hcju/isabelleps huggingface dataset:
ev_eq: shows "\<forall>\<^sub>F n in sequentially. flow0 ((flow0 x \<circ> t) n) ((\<tau> \<circ> (flow0 x \<circ> t)) n) = (flow0 x \<circ> s) n"
lowner_lub_def: shows "lowner_lub = The local.lowner_is_lub"
sdp_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "sub_distrib_pconj body"   and "nearly_healthy (wlp body)"   and "healthy (wp body)" shows "sub_distrib_pconj (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)"
fminus_cancel: fixes A :: "'a fset" shows "A |-| A = {||}"
qSwap_preserves_qAFresh_distinct: fixes ys :: "'d"   and zs :: "'d"   and y :: "'e"   and z1 :: "'e"   and z2 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" assumes "ys \<noteq> zs \<or> y \<notin> {z1, z2}" shows "qAFresh ys y (X #[[z1 \<and> z2]]_zs) = qAFresh ys y X"
simps_15: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (Iff p q) = (Ifm bbs bs p = Ifm bbs bs q)"
VSagree_UNIV_eq: fixes A :: "(real, 'a) vec"   and B :: "(real, 'a) vec" assumes "VSagree A B UNIV" shows "A = B"
hd_in_awalk_verts_2: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.apath u p v" shows "u \<in> set (pawalk_verts u p)"
a_coset_eq: shows "\<forall>x\<in>carrier G. H +> x = x <+ H"
thd3_simp: fixes a :: "'b"   and b :: "'c"   and c :: "'a" shows "thd3 (a, b, c) = c"
covering_space_lift_homotopic_paths: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and g1 :: "real \<Rightarrow> 'd"   and g2 :: "real \<Rightarrow> 'd"   and h1 :: "real \<Rightarrow> 'c"   and h2 :: "real \<Rightarrow> 'c" assumes "covering_space C p S"   and "path g1"   and "path_image g1 \<subseteq> S"   and "path g2"   and "path_image g2 \<subseteq> S"   and "homotopic_paths S g1 g2"   and "path h1"   and "path_image h1 \<subseteq> C"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h1 t) = g1 t"   and "path h2"   and "path_image h2 \<subseteq> C"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p (h2 t) = g2 t"   and "pathstart h1 = pathstart h2" shows "homotopic_paths C h1 h2"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"   and "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs type_wf known_ptr" shows "l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs known_ptr"
adj_int_im: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<sim> D"   and "f ` C \<noteq> f ` D" shows "f ` C \<inter> f ` D \<lhd> f ` C"
that: fixes gs :: "'a list" assumes "set gs \<subseteq> carrier G"   and "cycle gs"   and "is_idirprod (carrier G) (\<lambda>g. generate G {g}) (set gs)"   and "successively (dvd) (map ord gs)"   and "card (set gs) \<le> card gen"   and "\<one> \<notin> set gs" shows "thesis"
lengths2: shows "(\<forall>w\<in>set (Matrix_Legacy.transpose nr m). length w = ncc) \<and> length (Matrix_Legacy.transpose nr m) = nr"
integrable_const: fixes a :: "'c" shows "integrable P (\<lambda>x. a)"
collection_unop_type_det: fixes op :: "collection_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "collection_unop_type op \<tau> \<sigma>\<^sub>1"   and "collection_unop_type op \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
bangRel_def: shows "bangRel \<equiv> \<lambda>Rel. {(xa, x). bangRelp (\<lambda>x xa. (x, xa) \<in> Rel) xa x}"
box_axiom_def: shows "box_axiom \<equiv> \<langle> $\<alpha> vid1 \<rangle> Pc pid1 \<leftrightarrow> ! ([[$\<alpha> vid1]]! (Pc pid1))"
repeat_stabilize_induct: fixes P :: "'b \<Rightarrow> bool"   and m :: "'b"   and f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "P m"   and "\<And>m. P m \<Longrightarrow> P (f m)" shows "P (repeat_stabilize n f m)"
PP_def: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list" assumes "spectral_radius_1_jnf_max A n m lam n_as" shows "spectral_radius_1_jnf_max.PP A n_as = (SOME PP. similar_mat_wit (of_real_hom.mat_hom A) (jordan_matrix n_as) (fst PP) (snd PP))"
zip_map_fst_snd: fixes zs :: "('d \<times> 'e) list" shows "zip (map fst zs) (map snd zs) = zs"
elementLevelPrecedesMarkedElementLt: fixes M :: "('b \<times> bool) list"   and e :: "'b"   and d :: "'b" assumes "uniq (elements M)"   and "e \<noteq> d"   and "d \<in> set (markedElements M)"   and "precedes e d (elements M)" shows "elementLevel e M < elementLevel d M"
bis_prefix: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set" assumes "Bisim.mono Retr" shows "bis Retr \<subseteq> Retr (bis Retr)"
additive_def: fixes f :: "'c \<Rightarrow> 'd" shows "additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
closest_bound_f_def: fixes Q\<^sub>b :: "'a"   and Q :: "'a set"   and f :: "nat \<Rightarrow> 'a" shows "closest_bound_f Q\<^sub>b Q f \<equiv> is_bound_f Q\<^sub>b Q f \<and> (\<forall>Q\<^sub>b'. is_bound Q\<^sub>b' Q \<and> Q\<^sub>b' \<noteq> Q\<^sub>b \<longrightarrow> [[f 0 Q\<^sub>b Q\<^sub>b']])"
zero_base_lsb_imp_set_eq_as_bit_operation: fixes len :: "nat"   and base :: "'b word"   and a :: "'b word" assumes "mask (LENGTH('b) - len) && base = 0" shows "(base = ~~ mask (LENGTH('b) - len) && a) = (a \<in> {base..base || mask (LENGTH('b) - len)})"
adj_sub_prop: fixes \<sigma> :: "('b, 'c, 'a) subst"   and x2 :: "'a \<Rightarrow> ('b, 'a) trm"   and j :: "'a"   and x1 :: "'a" shows "(\<Union>i\<in>SIGT (x2 j). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF ($\<phi> x1 x2)))"
sg_subset: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<guillemotright> H " shows "H \<subseteq> carrier G"
indets_empty_imp_monomial: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "indets p = {}" shows "monomial (lookup p 0) 0 = p"
assms_1: shows "\<forall>a\<in>set as. irreducible G a"
simps_10: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and x3 :: "'b" shows "(case dRETURN x3 of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = f3 x3"
fcompose_uminus: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "fcompose (- p) q r = - fcompose p q r"
index_1_imp_mult_1: fixes bl :: "'a set" assumes "1 = 1"   and "bl \<in># \<B>"   and "2 \<le> int (card bl)" shows "multiplicity bl = 1"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and finite_3 :: "Enum.finite_3" shows "P (case finite_3 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3) = (\<not> (finite_3 = a\<^sub>1 \<and> \<not> P f1 \<or> finite_3 = a\<^sub>2 \<and> \<not> P f2 \<or> finite_3 = a\<^sub>3 \<and> \<not> P f3))"
DDH_\<A>'_def: fixes \<G> :: "'b cyclic_group"   and D_ddh :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf"   and a :: "'b"   and b :: "'b"   and c :: "'b" shows "ddh.DDH_\<A>' \<G> D_ddh a b c = D_ddh a b (c \<otimes>\<^bsub>\<G>\<^esub> \<^bold>g\<^bsub>\<G>\<^esub>)"
inj_fn: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "inj f" shows "inj (f ^^ n)"
wt_Unifier_if_Unifier: fixes s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m s"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<Gamma> s = \<Gamma> t"   and "Unifier \<delta> s t" shows "\<exists>\<theta>. Unifier \<theta> s t \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
r1_1: shows "s \<sim> ro s +\<^sub>r s'"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (Ring_Divisibility.mult_of R)"   and "y \<in> carrier (Ring_Divisibility.mult_of R)" shows "x \<otimes> y = y \<otimes> x"
numeral_times_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "num"   and n :: "num" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m * n)"
fi_map_alt: shows "fi_map = the_inv_into_map V f"
nat_pow_comm: fixes x :: "'a vec"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier_vec n" shows "[n] \<cdot>\<^bsub>V\<^esub> x + [m] \<cdot>\<^bsub>V\<^esub> x = [m] \<cdot>\<^bsub>V\<^esub> x + [n] \<cdot>\<^bsub>V\<^esub> x"
structCongBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim> Q"
zcount_update_zmultiset: fixes M :: "'a zmultiset"   and t :: "'a"   and n :: "int"   and t' :: "'a" shows "zcount (update_zmultiset M t n) t' = zcount M t' + (if t = t' then n else 0)"
set_mono_suffix: fixes xs :: "'b list"   and ys :: "'b list" assumes "suffix xs ys" shows "set xs \<subseteq> set ys"
nle: fixes e :: "real" assumes "0 < e" shows "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
inst: shows "instance_of\<^sub>l\<^sub>s (C\<^sub>2 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''2'' @ x))) C\<^sub>2"
sets_stream_space_in_sets: fixes N :: "'b stream measure"   and M :: "'b measure" assumes "space N = streams (space M)"   and "\<And>i. (\<lambda>x. x !! i) \<in> N \<rightarrow>\<^sub>M M" shows "sets (stream_space M) \<subseteq> sets N"
L2_set_nonneg: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "0 \<le> L2_set f A"
ide_src: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "D.ide (src\<^sub>D \<mu>)"
nth_coeffs_coeff: fixes i :: "nat"   and f :: "'a poly" assumes "i < length (coeffs f)" shows "coeffs f ! i = coeff f i"
vars_add: fixes p1 :: "'a mpoly"   and p2 :: "'a mpoly" shows "vars (p1 + p2) \<subseteq> vars p1 \<union> vars p2"
exp_def: shows "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R fact n)"
outpt_chosen: fixes s :: "state"   and s' :: "state" assumes "outpt s = outpt s'"   and "Inv2c s"   and "HNextPart s s'" shows "chosen s' = chosen s"
rsuml_lsumr_parallel_converter2: fixes conv1 :: "('h, 'k, 'n, 'q) converter"   and conv2 :: "('i, 'l, 'o, 'r) converter"   and conv3 :: "('j, 'm, 'p, 's) converter" shows "map_converter id id rsuml lsumr ((conv1 |\<^sub>= conv2) |\<^sub>= conv3) = map_converter rsuml lsumr id id (conv1 |\<^sub>= conv2 |\<^sub>= conv3)"
section_def: fixes f :: "'a" shows "B.section f \<equiv> \<exists>g. B.ide (g \<cdot>\<^sub>B f)"
ereal_mult_strict_right_mono: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a < b"   and "0 < c"   and "c < \<infinity>" shows "a * c < b * c"
read_only_dom: fixes \<S> :: "nat \<Rightarrow> bool option" shows "read_only \<S> \<subseteq> dom \<S>"
insertion_mult: fixes f :: "nat \<Rightarrow> 'b"   and p :: "'b mpoly"   and q :: "'b mpoly" shows "insertion f (p * q) = insertion f p * insertion f q"
simps_4: fixes S :: "'a formula \<Rightarrow> 'a formula"   and F :: "'a formula"   and G :: "'a formula" shows "freshnats.tseytin_tran1 S (F \<^bold>\<and> G) = [S (F \<^bold>\<and> G) \<^bold>\<leftrightarrow> S F \<^bold>\<and> S G] @ freshnats.tseytin_tran1 S F @ freshnats.tseytin_tran1 S G"
eq_iff: fixes x :: "int poly"   and y :: "int poly" shows "(of_int_poly x = of_int_poly y) = (x = y)"
p_antitone_iff: fixes y :: "'a"   and x :: "'a" shows "(- y \<le> x) = (- x \<le> y)"
set_n_lists: fixes n :: "nat"   and xs :: "'b list" shows "set (List.n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
divide_nonneg_neg: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "y < (0::'a)" shows "x / y \<le> (0::'a)"
stc_SComplex_eq: fixes x :: "complex star" assumes "x \<in> SComplex" shows "stc x = x"
\<chi>_is_universal_1: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "\<guillemotleft>\<psi> (a', a) (\<Psi>.FUN a' (\<iota> \<chi>')) : a' \<rightarrow> a\<guillemotright>"
a0: shows "v = Tensor.mat_of_cols_list 8 [[\<alpha> / 2, \<beta> / 2, \<beta> / 2, \<alpha> / 2, \<alpha> / 2, - \<beta> / 2, - \<beta> / 2, \<alpha> / 2]]"
wfp_sdrop: fixes AP' :: "'aprop set"   and \<pi> :: "('state \<times> 'aprop set) stream"   and i :: "nat" assumes "wfp AP' \<pi>" shows "wfp AP' (sdrop i \<pi>)"
insort_eq_ConsI: fixes cmp :: "'b comparator"   and xs :: "'b list"   and x :: "'b" assumes "Sorting_Algorithms.sorted cmp xs"   and "\<And>y. y \<in> set xs \<Longrightarrow> compare cmp x y \<noteq> Greater" shows "Sorting_Algorithms.insort cmp x xs = x # xs"
normal_upper_triangular_matrix_is_diagonal: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "upper_triangular A"   and "A * adjoint A = adjoint A * A" shows "diagonal_mat A"
UBset_def: fixes z :: "ant" shows "UBset z = {x. x \<le> z}"
ratfps_subdegree_def: shows "ratfps_subdegree \<equiv> map_fun Rep_ratfps id (\<lambda>x. poly_subdegree (numerator x))"
assms_2: shows "f2 = (\<lambda>x y z. a21 * x + a22 * y + a23 * z)"
radical_ideal_eq_UNIV_iff: fixes F :: "'c set" shows "(\<surd>ideal F = UNIV) = (ideal F = UNIV)"
conj_ac_2: fixes a :: "'a"   and b :: "'a" shows "a AND b = b AND a"
comb_def': shows "comb = c *\<^sub>R r + b *\<^sub>R q + a *\<^sub>R p"
intro: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('a, 'd, 'e) set_ops_scheme" assumes "StdSet ops1"   and "StdSet ops2" shows "g_set_xx_loc ops1 ops2"
mult_right_dist_sup: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "times (sup x y) z = sup (times x z) (times y z)"
le_less_linear: fixes x :: "'b"   and y :: "'b" shows "x \<le> y \<or> y < x"
cone_convex_hull: fixes S :: "'a set" assumes "cone S" shows "cone (convex hull S)"
RSpan_RSpan_closed: fixes ms :: "'m list"   and ns :: "'m list" assumes "set ms \<subseteq> M"   and "set ns \<subseteq> RSpan ms" shows "RSpan ns \<subseteq> RSpan ms"
elims_3: fixes x :: "('b \<times> color) tree" assumes "\<not> invc x" obtains l :: "('b \<times> color) tree" and a :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (a, c), r\<rangle>" and "\<not> ((c = Red \<longrightarrow> color l = Black \<and> color r = Black) \<and> invc l \<and> invc r)"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a4 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4) \<in> pr_algo_rel" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = push_effect f__ e" and "a2 = l__" and "a3 = f__" and "a4 = l__" and "Height_Bounded_Labeling c s t f__ l__" and "push_precond f__ l__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = relabel_effect f l u" and "a3 = f" and "a4 = l" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
transD: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "trans r"   and "x \<succeq>[r] y"   and "y \<succeq>[r] z" shows "x \<succeq>[r] z"
domSubset3: fixes x :: "((int \<times> int) set set, int) Combinators" shows "dom (C (DenyAll \<oplus> x)) = dom (C DenyAll)"
llist_all2_lnullD: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'f llist"   and ys :: "'g llist" assumes "llist_all2 P xs ys" shows "lnull xs = lnull ys"
rsapss_sign_help1_def: fixes em_nat :: "nat"   and e :: "nat"   and n :: "nat" shows "rsapss_sign_help1 em_nat e n = nat_to_bv_length (rsa_crypt em_nat e n) (length (nat_to_bv n))"
map_eqI: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "dom f = dom g"   and "\<forall>x\<in>dom f. the (f x) = the (g x)" shows "f = g"
subsumes_list_alt: 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" shows "subsumes_list Ls Ks \<sigma> = subsumes_list_modulo Ls Ks \<sigma>"
cInf_eq_Min: fixes X :: "'a set" assumes "finite X"   and "X \<noteq> {}" shows "\<Sqinter> X = Min X"
less_length_takeWhile_conv: fixes i :: "nat"   and P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "(i < length (takeWhile P l)) = (i < length l \<and> (\<forall>j\<le>i. P (l ! j)))"
inj_term_sym_simps_3: fixes v :: "Term.var" shows "In2 v = \<langle>v\<rangle>"
le_minus_iff: 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 a (uminus b) = less_eq b (uminus a)"
Lcm_empty: 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 Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<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" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm {} = one"
assms: shows "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
emp_gen: fixes P :: "'a set set"   and I :: "'a set" assumes "gen P I = {}" shows "I = {}"
nrows_not_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "0 \<noteq> nrows A"
SCrd: fixes p :: "('b, 'c) Combinators list" assumes "singleCombinators p" shows "singleCombinators (remdups p)"
set_map_1: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and v :: "('f, 'g) nondetT" shows "phantom_nondetT (map_nondetT' f1 f2 v) = f1 ` phantom_nondetT v"
scalar_vec_one: fixes v :: "'c vec" shows "(1::'c) \<cdot>\<^sub>v v = v"
per_drop_exp': fixes i :: "nat"   and p :: "'b list"   and e :: "'b list" assumes "i \<noteq> 0"   and "p \<le>p e \<^sup>@ i \<cdot> p" shows "p \<le>p e \<cdot> p"
add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> H"   and "y \<in> H" shows "h (x + y) = h x + h y"
position_if_emb_step_at: fixes p :: "dir list"   and d :: "dir"   and t :: "('a, 'b) tm"   and u :: "('a, 'b) tm" assumes "emb_step_at p d t = u"   and "t \<noteq> u" shows "position_of t (p @ [d])"
carrier_vs_is_self: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and N :: "'i set" assumes "module R M" shows "carrier (M\<lparr>carrier := N\<rparr>) = N"
wf_tcsigs_def: fixes sub :: "(String.literal \<times> String.literal) set"   and tcs :: "'a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" shows "wf_tcsigs sub tcs = (coregular_tcsigs sub tcs \<and> complete_tcsigs sub tcs \<and> consistent_length_tcsigs tcs \<and> all_normalized_and_ex_tcsigs sub tcs)"
a_Inj2_inv_V: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, Inj2 eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2" obtains e :: "Syntax.term" and eP :: "Syntax.term" where "ea__ = Inj2 e" and "ePa__ = Inj2 eP" and "\<Gamma> \<turnstile> e, eP, eV : \<tau>\<^sub>2"
abs_eq: shows "zero_pdevs = Abs_pdev (\<lambda>_. 0::'b)"
pCons_cases: fixes p :: "'c poly" obtains a :: "'c" and q :: "'c poly" where "p = pCons a q"
strict_subset_divisors_dvd: fixes a :: "'a"   and b :: "'a" shows "({c. c dvd a} \<subset> {c. c dvd b}) = (a dvd b \<and> \<not> b dvd a)"
simps_1: shows "wbt \<langle>\<rangle> = True"
hoare_mult: fixes p :: "'a"   and x :: "'b"   and r :: "'a"   and y :: "'b"   and q :: "'a" assumes "\<lbrace>\<iota> p\<rbrace> x \<lbrace>\<iota> r\<rbrace>"   and "\<lbrace>\<iota> r\<rbrace> y \<lbrace>\<iota> q\<rbrace>" shows "\<lbrace>\<iota> p\<rbrace> x \<cdot> y \<lbrace>\<iota> q\<rbrace>"
simp_filter: fixes x :: "'b Filters.filter" shows "ord_class.filter (Filters.filter.Rep_filter x)"
exec_def: shows "exec \<equiv> [\<lparr>states = \<lambda>p. s0, msgs = \<lambda>v. (If (<p0, inM True> = v) 1 \<union># If (<p1, inM True> = v) 1) 0\<rparr>]"
t\<^sub>p_sumofALGALGP: fixes s :: "'c list"   and qs :: "'c list"   and i :: "nat"   and a :: "nat \<times> nat list" assumes "distinct s"   and "qs ! i \<in> set s"   and "\<forall>l<length (snd a). Suc (snd a ! l) < length s" shows "t\<^sub>p s (qs ! i) a = (\<Sum>e\<in>set s. ALG e qs i (swaps (snd a) s, ())) + (\<Sum>(x, y)\<in>{(x, y). x \<in> set s \<and> y \<in> set s \<and> x < y}. ALG_P (snd a) x y s)"
less_eq_neg_nonpos: fixes 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 a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (uminus a) = less_eq a zero"
bal_def: fixes n :: "nat"   and xs :: "'b list" shows "bal n xs = Time_Monad.val (bal_tm n xs)"