hcju/isabelleps huggingface dataset:
reachable_AB: fixes x :: "'v"   and y :: "'v" assumes "reachable x"   and "(x, y) \<in> \<^bold>E" shows "reachable y"
Gamma_fact: fixes n :: "nat" shows "Gamma ((1::'b) + of_nat n) = fact n"
ccw'_subst_psi_disj: fixes t :: "real \<times> real"   and r :: "real \<times> real"   and s :: "real \<times> real"   and p :: "real \<times> real" assumes "coll t r s"   and "psi t r s \<or> psi t s r \<or> psi s r t"   and "s \<noteq> t"   and "ccw' t r p" shows "ccw' t s p"
return_def: fixes a :: "'f" shows "return a = State (Pair a)"
insert_not_empty: fixes a :: "'a"   and A :: "'a set" shows "set_ins a A \<noteq> {}"
convergence_U: fixes x :: "'s"   and y :: "'s"   and z :: "real" assumes "convergence_G x y z" shows "summable (\<lambda>n. u x y n * z ^ Suc n)"
det_1: shows "v * det (four_block_mat A1 A2 A3 A4) = det (four_block_mat A1 A2 B3 B4)"
cases: fixes a1 :: "'a list"   and a2 :: "'a list" assumes "initial_segment a1 a2" obtains xs :: "'a list" and ys :: "'a list" where "a1 = xs" and "a2 = xs @ ys"
elims_2: fixes x :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and xa :: "'c \<Rightarrow> 'b" assumes "wf_disc_sel x xa" obtains disc :: "'b \<Rightarrow> bool" and sel :: "'b \<Rightarrow> 'c" and C :: "'c \<Rightarrow> 'b" where "x = (disc, sel)" and "xa = C" and "(\<forall>a. disc a \<longrightarrow> C (sel a) = a) \<and> (\<forall>a. sel (C a) = a)"
nres_inequalities_1: fixes X :: "'c set" shows "FAIL \<noteq> RES X"
rev_min_pm: fixes b :: "nat"   and a :: "nat" shows "min b a + (a - b) = a"
orient2: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> set (filter (\<lambda>(x, y). weak_gt x y) (concat (map (\<lambda>x. map (Pair x) (concat (map (\<lambda>B. concat (map (\<lambda>i. map (\<lambda>j. B $$ (i, j)) [0..<n]) [0..<n])) (map snd ABs)))) (concat (map (\<lambda>A. concat (map (\<lambda>i. map (\<lambda>j. A $$ (i, j)) [0..<n]) [0..<n])) (map fst ABs))))))" shows "gt x y"
distinct_5: fixes x2 :: "t list" shows "Type x2 \<noteq> Bot"
right_Crypt: fixes K :: "nat"   and X :: "msg" shows "right (Crypt K X) = right X"
resOutputRight: fixes x :: "name"   and a :: "name"   and b :: "name"   and Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> b"   and "Id \<subseteq> Rel"   and "eqvt Rel" shows "a{b}.(<\<nu>x>P) \<leadsto>[Rel] <\<nu>x>a{b}.P"
w_def: shows "w = perron_frobenius.z (Finite_Cartesian_Product.transpose A)"
schroeder_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<sqinter> z = bot) = (z * y\<^sup>T \<sqinter> x = bot)"
least_Upper_cong_r: fixes A :: "'a set"   and A' :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=} A'" shows "is_lub L x A = is_lub L x A'"
cond_spmf_UNIV: fixes p :: "'a spmf" shows "cond_spmf p UNIV = mk_lossless p"
snd_iso_thme: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "subgroup N G"   and "subgroup H (G\<lparr>carrier := normalizer G N\<rparr>)" shows "G\<lparr>carrier := N <#> H\<rparr> Mod N \<cong> G\<lparr>carrier := H\<rparr> Mod H \<inter> N"
max_absorb1: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x" shows "max x y = x"
square_free_separable_GFp: fixes f :: "'a mod_ring poly" assumes "degree f < CARD('a)"   and "square_free f" shows "separable f"
wr: fixes s :: "'a"   and s' :: "'a"   and a :: "'b" assumes "P s"   and "P s'"   and "Q (s, a, s')"   and "I a" shows "R (s, a, s')"
no_loop_in_tree: fixes v :: "'v" shows "(v, v) \<notin> (tree_edges s)\<^sup>+"
le_mask_iff: fixes w :: "'b word"   and n :: "nat" shows "(w \<le> mask n) = (w >> n = 0)"
mcatch_def: shows "mcatch \<equiv> \<Lambda> body handler. case body of Nothing \<Rightarrow> handler | Just\<cdot>x \<Rightarrow> Just\<cdot>x"
norm_nat_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and y :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < n" shows "norm (nat_power n y) = real n powr inner y one"
assms: fixes x :: "'a" shows "dist (apply_bcontfun f x) (apply_bcontfun g x) \<le> b"
assoc: fixes a :: "'g"   and b :: "'g"   and c :: "'g" shows "a + b + c = a + (b + c)"
flush_flush_all_until_volatile_write_Write\<^sub>s\<^sub>b_volatile_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and m :: "nat \<Rightarrow> nat"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e" assumes "i < length ts"   and "ts ! i = (p, is, xs, Write\<^sub>s\<^sub>b True a sop v A L R W # sb, \<D>, \<O>, \<R>)"   and "\<forall>i<length ts. \<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>i, uu_, uu_, uu_) = ts ! i; (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in outstanding_refs is_Write\<^sub>s\<^sub>b sb\<^sub>i \<inter> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) = {})"   and "\<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))" shows "flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) ((flush_all_until_volatile_write ts m)(a := v)) = flush_all_until_volatile_write (ts[i := (p, is, xs, sb, \<D>', \<O>', \<R>')]) (m(a := v))"
at_bij1: fixes pi :: "('a \<times> 'a) list"   and x :: "'a"   and y :: "'a" assumes "at TYPE('a)"   and "pi \<bullet> x = y" shows "x = rev pi \<bullet> y"
monoid_add_isom_axioms: shows "monoid_add_isom hom"
valid_elim: fixes x :: "name"   and T :: "ty"   and \<Gamma> :: "(name \<times> ty) list" assumes "valid ((x, T) # \<Gamma>)" obtains "valid \<Gamma>" and "x \<sharp> \<Gamma>"
simps_1: fixes s :: "'a" shows "tFirst (tInit s) = s"
Proc_CFG_Call_nodes_eq: fixes prog :: "cmd"   and n :: "label"   and p :: "char list"   and es :: "expr list"   and rets :: "char list list"   and n' :: "label"   and p' :: "char list"   and es' :: "expr list"   and rets' :: "char list list"   and n'' :: "label" assumes "prog \<turnstile> n -CEdge (p, es, rets)\<rightarrow>\<^sub>p n'"   and "prog \<turnstile> n -CEdge (p', es', rets')\<rightarrow>\<^sub>p n''" shows "n' = n'' \<and> p = p' \<and> es = es' \<and> rets = rets'"
upper_triangularI: fixes A :: "'b mat" assumes "\<And>i j. \<lbrakk>j < i; i < dim_row A\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'b)" shows "upper_triangular A"
Inl_in_Plus: fixes x :: "'a"   and A :: "'a set"   and B :: "'b set" shows "(Inl x \<in> A <+> B) = (x \<in> A)"
parse_name_def: fixes s :: "char list" shows "parse_name s = (many_letters \<bind> (\<lambda>n. spaces \<bind> (\<lambda>_. if n = [] then Parser_Monad.error (''expected letter '' @ letters @ [CHR '' '', CHR ''b'', CHR ''u'', CHR ''t'', CHR '' '', CHR ''f'', CHR ''i'', CHR ''r'', CHR ''s'', CHR ''t'', CHR '' '', CHR ''s'', CHR ''y'', CHR ''m'', CHR ''b'', CHR ''o'', CHR ''l'', CHR '' '', CHR ''i'', CHR ''s'', CHR '' '', CHR 0x22] @ take 1 s @ [CHR 0x22]) else Parser_Monad.return n))) s"
intro: fixes alt :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and fail :: "'d" assumes "\<And>m. alt fail m = m"   and "\<And>m. alt m fail = m" shows "monad_fail_alt_axioms fail alt"
\<section>: fixes u :: "real" assumes "0 < u"   and "u \<le> 1" shows "g2 u \<in> (\<lambda>x. g2 (2 * x - 1)) ` ({0..1} \<inter> {x. \<not> x * 2 \<le> 1})"
LeastI2_wellorder: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
gState_progress_rel_channels_update: fixes prog :: "program"   and g :: "'a gState_scheme"   and c :: "channel"   and i :: "nat" assumes "gState_inv prog g"   and "channel_inv c"   and "i < length (gState.channels g)" shows "(g, gState.channels_update (\<lambda>cs. cs[i := c]) g) \<in> gState_progress_rel prog"
Iagree_comm: fixes A :: "('d, 'e, 'f) interp"   and B :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set" assumes "Iagree A B V" shows "Iagree B A V"
inorder_rotateL: fixes r :: "('b \<times> nat) tree"   and l :: "('b \<times> nat) tree"   and a :: "'b" assumes "r \<noteq> \<langle>\<rangle>" shows "Tree2.inorder (rotateL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
shifted_cs_eq_is_eq: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and k :: "nat"   and k' :: "nat"   and n :: "nat"   and n' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> n = cs\<^bsup>\<pi>' \<guillemotleft> k'\<^esup> n'" shows "cs\<^bsup>\<pi>\<^esup> (k + n) = cs\<^bsup>\<pi>'\<^esup> (k' + n')"
ipp_conditions: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "assoc_bpp (aug_pm_set z i {\<^sub>i- a}) f \<and> ipp_cond1 {\<^sub>i- a} i \<and> ipp_cond2 z {\<^sub>i- a} i f \<and> inv_ipp z i f {\<^sub>i- a} \<and> commute_bpp f (aug_pm_set z i {\<^sub>i- a}) \<and> zeroA z i f {\<^sub>i- a} z"
conjI: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "sb.eval cid t t' n \<gamma> \<and> sb.eval cid t t' n \<gamma>'" shows "sb.eval cid t t' n (\<gamma> \<and>\<^sup>b \<gamma>')"
Inf_lower: fixes Inf :: "'a::type set \<Rightarrow> 'a::type"   and Sup :: "'a::type set \<Rightarrow> 'a::type"   and inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and bot :: "'a::type"   and top :: "'a::type"   and x :: "'a::type"   and A :: "'a::type set" assumes "class.complete_lattice (Inf::'a::type set \<Rightarrow> 'a::type) (Sup::'a::type set \<Rightarrow> 'a::type) (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (bot::'a::type) (top::'a::type)"   and "(x::'a::type) \<in> (A::'a::type set)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) ((Inf::'a::type set \<Rightarrow> 'a::type) (A::'a::type set)) (x::'a::type)"
Sup_subset_mono: fixes A :: "('d \<times> 'e) set"   and B :: "('d \<times> 'e) set" assumes "A \<subseteq> B" shows "less_eq_prod (Sup_prod A) (Sup_prod B)"
facts_4: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "probGn p n (\<lambda>es. \<not> H \<sqsubseteq> edge_space.edge_ugraph n es) \<le> 1 / prob_space_with_indicators.\<mu> (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))}) + prob_space_with_indicators.\<Delta>\<^sub>d (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))}) / (prob_space_with_indicators.\<mu> (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))}))\<^sup>2"
nat_of_integer_def: shows "nat_of_integer \<equiv> map_fun int_of_integer id nat"
valuate_minus: fixes p1 :: "linear_poly"   and p2 :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "p1 - p2 \<lbrace> v \<rbrace> = p1 \<lbrace> v \<rbrace> - p2 \<lbrace> v \<rbrace>"
powsum_mpoly_infinite: fixes A :: "nat set"   and k :: "nat" assumes "infinite A" shows "powsum_mpoly A k = 0"
sub_list_id: fixes ps :: "('c, 'd) fm list" shows "sub_list id ps = ps"
ty_binding_existence: fixes X :: "tyvrs"   and a :: "binding" assumes "X \<in> tyvrs_of a" shows "\<exists>T. TVarB X T = a"
cp_defined: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" shows "(\<delta> X) \<tau> = (\<delta> (\<lambda>_. X \<tau>)) \<tau>"
B: shows "\<forall>b\<in>set B. \<exists>a\<in>set A. \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> P \<delta>"
ide_composite: shows "ide (\<nu> \<star> \<pi>)"
assms_1: shows "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B"
of_hypreal_def: shows "of_hypreal = *f* of_real"
trans_Id_on: fixes A :: "'b set" shows "trans (Id_on A)"
invertibleE: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c"   and thesis :: "bool" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "\<And>v. \<lbrakk>composition u v = unit \<and> composition v u = unit; v \<in> M\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> M" shows "thesis"
bij_to_nat: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "bij_betw (mod_type.to_nat Rep) UNIV {0..<CARD('b)}"
norm_ge_zero: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "0 \<le> norm x"
order_one_closed: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "(less_eq one one \<and> less_eq (times one one) one) \<and> less_eq (inf one (conv one)) one"
inj: fixes n :: "nat" shows "inj_on (\<F> n) S"
Bossy'_CD_on_Cd: fixes X :: "'a set" shows "Bossy'.CD_on ds X = Bossy'.Cd (Xd d1h1) X \<union> Bossy'.Cd (Xd d3h1) X"
I_1: shows "NominalSequents.mset C = \<LM> F \<nabla> [x].A  \<RM>"
region_unique_spec: fixes u :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set" assumes "u \<in> R"   and "R \<in> \<R>" shows "[u]\<^sub>\<R> = R"
F_def: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<sqinter>) ((c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice))"
diff_col_ex: fixes A :: "'p"   and B :: "'p" shows "\<exists>C. A \<noteq> C \<and> B \<noteq> C \<and> Col A B C"
relabel_nonsat_potential: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "Network.relabel_precond c t f l u" shows "Network.nonsat_potential c f (Network.relabel_effect c f l u) \<le> Network.nonsat_potential c f l + 2 * card (Graph.V c)"
algebraic_power: fixes x :: "'b"   and n :: "nat" assumes "algebraic x" shows "algebraic (x ^ n)"
reachArrLen: fixes s :: "Store"   and T :: "Arraytype"   and a :: "nat" shows "s\<turnstile> arrLenLoc T a reachable_from arrV T a"
exhaust_sel: fixes tree :: "'b tree" obtains "tree = Node (rank tree) (tree.root tree) (children tree)"
rk_increment_def: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat"   and A :: "nat \<Rightarrow> nat \<Rightarrow> real"   and b :: "nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real"   and h :: "real"   and t :: "real"   and x :: "'b" shows "rk_increment f s A b c h t x = (\<Sum>j = 1..s. b j *\<^sub>R rk_eval A c f t h x j)"
Rats_dense_in_real: fixes x :: "real"   and y :: "real" assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
nat_sub1: fixes b :: "nat"   and a :: "nat"   and d :: "nat"   and c :: "nat" assumes "b \<le> a"   and "d \<le> c" shows "a - b + (c - d) = a + c - (b + d) \<and> b + d \<le> a + c"
arr_runit: fixes a :: "'c" assumes "C.ide a" shows "C.arr \<r>\<^sub>C[a]"
step: shows "S t k \<turnstile> t ! k \<mapsto> S t (k + 1)"
cgenideal_pirreducible: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "pirreducible\<^bsub>K [X]\<^esub> K p"   and "pirreducible\<^bsub>K [X]\<^esub> K q"   and "q \<in> PIdl\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> p" shows "p \<sim>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q"
vend_fail: shows "value_gt (Some (Num 100)) (r $ 2) = trilean.true \<longrightarrow> P (Some 1, [], r)"
preorder_on_converse: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "preorder_on A (r\<inverse>) = preorder_on A r"
prv_PPf_decide: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "\<not> bprv (PPf n \<langle>\<phi>\<rangle>)" shows "bprv (B.neg (PPf n \<langle>\<phi>\<rangle>))"
singular_chain_imp_relboundary: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p (subtopology X S) c" shows "singular_relboundary p X S c"
skl1_step3_refines_a0i_running_skip_r: fixes A :: "agent"   and B :: "agent"   and Ra :: "fid_t"   and Nr :: "msg"   and gny :: "msg" shows "{R0sk1iar} (a0i_running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>)\<^sup>=, skl1_step3 Ra A B Nr gny {> R0sk1iar}"
prv_imp_neg_rev: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (imp \<phi> \<chi>)" shows "bprv (imp (neg \<chi>) (neg \<phi>))"
foldl_f_commute: fixes a :: "'a"   and b :: "'b"   and xs :: "'a list" shows "f a (foldl (\<lambda>a b. f b a) b xs) = foldl (\<lambda>a b. f b a) (f a b) xs"
disjoint_bind: fixes S :: "'a \<Rightarrow> bool"   and T :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" assumes "\<And>s t. S s \<and> T t \<Longrightarrow> f s \<inter> g t = {}" shows "({s. S s} \<bind> f) \<inter> ({t. T t} \<bind> g) = {}"
alternating_list_Suc_Cons: fixes k :: "nat"   and s :: "'b"   and t :: "'b" shows "alternating_list (Suc k) s t = s # alternating_list k t s"
scaleC_sum_left: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set"   and x :: "'b" shows "sum f A *\<^sub>C x = (\<Sum>a\<in>A. f a *\<^sub>C x)"
intros_2: fixes M :: "'a measure"   and N :: "'a measure" assumes "space M = space N"   and "sets M \<subset> sets N" shows "M \<le> N"
map_of_heap_def: fixes heap :: "heap"   and k :: "'a" shows "map_of_heap heap k = fst (the (execute (lookup k) heap))"
i_expand_i_take_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" shows "f \<odot> k \<Down> (n * k) = f \<Down> n \<odot> k"
IVT': fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and y :: "'a"   and b :: "'b" assumes "f a \<le> y"   and "y \<le> f b"   and "a \<le> b"   and "continuous_on {a..b} f" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
tag_Star_def: fixes A :: "'a list set" shows "tag_Star A \<equiv> \<lambda>x. {uu_. \<exists>u v. uu_ = \<approx>A `` {v} \<and> strict_prefix u x \<and> u \<in> A\<star> \<and> (u, v) \<in> Partitions x}"
affine_hull_Diff: fixes S :: "'b set"   and F :: "'b set" assumes "openin (top_of_set (affine hull S)) S"   and "finite F"   and "F \<subset> S" shows "affine hull (S - F) = affine hull S"
z_firstocc: shows "\<forall>j<k. cs ! j \<noteq> cs ! k"
add_seq_inf_par: fixes a :: "'c"   and b :: "'c" shows "a ; b + b ; a \<le>\<^sub>\<K> a * b"
permutes_subset: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and T :: "'c set" assumes "p permutes S"   and "S \<subseteq> T" shows "p permutes T"
iMODb_div_mod_gr0_is_0_not_ex0: fixes k :: "nat"   and m :: "nat"   and c :: "nat"   and r :: "nat" assumes "0 < k"   and "k < m"   and "0 < m mod k"   and "k \<le> c"   and "r mod k = 0" shows "\<nexists>r' m' c'. [ r', mod m', c' ] = [ r, mod m, c ] \<oslash> k"
le_Integer_numeral_simps_4: shows "le\<cdot>1\<cdot>1 = TT"
Some_eq_map_of_iff: fixes xys :: "('b \<times> 'c) list"   and y :: "'c"   and x :: "'b" assumes "distinct (map fst xys)" shows "(Some y = map_of xys x) = ((x, y) \<in> set xys)"
indicator_simps_1: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "indicator S x = (1::'b)"
intros_7: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F"   and "\<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> F \<^bold>\<and> G"
sum_pow2_image_Suc: fixes X :: "nat set" assumes "finite X" shows "sum ((^) 2) (Suc ` X) = 2 * sum ((^) 2) X"
WT1: shows "\<I> \<turnstile>g gpv \<surd>"
ground_less_less_kbo: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "ground s"   and "ground t"   and "s < t" shows "less_kbo s t"
nonempty_nsphere: fixes n :: "nat" shows "topspace (nsphere n) \<noteq> {}"
vifunion_vinsert_is: fixes j :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> (VLambda (vinsert j I) f `\<^sub>\<circ> vinsert j I) = f j \<union>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
pred_eqI: fixes P :: "'a pred"   and Q :: "'a pred" assumes "\<And>w. eval P w = eval Q w" shows "P = Q"
relabel_pres_height_bound: fixes u :: "nat" assumes "relabel_precond f (relabel_effect f l u) u" shows "Height_Bounded_Labeling c s t f (relabel_effect f (relabel_effect f l u) u)"