hcju/isabelleps huggingface dataset:
subst_dbtm_graph_aux_def: shows "subst_dbtm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_dbtm_aux u i. x1 = (u, i, DBZero) \<and> x2 = DBZero) \<or> (\<exists>subst_dbtm_aux u i name. x1 = (u, i, DBVar name) \<and> x2 = (if i = name then u else DBVar name)) \<or> (\<exists>subst_dbtm_aux u i j. x1 = (u, i, DBInd j) \<and> x2 = DBInd j) \<or> (\<exists>subst_dbtm_aux u i t1 t2. x1 = (u, i, DBEats t1 t2) \<and> x2 = DBEats (((\<lambda>uu_. uu_) \<circ> subst_dbtm_aux) (u, i, t1)) (((\<lambda>uu_. uu_) \<circ> subst_dbtm_aux) (u, i, t2)) \<and> p (u, i, t1) (((\<lambda>uu_. uu_) \<circ> subst_dbtm_aux) (u, i, t1)) \<and> p (u, i, t2) (((\<lambda>uu_. uu_) \<circ> subst_dbtm_aux) (u, i, t2))))"
cod_simps_2: shows "HH\<mu>\<nu>\<pi>.cod.leg0 = HH\<mu>\<nu>\<pi>.leg0"
all: shows "\<forall>as''. n -as''\<rightarrow>* (_Exit_) \<longrightarrow> n' \<in> set (sourcenodes as'')"
f_OK: fixes xc :: "'xc" assumes "invar xc"   and "\<alpha> xc \<in> S0" shows "f xc = f' (\<alpha> xc)"
element_ptr_casts_commute2: fixes element_ptr :: "(_) element_ptr" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr) = Some element_ptr"
sup_inf_distrib2_lgroup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'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 b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup" shows "sup (inf b c) a = inf (sup b a) (sup c a)"
Un_Diff_Diff: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> B - (A - B) = B"
lessThan_k: fixes k :: "nat" assumes "0 < k" shows "{..<k} = insert 0 {0<..<k}"
cSUP_upper: fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># Sup (f ` A)"
Abs_errorable_inject: fixes x :: "'b option"   and y :: "'b option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_errorable x = Abs_errorable y) = (x = y)"
lowdim_subset_hyperplane: fixes S :: "'c set" assumes "dim S < DIM('c)" shows "\<exists>a. a \<noteq> (0::'c) \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
ilist_Previous_Cons: fixes f :: "nat \<Rightarrow> 'd"   and init :: "'d"   and n :: "nat" shows "f\<^bsup>\<leftarrow> init\<^esup> n = ([init] \<frown> f) n"
split_block_def: fixes A :: "'a mat"   and sr :: "nat"   and sc :: "nat" shows "split_block A sr sc = (let nr = dim_row A; nc = dim_col A; nr2 = nr - sr; nc2 = nc - sc; A1 = Matrix.mat sr sc (($$) A); A2 = Matrix.mat sr nc2 (\<lambda>(i, j). A $$ (i, j + sc)); A3 = Matrix.mat nr2 sc (\<lambda>(i, j). A $$ (i + sr, j)); A4 = Matrix.mat nr2 nc2 (\<lambda>(i, j). A $$ (i + sr, j + sc)) in (A1, A2, A3, A4))"
GSMP_hom: fixes t :: "('fun, 'var) Term.term" assumes "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"
bounded_const: fixes c :: "real" shows "Expectations.bounded (\<lambda>x. c)"
Keys_to_list_eq_pps_to_list: fixes ps :: "('a \<Rightarrow>\<^sub>0 'd) list" shows "punit.Keys_to_list ps = punit.pps_to_list (Keys (set ps))"
subt: fixes s :: "('a, 'b) Term.term"   and ss :: "('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term"   and f :: "'a" assumes "s \<in> set ss"   and "s \<rhd> t" shows "Fun f ss \<rhd> t"
church_rosser_equiv: fixes y :: "'a"   and x :: "'a" shows "(y\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>) = ((x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>)"
ereal_infty_less_eq_1: fixes x :: "ereal" shows "(\<infinity> \<le> x) = (x = \<infinity>)"
max_norm_def: fixes A :: "((real, 'd) vec, 'c) vec" shows "\<parallel>A\<parallel>\<^sub>m\<^sub>a\<^sub>x = Max {\<bar>A $ i $ j\<bar> |i j. i \<in> UNIV \<and> j \<in> UNIV}"
is_open_from_ind_is_open: fixes U :: "'a set" assumes "is_open S"   and "ind_is_open U" shows "is_open U"
eqvt_at_apply'': fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and p :: "perm" assumes "eqvt_at f x" shows "(p \<bullet> f) (p \<bullet> x) = f (p \<bullet> x)"
has_integral_integral_lborel: fixes f :: "'a \<Rightarrow> 'b" assumes "integrable lborel f" shows "(f has_integral integral\<^sup>L lborel f) UNIV"
A: shows "A \<in> carrier_mat (k + n) (k + n)"
functions_of_def: fixes P :: "'c set set"   and A :: "'c set"   and B :: "'d set" shows "functions_of P A B = {f \<in> A \<rightarrow>\<^sub>E B. (\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = P}"
cont_apply: fixes t :: "'g \<Rightarrow> 'h"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" assumes "cont t"   and "\<And>x. cont (f x)"   and "\<And>y. cont (\<lambda>x. f x y)" shows "cont (\<lambda>x. f x (t x))"
evals\<^sub>1_preserves_len: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and es\<^sub>0 :: "nat exp list"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>0 :: "val list"   and es\<^sub>1 :: "nat exp list"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>1 :: "val list" assumes "P \<turnstile>\<^sub>1 \<langle>es\<^sub>0,(h\<^sub>0, ls\<^sub>0)\<rangle> [\<Rightarrow>] \<langle>es\<^sub>1,(h\<^sub>1, ls\<^sub>1)\<rangle>" shows "length ls\<^sub>0 = length ls\<^sub>1"
real0: shows "0 < real k * (2 * pi) / real n"
m'_def: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" shows "m' N M y = distr (m N M y) (stream_space (count_space UNIV)) ((##) y)"
simps: fixes G :: "('k, 'l) graph_scheme"   and A :: "'k set"   and B :: "'k set"   and S :: "'k set" shows "separating_gen G A B S = (\<forall>x xa xb. x \<in> A \<longrightarrow> xa \<in> B \<longrightarrow> path G x xb xa \<longrightarrow> (\<exists>z\<in>set xb. z \<in> S) \<or> x \<in> S)"
zero_in_Ker: shows "(0::'g) \<in> Ker"
shadow_root_ptr_casts_commute3: fixes document_ptr :: "(_) document_ptr" assumes "is_shadow_root_ptr_kind\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr" shows "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)) = document_ptr"
simps_3: fixes x :: "brick list"   and y :: "brick list"   and ys :: "wall" shows "basic x \<otimes> y * ys = (if codomain_block x = 0 then (x \<otimes> y) * ys else (x \<otimes> y) * (basic (make_vert_block (nat (codomain_block x))) \<otimes> ys))"
integral_pmf: fixes X :: "'a set"   and M :: "'a pmf" shows "integral\<^sup>L (count_space X) (pmf M) = measure_pmf.prob M X"
qf3_oddprimedivisor_not: fixes P :: "int"   and Q :: "int" assumes "prime P \<and> odd P \<and> 0 < Q \<and> is_qfN (P * Q) 3 \<and> \<not> is_qfN P 3" shows "\<exists>R. prime R \<and> odd R \<and> R dvd Q \<and> \<not> is_qfN R 3"
scalar_matrix_mult_row_code: fixes c :: "'b"   and A :: "(('b, 'a) vec, 'c) vec"   and i :: "'c" shows "($) (scalar_matrix_mult_row c A i) = (\<lambda>j. c * A $ i $ j)"
Rep_phi: fixes x :: "('c, 'd) phi" shows "Rep_phi x \<in> {f. bounded_lattice_homomorphism f}"
Collect_empty_eq: fixes P :: "'b::type \<Rightarrow> bool" shows "(Collect (P::'b::type \<Rightarrow> bool) = {}) = (\<forall>x::'b::type. \<not> P x)"
tendsto_imp_eventually_ne: fixes "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and c :: "'e"   and F :: "'f filter"   and c' :: "'e" assumes "class.t1_space open"   and "filterlim f (topological_space.nhds open c) F"   and "c \<noteq> c'" shows "\<forall>\<^sub>F z in F. f z \<noteq> c'"
refine_map: fixes R1 :: "('c \<times> 'e) set"   and R2 :: "('d \<times> 'f) set" shows "(map, map) \<in> (R1 \<rightarrow> R2) \<rightarrow> \<langle>R1\<rangle>list_rel \<rightarrow> \<langle>R2\<rangle>list_rel"
res1: shows "1 \<le> |restrict v1 (res ts) c|"
thmD2: fixes X :: "symbol"   and p :: "(symbol \<times> character list) list" assumes "is_terminal X"   and "p \<in> \<PP>"   and "terminals p @ [X] \<in> \<L>\<^sub>P" shows "\<exists>x. pvalid p x \<and> next_symbol x = Some X"
splay_bstR: fixes t :: "'a tree"   and a :: "'a"   and l :: "'a tree"   and e :: "'a"   and r :: "'a tree"   and x :: "'a" assumes "bst t"   and "splay a t = \<langle>l, e, r\<rangle>"   and "x \<in> set_tree r" shows "a < x"
strict_trans1: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a \<le>s b"   and "b <s c" shows "a <s c"
simps_2: fixes a :: "('b, 'c) Combinators"   and x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "FWNormalisationCore.insort a (x # xs) l = (if smaller a x l then a # x # xs else x # FWNormalisationCore.insort a xs l)"
subcls_irrefl: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "(C, D) \<in> (subcls1 P)\<^sup>+" shows "C \<noteq> D"
A: shows "(f, g) \<in> [P]\<^sub>a RR' \<rightarrow> S"
\<R>\<^sub>2_sym: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sym (sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>)"
simps_4: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and x1 :: "'c" shows "(case bot.Value x1 of bot.Value x \<Rightarrow> f1 x | bot.Bot \<Rightarrow> f2) = f1 x1"
iffs: fixes nothing :: "unit"   and more :: "'m"   and nothing' :: "unit"   and more' :: "'m" shows "(\<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing', \<dots> = more'\<rparr>) = (nothing = nothing' \<and> more = more')"
poly_minus: fixes p :: "'b poly"   and x :: "'b" shows "poly (- p) x = - poly p x"
size_int_def: shows "index_class.size \<equiv> int_index_def.size"
elims: fixes x :: "'a ltl"   and xa :: "'a set"   and y :: "'a ltl" assumes "af_letter_opt x xa = y" obtains \<phi> :: "'a ltl" and \<nu> :: "'a set" where "x = \<phi>" and "xa = \<nu>" and "y = Unf (step \<phi> \<nu>)"
Cons: fixes vs :: "'a list" shows "distinct (map f (remdups_wrt_rev f xs vs))"
assms_3: shows "g \<equiv> \<lambda>x. 1 / of_nat k * (\<Sum>m<k. f m * unity_root k (- int x * int m))"
symclpI1: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "r x y" shows "symclp r x y"
intro: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and fi\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and fi\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "cancel_comm_semigroup f\<^sub>a fi\<^sub>a"   and "cancel_comm_semigroup f\<^sub>b fi\<^sub>b" shows "cancel_comm_semigroup_pair f\<^sub>a fi\<^sub>a f\<^sub>b fi\<^sub>b"
simps_1: fixes T :: "nat set" shows "data_dependency_consistent_instrs T [] = True"
gtr_empty_conf1: fixes w :: "'b list"   and c' :: "'a multiset"   and S :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" shows "(({#}, w, c') \<in> trcl (gtr S)) = (w = [] \<and> c' = {#})"
doesnt_read_mutually_exclusive': fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and c :: "'a"   and x :: "'b"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'c"   and c' :: "'a"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.doesnt_read eval c x"   and "(((c, mds), mem), (c', mds'), mem') \<in> eval"   and "\<forall>v. (((c, mds), mem(x := v)), (c', mds'), mem') \<in> eval" shows "\<not> (\<forall>v. (((c, mds), mem(x := v)), (c', mds'), mem'(x := v)) \<in> eval)"
less_add_same_cancel1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus a b) = less zero b"
IH: fixes vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "vars_change as vs s = []" shows "fmrestrict_set vs (exec_plan s as) = fmrestrict_set vs s"
intros_1: fixes S :: "'c set"   and B :: "'c set set" shows "generated_topology S B S"
pi_disjoint: shows "path_image c \<inter> outside (path_image c1 \<union> path_image c2) = {}"
Map_in_Hom: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "Map f \<in> Hom (local.Dom f) (local.Cod f)"
lift_spmf_inject: fixes p :: "'d spmf"   and q :: "'d spmf" shows "(lift_spmf p = lift_spmf q) = (p = q)"
is_gcd_def: fixes gcd' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "is_gcd gcd' = (\<forall>a b. gcd' a b dvd a \<and> gcd' a b dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd gcd' a b))"
set_category_given_img_axioms: shows "set_category_given_img (\<cdot>) some_img (cardSuc (cmax |UNIV| natLeq))"
fin1: shows "\<not> is_final (step0 (steps0 c A n) A)"
assms_2: shows "\<S>' = update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)"
set_child_nodes_writes: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr) (set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr children) h h'"
ext: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
simps_2: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and A :: "('a, 'b) stateful_strand_step list"   and I :: "'b \<Rightarrow> ('a, 'b) Term.term"   and D :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t (insert\<langle>t,s\<rangle> # A) I D = db'\<^sub>s\<^sub>s\<^sub>t A I (List.insert ((t, s) \<cdot>\<^sub>p I) D)"
init_segment_def: fixes S :: "nat set"   and T :: "nat set" shows "init_segment S T \<equiv> \<exists>S'. T = S \<union> S' \<and> S \<lless> S'"
trigonometric_set_def: fixes n :: "nat" shows "trigonometric_set n \<equiv> if n = 0 then \<lambda>x. 1 / sqrt (2 * pi) else if odd n then \<lambda>x. sin (real (Suc (n div 2)) * x) / sqrt pi else (\<lambda>x. cos (real n / 2 * x) / sqrt pi)"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P p Q,A"
Domain_mono: fixes r :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "r \<subseteq> s" shows "r2s r \<subseteq> r2s s"
kmp2_def: fixes s :: "'a list"   and t :: "'a list" shows "kmp2 s t \<equiv> ASSERT (s \<noteq> []) \<bind> (\<lambda>_. let i = 0; j = 0; pos = None in compute_butlast_\<ff>s s \<bind> (\<lambda>\<ff>s. WHILE\<^sub>T\<^bsup>I_outer s t\<^esup> (\<lambda>(i, j, pos). i + length s \<le> length t \<and> pos = None) (\<lambda>(i, j, pos). ASSERT (i + length s \<le> length t \<and> pos = None) \<bind> (\<lambda>_. WHILE\<^sub>T\<^bsup>I_in_na s t i\<^esup> (\<lambda>(j, pos). t ! (i + j) = s ! j \<and> pos = None) (\<lambda>(j, pos). let j = j + 1 in if j = length s then RETURN (j, Some i) else RETURN (j, None)) (j, pos) \<bind> (\<lambda>(j, pos). if pos = None then ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let i = i + (j - \<ff>s ! j + 1); j = max 0 (\<ff>s ! j - 1) in RETURN (i, j, None)) else RETURN (i, j, Some i)))) (i, j, pos) \<bind> (\<lambda>(uu_, uu_, pos). RETURN pos)))"
leq_lcs_fun_1: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "s \<le> lcs s t"
CBlinfun_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. bounded_clinear f}" shows "cblinfun_apply (CBlinfun y) = y"
q: shows "extended_to_set q \<subseteq> I ` set s"
fps_X_power_nth: fixes k :: "nat"   and n :: "nat" shows "fps_X ^ k $ n = (if n = k then 1::'a else (0::'a))"
has_derivative_at2: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "(f has_derivative f') (at x) = (bounded_linear f' \<and> (\<lambda>y. (1 / norm (y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) \<midarrow>x\<rightarrow> (0::'b))"
no_spoofing_def: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and rs :: "'b common_primitive rule list" shows "no_spoofing TYPE('a) (ipassmt::iface \<Rightarrow> ('b word \<times> nat) list option) (rs::'b common_primitive rule list) \<equiv> \<forall>iface::iface\<in>dom ipassmt. \<forall>p::('b, 'a) tagged_packet_scheme. (common_matcher, in_doubt_allow),p\<lparr>p_iiface := iface_sel iface\<rparr>\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<longrightarrow> p_src p \<in> ipcidr_union_set (set (the (ipassmt iface)))"
inject: fixes x3 :: "nat \<times> nat"   and y3 :: "nat \<times> nat" shows "(SAT_PUSH' x3 = SAT_PUSH' y3) = (x3 = y3)"
true_cls_replicate_mset: fixes I :: "'a set"   and n :: "nat"   and L :: "'a literal" shows "(I \<Turnstile> replicate_mset n L) = (n \<noteq> 0 \<and> I \<Turnstile>l L)"
must_be_an_update: fixes U1 :: "('a \<times> 'b) list"   and r :: "'a"   and U2 :: "('a \<times> 'b) list" assumes "U1 \<noteq> []"   and "fst (hd U1) = r \<and> tl U1 = U2" shows "\<exists>u. U1 = (r, u) # U2"
divides_unit: fixes a :: "'a"   and u :: "'a" assumes "a divides u"   and "a \<in> carrier G"   and "u \<in> Units G" shows "a \<in> Units G"
fixp_induct: fixes P :: "(('b, 'call, 'ret) gpv \<Rightarrow> 's \<Rightarrow> ('b \<times> 's + 'call \<times> 'call' \<times> ('ret \<times> 's, 'call', 'ret') rpv \<times> ('b, 'call, 'ret) rpv) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>inline1'. P (curry inline1'))"   and "P (\<lambda>inline1' gpv. return_pmf None)"   and "\<And>s. P s \<Longrightarrow> P (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (Inl (x, aa)) | IO out rpv \<Rightarrow> the_gpv (callee aa out) \<bind> (\<lambda>a. case a of Pure (x, y) \<Rightarrow> s (rpv x) y | IO out' rpv' \<Rightarrow> return_spmf (Inr (out, out', rpv', rpv)))))" shows "P local.inline1'"
permutes_inverses_1: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "p permutes S" shows "p (inv p x) = x"
interp_ratfps_solution_def: shows "interp_ratfps_solution = (\<lambda>(p, cs) n. coeff p n + (\<Sum>(q, c)\<leftarrow>cs. poly q (of_nat n) * c ^ n))"
xy_1: shows "x\<^sup>2 + y\<^sup>2 = p"
add_canc1: fixes x :: "'a"   and y :: "'a" assumes "D x y" shows "rquot (x \<oplus> y) x = y"
word_rotr_rev: fixes n :: "nat"   and w :: "'b word" shows "word_rotr n w = word_reverse (word_rotl n (word_reverse w))"
create_element_preserves_wellformedness_1: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and tag :: "char list"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>h h'"   and "type_wf h"   and "known_ptrs h" shows "heap_is_wellformed h'"
div_free_DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P: fixes A :: "'b set" shows "D (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {}"
monad_state_alt_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state_alt return bind get put alt" shows "monad_state_alt (return_env return) (bind_env bind) (get_env get) (put_env put) (alt_env alt)"
mult_row_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and a :: "'f"   and q :: "'d" shows "mult_row A a q = (\<chi>i j. if i = a then q * A $h a $h j else A $h i $h j)"
eq_qe_forwards: fixes eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list" assumes "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
surjective: fixes r :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) NatTrans_scheme" shows "r = \<lparr>NTDom = NTDom r, NTCod = NTCod r, NatTransMap = NatTransMap r, \<dots> = NatTrans.more r\<rparr>"
neq_label_nested: fixes n :: "nat"   and les :: "e list"   and es :: "e list" shows "[Label n les es] \<noteq> es"
evaln_no_abrupt: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and n :: "nat"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w, s')"   and "normal s'" shows "normal s"
monoms_add_poly_l'D: fixes aa :: "char list list"   and ba :: "int"   and x :: "(char list list \<times> int) list \<times> (char list list \<times> int) list" assumes "(aa, ba) \<in> set (add_poly_l' x)" shows "aa \<in> mononoms (fst x) \<or> aa \<in> mononoms (snd x)"
adj_correct_aux: fixes g :: "'m"   and u :: "'v" assumes "invar g" shows "set (adj g u) = {(v, d). (u, v) \<in> edges (\<alpha>g g) \<and> \<alpha>w g {u, v} = d}"
RSpan_append: fixes ms :: "'m list"   and ns :: "'m list" shows "RSpan (ms @ ns) = RSpan ms + RSpan ns"
prod: fixes g :: "'b \<Rightarrow> nat"   and A :: "'b set" shows "f (prod g A) = (\<Prod>x\<in>A. f (g x))"