hcju/isabelleps huggingface dataset:
l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes 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 set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf create_character_data known_ptr \<equiv> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_val type_wf set_val set_val_locs \<and> l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_val create_character_data known_ptr"
adj_int_im: fixes C :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "C \<sim> D"   and "f ` C \<noteq> f ` D" shows "f ` C \<inter> f ` D \<lhd> f ` C"
rule0: fixes a :: "'h" shows "a \<equiv> a + (0::'h)"
ipresSwapAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'd, 'e, 'j) model_scheme" shows "ipresSwapAll h hA MOD \<equiv> ipresSwap h MOD \<and> ipresSwapAbs hA MOD"
subc_sub_0_new: fixes c :: "'a"   and t :: "'a FOL_Fitting.term"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form" assumes "new_term c t" shows "subc c s (p[t/0]) = subc c (liftt s) p[t/0]"
eq2: shows "proj_psi_l R * (post - proj_psi_l R) = 0\<^sub>m N N"
similar_matrix_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "similar_matrix A B = (\<exists>P Q. similar_matrix_wit A B P Q)"
cast\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes shadow_root :: "('m, 'n, 'o, 'p option) RShadowRoot_scheme" shows "cast shadow_root = RDocument.extend (RDocument.truncate shadow_root) (Some (Inl (RDocument.more shadow_root)))"
simps_2: fixes p :: "nat"   and x :: "real"   and n :: "nat" shows "power_down p x (Suc n) = (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))\<^sup>2) else truncate_down (Suc p) (x * power_down p x n))"
UP_mult_closed: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "p \<otimes>\<^bsub>UP R\<^esub> q \<in> carrier (UP R)"
mem_in_image2: fixes a :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "a \<in> A" shows "f a \<in> f ` A"
prj_sfun_def: shows "PRJ('e \<rightarrow>! 'f) = sfun_map\<cdot>EMB('e)\<cdot>PRJ('f) oo sfun_prj"
Just_def: shows "Just \<equiv> \<Lambda> a. maybe_abs\<cdot>(sinr\<cdot>(up\<cdot>a))"
sub_rev_sub: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "Cring_Poly.compose R f g = rev_compose R g f"
assms_3: shows "a \<in>\<^sub>\<circ> A"
aff_dim_affine: fixes S :: "'a set"   and L :: "'a set" assumes "S \<noteq> {}"   and "affine S"   and "subspace L"   and "affine_parallel S L" shows "aff_dim S = int (dim L)"
poly_divides_refl: fixes p :: "'a list" shows "p divides p"
ex_cos_gt: fixes \<alpha> :: "real"   and a :: "real" assumes "- pi / 2 \<le> \<alpha> \<and> \<alpha> \<le> pi / 2"   and "a < 1" shows "\<exists>\<alpha>'\<ge>- pi / 2. \<alpha>' \<le> pi / 2 \<and> \<alpha>' \<noteq> \<alpha> \<and> a < cos (\<alpha> - \<alpha>')"
quote_all_MonPon2_PfP_ssubst: fixes \<alpha>1 :: "fm"   and \<alpha>2 :: "fm"   and \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> \<alpha>1 IMP \<alpha>2 IMP \<beta>"   and "V \<subseteq> V'"   and "supp \<alpha>1 \<subseteq> atom ` V'"   and "supp \<alpha>2 \<subseteq> atom ` V'"   and "supp \<beta> \<subseteq> atom ` V'" shows "quote_all p' V \<turnstile> PfP (ssubst \<lfloor>\<alpha>1\<rfloor>V V F') IMP PfP (ssubst \<lfloor>\<alpha>2\<rfloor>V V F') IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F')"
Extend_satisfies_decr: fixes k :: "'k"   and i :: "nat"   and \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and P :: "'v" assumes "\<not> local.find k i \<phi>"   and "LEQ k i (#\<^sub>V \<AA>)"   and "lformula \<phi>" shows "(Extend k i \<AA> P \<Turnstile> \<phi>) = (\<AA> \<Turnstile> decr k i \<phi>)"
vimage_comp: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and x :: "'d set" shows "f -` g -` x = (g \<circ> f) -` x"
sub_minus_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.subset_extended minus uminus inf sup bot less_eq less top" shows "minus (uminus x) (uminus y) = uminus (sup (uminus (uminus x)) (uminus y))"
induct: fixes P :: "('a option list set \<Rightarrow> 'a option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a option list set list \<times> 'a option list set list \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "'a option list set \<Rightarrow> 'a option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "'b"   and a2 :: "'a option list set list \<times> 'a option list set list"   and a3 :: "nat list" assumes "\<And>f z x xs y ys t ts. P f (f x y t z) (xs, ys) ts \<Longrightarrow> P f z (x # xs, y # ys) (t # ts)"   and "\<And>f z ys ts. P f z ([], ys) ts"   and "\<And>f z xs ts. P f z (xs, []) ts"   and "\<And>f z xs ys. P f z (xs, ys) []" shows "P a0 a1 a2 a3"
max_add_distrib_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus x (ord.max less_eq y z) = ord.max less_eq (plus x y) (plus x z)"
wp_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp c Q = (\<lambda>s. \<forall>t. (c, s) \<Rightarrow> t \<longrightarrow> Q t)"
prob_space_stream_space: shows "prob_space (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x))))"
p: shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
select_convs_2: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getF \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getF"
F2: shows "\<forall>u. (b, u) \<in> R \<longrightarrow> (a, u) \<in> R\<^sup>*"
comm_monoid_mult_isom_axioms: shows "comm_monoid_mult_isom hom"
matches_dest_2: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and m' :: "'c match_expr"   and a :: "action"   and p :: "'d" assumes "matches \<gamma> (MatchAnd m m') a p"   and "\<not> matches \<gamma> m' a p" shows "False"
not_lexs: fixes x :: "real \<times> real"   and y :: "real \<times> real" shows "(\<not> lexs x y) = Counterclockwise_2D_Arbitrary.lex y x"
sup1: shows "P \<turnstile> last Cs' \<prec>\<^sup>1 last Cs''"
f_preserves: fixes as_pos :: "'b list"   and as_neg :: "'b list" assumes "f [] = Some (as_pos, as_neg)" shows "as_pos = [] \<and> as_neg = []"
unit_factor_snd_quot_of_ratfps: fixes x :: "'b ratfps" shows "unit_factor (snd (quot_of_ratfps x)) = 1"
H': fixes x :: "'a \<times> 'b" assumes "x \<in> S" shows "(H has_derivative blinfun_apply (H' x)) (at x)"
add_integral_ivl_bound: fixes t0 :: "real"   and h :: "real"   and f :: "real \<Rightarrow> 'e"   and l :: "'e"   and x0 :: "'e"   and u :: "'e" assumes "\<And>x h'. \<lbrakk>h' \<in> {t0..h}; x \<in> {t0..h}\<rbrakk> \<Longrightarrow> (h' - t0) *\<^sub>R f x \<in> {l - x0..u - x0}"   and "t0 \<le> h"   and "f integrable_on {t0..h}" shows "x0 + integral {t0..h} f \<in> {l..u}"
received_msg_inv: fixes i :: "nat"   and P :: "msg \<Rightarrow> bool" shows "paodv i \<TTurnstile> (recvmsg P \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:1} \<longrightarrow> P (msg \<xi>))"
simps_2: fixes d :: "int" shows "intv'_const (Const' d) = d"
unfoldl_inductl_implies_unfoldr: fixes rtc :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "\<And>x. 1' + x ; rtc x \<le> rtc x"   and "\<And>x y z. x + y ; z \<le> z \<Longrightarrow> rtc y ; x \<le> z" shows "1' + rtc x ; x \<le> rtc x"
assms_1: shows "list_all (\<lambda>r. length r = nc) rs"
prules_def: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> pterm) fset" shows "prules C_info rs \<equiv> constants C_info (fst |`| rs) \<and> prules_axioms C_info rs"
reduc_saturated_def: fixes N :: "'f set" shows "reduc_saturated N = (Inf_from (N - Red_F N) \<subseteq> Red_I N)"
equalityCE: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
unlabel_append: fixes A :: "('e \<times> 'd) list"   and B :: "('e \<times> 'd) list" shows "unlabel (A @ B) = unlabel A @ unlabel B"
le_inf_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> inf y z) = (x \<le> y \<and> x \<le> z)"
unit_in_vhom\<^sub>P\<^sub>B\<^sub>U: fixes a :: "'a" assumes "weak_unit a" shows "\<guillemotleft>\<i>[a] : a \<star> a \<Rightarrow> a\<guillemotright>"
neq_no_wait_locks_conv: fixes ln :: "'b \<Rightarrow>f nat" shows "(ln \<noteq> no_wait_locks) = (\<exists>l. 0 < ln $ l)"
fls_base_factor_to_fps_shift: fixes m :: "int"   and f :: "'b fls" shows "fls_base_factor_to_fps (fls_shift m f) = fls_base_factor_to_fps f"
ereal_Inf_uminus_image_eq: fixes S :: "ereal set" shows "Inf (uminus ` S) = - Sup S"
dbm_entry_val_mono_2: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and b :: "'d DBMEntry"   and b' :: "'d DBMEntry" assumes "dbm_entry_val u None (Some c) b"   and "b \<preceq> b'" shows "dbm_entry_val u None (Some c) b'"
GreenThm_typeI_twoCube_2: shows "\<forall>(k, \<gamma>)\<in>boundary twoC. line_integral_exists F {i} \<gamma>"
complex_inner_1_right: fixes x :: "complex" shows "x \<bullet>\<^sub>C 1 = cnj x"
pmf_of_list_def: fixes xs :: "('b \<times> real) list" shows "pmf_of_list xs = embed_pmf (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
ltl_lang_equiv_equivp: shows "equivp (\<sim>\<^sub>L)"
bot_option_def: shows "UML_Types.bot_class.bot \<equiv> \<bottom>"
sp_Cons: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and e :: "('d \<times> nat) edge"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.subpath g v1 (e # es) v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs\<^sup>+) \<and> e \<in> edges g \<and> SubRel.subpath g (tgt e) es v2 subs)"
rep_eq: fixes x :: "'a dual"   and xa :: "'a dual" shows "(x \<le> xa) = (undual xa \<le> undual x)"
simps_1: fixes at :: "('c, 'd) atm" shows "varsL (Pos at) = varsA at"
assms_1: shows "\<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace>"
valid_sops_nth_update: fixes i :: "nat"   and sb :: "'a memref list"   and "is" :: "instr list"   and p :: "'a"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "\<forall>sop\<in>write_sops sb. valid_sop sop"   and "\<forall>sop\<in>store_sops is. valid_sop sop" shows "valid_sops (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
finite_pred_imp_se_updatable: fixes c :: "('d, 'e) conf" assumes "finite (Bexp.vars (conjunct (pred c)))" shows "updatable c"
update_access_same: fixes l :: "Location"   and s :: "Store"   and x :: "Value" assumes "alive (ref l) s"   and "alive x s"   and "\<tau> x \<le> ltype l" shows "s\<langle>l := x\<rangle>@@l = x"
assms_3: shows "cd = [c, d]\<^sub>\<circ>"
infsetsum_reindex_bij_betw: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "bij_betw g A B" shows "(\<Sum>\<^sub>ax\<in>A. f (g x)) = infsetsum f B"
IH_2: shows "((l, receive\<langle>t\<rangle>) \<in> set A) = ((l, send\<langle>t\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
infsetsum_nonneg: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "0 \<le> infsetsum f A"
tree1Eq: fixes n :: "nat"   and Y :: "'b"   and subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b" shows "((Suc n, Y) \<in> tree subs gamma) = (\<exists>sigma\<in>subs gamma. (n, Y) \<in> tree subs sigma)"
synth_simps_5: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "(Crypt K X \<in> synth H) = (Crypt K X \<in> H \<or> X \<in> synth H \<and> Key K \<in> H)"
inverse_cancel: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)" shows "((\<lambda>x. inverse (f x)) \<in> L F (\<lambda>x. inverse (g x))) = (g \<in> L F f)"
AE_def: shows "AE = ASSERT_LIVE \<circ> HAVOC"
reduce_alist_code: fixes xs :: "('a \<times> bool) list"   and t :: "'a ifex" shows "reduce (AList_Mapping.Mapping xs) t = reduce_alist xs t"
hom_component_plus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "hom_component (p + q) n = hom_component p n + hom_component q n"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "H \<le> G"   and "a \<in> H" shows "inv a \<in> H"
livenessE: fixes A :: "'b set"   and P :: "'b llist set"   and s :: "'b llist" assumes "liveness A P" obtains t :: "'b llist" where "t \<in> A\<^sup>\<infinity>" and "s @@ t \<in> P"   | "s \<notin> A\<^sup>\<star>"
pdivides_imp_root_sharing: fixes p :: "'a list list"   and q :: "'a list list"   and a :: "'a list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "p pdivides\<^bsub>poly_ring R\<^esub> q"   and "a \<in> carrier (poly_ring R)"   and "UP.eval p a = \<zero>\<^bsub>poly_ring R\<^esub>" shows "UP.eval q a = \<zero>\<^bsub>poly_ring R\<^esub>"
slim_is_slim: shows "pair_pre_digraph.is_slim slim (verts3 (with_proj G))"
simps_11: fixes p :: "fm" shows "bound0 (NOT p) = bound0 p"
ddh_1_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> bool spmf" shows "ddh_1 \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. sample_uniform (order \<G>) \<bind> (\<lambda>z. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) (\<^bold>g [^] z))))"
access_update_other: fixes l :: "Location"   and m :: "Location"   and s :: "Store"   and x :: "Value" assumes "l \<noteq> m" shows "s\<langle>l := x\<rangle>@@m = s@@m"
listrel1I2: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set"   and x :: "'g" assumes "(xs, ys) \<in> listrel1 r" shows "(x # xs, x # ys) \<in> listrel1 r"
Class_self: fixes a :: "'a" assumes "a \<in> S" shows "a \<in> Class a"
class_of_def: fixes U :: "'a set"   and s :: "'b" shows "stk2.class_of U s \<equiv> e2.Class (U, s)"
vec_eq_iff: fixes x :: "'a Matrix.vec"   and y :: "'a Matrix.vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $ i = y $ i))"
e_val: fixes v :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "isval v" shows "\<exists>v'. v' \<in> E v \<rho>"
set_subset_length_eq: fixes A :: "'b list"   and B :: "'b list" assumes "set A \<subseteq> set B"   and "length B \<le> length A"   and "distinct A"   and "distinct B" shows "set A = set B"
nrename_AndR: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and M :: "trm"   and d :: "coname"   and N :: "trm"   and e :: "coname" assumes "R[x\<turnstile>n>y] = AndR <c>.M <d>.N e"   and "c \<sharp> (d, e, N, R)"   and "d \<sharp> (c, e, M, R)" shows "\<exists>M' N'. R = AndR <c>.M' <d>.N' e \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N \<and> c \<sharp> N' \<and> d \<sharp> M'"
dropWhile_all_False_conv: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "dropWhile P xs = xs"
cfun_beta_Pair: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and y :: "'b" assumes "cont (\<lambda>p. f (fst p) (snd p))" shows "(\<Lambda> (a, b). f a b)\<cdot>(x, y) = f x y"
comp_arr_inv': fixes f :: "'a" assumes "C.iso f" shows "f \<cdot> C.inv f = C.cod f"
continuous_trivial_limit: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "net = bot" shows "continuous net f"
inner_minus_right: fixes x :: "'b"   and y :: "'b" shows "inner x (- y) = - inner x y"
eq_LConsD: fixes xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" assumes "xs = y % ys" shows "xs \<noteq> <> \<and> lhd xs = y \<and> ltl xs = ys"
Zl_mI_mem_K: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and l :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "l \<le> n" shows "Zl_mI K P I l \<in> carrier K"
simps_2: fixes a :: "('fsym, 'psym) atm" shows "wtL (Neg a) = wtA a"
zero_less_divide_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
smult_r_null: fixes a :: "'a" assumes "a \<in> carrier R" shows "a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
lunit_naturality: fixes f :: "'d" assumes "D.arr f" shows "D.lunit (D.cod f) \<cdot>\<^sub>D D.tensor D.unity f = f \<cdot>\<^sub>D D.lunit (D.dom f)"
digraphE: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" obtains "\<Z> \<alpha>" and "vfsequence \<CC>" and "vsv (\<CC>\<lparr>Dom\<rparr>)" and "vsv (\<CC>\<lparr>Cod\<rparr>)" and "vcard \<CC> = 4\<^sub>\<nat>" and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>" and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>" and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "\<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>" and "\<And>A B. \<lbrakk>A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>; B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>; A \<in>\<^sub>\<circ> Vset \<alpha>; B \<in>\<^sub>\<circ> Vset \<alpha>\<rbrakk> \<Longrightarrow> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>"
fres_iso: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "less_eq_prod (prod.fres x z) (prod.fres y z)"
mismatchId: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "[a\<noteq>b]P \<sim>\<^sub>l P"
distinct_5: fixes x2 :: "'f"   and x31 :: "'e"   and x32 :: "'f" shows "Skip x2 \<noteq> Yield x31 x32"
assms_6: shows "\<forall>b\<in>basis. \<forall>(k, \<gamma>2)\<in>one_chain. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
vec_lambda_beta: fixes g :: "'c \<Rightarrow> 'b"   and i :: "'c" shows "vec_lambda g $h i = g i"
next_def: fixes A :: "'e \<times> 'd \<times> 'f" shows "next A = fst (snd A)"
ncols_diagonal_step: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat"   and c :: "'a"   and v :: "'a" shows "ncols (diagonal_step A i j c v) = ncols A"
perm_empty: fixes M2 :: "('e, 'f, 'g) FSM"   and V :: "'e list set"   and V'' :: "('e \<times> 'h) list set"   and M1 :: "('e, 'h, 'i) FSM" assumes "is_det_state_cover M2 V"   and "V'' \<in> Perm V M1" shows "[] \<in> V''"
collapse_2: fixes "term" :: "('a, 'b) Term.term" assumes "is_Fun term" shows "Fun (the_Fun term) (args term) = term"
nextl: fixes q :: "hf"   and u :: "'a list" assumes "q \<in> states M" shows "h (M.nextl q u) = N.nextl (h q) u"
outstanding_refs_subsets_1: fixes sb :: "'b memref list" shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<subseteq> outstanding_refs is_Write\<^sub>s\<^sub>b sb"
boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<squnion> c \<le> a" obtains "b \<le> a" and "c \<le> a"
subst_type_term_simulates_subst_tsubst_gen': fixes lty :: "(variable \<times> String.literal set) list"   and t :: "Core.term"   and lt :: "(variable \<times> typ) list"   and \<rho>ty :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ"   and \<rho>t :: "variable \<Rightarrow> typ \<Rightarrow> Core.term" assumes "distinct lty"   and "tvs t \<subseteq> set lty"   and "distinct lt"   and "fv (tsubst t \<rho>ty) \<subseteq> set lt" shows "subst (tsubst t \<rho>ty) \<rho>t = subst_type_term (map (\<lambda>(x, y). ((x, y), \<rho>ty x y)) lty) (map (\<lambda>(x, y). ((x, y), \<rho>t x y)) lt) t"
remap_dta: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option"   and f :: "'a \<Rightarrow> 'c" assumes "det_tree_automaton TA A"   and "inj_on f (ta_rstates TA)" shows "det_tree_automaton (ta_remap f TA) A"
1: shows "subst (subst Pf (encPf prf) yy) \<langle>\<phi>\<rangle> xx = subst (subst Pf \<langle>\<phi>\<rangle> xx) (substT (encPf prf) \<langle>\<phi>\<rangle> xx) yy"
intros_3: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "u $ s \<rightarrow>\<^sub>\<beta> u $ t"