hcju/isabelleps huggingface dataset:
wordinterval_CIDR_split1_some_r_ne: fixes r :: "'b wordinterval"   and s :: "'b prefix_match"   and u :: "'b wordinterval" assumes "wordinterval_CIDR_split1 r = (Some s, u)" shows "\<not> wordinterval_empty r"
set_inner_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "apath u p v" shows "set (inner_verts p) = set (pawalk_verts u p) - {u, v}"
fresh_2: fixes a :: "name"   and x1 :: "name"   and x2 :: "'d frame" shows "a \<sharp> \<lparr>\<nu>x1\<rparr>x2 = a \<sharp> [x1].x2"
left_commute: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" shows "signed.min b (signed.min a c) = signed.min a (signed.min b c)"
shift2_invI: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and d :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs"   and "\<And>d0. d0 < d \<Longrightarrow> card {q \<in> set qs. poly_deg (fst q) = d0 \<and> m < card (snd q)} \<le> 1" shows "shift2_inv d qs"
indep_set_sigma_sets: fixes A :: "'b set set"   and B :: "'b set set" assumes "X.indep_set A B"   and "Int_stable A"   and "Int_stable B" shows "X.indep_set (sigma_sets (space (distr M S X)) A) (sigma_sets (space (distr M S X)) B)"
n_distrib_alt': fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "!x \<cdot> !z = !y \<cdot> !z"   and "!x + !z = !y + !z" shows "!x = !y"
map_family_triangle3: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and S1 :: "('h, 'i) aGroup_scheme"   and f :: "'c \<Rightarrow> 'f \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'h \<Rightarrow> 'd"   and h1 :: "'h \<Rightarrow> 'f"   and h2 :: "'f \<Rightarrow> 'h" assumes "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "aGroup S1"   and "\<forall>j\<in>I. f j \<in> aHom S (A j)"   and "\<forall>j\<in>I. g j \<in> aHom S1 (A j)"   and "h1 \<in> aHom S1 S"   and "h2 \<in> aHom S S1"   and "\<forall>j\<in>I. compos S (g j) h2 = f j"   and "\<forall>j\<in>I. compos S1 (f j) h1 = g j" shows "\<forall>j\<in>I. compos S (f j) (compos S h1 h2) = f j"
i_Exec_Stream_Acc_LocalState__State_Idle_all: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and xs :: "nat \<Rightarrow> 'g message_af"   and i :: "nat" assumes "0 < l"   and "l \<le> k"   and "State_Idle localState output_fun trans_fun (localState c)"   and "\<And>c m. State_Idle localState output_fun trans_fun (localState c) \<Longrightarrow> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun (m # NoMsg\<^bsup>l - Suc 0\<^esup>) c))" shows "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c i)"
set_disconnected_nodes_get_disconnected_nodes: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "h \<turnstile> set_disconnected_nodes document_ptr disc_nodes \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"
cut_ge_singleton: fixes a :: "'a"   and t :: "'a" shows "{a} \<down>\<ge> t = (if t \<le> a then {a} else {})"
st_inv_exec: fixes s :: "'b status" assumes "st_inv s" shows "st_inv (exec s)"
times_left_anti_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq y x"   and "less_eq z zero" shows "less_eq (times x z) (times y z)"
Sup_trans_upper2: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set"   and b :: "real"   and P :: "'a \<Rightarrow> real" assumes "t \<in> S"   and "\<forall>u\<in>S. \<forall>P. nneg P \<and> bounded_by b P \<longrightarrow> nneg (u P) \<and> bounded_by b (u P)"   and "nneg P"   and "bounded_by b P" shows "t P \<tturnstile> Sup_trans S P"
a1: shows "I \<subseteq> {sa. (\<exists>i\<in>I. i \<rightarrow>\<^sub>i* sa) \<and> sa \<in> AG s}"
diff_fun_add: fixes a :: "'a \<Rightarrow> 'e"   and b :: "'a \<Rightarrow> 'e" assumes "diff_fun k charts a"   and "diff_fun k charts b" shows "diff_fun k charts (a + b)"
period: fixes l :: "nat" shows "periodic_arithmetic (\<lambda>x. unity_root k (int x)) (k * l)"
sinks_interference_eq: fixes u :: "'c"   and D :: "'d \<Rightarrow> 'c"   and x :: "'d"   and I :: "('c \<times> 'c) set"   and xs :: "'d list" shows "((u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I)) = (D x \<in> sinks I D u (xs @ [x]))"
fmadd_fmupd: fixes m :: "('c, 'd) fmap"   and a :: "'c"   and b :: "'d"   and n :: "('c, 'd) fmap" shows "m ++\<^sub>f n(a $$:= b) = (m ++\<^sub>f n)(a $$:= b)"
create_shadow_root_known_ptr: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> attach_shadow_root element_ptr shadow_root_mode \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "known_ptr (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr)"
plusinf_inf: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list" assumes "islin p" shows "\<exists>z. \<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p"
rel_mono_strong: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_option R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_option x; yb \<in> set_option y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_option Ra x y"
Node_6: shows "r (Node i j a t1 t2) \<le> length as"
frontier_straddle: fixes a :: "'a"   and S :: "'a set" shows "(a \<in> frontier S) = (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
converse_rtrancl_induct2: fixes ax :: "'b"   and ay :: "'c"   and bx :: "'b"   and "by" :: "'c"   and r :: "(('b \<times> 'c) \<times> 'b \<times> 'c) set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>*"   and "P bx by"   and "\<And>a b aa ba. \<lbrakk>((a, b), aa, ba) \<in> r; ((aa, ba), bx, by) \<in> r\<^sup>*; P aa ba\<rbrakk> \<Longrightarrow> P a b" shows "P ax ay"
sl6: shows "- abscissa C * (ordinate C - ordinate D) + ordinate C * (abscissa C - abscissa D) \<in> radical_sqrt"
modular_1': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z = x ; (y \<cdot> x\<^sup>T ; z) \<cdot> z"
simps_5: fixes var :: "'f"   and p :: "('d, 'e, 'f) formula" shows "SIGF (Exists var p) = SIGF p"
verit_comp_simplify_5: fixes n :: "num" shows "(1::'a) \<le> numeral n"
O_mono2: fixes R :: "('b \<times> 'c) set"   and R' :: "('b \<times> 'c) set"   and T :: "('c \<times> 'd) set" assumes "R \<subseteq> R'" shows "R O T \<subseteq> R' O T"
the_inv_into_image_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and Y :: "'c set" assumes "inj_on f A"   and "Y \<subseteq> f ` A" shows "the_inv_into A f ` Y = f -` Y \<inter> A"
wf_def: fixes r :: "('a \<times> 'a) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
addpreRT_partly_welldefined: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:16..PRreq-:18} \<union> {PRrep-:1..PRrep-:5} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l \<in> {PRreq-:3..PRreq-:17} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>)))"
cond_spmf_fst_def: fixes p :: "('e \<times> 'd) spmf"   and a :: "'e" shows "cond_spmf_fst p a = map_spmf snd (cond_spmf p ({a} \<times> UNIV))"
simps: fixes n :: "char list"   and params :: "(variable \<times> PDDL_STRIPS_Semantics.type) list"   and pre :: "PDDL_STRIPS_Semantics.term atom formula"   and eff :: "PDDL_STRIPS_Semantics.term ast_effect"   and args :: "object list" shows "instantiate_action_schema (Action_Schema n params pre eff) args = (let tsubst = subst_term ((the \<circ>\<circ> map_of) (zip (map fst params) args)); pre_inst = (map_formula \<circ> map_atom) tsubst pre in Let (map_ast_effect tsubst eff) (Ground_Action pre_inst))"
seq_char: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" shows "seq g f = (arr f \<and> arr g \<and> Dom g = Cod f)"
lprefix_conv_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lprefix xs ys = (\<exists>zs. ys = lappend xs zs)"
completeness_def: shows "Schnorr_\<Sigma>.completeness \<equiv> \<forall>h w e. (h, w) \<in> R_DL \<longrightarrow> e \<in> challenge_space \<longrightarrow> spmf (Schnorr_\<Sigma>.completeness_game h w e) True = 1"
simps_1: shows "height \<langle>\<rangle> = 0"
cong_mult_trans_1: fixes a :: "'b"   and b :: "'b"   and x :: "'b"   and n :: "'b"   and y :: "'b" assumes "[a = b * x] (mod n)"   and "[x = y] (mod n)" shows "[a = b * y] (mod n)"
nn_transfer_operator_def: fixes f :: "'a \<Rightarrow> ennreal" shows "nn_transfer_operator f = (if f \<in> borel_measurable M then RN_deriv M (distr (density M f) M T) else (\<lambda>_. 0))"
lnths_split: fixes xs :: "'b llist"   and A :: "nat set"   and n :: "nat" shows "lselect A xs = lselect A (ltake (enat n) xs) $ lselect {m. n + m \<in> A} (ldropn n xs)"
qgood: shows "qGood (pick X) \<and> qGood (pick X')"
nth_mem: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "xs ! n \<in> set xs"
agunit_gone: shows "\<one>\<^bsub>b_ag A\<^esub> = \<zero>"
ContractsWithUnilateralSubstitutes_axioms_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithUnilateralSubstitutes_axioms Xd Ch \<equiv> \<forall>h. Contracts.unilateral_substitutes_on Xd UNIV (Ch h)"
imp_vacuous: fixes u :: "'a upred" shows "(false \<Rightarrow> u) = true"
rsp: shows "eq_onp terminates (\<lambda>_. Done) (\<lambda>_. Done)"
SN_on_restrict: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and S :: "'b set" assumes "SN_on r A" shows "SN_on (restrict r S) A"
vimage_restr_def: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set" shows "f --` A \<equiv> f -` (A \<inter> space M) \<inter> space M"
de_Morgan_conj: fixes x :: "'a"   and y :: "'a" shows "\<sim> (x \<^bold>\<sqinter> y) = \<sim> x \<^bold>\<squnion> \<sim> y"
not_eq_extremum: fixes a :: "'k word" shows "(a \<noteq> 0) = (0 < a)"
ap_sum_ichng: fixes c :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "('g \<Rightarrow> 'e) + 'f"   and x :: "'g" shows "ap_sum c f (pure x) = ap_sum c (pure (\<lambda>f. f x)) f"
fw_upd_out_of_bounds2: fixes j :: "nat"   and j' :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and i' :: "nat" assumes "j < j'" shows "fw_upd M k i j i' j' = M i' j'"
L4: fixes f :: "'b \<Rightarrow> 'c" shows "map f \<circ> wrap = wrap \<circ> f"
linear_scale: fixes f :: "'a \<Rightarrow> 'b"   and r :: "complex"   and b :: "'a" assumes "clinear f" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
case_eq_if: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (if isl sum then f1 (projl sum) else f2 (projr sum))"
in_proj_in_GSMP: fixes t :: "('fun, 'var) Term.term"   and n :: "'a"   and A :: "('a strand_label \<times> ('fun, 'var) strand_step) list" assumes "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))" shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)"
nodes_ins_edge: fixes e :: "'b \<times> 'b"   and g :: "'b ugraph" shows "nodes (ins_edge e g) = {fst e, snd e} \<union> nodes g"
algebra_single_set: fixes X :: "'c set"   and S :: "'c set" assumes "X \<subseteq> S" shows "algebra S {{}, X, S - X, S}"
proper_map_imp_subset_topspace: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "proper_map X Y f" shows "f ` topspace X \<subseteq> topspace Y"
S1: shows "nat_to_pr n x = sch_to_pr (nat_to_sch n) x"
ultr_def: fixes T :: "'b set set"   and F :: "'b set set" shows "ultra\<^bsub>T\<^esub> F = (\<forall>A\<subseteq>carrier\<^bsub>T\<^esub>. A \<in> F \<or> carrier\<^bsub>T\<^esub> - A \<in> F)"
Sigma: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "(\<Sum>x\<in>A. sum (g x) (B x)) = (\<Sum>(x, y)\<in>Sigma A B. g x y)"
submatrix_mult: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'c mat"   and B :: "'c mat"   and I :: "nat set"   and J :: "nat set" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "submatrix (A * B) I J = submatrix A I UNIV * submatrix B UNIV J"
assms_2: shows "x \<in> cl_in \<E> X"
nle: fixes n :: "nat"   and x :: "'a"   and h :: "'a" assumes "N \<le> n"   and "x \<in> S" shows "norm (f' n x h - g' x h) \<le> e * norm h"
game_hash_set_def: fixes W :: "'domain set" shows "game_hash_set W = seed_gen \<bind> (\<lambda>seed. return_spmf (\<not> inj_on (hash seed) W))"
cont_Atm: fixes atm :: "'atom"   and s :: "'state"   and i :: "nat" shows "local.cont (Atm atm) s i = Done"
rel_list_all2_Cons2: fixes S :: "('a \<times> 'b) set"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list" shows "list_all2 (fun_of S) xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> (z, y) \<in> S \<and> list_all2 (fun_of S) zs ys)"
FAccReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<rangle> \<rightarrow>* \<langle>e'\<bullet>F{D},s'\<rangle>"
u_simps_2: shows "arr r\<^sub>0s\<^sub>1.p\<^sub>0"
inv_chart_inverse: fixes x :: "'c"   and c :: "('c, 'd) chart" assumes "x \<in> domain c" shows "inv_chart c (apply_chart c x) = x"
iso_eq: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme"   and f' :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "f \<in> Group.iso G H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> f' x = f x" shows "f' \<in> Group.iso G H"
narrow: shows "G\<turnstile>S\<succ>PrimT Boolean"
strict_unit: shows "\<iota>\<^sub>D = \<I>\<^sub>D"
effective_row_col_match_1: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "matrix_match A1 A2 B1 B2" shows "\<forall>i j. i < row_length A1 * row_length B1 \<and> j < length A2 * length B2 \<longrightarrow> length (row A1 (i div row_length B1)) = length (col A2 (j div length B2))"
trm_tensor: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<t>[a \<otimes> b] = \<iota> \<cdot> (\<t>[a] \<otimes> \<t>[b])"
vars_subset_dim_vec_to_lpoly_dim: fixes v :: "rat vec" shows "vars (vec_to_lpoly v) \<subseteq> {0..<dim_vec v}"
take_set: fixes n :: "nat"   and l :: "'a list" shows "set (take n l) = {l ! i |i. i < n \<and> i < length l}"
seqE: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and g :: "'ag"   and f :: "'ag" assumes "category C"   and "partial_magma.arr C (C g f)" obtains "partial_magma.arr C f" and "partial_magma.arr C g" and "partial_magma.dom C g = partial_magma.cod C f"
monoidal_category_axioms: shows "monoidal_category (\<cdot>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu>) \<alpha> \<omega>"
hyps: fixes x :: "'a"   and acc :: "'a list" assumes "funpow_dist f (f xa__) y = funpow_dist1 f x y"   and "y \<in> orbit f x" shows "orbit_list_impl f y acc x = rev acc @ iterate 0 (funpow_dist1 f x y) f x"
infI: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<And>i. is_glb L i A \<Longrightarrow> P i"   and "A \<subseteq> carrier L" shows "P (\<Sqinter>A)"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : tab\<^sub>0 g \<star> w \<rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : tab\<^sub>0 g \<star> w' \<rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : tab\<^sub>1 g \<star> w \<rightarrow> tab\<^sub>1 g \<star> w'\<guillemotright>"   and "(g \<star> \<theta>) \<cdot> \<a>[g, tab\<^sub>0 g, w] \<cdot> (g.tab \<star> w) = ((g \<star> \<theta>') \<cdot> \<a>[g, tab\<^sub>0 g, w'] \<cdot> (g.tab \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<rightarrow> w'\<guillemotright> \<and> \<beta> = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)"
streams_iff_snth: fixes s :: "'b stream"   and X :: "'b set" shows "(s \<in> streams X) = (\<forall>n. s !! n \<in> X)"
SeqWRP_imp_OrdP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqWRP s k t} \<turnstile> OrdP k"
sing_pow_empty: fixes a :: "'b"   and n :: "nat" shows "([a] \<^sup>@ n = \<epsilon>) = (n = 0)"
lm079: fixes P :: "('d \<times> 'c) set"   and Q :: "('d \<times> 'c) set"   and X :: "'d set" shows "(P +< Q) `` (Domain Q \<inter> X) = Q `` (Domain Q \<inter> X)"
assms_2: shows "z1 \<noteq> \<infinity>\<^sub>h"
cos_ms_aux'_def: fixes xs :: "('a \<times> real) msllist" shows "cos_ms_aux' xs = powser_ms_aux (msllist_of_msstream cos_series_stream) (times_ms_aux xs xs)"
rename_Join: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and G :: "'c program" assumes "bij h" shows "rename h (F \<squnion> G) = rename h F \<squnion> rename h G"
local_time_def: fixes A :: "'a set"   and n :: "nat"   and x :: "'a" shows "local_time A n x = card {i \<in> {..<n}. (T ^^ i) x \<in> A}"
assms: shows "eval the_singleton t t' n [\<phi>]\<^sub>b"
terminates_flatten_raw: fixes g'' :: "'e \<Rightarrow> ('f, 'e) step"   and g :: "'g \<Rightarrow> ('h, 'g) step"   and f :: "'h \<Rightarrow> 'e" assumes "terminates g''"   and "terminates g" shows "terminates (flatten_raw f g'' g)"
sub2_preserves_dimensions: fixes d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub2 d l (r, A) = (r', A')" shows "dim_row A' = dim_row A \<and> dim_col A' = dim_col A"
p_top: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus top = bot"
lt_eq_tt_monomial: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" shows "lt p = tt p"
StartBallot_HInv3_R: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "StartBallot s s' t"   and "HInv3_R s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_R s' p q d"
relpow_add: fixes m :: "nat"   and n :: "nat"   and R :: "('a \<times> 'a) set" shows "R ^^ (m + n) = R ^^ m O R ^^ n"
simps_2: fixes x :: "'a"   and P :: "'a pred" shows "pred_of_seq (Insert x P) = single x \<squnion> P"
findIndices_def: shows "findIndices = (\<Lambda> P xs. map\<cdot>snd\<cdot>(filter\<cdot>(\<Lambda> \<langle>x, i\<rangle>. P\<cdot>x)\<cdot>(zip\<cdot>xs\<cdot>[0..])))"
simps: fixes rg :: "recf"   and rc :: "recf"   and rgcs :: "(recf \<times> recf) list" shows "rec_embranch ((rg, rc) # rgcs) = Let (arity rg) (rec_embranch' ((rg, rc) # rgcs))"
freess_def: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees = ffUnion \<circ> fset_of_list \<circ> map frees"
butlast_tl: fixes xs :: "'a list"   and ys :: "'a list" assumes "last xs = hd ys"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "butlast xs @ ys = xs @ tl ys"
Red_I_of_Inf_to_N_F: fixes \<iota> :: "'f inference"   and N :: "'f set" assumes "\<iota> \<in> Inf_F"   and "concl_of \<iota> \<in> N" shows "\<iota> \<in> Red_I_\<G> N"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) lit" assumes "\<not> Ik.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> Ik.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> Ik.satA \<xi> a"
geseqE: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and A :: "'b set" assumes "(f, g) \<in> mbs.geseq A" obtains "\<forall>i. f i \<in> A" and "\<forall>i. g i \<in> A" and "f = g"   | i :: "nat" where "\<forall>i. f i \<in> A" and "\<forall>i. g i \<in> A" and "size (g i) < size (f i)" and "\<forall>j<i. f j = g j"
set_val_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_val_locs character_data_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
zero: shows "(\<lambda>x. 0) \<in> PrimRec1"
derivation_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "fixed_point_proof A f" shows "fixed_point_proof.derivation A less_eq f X \<equiv> X \<subseteq> A \<and> well_ordered_set X less_eq \<and> (\<forall>x\<in>X. let Y = {y \<in> X. asympartp less_eq y x} in (\<exists>y. extreme Y less_eq y \<and> x = f y) \<or> f ` Y \<subseteq> Y \<and> extreme_bound A less_eq Y x)"
abs_prod: 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 f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "abs (comm_monoid_mult.prod times one f A) = comm_monoid_mult.prod times one (\<lambda>x. abs (f x)) A"
\<psi>_def: shows "\<psi> = fold fst fgs"
cobounded1: fixes a :: "'b::semilattice_sup"   and b :: "'b::semilattice_sup" shows "(a::'b::semilattice_sup) \<le> a \<squnion> (b::'b::semilattice_sup)"
circ_increasing: fixes x :: "'a" shows "x \<le> x\<^sup>\<circ>"
paths_def: fixes u :: "'v"   and v :: "'v" shows "paths u v \<equiv> {p. path u p v}"