "\<And>n. (2::'a) ^ n \<noteq> (0::'a) \<Longrightarrow> bit a n = bit b n" shows "a = b"
present_def: fixes h :: "heap"   and r :: "'a ref" shows "Ref.present h r = (addr_of_ref r < lim h)"
tendsto_cong_limit: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter"   and k :: "'b" assumes "(f \<longlongrightarrow> l) F"   and "k = l" shows "(f \<longlongrightarrow> k) F"
D: fixes a :: "'a" shows "a \<le> (1::'a)"
Let_1: shows "atom ` domA \<Delta> \<sharp>* \<Gamma>"
suffix_tl: fixes xs :: "'c list" shows "suffix (tl xs) xs"
bounded_insert: fixes x :: "'a"   and S :: "'a set" shows "bounded (insert x S) = bounded S"
LID\<^sub>S_def: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" shows "LID\<^sub>S \<sigma> \<equiv> dom \<sigma> \<union> \<Union> (LID\<^sub>V ` ran \<sigma>)"
RSR: shows "range (snd \<circ> r) \<subseteq> sa.V"
cases: fixes a :: "('s, 'v) tm" assumes "wary a" obtains \<zeta> :: "('s, 'v) hd" where "a = Hd \<zeta>"   | s :: "('s, 'v) tm" and t :: "('s, 'v) tm" where "a = App s t" and "wary s" and "wary t" and "enat (num_args s) < arity_hd (head s)"
assms_1: shows "f\<cdot>r \<noteq> \<bottom>"
fresh_3: fixes a :: "name"   and x1 :: "name"   and x2 :: "trm" shows "a \<sharp> Lam [x1].x2 = a \<sharp> [x1].x2"
I_Q: shows "I \<inter> - b \<subseteq> Q"
holomorphic_convex_primitive': fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "convex S"   and "open S"   and "f holomorphic_on S" obtains g :: "complex \<Rightarrow> complex" where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
distinct_3: fixes x31 :: "'a list"   and x32 :: "'a list" shows "NO_CYC \<noteq> CIRC x31 x32"
hyps: shows "!Q \<longmapsto>\<alpha> \<prec> Q'"
leg0_simps_5: shows "cod t\<^sub>0 = t\<^sub>0"
diff_le_0_iff_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (minus a b) zero = less_eq a b"
carrierE: fixes x :: "'a" assumes "x \<in> carrier" obtains t :: "'a set" where "t open" and "x \<in> t"
assms: shows "R[a\<turnstile>c>b] = OrR1 <c>.M e"
fg: shows "f = g \<or> (\<exists>a b d. (a, d) \<in> c\<^sup>+ \<and> (b, d) \<in> c\<^sup>+ \<and> f = Abs a \<and> g = Abs b)"
ZFCartProd: fixes x :: "ZF"   and A :: "ZF"   and B :: "ZF" assumes "x |\<in>| A |\<times>| B" shows "Fst x |\<in>| A \<and> Snd x |\<in>| B \<and> isOpair x"
l_comb_Suc: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and A :: "'f set"   and s :: "nat \<Rightarrow> 'f"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e" assumes "R module M"   and "H \<subseteq> carrier M"   and "ideal R A"   and "s \<in> {j. j \<le> Suc n} \<rightarrow> carrier R"   and "f \<in> {j. j \<le> Suc n} \<rightarrow> H" shows "l_comb R M (Suc n) s f = l_comb R M n s f \<plusminus>\<^bsub>M\<^esub> s (Suc n) \<cdot>\<^sub>s\<^bsub>M\<^esub> f (Suc n)"
CastThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
main: fixes z :: "complex"   and y :: "complex"   and u :: "complex"   and f :: "complex" assumes "e \<noteq> 0 \<Longrightarrow> z\<^sup>2 - y * z - e / 3 = 0"   and "e \<noteq> 0 \<Longrightarrow> u = z ^ 3" shows "(y ^ 3 + e * y + f = 0) = (u\<^sup>2 + f * u + v = 0)"
wf_typeE1: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a" assumes "yoneda_functor C S \<phi>"   and "yoneda_functor_fixed_object_axioms C a" shows "yoneda_functor_fixed_object C S \<phi> a"
tranclE: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set" assumes "(a, b) \<in> r\<^sup>+" obtains "(a, b) \<in> r"   | c :: "'e" where "(a, c) \<in> r\<^sup>+" and "(c, b) \<in> r"
grow_mono: fixes bc :: "nat \<Rightarrow> nat"   and mining :: "nat \<Rightarrow> bool"   and n :: "nat"   and n' :: "nat" assumes "honest_growth bc mining"   and "n \<le> n'" shows "bc n \<le> bc n'"
supr_under: fixes a :: "'a" assumes "a \<in> Field r" shows "a = supr (local.under a)"
sorted_wrt_subsumes_list_subsumes_list_filter: fixes Ls :: "('a, 'b) Term.term literal list"   and Ks :: "('a, 'b) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option" assumes "sorted_wrt leq_lit Ls" shows "subsumes_list Ls Ks \<sigma> = subsumes_list_filter Ls Ks \<sigma>"
QTrans: shows "Q \<Longrightarrow>\<^sub>ly in Q''\<rightarrow>a<x> \<prec> Q'"
zip_same_conv_map: fixes xs :: "'a list" shows "zip xs xs = map (\<lambda>x. (x, x)) xs"
\<alpha>'_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set" assumes "arr f"   and "arr g"   and "arr h" shows "\<alpha>' (f, g, h) = tensor (tensor f g) h \<cdot> assoc' (local.dom f) (local.dom g) (local.dom h)"