hcju/isabelleps huggingface dataset:
simps: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.siso tval aval indis a = (\<exists>c. a = c \<and> (\<forall>x xa xb. PL.transC tval aval (c, x) (xa, xb) \<longrightarrow> PL_Indis.siso tval aval indis xa) \<and> (\<forall>x xa xb xc. (x, xa) \<in> indis \<longrightarrow> PL.transC tval aval (c, x) (xb, xc) \<longrightarrow> (\<exists>t'. (xc, t') \<in> indis \<and> PL.transC tval aval (c, xa) (xb, t'))) \<and> (\<forall>x xa xb. (x, xa) \<in> indis \<longrightarrow> PL.transT tval aval (c, x) xb \<longrightarrow> (\<exists>t'. (xb, t') \<in> indis \<and> PL.transT tval aval (c, xa) t')))"
not_ev_iff: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "(\<not> ev P \<omega>) = alw (\<lambda>xs. \<not> P xs) \<omega>"
p2r_neg_hom: fixes P :: "'a \<Rightarrow> bool" shows "rel_ad \<lceil>P\<rceil> = \<lceil>- P\<rceil>"
total_on_singleton: fixes x :: "'d" shows "total_on {x} {(x, x)}"
UN_UN_lenset: fixes M :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a set"   and n :: "nat" shows "(\<Union>b. \<Union> (M b ` {x. length x = n})) = (\<Union>bs\<in>{x. length x = Suc n}. M (last bs) (butlast bs))"
prems: fixes x :: "'c" assumes "x \<in> insert a A" shows "f x = \<one>"
lookup_monom_mult_eq_zero: fixes s :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" assumes "s \<oplus> lt p \<prec>\<^sub>t v" shows "lookup (monom_mult c s p) v = (0::'c)"
cases: fixes r :: "('d, 'c) TopoS_packed" obtains nm_name :: "char list" and nm_receiver_violation :: "bool" and nm_default :: "'c" and nm_sinvar :: "'d list_graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> bool" and nm_offending_flows :: "'d list_graph \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<times> 'd) list list" and nm_node_props :: "('d, 'c) TopoS_Params \<Rightarrow> 'd \<Rightarrow> 'c" and nm_eval :: "'d list_graph \<Rightarrow> ('d, 'c) TopoS_Params \<Rightarrow> bool" where "r = \<lparr>nm_name = nm_name, nm_receiver_violation = nm_receiver_violation, nm_default = nm_default, nm_sinvar = nm_sinvar, nm_offending_flows = nm_offending_flows, nm_node_props = nm_node_props, nm_eval = nm_eval\<rparr>"
f1: shows "Tensor.mat_of_cols_list (dim_row A * dim_row C) (mat_mult (mult.row_length (mat_to_cols_list M)) (mat_to_cols_list M) (mat_to_cols_list N)) = Tensor.mat_of_cols_list (dim_row A * dim_row C) (mat_mult (mult.row_length (mult.Tensor (*) A' C')) (mult.Tensor (*) A' C') (mult.Tensor (*) B' D'))"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "ty list" assumes "P,a1 \<turnstile> a2 [::] a3" obtains E :: "char list \<Rightarrow> ty option" where "a1 = E__" and "a2 = []" and "a3 = []"   | E :: "char list \<Rightarrow> ty option" and e :: "expr" and T :: "ty" and es :: "expr list" and Ts :: "ty list" where "a1 = E" and "a2 = e # es" and "a3 = T # Ts" and "P,E \<turnstile> e :: T" and "P,E \<turnstile> es [::] Ts"
empty_correct: fixes q :: "('a, 'b) BinomialHeap" shows "(BinomialHeap_loc.to_mset q = {#}) = (q = BinomialHeap_loc.empty)"
136: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> y) \<sqinter> - - x = y \<sqinter> - - x"
\<A>ss_OR_alt_def: fixes x0 :: "'g"   and x1 :: "'a"   and a0 :: "'j"   and a1 :: "'d"   and s :: "'e"   and e0 :: "'e"   and z0 :: "'k"   and e1 :: "'e"   and z1 :: "'f"   and s' :: "'e"   and e0' :: "'e"   and z0' :: "'k"   and e1' :: "'e"   and z1' :: "'f" shows "\<A>ss_OR (x0, x1) ((a0, a1), s, (e0, z0), e1, z1) ((a0, a1), s', (e0', z0'), e1', z1') = (if e0 \<noteq> e0' then \<A>ss0 x0 (a0, e0, z0) (a0, e0', z0') \<bind> (\<lambda>w0. return_spmf (Inl w0)) else \<A>ss1 x1 (a1, e1, z1) (a1, e1', z1') \<bind> (\<lambda>w1. return_spmf (Inr w1)))"
subset_iff_psubset_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (A \<subset> B \<or> A = B)"
cd_suc: shows "\<forall>k k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k' \<and> l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k \<longrightarrow> \<pi> (Suc k) = \<pi>' (Suc k')"
outstanding_refs_is_Write\<^sub>s\<^sub>b_union: fixes xs :: "'b memref list" shows "outstanding_refs is_Write\<^sub>s\<^sub>b xs = outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<union> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b xs"
commute: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.min a b = ord_term_lin.min b a"
interaction_bound_restrict_gpv: fixes "consider" :: "'out \<Rightarrow> bool"   and gpv :: "('b, 'out, 'in) gpv" shows "interaction_bound consider (local.restrict_gpv gpv) \<le> interaction_bound consider gpv"
step_a_z_dbm: fixes l :: "'e"   and g :: "('f, 'g) cconstraint"   and a :: "'h"   and r :: "'f list"   and l' :: "'e"   and A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and v :: "'f \<Rightarrow> nat"   and n :: "nat" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', And (reset' (And D (abstr g (\<lambda>i j. \<infinity>) v)) n r v (0::'g)) (abstr (inv_of A l') (\<lambda>i j. \<infinity>) v)\<rangle>"
zcount_diff: fixes M :: "'b zmultiset"   and N :: "'b zmultiset"   and a :: "'b" shows "zcount (M - N) a = zcount M a - zcount N a"
mc'_red: shows "reduce Ds C (M # C') \<noteq> M # C'"
bij_betw_from_nat_into_finite: fixes S :: "'b set" assumes "finite S" shows "bij_betw (from_nat_into S) {..<card S} S"
weakBisimBangRel: shows "bangRel weakBisimulation \<subseteq> weakBisimulation"
a: shows "M \<longrightarrow>\<^sub>a* M'"
divide_less_eq_numeral1_1: fixes b :: "'b"   and w :: "num"   and a :: "'b" shows "(b / numeral w < a) = (b < a * numeral w)"
polypow_norm: fixes p :: "poly"   and k :: "nat" assumes "isnpoly p" shows "isnpoly (p ^\<^sub>p k)"
R0_def: shows "R0 = {(S1, S2). \<exists>c1 c1' c2 c2' W W'. S1 = (c1;c2) # W \<and> S2 = (c1';c2') # W' \<and> [c1] \<approx>\<^bsub>d\<^esub> [c1'] \<and> [c2] \<approx>\<^bsub>d\<^esub> [c2'] \<and> W \<approx>\<^bsub>d\<^esub> W'}"
ord_le_eq_trans: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
Max_mono_on2: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "SetInterval2.mono_on f A"   and "A \<noteq> {}"   and "finite A" shows "Max (f ` A) = f (Max A)"
inv_in_iff: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "subgroup H G"   and "x \<in> carrier G"   and "Group.group G" shows "(inv\<^bsub>G\<^esub> x \<in> H) = (x \<in> H)"
assms_2: shows "\<not> new p a ((ps, a) # branch)"
elims: fixes x :: "'d \<Rightarrow> 'c"   and xa :: "'d list"   and xb :: "'c list"   and y :: "'c list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'d \<Rightarrow> 'c" and bs :: "'c list" where "x = f__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'d \<Rightarrow> 'c" and a :: "'d" and as :: "'d list" and bs :: "'c list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
accpI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y" shows "Wellfounded.accp r x"
negate_f: assumes "has_disc_negated disc False m" shows "\<forall>pts. \<not> has_disc disc2 (negate_f C pts)"
aff_dim_open: fixes S :: "'a set" assumes "open S"   and "S \<noteq> {}" shows "aff_dim S = int DIM('a)"
return_cong: fixes A :: "'c measure"   and B :: "'c measure"   and x :: "'c" assumes "sets A = sets B" shows "return A x = return B x"
hyps_1: fixes mr :: "mregex"   and ms :: "mregex" assumes "r = from_mregex mr \<phi>s"   and "safe_regex Futu g (from_mregex mr \<phi>s)"   and "ms \<in> LPD mr" shows "safe_regex Futu g (from_mregex ms \<phi>s)"
comm_monoid_list_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "comm_monoid_list f z \<equiv> comm_monoid f z \<and> monoid_list f z"
mvt_simple: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real" assumes "a < b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x within {a..b})" shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
Ramsey: fixes Z :: "'a set"   and r :: "nat"   and f :: "'a set \<Rightarrow> nat"   and s :: "nat" assumes "infinite Z"   and "\<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = r \<longrightarrow> f X < s" shows "\<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s \<and> (\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = r \<longrightarrow> f X = t)"
R_def: fixes h :: "'g \<times> 'a"   and w :: "'h \<times> 'b"   and e :: "'e" shows "\<Sigma>_AND.R h w e = init_AND h w \<bind> (\<lambda>(r, a). response_AND r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
simps_4: fixes v :: "term" shows "linear (\<Lambda> v) = False"
UN_UN_flatten: fixes C :: "'c \<Rightarrow> 'b set"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" shows "\<Union> (C ` \<Union> (B ` A)) = (\<Union>y\<in>A. \<Union> (C ` B y))"
asym: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry" assumes "b \<prec> a"   and "a \<prec> b" shows "False"
tau_par: fixes x :: "'a"   and y :: "'a" shows "\<tau> (x \<parallel> y) = \<tau> x \<parallel> \<tau> y"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
borel_eq_PiF_borel: shows "borel = Fin_Map.Pi\<^sub>F (Collect finite) (\<lambda>_. borel)"
fun_ord_conv_rel_fun: shows "fun_ord = rel_fun (=)"
dependent_live_vars_cdep_empty_fst: fixes n'' :: "'node"   and as :: "'edge list"   and n' :: "'node"   and V' :: "'var" assumes "n'' -as\<rightarrow>\<^sub>c\<^sub>d n'"   and "V' \<in> Use n''" shows "(V', [], as) \<in> dependent_live_vars n'"
apply_guards_def: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "apply_guards G s = (\<forall>g\<in>set (map (\<lambda>g. gval g s) G). g = true)"
f_last_message_hold_def: fixes xs :: "'b message_af list"   and k :: "nat" shows "xs \<longmapsto> k \<equiv> concat (map last_message_hold (list_slice2 xs k))"
comm_monoid_axioms: shows "comm_monoid (\<sqinter>) top"
card_insert_disjoint: fixes A :: "'q set"   and x :: "'q" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
FINITE_2: shows "finite (P\<^sub>2 \<union> wrap B\<^sub>2)"
pw_eq_on_def: fixes S :: "'c set"   and w :: "'d \<Rightarrow> 'c set"   and w' :: "'d \<Rightarrow> 'c set" shows "pw_eq_on S w w' \<equiv> \<forall>i. w i \<inter> S = w' i \<inter> S"
xor_int_def: fixes k :: "int"   and l :: "int" shows "k XOR l = k AND NOT l OR NOT k AND l"
sel_2: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "l (Node x21 x22 x23 x24 x25) = x21"
r_neg: fixes G :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "abelian_group G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
split_block_def: fixes A :: "'a mat"   and sr :: "nat"   and sc :: "nat" shows "split_block A sr sc = (let nr = dim_row A; nc = dim_col A; nr2 = nr - sr; nc2 = nc - sc; A1 = mat sr sc (($$) A); A2 = mat sr nc2 (\<lambda>(i, j). A $$ (i, j + sc)); A3 = mat nr2 sc (\<lambda>(i, j). A $$ (i + sr, j)); A4 = mat nr2 nc2 (\<lambda>(i, j). A $$ (i + sr, j + sc)) in (A1, A2, A3, A4))"
cut_less_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>< t = {}) = (\<forall>x\<in>I. t \<le> x)"
get_ancestors_di_never_empty: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "h \<turnstile> get_ancestors_di child \<rightarrow>\<^sub>r ancestors" shows "ancestors \<noteq> []"
index_minus_vec_2: fixes v\<^sub>1 :: "'d Matrix.vec"   and v\<^sub>2 :: "'d Matrix.vec" shows "dim_vec (v\<^sub>1 - v\<^sub>2) = dim_vec v\<^sub>2"
select_index_belongs: fixes x :: "'a" assumes "x \<in> U" shows "select_index x \<in> I"
prob_space_distrD: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "prob_space (distr M N f)" shows "prob_space M"
False: shows "f \<noteq> (\<lambda>_. 0::'b)"
equiv_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x \<approx> y"
rev_predicate2D: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "P x y"   and "P \<le> Q" shows "Q x y"
P_widen: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "P C1"   and "P C2" shows "P (C1 \<nabla> C2)"
altc_writerT_def: shows "altc = altc_writer altc"
lsl_Inf_closed: shows "Inf_closed_set (Fix \<nu>)"
bij_region_coregion: shows "bij_lens (\<V> +\<^sub>L \<C>)"
simps_462: shows "CondOr \<noteq> CondAnd"
multiplicity_lessI: fixes n :: "nat" assumes "\<not> p ^ n dvd x" shows "multiplicity p x < n"
min_enat_simps_4: fixes q :: "enat" shows "min q \<infinity> = q"
L2_set_empty: fixes f :: "'a \<Rightarrow> real" shows "L2_set f {} = 0"
relation_of_Skip: shows "action.relation_of Skip = R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A = alpha_rp.more A')"
nfa_emptybdd_def: fixes n :: "nat" shows "nfa_emptybdd n = Leaf (replicate n False)"
lprod: assumes "R \<subseteq> R\<^sup>+" shows "wf (lprod R)"
q2_def: shows "q2 = [:\<i>, 1:] ^ degree p"
Ord_mem_iff_less_TC: fixes l :: "V"   and k :: "V" assumes "Ord l" shows "(k \<in>\<^sub>\<circ> l) = (k \<sqsubset> l)"
P\<^sub>2_B\<^sub>2_lower_bound_P\<^sub>1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2)"   and "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))" shows "2 * card (P\<^sub>2 \<union> wrap B\<^sub>2) \<le> card P\<^sub>1 + 1"
closed_Union_compact_subsets: fixes S :: "'b set" assumes "closed S" obtains F :: "nat \<Rightarrow> 'b set" where "\<And>n. compact (F n)" and "\<And>n. F n \<subseteq> S" and "\<And>n. F n \<subseteq> F (Suc n)" and "\<Union> (range F) = S" and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> F n"
step_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and \<sigma>' :: "jvm_state"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"   and "\<lfloor>\<sigma>'\<rfloor> = exec (P, None, h, (stk, loc, C, M, pc, ics) # frs, sh)"   and "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]" shows "P,\<Phi> |- \<sigma>' [ok]"
LNil_eq_iff_lnull: fixes xs :: "'c llist" shows "(LNil = xs) = lnull xs"
max2_equals1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(local.max2 a b = a) = ((b, a) \<in> r)"
lnth_cons_def: fixes n :: "nat"   and s :: "'s" shows "local.lnth_cons n s = lnth (lunstream g s) n"
seqSubst_def: fixes M :: "'a"   and \<sigma> :: "(name list \<times> 'b list) list" shows "M[<\<sigma>>] \<equiv> foldl (\<lambda>N (xvec, Tvec). N[xvec::=Tvec]) M \<sigma>"
"1_2": fixes n :: "'c"   and A :: "('c strand_label \<times> ('a, 'b) stateful_strand_step) list" shows "setops\<^sub>s\<^sub>s\<^sub>t (proj_unl n A) \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
ord_less_ODord: fixes D :: "'c Order"   and c :: "'c"   and d :: "'c"   and a :: "'c Order set"   and b :: "'c Order set" assumes "Worder D"   and "c \<in> carrier D"   and "d \<in> carrier D"   and "a = ordinal_number (Iod D (segment D c))"   and "b = ordinal_number (Iod D (segment D d))" shows "c \<prec>\<^bsub>D\<^esub> d = a \<sqsubset> b"
inner_Basis_eq_zero_absI: fixes x :: "'a"   and u :: "'a" assumes "x \<in> Basis"   and "\<bar>u\<bar> \<in> Basis"   and "x \<noteq> \<bar>u\<bar>"   and "x \<noteq> (0::'a)" shows "u \<bullet> x = 0"
geodesic_segment_split_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<in> {x--z}" shows "{x--y} \<inter> {y--z} = {y}"
add_le_cancel_left0: fixes x :: "V"   and z :: "V" shows "x \<subseteq>\<^sub>\<circ> x + z"
box_def: fixes a :: "'a"   and b :: "'a" shows "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
closed_segment_translation_eq: fixes d :: "'b"   and x :: "'b"   and a :: "'b"   and b :: "'b" shows "(d + x \<in> closed_segment (d + a) (d + b)) = (x \<in> closed_segment a b)"
ldropn_lnull: fixes xs :: "'b llist"   and n :: "nat" assumes "lnull xs" shows "ldropn n xs = <>"
coeff_nonzero_polyn_nonzero: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "(polyn_expr R X k c \<noteq> \<zero>) = (\<exists>j\<le>k. snd c j \<noteq> \<zero>\<^bsub>S\<^esub>)"
set_sel_2: fixes xb :: "'c \<times> 'c list_R1.list_R1"   and a :: "'c list_R1.list_R1"   and xd :: "'c list_R1.list_R1"   and xe :: "'c" assumes "xb \<in> setr (list_R1.unR a)"   and "xd \<in> snds xb"   and "xe \<in> list_R1.set_list_R1 xd" shows "xe \<in> list_R1.set_list_R1 a"
winning_regionI: fixes v :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> V"   and "strategy p \<sigma>"   and "winning_strategy p \<sigma> v" shows "v \<in> winning_region p"
poly_weak_mono_all: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" assumes "poly_weak_mono_all p" shows "poly_weak_mono p v"
kE_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "I.wtE (kE \<xi>)" shows "F.satPB \<xi> \<Phi>"
C0_ms_ss_C0: shows "map (\<lambda>w. w `\<rightarrow> C0) (sums ss) = C0 # Ms @ [sum_list ss `\<rightarrow> C0]"
approx_SReal_mult_cancel_zero: fixes a :: "real star"   and x :: "real star" assumes "a \<in> \<real>"   and "a \<noteq> 0"   and "a * x \<approx> 0" shows "x \<approx> 0"
mset_subset_eq_insertD: fixes x :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "x, A \<subseteq># B" shows "x \<in># B \<and> A \<subset># B"
assms_2: shows "oid \<notin> fst ` set xs"
distinct_27: fixes x2 :: "nat"   and x5 :: "ephkey" shows "cNumber x2 \<noteq> cEphK x5"
decreasing_sum_le_integral_2: fixes f :: "real \<Rightarrow> real"   and I :: "real" assumes "\<And>x. 0 \<le> x \<Longrightarrow> 0 \<le> f x"   and "\<And>x y. \<lbrakk>0 \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> f y \<le> f x"   and "(f has_integral I) {0..}" shows "(\<Sum>i. f (real (Suc i))) \<le> I"
map_prod_imageI: fixes a :: "'a"   and b :: "'b"   and R :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "(a, b) \<in> R" shows "(f a, g b) \<in> map_prod f g ` R"
cs_sym: fixes x :: "'a"   and y :: "'a" assumes "x \<down>\<^sup>* y" shows "y \<down>\<^sup>* x"
exec_move_AAccI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<lfloor>e'\<rceil>) h s ta h' s'"
set1_FGf_def: fixes x :: "('p, 'q, 'r, 's, 't, 'u, 'v, 'w, 'x, ('y, 'r, 'z, 'u, 'aa, 'x, 'ab) G) F" shows "set1_FGf x = set1_F x"
rfb_Sup_dual_var: fixes \<phi> :: "('d \<times> 'c) set set" shows "fb\<^sub>\<R> (\<mu> \<phi>) = \<Sqinter> \<P> fb\<^sub>\<R> \<phi>"
simps_4: fixes n :: "nat"   and m :: "nat"   and y :: "nat"   and x :: "nat" assumes "n \<noteq> 1"   and "odd n" shows "mod_exp_nat.efficient_power m y x n = mod_exp_nat.efficient_power m (x * y mod m) (x * x mod m) (n div 2)"
zero_left: fixes b :: "'c" shows "0 *\<^sub>v b = (0::'b)"
is_unit_gcd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = algebraic_semidom.coprime times one a b"
outer_loop_rel_def: fixes src :: "nat" shows "outer_loop_rel src \<equiv> inv_image (less_than_bool <*lex*> greater_bounded (max_dist src + 1) <*lex*> finite_psubset) (\<lambda>(f, PRED, C, N, d). (\<not> f, d, C))"
derived_set_in_carrier: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "derived_set G H \<subseteq> carrier G"
continuous_map_real_mult_left: fixes X :: "'b topology"   and f :: "'b \<Rightarrow> real"   and c :: "real" assumes "continuous_map X euclideanreal f" shows "continuous_map X euclideanreal (\<lambda>x. c * f x)"
variant_variable_adds: fixes s :: "variable"   and S :: "variable set" shows "snd (variant_variable s S) = insert (fst (variant_variable s S)) S"
SystemEquivCut_def: shows "SystemEquivCut = ((TT && Geq (f0 fid1) (Const 0)) \<rightarrow> [[DiffAssign vid1 (f0 fid1)]]Geq (Differential (Var vid1)) (Differential (Const 0)) \<leftrightarrow> ((TT && Geq (f0 fid1) (Const 0)) \<rightarrow> [[DiffAssign vid1 (f0 fid1)]]Geq ($' vid1) (Const 0)))"
SpanningForest_def: fixes X :: "(nat \<times> int \<times> nat) set" shows "s.SpanningForest X = ((forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>) \<and> (\<forall>x\<in>E - X. \<not> (forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>)))"
prems_1: shows "wbt \<langle>l, a, r\<rangle>"
exec_inc_imp_trace_inc: fixes B :: "('c, 'b) ioa"   and A :: "('d, 'b) ioa" assumes "externals (ioa.asig B) = externals (ioa.asig A)"   and "\<And>e_B. is_exec_of B e_B \<Longrightarrow> \<exists>e_A. is_exec_of A e_A \<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B" shows "local.traces B \<subseteq> local.traces A"
Compl_lemma: fixes n :: "nat"   and l :: "nat" assumes "n \<le> l" shows "\<exists>m\<le>l. n = l - m"
\<pi>_Fbcc_respect: shows "rel_fun Ebc Ec \<pi>_Fbcc \<pi>_Fbcc"
infinite: fixes A :: "'c set" assumes "infinite A" shows "sorted_list_of_set A = []"
trivial_derived_seq_imp_solvable: fixes H :: "'a set"   and n :: "nat" assumes "subgroup H G"   and "(derived G ^^ n) H = {\<one>}" shows "solvable_seq G H"
AbstForm_def: fixes v :: "hf"   and i :: "hf"   and x :: "hf"   and x' :: "hf" shows "AbstForm v i x x' \<equiv> is_Var v \<and> Ord i \<and> (\<exists>s k. SeqAbstForm v i x x' s k)"