hcju/isabelleps huggingface dataset:
T_mtf_eq_lem: shows "sum t_mtf {0..<length rs} = int (T (s_mtf 0) (drop 0 rs) (drop 0 (off MTF s0 rs)))"
degen_infs: fixes cs :: "('d \<Rightarrow> bool) list"   and w :: "'d stream"   and k :: "nat" shows "infs (degen cs) (w ||| k ## sscan (count cs) w k) = gen infs cs w"
IsNullAngaP_def: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "IsNullAngaP a \<equiv> QCongAACute a \<and> (\<exists>A B C. a A B C \<and> B Out A C)"
sbintrunc_bintrunc_lt: fixes n :: "nat"   and m :: "nat"   and w :: "int" assumes "n < m" shows "signed_take_bit n (take_bit m w) = signed_take_bit n w"
itrm_brackets_Cons_SomeE: fixes v :: "nat"   and vs :: "nat list"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_brackets (v # vs) x = Some x'" obtains y' :: "nat itrm" where "itrm_brackets vs x = Some y'" and "itrm_bracket v y' = Some x'"
map_gpv'_map_gpv_swap: fixes f :: "'k \<Rightarrow> 'h"   and g :: "'l \<Rightarrow> 'i"   and h :: "'j \<Rightarrow> 'm"   and f' :: "'n \<Rightarrow> 'k"   and gpv :: "('n, 'l, 'm) gpv" shows "map_gpv' f g h (map_gpv f' id gpv) = map_gpv (f \<circ> f') id (map_gpv' id g h gpv)"
partn_lst_greater_resource: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and \<alpha> :: "V list"   and n :: "nat"   and C :: "'a set" assumes "Partitions.partn_lst r B \<alpha> n"   and "B \<subseteq> C" shows "Partitions.partn_lst r C \<alpha> n"
simps: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" shows "a1 cd-a2\<rightarrow>\<^sub>d* a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> valid_SDG_node n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n cd-ns\<rightarrow>\<^sub>d* n'' \<and> n'' \<longrightarrow>\<^bsub>cd\<^esub> n'))"
nonpos_Reals_cases: fixes x :: "'a" assumes "x \<in> \<real>\<^sub>\<le>\<^sub>0" obtains r :: "real" where "x = of_real r" and "r \<le> 0"
dy_fake_msg_insert_chan: fixes x :: "tag"   and M :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and A :: "agent"   and B :: "agent"   and CH :: "chan set" assumes "x = insec \<or> x = auth" shows "M \<in> dy_fake_msg bad IK (insert (Chan x A B M) CH)"
open_restricted_space: fixes P :: "'d set \<Rightarrow> bool" shows "open {m. P (domain m)}"
sum_Suc_reindex: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum f {0..n} = f 0 - f (Suc n) + (\<Sum>i = 0..n. f (Suc i))"
le_simps_1: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "m \<le> n"
sum2sq_int_nat_eq: fixes a :: "nat"   and b :: "nat" shows "int (sum2sq_nat a b) = sum2sq_int (int a, int b)"
Int: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and a :: "'c set"   and b :: "'c set" assumes "semiring_of_sets \<Omega> M"   and "a \<in> M"   and "b \<in> M" shows "a \<inter> b \<in> M"
cofactor_def: fixes A :: "(('c, 'd) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "cofactor A i j = det (minorM A i j)"
uconnected_transcl: fixes E :: "'b uprod set" shows "(uconnected E)\<^sup>* = uconnected E"
restrict_Pi_cancel: fixes x :: "'e \<Rightarrow> 'f"   and I :: "'e set"   and A :: "'e \<Rightarrow> 'f set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
select_convs_6: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "sports \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = sports"
cs_split_shift_cd: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and j :: "nat"   and k :: "nat" assumes "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"   and "j < k"   and "k < n"   and "\<forall>j'<k. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j' \<longrightarrow> j' \<le> j" shows "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> j @ cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> (n - k)"
ipurge_ref_distrib_union: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list"   and X :: "'e set"   and Y :: "'e set" shows "ipurge_ref I D u xs (X \<union> Y) = ipurge_ref I D u xs X \<union> ipurge_ref I D u xs Y"
locally_path_connected_imp_locally_connected: fixes S :: "'a set" assumes "locally path_connected S" shows "locally connected S"
relcompI: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set"   and c :: "'d"   and s :: "('c \<times> 'd) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
dyn_override_Package: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old"   and "accmodi old = Package"   and "accmodi new = Package"   and "wf_prog G" shows "pid (declclass old) = pid (declclass new)"
UV: shows "\<forall>x\<in>S. openin X (U x) \<and> openin X (V x) \<and> x \<in> U x \<and> T \<subseteq> V x \<and> disjnt (U x) (V x)"
interpret_floatarith_one: fixes xs :: "real list" shows "interpret_floatarith 1 xs = 1"
jordan_matrix_carrier: fixes n_as :: "(nat \<times> 'a) list" shows "jordan_matrix n_as \<in> carrier_mat (sum_list (map fst n_as)) (sum_list (map fst n_as))"
insert_3: fixes X :: "nat set"   and Y :: "nat set" assumes "finite X"   and "finite Y"   and "X \<inter> Y = {}"   and "S \<inter> (X \<union> Y) = {}"   and "S \<union> X \<union> Y = {0..<m}" shows "card {xs. (\<forall>a\<in>X. xs ! a) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m} = 2 ^ (m - card X - card Y)"
keys_single: fixes k :: "'b"   and v :: "'c" shows "keys (monomial v k) = (if v = (0::'c) then {} else {k})"
OR2: fixes evs2 :: "event list"   and NB :: "nat"   and B :: "agent"   and M :: "nat"   and A :: "agent"   and X :: "msg" assumes "evs2 \<in> orb"   and "Nonce NB \<notin> used evs2"   and "Gets B \<lbrace>Nonce M, Agent A, Agent B, X\<rbrace> \<in> set evs2" shows "Says B Server \<lbrace>Nonce M, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>Nonce NB, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> # evs2 \<in> orb"
\<MM>_minus_ln_limit: obtains c :: "real" where "((\<lambda>x. \<MM> x - ln x) \<longlongrightarrow> c) at_top"
trans: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Omega>[F](g)"   and "g \<in> \<Omega>[F](h)" shows "f \<in> \<Omega>[F](h)"
hyps_2: assumes "ordered rt var"   and "not \<le> rt" shows "ordered not var"
snd_convol: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'e" shows "snd \<circ> BNF_Def.convol f g = g"
WCFG_sourcelabel_less_num_nodes: fixes prog :: "cmd"   and l :: "nat"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and n' :: "w_node" assumes "prog \<turnstile> (_ l _) -et\<rightarrow> n'" shows "l < #:prog"
int_pow_neg_int: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "x [^] - int n = inv (x [^] n)"
simps_3: fixes Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Length :: "'b \<Rightarrow> nat"   and satisfies0 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'b"   and \<phi> :: "('d, 'a) aformula" shows "Formula_Operations.satisfies_gen Extend Length satisfies0 r \<AA> (FNot \<phi>) = (\<not> Formula_Operations.satisfies_gen Extend Length satisfies0 r \<AA> \<phi>)"
times_ms_aux_MSLNil_1: fixes G :: "('a \<times> real) msllist" shows "times_ms_aux MSLNil G = MSLNil"
height_gt_0_in_alphabet_imp_sibling_left: fixes t\<^sub>1 :: "'a tree"   and a :: "'a"   and w :: "nat"   and t\<^sub>2 :: "'a tree" assumes "0 < height t\<^sub>1"   and "a \<in> alphabet t\<^sub>1" shows "sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>1 a"
eval_poly_poly: fixes p :: "real poly"   and x :: "real" shows "eval_poly of_real p (of_real x) = of_real (poly p x)"
item_rhs_of_inc_item: fixes x :: "item"   and k :: "nat" shows "item_rhs (inc_item x k) = item_rhs x"
U_Model: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "M_FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "CU.Model wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) \<Phi> (intT any) intF intP"
plus_p_integer_def: fixes p :: "integer"   and x :: "integer"   and y :: "integer" shows "plus_p_integer p x y \<equiv> let z = x + y in if p \<le> z then z - p else z"
eq1: shows "1\<^sub>m N * (1\<^sub>m N - post) = 1\<^sub>m N - post"
simps_1: fixes m :: "Keys" shows "KS2Expression (kKS m) = kE m"
position_of_Nil: fixes uu :: "('c, 'd) tm" shows "position_of uu [] = True"
S2P_app: fixes r :: "'c set"   and x :: "'c" shows "S2P r x = (x \<in> r)"
Standard_diff: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> Standard"   and "y \<in> Standard" shows "x - y \<in> Standard"
reflp_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
WR_quot_Var: fixes e :: "(name, hf) finfun"   and x :: "name" shows "WR \<lbrakk>\<guillemotleft>Var x\<guillemotright>\<rbrakk>e \<lbrakk>\<guillemotleft>\<guillemotleft>Var x\<guillemotright>\<guillemotright>\<rbrakk>e"
remdups_append: fixes xs :: "'c list"   and ys :: "'c list" shows "remdups (xs @ ys) = remdups (filter (\<lambda>x. x \<notin> set ys) xs) @ remdups ys"
list_update_id: fixes xs :: "'e list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
vars_partitioned_var_disjoint: fixes Cs :: "('a, 'b) Term.term literal multiset list" assumes "vars_partitioned Cs" shows "var_disjoint Cs"
swaps_eq_nth_take_drop: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "swaps [0..<i] xs = xs ! i # take i xs @ drop (Suc i) xs"
ap_state_rev_def: shows "(\<diamondop>) = ap_dual pure (\<diamondop>) (\<diamondop>)"
axioms_2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "term_struct_rel_strong P" shows "term_struct_rel_strong_axioms P"
nth_last: fixes i :: "nat"   and xs :: "'a list" assumes "Suc i = |xs|" shows "xs ! i = last xs"
repeatable_many_calls: fixes x :: "'a" shows "repeatable (many_calls x)"
taut_test: fixes b :: "'c bexp" shows "taut_test Boolean_Expression_Example.ifex_of b = (\<forall>s. bval b s)"
bij_to_nat: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'c \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'c" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "bij_betw (mod_type.to_nat Rep) UNIV {0..<CARD('c)}"
C1l: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<odot> y \<le> y \<odot> z" shows "x\<^sup>\<star> \<odot> y \<le> y \<odot> z\<^sup>\<star>"
measurable_restrict_subset: fixes J :: "'b set"   and L :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" assumes "J \<subseteq> L" shows "(\<lambda>f. restrict f J) \<in> Pi\<^sub>M L M \<rightarrow>\<^sub>M Pi\<^sub>M J M"
prob_space_imp_subprob_space: fixes M :: "'a measure" assumes "prob_space M" shows "subprob_space M"
implies_galois_decreasing: fixes y :: "'a"   and x :: "'a" shows "(y \<leadsto> x) \<sqinter> y \<le> x"
mkCone_cone: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b"   and a :: "'b"   and \<chi> :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'b" assumes "parallel_pair_diagram C f0 f1"   and "Limit.cone (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) C (parallel_pair_diagram.map C f0 f1) a \<chi>" shows "parallel_pair_diagram.mkCone C f0 (\<chi> quiver.Zero) = \<chi>"
Iter_def: fixes X :: "'b list set"   and ES :: "('b list set \<times> 'b trm set) set" shows "Iter X ES \<equiv> let (Y, yrhs) = SOME (Y, yrhs). (Y, yrhs) \<in> ES \<and> X \<noteq> Y in Remove ES Y yrhs"
iso_i_1: shows "\<guillemotleft>i : \<I> \<rightarrow> \<I>\<^sub>1\<guillemotright>"
prefix_append_subsequence: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool"   and i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'b" assumes "af_congruent eq"   and "i \<le> j" shows "prefix i w @ w [i \<rightarrow> j] = prefix j w"
simps_2: fixes n :: "nat"   and s :: "'d stream" shows "sdrop (Suc n) s = sdrop n (stl s)"
bounded_linear_prod_right: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_bilinear prod" shows "bounded_linear (bounded_bilinear.prod_right prod)"
l_set_tag_name_get_disconnected_nodes_axioms_def: fixes set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_disconnected_nodes_axioms set_tag_name_locs get_disconnected_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
conv_radius_cong_bigtheta: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> \<Theta>(g)" shows "conv_radius f = conv_radius g"
twolem: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<otimes> (S \<parallel> Multirelations.U) \<subseteq> R \<cdot> S \<parallel> Multirelations.U"
ext_inject: fixes nodes :: "'a set"   and edges :: "('a \<times> 'a) set"   and more :: "'b"   and nodes' :: "'a set"   and edges' :: "('a \<times> 'a) set"   and more' :: "'b" shows "(\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes', edges = edges', \<dots> = more'\<rparr>) = (nodes = nodes' \<and> edges = edges' \<and> more = more')"
length_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "length (map f xs) = length xs"
min_gallery_in_apartment: fixes Y :: "'a set set"   and Cs :: "'a set list" assumes "Y \<in> \<A>"   and "min_gallery Cs"   and "order.greater_eq Y (set Cs)" shows "SimplicialComplex.min_maxsimpchain Y Cs"
Least_equality: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y" shows "ord.Least less_eq P = x"
div_power: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b a" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
psi_l_Suc_l_derive: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "2 * (grover_state.alpha_l n f l * complex_of_real (cos (grover_state.\<theta> n f / 2)) - grover_state.beta_l n f l * complex_of_real (sin (grover_state.\<theta> n f / 2))) \<cdot>\<^sub>v grover_state.\<psi> n - grover_state.psi'_l n f l = grover_state.psi_l n f (l + 1)"
esing_simps_1: fixes x :: "'b"   and n :: "'a" shows "(esing x\<cdot>n) x = n"
in_indets_poly_substE: fixes x :: "'c"   and f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "x \<in> indets (poly_subst f p)" obtains y :: "'e" where "y \<in> indets p" and "x \<in> indets (f y)"
rel_restrict_notR_2: fixes x :: "'a"   and y :: "'a"   and A :: "('a \<times> 'a) set"   and R :: "'a set" assumes "(x, y) \<in> rel_restrict A R" shows "y \<notin> R"
linear_cmul: fixes f :: "'b \<Rightarrow> 'c"   and r :: "real"   and b :: "'b" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
gt_unboundedD: fixes u :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set"   and c :: "'c" assumes "u \<in> R"   and "R \<in> \<R>"   and "c \<in> \<X>"   and "real (k c) < u c" shows "unbounded c R"
beh_consecutive_in_trans: fixes b :: "'a list"   and TS :: "('a, 'b) TS_scheme"   and i :: "nat"   and s :: "'a"   and t :: "'a" assumes "b \<in> beh TS"   and "Suc i < length b"   and "s = b ! Suc i"   and "t = b ! i" shows "(s, t) \<in> TS.trans TS"
stereographic_inv_stereographic: fixes w :: "complex_homo" shows "stereographic (inv_stereographic w) = w"
Dijkstra_Impl_Defs_axioms: shows "Dijkstra_Impl_Defs G.invar G_succ G_empty G_add_edge RBT_Set.empty RBT_Map.update RBT_Map.delete lookup M.invar PST_RBT.empty PST_RBT.update PST_RBT.delete PM.invar lookup rbt_is_empty pst_getmin G.\<alpha>"
assms_1: shows "dim_vec c = dim_row A\<^sup>T"
up_to: fixes J :: "nat set" assumes "finite J" shows "J \<subseteq> {0..<up_to J}"
nat_pow_mult: fixes x :: "'a list"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (poly_ring R)" shows "x [^]\<^bsub>poly_ring R\<^esub> n \<otimes>\<^bsub>poly_ring R\<^esub> x [^]\<^bsub>poly_ring R\<^esub> m = x [^]\<^bsub>poly_ring R\<^esub> (n + m)"
1: fixes i :: "'a" assumes "i \<in> J" shows "f i \<noteq> 0"
pivot_lvars: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "lvars (\<T> (pivot x\<^sub>i x\<^sub>j s)) = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"
in_idealE_Polys_finite: fixes B :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "finite B"   and "B \<subseteq> P[X]"   and "p \<in> P[X]"   and "p \<in> ideal B" obtains q :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" where "\<And>b. q b \<in> P[X]" and "p = (\<Sum>b\<in>B. q b * b)"
option_rel_def: fixes R :: "('d \<times> 'e) set" shows "\<langle>R\<rangle>option_rel \<equiv> {(Some a, Some a') |a a'. (a, a') \<in> R} \<union> {(None, None)}"
transitive_acyclic_irreflexive: fixes x :: "'a" assumes "transitive x" shows "pd_kleene_allegory_class.acyclic x = irreflexive x"
nat_pow_Suc2: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier P" shows "x [^]\<^bsub>P\<^esub> Suc n = x \<otimes>\<^bsub>P\<^esub> x [^]\<^bsub>P\<^esub> n"
simps: fixes a1 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a2 :: "'a expr \<times> String.literal list \<times> 'a OCL_Types.type option \<times> 'a expr"   and a3 :: "'a OCL_Types.type \<times> 'a OCL_Types.type \<times> 'a OCL_Types.type" shows "(a1 \<turnstile>\<^sub>I a2 : a3) = (\<exists>\<Gamma> src \<tau> \<sigma> its_ty its body \<rho>. a1 = \<Gamma> \<and> a2 = (src, its, Some its_ty, body) \<and> a3 = (\<tau>, \<sigma>, \<rho>) \<and> \<Gamma> \<turnstile>\<^sub>E src : \<tau> \<and> element_type \<tau> \<sigma> \<and> \<sigma> \<le> its_ty \<and> \<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, its_ty)) its) \<turnstile>\<^sub>E body : \<rho>)"
axioms_1: fixes G :: "('c, 'd) graph" assumes "tree G" shows "forest G"
distinct_24: fixes ccs' :: "ccs"   and ccs1 :: "ccs"   and ccs2 :: "ccs" shows "!ccs' \<noteq> ccs1 \<oplus> ccs2"
f_2: shows "\<forall>(k, \<gamma>)\<in>one_chain1. if k = fst (f (k, \<gamma>)) then reparam_weak \<gamma> (snd (f (k, \<gamma>))) else reparam_weak \<gamma> (reversepath (snd (f (k, \<gamma>))))"
simps: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool"   and a1 :: "Value"   and a2 :: "CValue" shows "similar'_step s a1 a2 = (a1 = \<bottom> \<and> a2 = \<bottom> \<or> (\<exists>b. a1 = B\<cdot>b \<and> a2 = CB\<cdot>b) \<or> (\<exists>f g. a1 = Fn\<cdot>f \<and> a2 = CFn\<cdot>g \<and> (\<forall>x xa. s x (xa\<cdot>C\<^sup>\<infinity>) \<longrightarrow> s (f\<cdot>x) (g\<cdot>xa\<cdot>C\<^sup>\<infinity>))))"
assms_1: shows "Lam [x]. e = Lam [x']. e'"
intros_1: fixes xs :: "'a list" shows "append2 [] xs xs"
that_1: assumes "y = x"   and "y \<notin> S" shows "thesis"
assms: shows "varset_action a (vs'' - (vs' \<union> vs))"
dom_lhs: shows "\<D>\<^sub>\<circ> \<CC> = 4\<^sub>\<nat>"
one_div_two_eq_zero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide one (numeral.numeral one plus (num.Bit0 num.One)) = zero"
bind_cont_def: fixes m :: "('d, 'e) contT"   and f :: "'d \<Rightarrow> ('d, 'e) contT" shows "bind_cont m f = ContT (\<lambda>k. run_cont m (\<lambda>x. run_cont (f x) k))"
simps_4: fixes e1 :: "exp"   and e2 :: "exp" shows "FV (EApp e1 e2) = FV e1 \<union> FV e2"
hyps_1: shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>1, \<A>1)"
rsp: shows "eq_onp (\<lambda>alg. \<forall>f fs. square_free f \<longrightarrow> 0 < degree f \<longrightarrow> alg f = fs \<longrightarrow> f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)) LLL_factorization LLL_factorization"
simps_5: fixes z :: "octo" shows "Im4 (Octonions.cnj z) = - Im4 z"
ereal_dense3: fixes x :: "ereal"   and y :: "ereal" assumes "x < y" shows "\<exists>r. x < ereal (real_of_rat r) \<and> ereal (real_of_rat r) < y"
splemma: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<le> ys" shows "xs \<noteq> ys \<longrightarrow> length xs < length ys"
u3_4_4: shows "b \<bullet> u mod a ! k = 0"
tranclD2: fixes x :: "'d"   and y :: "'d"   and R :: "('d \<times> 'd) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
FrR: shows "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"
plus_eint_def: fixes m :: "eint"   and n :: "eint" shows "m + n = (case m of eint m \<Rightarrow> case n of eint n \<Rightarrow> eint (m + n) | \<infinity> \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>)"
rrt: shows "Dag ((rep \<propto> high) q) (rep \<propto> low) (rep \<propto> high) rrt"
P''''Trans: shows "P'''' \<longmapsto> \<alpha> \<prec> P'''"
Cons_1: fixes L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "l \<in> set L" shows "\<exists>l'\<in>set L'. convert_atom var l (xs' @ x # xs) = Some l'"
EVAL_def: fixes x :: "'j" shows "EVAL x \<equiv> RETURN x"
prob: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1"
msgrel_imp_eq_freediscrim: fixes U :: "freemsg"   and V :: "freemsg" assumes "U \<sim> V" shows "freediscrim U = freediscrim V"
elims_3: fixes x :: "real_max"   and xa :: "real_max" assumes "\<not> x \<le> xa" obtains x :: "real" and y :: "real" where "xa__ = Rmax x" and "xa = Rmax y" and "\<not> x \<le> y"
Cons_eq_append_conv: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
inject: fixes x1 :: "'a"   and x2 :: "'a"   and y1 :: "'a"   and y2 :: "'a" shows "(Interval x1 x2 = Interval y1 y2) = (x1 = y1 \<and> x2 = y2)"