hcju/isabelleps huggingface dataset:
nodes_target_elim: fixes q :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" assumes "q \<in> gnodes A w V" obtains r :: "(nat \<times> 'c) list" and p :: "nat \<times> 'c" where "p \<in> ginitial A \<inter> V" and "gpath A w V r p" and "q = gtarget r p"
product_swap: fixes A :: "'f set"   and B :: "'e set" shows "prod.swap ` (A \<times> B) = B \<times> A"
vector_sub_project_orthogonal: fixes b :: "'b"   and x :: "'b" shows "b \<bullet> (x - (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b) = 0"
map_id: fixes t :: "'c blindable\<^sub>h" shows "map_blindable\<^sub>h id t = t"
before_r1: fixes vs :: "'a list"   and r1 :: "'a"   and r2 :: "'a" assumes "before vs r1 r2" shows "r1 \<in> set vs"
d: fixes p :: "nat"   and x :: "int" shows "root_int_ceiling_pos p x = (if p = 0 then 0 else case root_int_main p x of (y, b) \<Rightarrow> if b then y else y + 1)"
group_eq_subgrp_rcoset_un: fixes G :: "'a set"   and H :: "'a set"   and gs :: "'a list" assumes "Group G"   and "Group H \<and> H \<subseteq> G"   and "Group.is_rcoset_replist G H gs" shows "G = (\<Union>g\<in>set gs. H + {g})"
less_induct: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>x. (\<And>y. x < y \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
kummers_test_convergence: fixes f :: "nat \<Rightarrow> real"   and p :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. 0 < f n"   and "\<forall>\<^sub>F n in sequentially. 0 \<le> p n"   and "0 < liminf (\<lambda>n. ereal (p n * f n / f (Suc n) - p (Suc n)))" shows "summable f"
mset_factors_imp_nonzero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and F :: "'a multiset"   and p :: "'a" assumes "class.semidom plus minus zero times one"   and "comm_semiring_1.mset_factors times one zero F p" shows "p \<noteq> zero"
a13: shows "{(t, u). s setinterleaves ((t, u), A) \<and> (\<exists>a list. t = a # list \<and> a \<notin> A) \<and> u = []} \<subseteq> {(hd s # list, []) |list. tl s setinterleaves ((list, []), A)}"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"
val_Zp_dist_def: fixes p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "padic_integers p" shows "padic_integers.val_Zp_dist p a b \<equiv> padic_integers.val_Zp p (a \<ominus>\<^bsub>padic_int p\<^esub> b)"
assms: shows "(sig_red sing_reg (\<prec>) F)\<^sup>*\<^sup>* p q"
fold_fold: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and t :: "('e, 'f) RBT.rbt" shows "RBT.fold f t = fold (\<lambda>(x, y). f x y) (RBT.entries t)"
d2_def: shows "ps2_P.d2 = prod_list ps2_P.dims2"
toN_def: fixes n :: "nat" shows "toN n \<equiv> [0..<Suc n]"
items_le_\<pi>_swap: fixes I :: "item set"   and T :: "(symbol \<times> character list) set"   and k :: "nat" assumes "wellformed_items I"   and "T \<subseteq> \<X> k" shows "items_le k (\<pi> k T I) = \<pi> k (empty_tokens T) (items_le k I)"
subdensity: fixes T :: "'c \<Rightarrow> 'd"   and P :: "'c measure"   and Q :: "'d measure"   and M :: "'e measure"   and X :: "'e \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> ennreal"   and Y :: "'e \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> ennreal" assumes "T \<in> P \<rightarrow>\<^sub>M Q"   and "distributed M P X f"   and "distributed M Q Y g"   and "Y = T \<circ> X" shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
validCE_cont: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i. active cid (t i)"   and "\<not> (\<exists>i'\<ge>n. active cid (t i'))"   and "dynamic_component.eval tCMP active cid t t' n \<gamma>" shows "\<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) (dynamic_component.cnf2bhv tCMP active cid t n)"
B: shows "\<rho>(x := fst (extract_pair v), y := snd (extract_pair v)) \<in> space (state_measure (set vs' \<union> {x, y}) \<Gamma>)"
assms: shows "P \<in> subgroups_of_size (p ^ a)"
assms: fixes i :: "nat" shows "poly.coeff p i \<in> A"
smaller_than_small: fixes A :: "'a set"   and B :: "'a set" assumes "small A"   and "B \<subseteq> A" shows "small B"
has_fields_is_class: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "P \<turnstile> C has_fields FDTs" shows "is_class P C"
igba_impl_rel_eext_internal_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rl :: "('h \<times> 'k) set" shows "igba_impl_rel_eext Rm Rv Rl \<equiv> \<langle>Rm, Rv \<rightarrow> Rl \<rightarrow> bool_rel\<rangle>gen_igba_impl_rel_eext"
env_restr_add: fixes S :: "'c set"   and m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd"   and S2 :: "'c set" shows "(m1 ++\<^bsub>S2\<^esub> m2) f|` S = m1 f|` S ++\<^bsub>S2\<^esub> m2 f|` S"
landau_gt_1_cong: fixes L :: "real filter \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) set"   and L' :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) set"   and Lr :: "real filter \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) set"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "landau_symbol L L' Lr"   and "\<And>x. 1 < x \<Longrightarrow> f x = g x" shows "L at_top f = L at_top g"
success_def: fixes f :: "'b Heap"   and h :: "heap" shows "success f h = (execute f h \<noteq> None)"
locally_sound_respects_guarantees: fixes cm :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val" assumes "locally_sound_mode_use (cm, mem)" shows "respects_own_guarantees cm"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "DocumentClass.known_ptr ptr = (CharacterDataClass.known_ptr ptr \<or> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
type_less_x_Bag_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Bag \<rho>"   and "\<rho> < \<sigma>" shows "\<tau> < Bag \<sigma>"
fpxs_val_minus_commute: fixes f :: "'b fpxs"   and g :: "'b fpxs" shows "fpxs_val (f - g) = fpxs_val (g - f)"
blocks_Cons: fixes V :: "char list"   and Vs :: "char list list"   and T :: "ty"   and Ts :: "ty list"   and v :: "val"   and vs :: "val list"   and e :: "expr" shows "blocks (V # Vs, T # Ts, v # vs, e) = {V:T; V:=Val v;; blocks (Vs, Ts, vs, e)}"
splitLeaf_huffman_commute: fixes ts :: "'a tree list"   and a :: "'a"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat"   and b :: "'a" assumes "consistent\<^sub>F ts"   and "ts \<noteq> []"   and "a \<in> alphabet\<^sub>F ts"   and "freq\<^sub>F ts a = w\<^sub>a + w\<^sub>b" shows "splitLeaf (huffman ts) w\<^sub>a a w\<^sub>b b = huffman (splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b)"
strict_mono_exceeds: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono f" shows "\<exists>k. n < f k"
at_ds8_aux: fixes pi :: "('a \<times> 'a) list"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "at TYPE('a)" shows "pi \<bullet> swap (a, b) c = swap (pi \<bullet> a, pi \<bullet> b) (pi \<bullet> c)"
d_def: shows "d \<equiv> sgn (lead_coeff f) * content f"
left_commute: fixes b :: "'e multiset"   and a :: "'e multiset"   and c :: "'e multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "x \<le> xa" obtains x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some ya__" and "x \<le> ya__"   | y :: "'a option" where "xa__ = None" and "xa = y"
eq_solution: fixes s :: "'a"   and U :: "'a \<Rightarrow> real set"   and t :: "real"   and X :: "real \<Rightarrow> 'a" assumes "s \<in> S"   and "is_interval (U s)"   and "U s \<subseteq> T"   and "t \<in> U s"   and "X \<in> Sols (\<lambda>t. f) U S 0 s" shows "X t = \<phi> t s"
ifp_def: fixes a :: "partition_id_t"   and b :: "partition_id_t" shows "Policy.ifp a b \<equiv> Policy.sp_spec_subj_subj a b \<or> Policy.sp_spec_subj_subj b a \<or> (\<exists>c y. Policy.sp_spec_subj_subj a c \<and> Policy.sp_spec_subj_obj c (PAGE y) WRITE \<and> Policy.sp_spec_subj_obj b (PAGE y) READ) \<or> a = b"
aff_dim_parallel_subspace: fixes V :: "'a set"   and L :: "'a set" assumes "V \<noteq> {}"   and "subspace L"   and "affine_parallel (affine hull V) L" shows "aff_dim V = int (dim L)"
eventually_at_top_mono: fixes P :: "'a \<Rightarrow> bool"   and z :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "eventually P at_top"   and "\<And>x. \<lbrakk>z \<le> x; \<forall>y\<ge>x. P y\<rbrakk> \<Longrightarrow> Q x" shows "eventually Q at_top"
padic_int_def: fixes p :: "int" shows "padic_int p \<equiv> \<lparr>carrier = padic_set p, monoid.mult = padic_mult p, one = padic_one p, zero = padic_zero p, add = padic_add p\<rparr>"
AssertBound_def: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" shows "AssertBound assert_bound \<equiv> ((\<forall>s a s'. \<not> \<U> s \<longrightarrow> \<Turnstile> s \<longrightarrow> \<triangle> (\<T> s) \<longrightarrow> \<nabla> s \<longrightarrow> assert_bound a s = s' \<longrightarrow> \<T> s' = \<T> s \<and> \<nabla> s') \<and> (\<forall>s a s'. \<not> \<U> s \<longrightarrow> \<Turnstile> s \<longrightarrow> \<triangle> (\<T> s) \<longrightarrow> \<nabla> s \<longrightarrow> assert_bound a s = s' \<longrightarrow> \<not> \<U> s' \<longrightarrow> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s' \<and> \<diamond> s') \<and> (\<forall>s ats a. \<not> \<U> s \<longrightarrow> \<Turnstile> s \<longrightarrow> \<triangle> (\<T> s) \<longrightarrow> \<nabla> s \<longrightarrow> (snd ` ats) \<doteq> \<B> s \<longrightarrow> (snd ` (ats \<union> {a})) \<doteq> \<B> (assert_bound a s))) \<and> (\<forall>s ats a. \<not> \<U> s \<longrightarrow> \<Turnstile> s \<longrightarrow> \<triangle> (\<T> s) \<longrightarrow> \<nabla> s \<longrightarrow> ats \<Turnstile>\<^sub>i \<B>\<I> s \<longrightarrow> insert a ats \<Turnstile>\<^sub>i \<B>\<I> (assert_bound a s)) \<and> (\<forall>s as a s'. \<not> \<U> s \<longrightarrow> \<Turnstile> s \<longrightarrow> \<triangle> (\<T> s) \<longrightarrow> \<nabla> s \<longrightarrow> index_valid as s \<longrightarrow> assert_bound a s = s' \<longrightarrow> \<U> s' \<longrightarrow> minimal_unsat_state_core s') \<and> (\<forall>s as a. \<not> \<U> s \<longrightarrow> \<Turnstile> s \<longrightarrow> \<triangle> (\<T> s) \<longrightarrow> \<nabla> s \<longrightarrow> index_valid as s \<longrightarrow> index_valid (insert a as) (assert_bound a s))"
t2: shows "{l (- dr)--l dr} \<subseteq> X"
16: shows "forest_components f \<le> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f)"
eqpoll_def: fixes A :: "'b set"   and B :: "'c set" shows "A \<approx> B \<equiv> \<exists>f. bij_betw f A B"
simps_1: fixes f :: "nat \<Rightarrow> nat" shows "inf_concat_simple f 0 = (0, 0)"
simps_5: fixes n :: "nat"   and p :: "pf"   and q :: "pf" shows "dfa_of_pf' n (Imp p q) = imp_dfa (dfa_of_pf' n p) (dfa_of_pf' n q)"
ex_cong1: fixes P :: "'c \<Rightarrow> bool"   and P' :: "'c \<Rightarrow> bool" assumes "\<And>x. P x = P' x" shows "(\<exists>x. P x) = (\<exists>x. P' x)"
vimage_Inr_Plus: fixes A :: "'f set"   and B :: "'e set" shows "Inr -` (A <+> B) = B"
case_eq_if: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and prod :: "'g \<times> 'h" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
length_zip: fixes xs :: "'c list"   and ys :: "'d list" shows "length (xs || ys) = min (length xs) (length ys)"
eq: shows "map_component (\<lambda>k. k - n + n) (lt p) = lt p"
FrameImp_def: fixes F :: "'b frame"   and \<Phi> :: "'c" shows "F \<turnstile>\<^sub>F \<Phi> = (\<exists>A\<^sub>F \<Psi>\<^sub>F. F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> \<and> A\<^sub>F \<sharp>* \<Phi> \<and> \<Psi>\<^sub>F \<turnstile> \<Phi>)"
comp_assoc: fixes f :: "'e \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'e"   and h :: "'a \<Rightarrow> 'd" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
Re: shows "\<forall>n. 0 \<le> Re (trace (f n)) \<and> Im (trace (f n)) = 0"
union_inclusion: fixes A :: "'a set"   and A' :: "'a set"   and B :: "'a set"   and B' :: "'a set" assumes "A \<subseteq> A'"   and "B \<subseteq> B'" shows "A \<union> B \<subseteq> A' \<union> B'"
routing_no_wildcards: fixes rtbl :: "'i routing_rule list" assumes "rtblo = Some rtbl" shows "ipassmt_sanity_nowildcards (map_of (routing_ipassmt rtbl))"
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "mpt M T"   and "finite_measure M" shows "fmpt M T"
assms_3: fixes x :: "'a" assumes "P x" shows "f x \<in> B"
len_lexext_compat_list: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and xs :: "'b list"   and xs' :: "'b list" assumes "gt y x" shows "len_lexext gt (xs @ y # xs') (xs @ x # xs')"
pr_id2_1': shows "pr_conv_2_to_1 (\<lambda>x y. x) \<in> PrimRec1'"
assms_1: shows "t - s \<in> existence_ivl 0 x"
wfp_on_ord_term_strict: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfp_on (\<prec>\<^sub>t) (pp_of_term -` dgrad_set d m)"
WT_converter_of_callee: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and callee :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h, 'k, 'l) gpv"   and s :: "'h" assumes "\<And>s q. q \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s q \<surd>"   and "\<And>s q r s'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, s') \<in> results_gpv \<I>' (callee s q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q" shows "\<I>, \<I>' \<turnstile>\<^sub>C CNV callee s \<surd>"
UNIV_I: fixes x :: "'g" shows "x \<in> UNIV"
makeCF_def: shows "makeCF \<equiv> \<lparr>fileSize = 0, nextFreeBlock = 0, data = makeCArray numBlocks (makeCArray blockSize fillByte)\<rparr>"
length_drop: fixes n :: "nat"   and xs :: "'a list" shows "length (drop n xs) = length xs - n"
simps_2: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" shows "PrimRecOp_last g h x (Suc y) = h x (PrimRecOp_last g h x y) y"
continuous_constant_on_closure: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and x :: "'c" assumes "continuous_on (closure S) f"   and "\<And>x. x \<in> S \<Longrightarrow> f x = a"   and "x \<in> closure S" shows "f x = a"
is_natural_2: fixes f :: "'a" assumes "A.arr f" shows "\<sigma> (A.cod f) \<cdot>\<^sub>B F f = \<sigma> f"
ncoplanar_perm_4: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Coplanar A D B C"
at_def: shows "at TYPE('a) \<equiv> (\<forall>x. [] \<bullet> x = x) \<and> (\<forall>a b pi x. ((a, b) # pi) \<bullet> x = swap (a, b) (pi \<bullet> x)) \<and> (\<forall>a b c. swap (a, b) c = (if a = c then b else if b = c then a else c)) \<and> infinite UNIV"
factorization: shows "is_alt_factorization_of fctrs (lr_fps_denominator cs)"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs G" shows "edge_rev GM a = a"
Lim_at: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and a :: "'b" shows "f \<midarrow>a\<rightarrow> l = (\<forall>e>0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
normalize_quotI: fixes a :: "'b"   and d :: "'b"   and b :: "'b"   and c :: "'b" assumes "a * d = b * c"   and "b \<noteq> (0::'b)"   and "(c, d) \<in> normalized_fracts" shows "normalize_quot (a, b) = (c, d)"
af_trans_imp_wf: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "almost_full_on P A"   and "transp_on P A" shows "wfp_on (strict P) A"
r: shows "r \<rightarrow>\<^sub>\<beta>\<^sup>* r'"
mult_DirProd': fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "x \<otimes>\<^bsub>G \<times>\<times> H\<^esub> y = (fst x \<otimes>\<^bsub>G\<^esub> fst y, snd x \<otimes>\<^bsub>H\<^esub> snd y)"
elims: fixes x :: "pnat"   and xa :: "pnat"   and y :: "pnat" assumes "pnat_min x xa = y" obtains x :: "nat" and y :: "nat" where "xa__ = pnat xb__" and "xa = pnat y" and "ya__ = pnat (min xb__ y)"   | x :: "pnat" where "xa__ = x" and "xa = pnat.PInfty" and "ya__ = x"   | v :: "nat" where "xa__ = pnat.PInfty" and "xa = pnat v" and "ya__ = pnat v"
increasing_tendsto: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in F. less_eq (f n) l"   and "\<And>x. less x l \<Longrightarrow> \<forall>\<^sub>F n in F. less x (f n)" shows "filterlim f (topological_space.nhds open l) F"
assms_2: shows "S \<subseteq> 1\<^sub>\<sigma>"
sub_mat_def: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list" shows "sub_mat nr nc m = map (sub_vec nr) (take nc m)"
inputFreshChainDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "xvec \<sharp>* P"   and "xvec \<sharp>* N" shows "xvec \<sharp>* P'"
All2_Eats_E2: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" assumes "atom i \<sharp> (t, u)"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (All2 i (Eats t u) A) H \<turnstile> A(i::=u)"
read_state_reg_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (read_state_reg_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
catch_env_def: fixes m1 :: "('c, 'm) envT"   and m2 :: "('c, 'm) envT" shows "local.catch_env m1 m2 = EnvT (\<lambda>r. catch (run_env m1 r) (run_env m2 r))"
lub_least: fixes A :: "'b llist set"   and z :: "'b llist" assumes "Complete_Partial_Order.chain lprefix A"   and "\<And>x. x \<in> A \<Longrightarrow> lprefix x z" shows "lprefix (lSup A) z"
rep_eq: fixes x :: "'a"   and xa :: "nat" shows "coeff (monom x xa) = (\<lambda>n. if xa = n then x else (0::'a))"
mul_ext_order_pair: shows "order_pair (s_mul_ext (ns_mul_ext NS S) (s_mul_ext NS S)) (ns_mul_ext (ns_mul_ext NS S) (s_mul_ext NS S))"
singletonD: fixes b :: "'g"   and a :: "'g" assumes "b \<in> {a}" shows "b = a"
a_in_principal: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" assumes "Ring R"   and "a \<in> carrier R" shows "a \<in> R \<diamondsuit>\<^sub>p a"
returns_result_def: fixes h :: "'l"   and p :: "('l, 'm, 'n) prog"   and r :: "'n" shows "h \<turnstile> p \<rightarrow>\<^sub>r r = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr (r', x) \<Rightarrow> r = r')"
finite_image_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
cs: shows "closed_segment X (X + H) \<subseteq> S"
sigma_finite_incseq: fixes M :: "'b measure" assumes "sigma_finite_measure M" obtains A :: "nat \<Rightarrow> 'b set" where "range A \<subseteq> sets M" and "\<Union> (range A) = space M" and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and "incseq A"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "K \<in> symKeys"   and "Key K \<notin> used evs"   and "evs \<in> set_cr" shows "K \<notin> keysFor (parts (knows Spy evs))"
stationary_distribution_exists: fixes A :: "'b st_mat" shows "\<exists>v. A *st v = v"
guaranteesD: fixes F :: "'b program"   and X :: "'b program set"   and Y :: "'b program set"   and G :: "'b program" assumes "F \<in> X guarantees Y"   and "F ok G"   and "F \<squnion> G \<in> X" shows "F \<squnion> G \<in> Y"
distinct_1: fixes x2 :: "'a" shows "None \<noteq> Some x2"
trivial_measure_def: fixes X :: "'a set" shows "trivial_measure X = sigma X {{}, X}"
pseudo_proj_False_preimage_stake: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_False n w" shows "pseudo_proj_False n -` {w} = {x. stake n x = stake n w}"
le_list_Cons2_iff: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "(x # xs \<le> y # ys) = (if x = y then xs \<le> ys else x # xs \<le> ys)"
attach_shadow_root_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   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 set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and attach_shadow_root :: "(_) element_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr) prog"   and type_wf :: "(_) 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_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 element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode" assumes "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root type_wf get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs" shows "attach_shadow_root element_ptr shadow_root_mode = Heap_Error_Monad.bind (get_tag_name element_ptr) (\<lambda>tag. Heap_Error_Monad.bind (if tag \<notin> safe_shadow_root_element_types then error NotSupportedError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>prev_shadow_root. Heap_Error_Monad.bind (if prev_shadow_root \<noteq> None then error NotSupportedError else return ()) (\<lambda>_. Heap_Error_Monad.bind new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M (\<lambda>new_shadow_root_ptr. Heap_Error_Monad.bind (set_mode new_shadow_root_ptr shadow_root_mode) (\<lambda>_. Heap_Error_Monad.bind (set_shadow_root element_ptr (Some new_shadow_root_ptr)) (\<lambda>_. return new_shadow_root_ptr)))))))"
sublist_def: fixes xs :: "'c list"   and i :: "nat"   and j :: "nat" shows "WB_Sort.sublist xs i j \<equiv> take (Suc j - i) (drop i xs)"
C_1: fixes n :: "nat" shows "B (Suc n) \<le> C * Max {B i |i. i \<le> n}"
ab_sum: shows "summable (\<lambda>j. real_of_int (b j) / real_of_int (a j))"
surjective: fixes r :: "('d, 'e, 'f, 'g) interp_scheme" shows "r = \<lparr>Functions = Functions r, Predicates = Predicates r, Contexts = Contexts r, Programs = Programs r, ODEs = ODEs r, ODEBV = ODEBV r, \<dots> = more r\<rparr>"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "S.cod (g \<cdot>\<^sub>S f) = S.cod g"