hcju/isabelleps huggingface dataset:
nn_integral_map_pmf: fixes g :: "'b \<Rightarrow> 'a"   and M :: "'b pmf"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf (map_pmf g M)) f = \<integral>\<^sup>+ x. f (g x) \<partial>measure_pmf M"
poincare_distance_I: fixes u :: "complex_homo"   and v :: "complex_homo"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "ideal_points (poincare_line u v) = {i1, i2}" shows "poincare_distance u v = calc_poincare_distance u i1 v i2"
alphaInp_imp_asInp_equal: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and qinp' :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" assumes "qGoodInp qinp"   and "(qinp, qinp') \<in> alphaInp" shows "asInp qinp = asInp qinp'"
dist_inf_mono: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "mono (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"
f_def: shows "f = Cn 3 r_and [Cn 3 r_less [recf.Id 3 2, Cn 3 r_length [Cn 3 r_pdec1 [Cn 3 r_state [recf.Id 3 0, recf.Id 3 1]]]], Cn 3 r_neq [recf.Id 3 2, Cn 3 r_pdec2 [Cn 3 r_state [recf.Id 3 0, recf.Id 3 1]]]]"
median: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "length xs \<le> d" shows "is_median (med xs) xs"
get_def: fixes h :: "heap"   and a :: "'a array" shows "Array.get h a = map from_nat (arrays h TYPEREP('a) (addr_of_array a))"
push_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "push_precond f l \<equiv> \<lambda>(u, v). (0::'a) < excess f u \<and> (u, v) \<in> cfE_of f \<and> l u = l v + 1"
Units_one_closed: shows "\<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> \<in> Units \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
sub1b: shows "{flow0 x t1--b} \<subseteq> {a--b}"
atLeastAtMost_subseteq_atLeastLessThan_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a..b} \<subseteq> {c..<d}) = (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
domains_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.domains (g \<cdot> f) = C.domains f"
eventually_at_rightI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.order_topology less_eq less open"   and "\<And>x. x \<in> ord.greaterThanLessThan less a b \<Longrightarrow> P x"   and "less a b" shows "eventually P (topological_space.at_within open a (ord.greaterThan less a))"
liveVal_in_allVars: fixes g :: "'g"   and v :: "'val" assumes "liveVal g v" shows "v \<in> allVars g"
less1: shows "\<^bold>|u \<and>\<^sub>p v\<^bold>| < \<^bold>|u\<^bold>|"
continuous_at_split: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "isCont f x = (continuous (at_left x) f \<and> continuous (at_right x) f)"
simps_10: shows "cospan_shape.comp cospan_shape.BB cospan_shape.AA = cospan_shape.Null"
LIM_zero_cancel: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'b)) F" shows "(f \<longlongrightarrow> l) F"
fin_I: shows "finite {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}}"
add_le_add_imp_diff_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and i :: "'a"   and k :: "'a"   and n :: "'a"   and j :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n"   and "less_eq n (plus j k)"   and "less_eq (plus i k) n"   and "less_eq n (plus j k)" shows "less_eq (minus n k) j"
ParEnv: fixes Ps :: "'b com option list"   and s :: "'b"   and t :: "'b" shows "(Ps, s) -pe\<rightarrow> (Ps, t)"
sexp_rec_unfold: fixes a :: "('d, unit) Old_Datatype.node set"   and c :: "'d \<Rightarrow> 'c"   and d :: "nat \<Rightarrow> 'c"   and e :: "('d, unit) Old_Datatype.node set \<Rightarrow> ('d, unit) Old_Datatype.node set \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "sexp_rec a c d e = sexp_case c d (\<lambda>N1 N2. e N1 N2 (cut (\<lambda>M. sexp_rec M c d e) pred_sexp a N1) (cut (\<lambda>M. sexp_rec M c d e) pred_sexp a N2)) a"
list_mono: fixes A :: "'a set"   and B :: "'a set"   and n :: "nat" assumes "A \<subseteq> B" shows "list n A \<subseteq> list n B"
affine_parallel_slice: fixes S :: "'b set"   and a :: "'b"   and b :: "real" assumes "affine S"   and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"   and "\<not> S \<subseteq> {x. a \<bullet> x \<le> b}" obtains a' :: "'b" and b' :: "real" where "a' \<noteq> (0::'b)" and "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}" and "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}" and "\<And>w. w \<in> S \<Longrightarrow> w + a' \<in> S"
coboundedI2: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "b \<le> c" shows "min a b \<le> c"
less_eq_parity_def: fixes x :: "parity"   and y :: "parity" shows "(x \<le> y) = (y = Either \<or> x = y)"
getFresh_def: fixes P :: "pvar set" shows "getFresh P \<equiv> SOME p. p \<notin> P"
distinct_13: fixes x5 :: "nat" shows "Null \<noteq> Addr x5"
facet_of_imp_face_of: fixes F :: "'b set"   and S :: "'b set" assumes "F facet_of S" shows "F face_of S"
nodisc: shows "\<not> has_disc_negated disc2 False m"
B_ncong_p: fixes x :: "int" assumes "x \<in> B" shows "[x \<noteq> 0] (mod int p)"
project_unless: fixes G :: "'c program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set" assumes "G \<in> stable C"   and "project h C G \<in> A unless B" shows "G \<in> C \<inter> extend_set h A unless extend_set h B"
to_bl_upt: fixes x :: "'a word" shows "to_bl x = rev (map ((!!) x) [0..<size x])"
set_vzero_eq_ord_of_nat_vone: shows "ZFC_in_HOL.set {[]\<^sub>\<circ>} = 1\<^sub>\<nat>"
induct: fixes P1 :: "trm \<Rightarrow> bool"   and P3 :: "(char list \<times> trm) list \<Rightarrow> bool"   and P2 :: "char list \<times> trm \<Rightarrow> bool"   and char_list_trm_prod :: "char list \<times> trm" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lambda>:x1. x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lambda><:x1. x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<bullet> x2)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (x1 \<bullet>\<^sub>\<tau> x2)"   and "\<And>x. P3 x \<Longrightarrow> P1 (Rcd x)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (x1..x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (LET x1 = x2 IN x3)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P2 char_list_trm_prod"
nth_mem: fixes n :: "nat"   and xs :: "'a list" assumes "n \<sqsubset> length xs" shows "xs ! n \<in> set xs"
path2_def: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" shows "g \<turnstile> n-ns\<rightarrow>m \<equiv> old.path g ns \<and> n = hd ns \<and> m = last ns"
Tag_Exp: fixes X :: "tag"   and X' :: "msg"   and Y' :: "msg" shows "Tag X \<noteq> Exp X' Y'"
upper_inv_eq: fixes y :: "'b" assumes "y \<in> carrier \<Y>" shows "\<pi>\<^sup>* (\<pi>\<^sub>* y) = y"
ex_label_retraction: shows "\<exists>\<phi>. label_wrt C0 \<phi> \<and> fixespointwise \<phi> C0"
monotone_id': fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "monotone ord ord (\<lambda>x. x)"
assms_1: shows "f \<in> F s \<rightarrow>\<^sub>M N"
eucl_ivals_3: fixes a :: "'c" shows "{..a} \<in> sets borel"
univ_poly_carrier: fixes R :: "('d, 'e) ring_scheme"   and K :: "'d set"   and p :: "'d list" shows "polynomial\<^bsub>R\<^esub> K p = (p \<in> carrier (K [X]\<^bsub>R\<^esub>))"
invertible_matrix_iff_invertible_lf': fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "linear (*s) (*s) f" shows "invertible (matrix f) = invertible_lf (*s) (*s) f"
ide_implies_incl: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b" assumes "set_category S \<AA>"   and "partial_magma.ide S a" shows "set_category.incl S \<AA> a"
K'_1: fixes n :: "nat" shows "emeasure (P (J n)) (B n) - emeasure (P' n) (K' n) \<le> ennreal (2 powr - real n) * (INF i. emeasure (P (J i)) (B i))"
prems_3: shows "list.set ms \<subseteq> {0<..}"
distinct_19: fixes x21 :: "POPLmarkRecord.type"   and x22 :: "trm"   and x51 :: "trm"   and x52 :: "POPLmarkRecord.type" shows "(\<lambda>:x21. x22) \<noteq> x51 \<bullet>\<^sub>\<tau> x52"
hyps: assumes "is_subobj P (C, Cs @ [D, D'] @ rev Cs'')" shows "P \<turnstile> D \<prec>\<^sub>R D'"
addrow_mat_def: fixes n :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l = Matrix.mat n n (\<lambda>(i, j). (if k = i \<and> l = j then (+) a else id) (if i = j then 1::'a else (0::'a)))"
of_real_numeral: fixes w :: "num" shows "of_real (numeral w) = numeral w"
elims_2: fixes x :: "heap \<times> nat set \<Rightarrow> bool"   and xa :: "heap \<times> nat set \<Rightarrow> bool"   and xb :: "heap \<times> nat set" assumes "times_assn_raw x xa xb" obtains P :: "heap \<times> nat set \<Rightarrow> bool" and Q :: "heap \<times> nat set \<Rightarrow> bool" and h :: "heap" and as :: "nat set" where "x = P" and "xa = Q" and "xb = (h, as)" and "\<exists>as1 as2. as = as1 \<union> as2 \<and> as1 \<inter> as2 = {} \<and> P (h, as1) \<and> Q (h, as2)"
2: shows "j < length M2 \<longrightarrow> (v # M) ! (j div length M2) = v"
eigenvector_char_matrix: fixes A :: "'b mat"   and n :: "nat"   and v :: "'b vec"   and e :: "'b" assumes "A \<in> carrier_mat n n" shows "eigenvector A v e = (v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> char_matrix A e *\<^sub>v v = 0\<^sub>v n)"
times_gt_mono: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<succ> z"   and "(1::'a) \<le> x" shows "y * x \<succ> z * x"
g_def: fixes x :: "'a" shows "g x = (if x \<in> S then - \<infinity> else \<infinity>)"
interaction_bound_restrict_gpv: fixes "consider" :: "'d \<Rightarrow> bool"   and \<I> :: "('d, 'f) \<I>"   and gpv :: "('e, 'd, 'f) gpv" shows "interaction_bound consider (restrict_gpv \<I> gpv) \<le> interaction_bound consider gpv"
ts_process_traces_implies_d: fixes P :: "'a process" assumes "ts_process (traces P) = P" shows "deterministic P"
list_assn_emp: fixes L :: "'a list" shows "list_assn id_assn L L = emp"
assms_3: fixes s' :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "modifies mods s' s"   and "Q s s'" shows "Q' s'"
plus_of_int: fixes x :: "'a"   and n :: "int" shows "f (g x (of_int n)) = f x"
cl2: fixes x :: "'a" shows "1\<^sub>\<pi> \<sqinter> (x + nc) = x \<cdot> (0::'a)"
arr_tensor: fixes f :: "'c"   and g :: "'c" assumes "C.arr f"   and "C.arr g" shows "C.arr (f \<otimes>\<^sub>C g)"
croots4_def: shows "croots4 = roots4_generic solve_depressed_quartic_complex"
norm_fds_nth_newman: fixes n :: "nat" shows "cmod (fds_nth fds_newman n) = \<MM> (real n)"
insertI1: fixes a :: "'k"   and B :: "'k set" shows "a \<in> insert a B"
intros: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and R :: "'b set" assumes "stellar slices W"   and "p \<in> W"   and "\<forall>Sl\<in>slices p. \<exists>q\<in>Sl \<inter> W. q \<in> R \<or> stellar.blocking_min slices W R q" shows "stellar.blocking_min slices W R p"
list_emb_code_1: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c list" shows "list_emb P [] ys = True"
distance_upper_bound: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "Digraph.distance G v w \<le> length xs"
const5_def: fixes x :: "'l" shows "const5 x \<equiv> \<lambda>_ _ _ _ _. x"
get_matrix_R_def: fixes data :: "rat mat \<times> (nat list \<times> nat list) list \<times> rat list list" shows "get_matrix_R data = fst data"
sum_moebius_mu_bound: fixes x :: "real" assumes "0 \<le> x" shows "\<bar>M x\<bar> \<le> x"
Units_r_inv: fixes x :: "'a" assumes "x \<in> Units \<G>" shows "x \<otimes> inv x = \<one>"
set_spmf_mk_lossless: fixes p :: "'c spmf" shows "set_spmf (mk_lossless p) = set_spmf p"
assms: shows "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
set_add_hom: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and I :: "'a set"   and J :: "'a set" assumes "h \<in> ring_hom R S"   and "I \<subseteq> carrier R"   and "J \<subseteq> carrier R" shows "h ` (I <+>\<^bsub>R\<^esub> J) = h ` I <+>\<^bsub>S\<^esub> h ` J"
UN_I: fixes a :: "'a"   and A :: "'a set"   and b :: "'c"   and B :: "'a \<Rightarrow> 'c set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
pivot_set_alt: shows "\<exists>f. pivot_fun A f (dim_col A) \<and> set (pivot_positions A) = {(i, f i) |i. i < dim_row A \<and> row A i \<noteq> 0\<^sub>v (dim_col A)}"
P: fixes z :: "real" assumes "z \<noteq> x"   and "\<bar>z - x\<bar> < s" shows "\<bar>(f z - f x) / (z - x) - f' x\<bar> < e / 2"
star_closed_unfold: fixes x :: "('a \<times> 'a) set" assumes "x\<^sup>* = x" shows "x = Id \<union> x O x"
GrD2: fixes x :: "'e"   and fx :: "'f"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "f x = fx"
ccProd_insert2: fixes S :: "Vars.var set"   and x :: "Vars.var"   and S' :: "Vars.var set" shows "S G\<times> insert x S' = S G\<times> {x} \<squnion> S G\<times> S'"
mod_mult_self2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times b c)) b = modulo a b"
strict_implies_order: fixes b :: "'c word"   and a :: "'c word" assumes "b <s a" shows "b \<le>s a"
quote_all_Mem_imp_All2: fixes i :: "name"   and i' :: "name"   and \<alpha> :: "fm"   and Fi :: "name \<Rightarrow> tm"   and j :: "name"   and pi :: "perm" assumes "insert (QuoteP (Var i) (Var i')) (quote_all p V) \<turnstile> \<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i V) (insert i V) Fi)"   and "supp (All2 i (Var j) \<alpha>) \<subseteq> atom ` V"   and "atom j \<sharp> (i, \<alpha>)"   and "atom i \<sharp> p"   and "atom i' \<sharp> (i, p, \<alpha>)"   and "pi = (atom i \<rightleftharpoons> atom i') + p"   and "Fi = make_F (insert i V) pi" shows "insert (All2 i (Var j) \<alpha>) (quote_all p V) \<turnstile> PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>V V F)"
in_measure: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<Rightarrow> nat" shows "((x, y) \<in> measure f) = (f x < f y)"
poly_rel_eq: shows "(poly_rel ===> poly_rel ===> (=)) (=) (=)"
prems_2: shows "\<forall>c\<in>insert p P. [a = b] (mod c)"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> orbit.Partition" shows "orbit.Class a = A"
ivl_disj_un_3: fixes l :: "'g"   and u :: "'g" assumes "l < u" shows "{l} \<union> {l<..<u} = {l..<u}"
Rep_Node_inject: fixes x :: "('b, 'a) node"   and y :: "('b, 'a) node" shows "(Rep_Node x = Rep_Node y) = (x = y)"
less_le_trans: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term"   and z :: "('a, 'b) Term.term" assumes "x \<sqsubset> y"   and "y \<sqsubseteq> z" shows "x \<sqsubset> z"
subterm_wpo_ns: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" assumes "s \<unrhd> t" shows "s \<succeq> t"
rsemimodule2: fixes s :: "'b"   and a :: "'a"   and b :: "'a" shows "\<lambda> (s, a + b) = \<lambda> (s, a) \<oplus> \<lambda> (s, b)"
prefix_subst1: fixes m :: "'c list"   and l :: "'c list"   and t :: "'c list" assumes "m = l @ t" shows "m - l = t"
compl_top_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus top = bot"
SimpleLevi: fixes p :: "'a list"   and s :: "'a list"   and a :: "'a"   and list :: "'a list" shows "(p @ s = a # list) = (p = [] \<and> s = a # list \<or> (\<exists>t. p = a # t \<and> t @ s = list))"
IH_1: fixes u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "set us \<subseteq> uses g n"   and "Mapping.lookup u n = None"   and "foldr (aux_1 g n) us (u, p) = (u', p')"   and "finite (Mapping.keys p)"   and "\<And>n v. (n, v) \<in> Mapping.keys p \<Longrightarrow> length (predecessors g n) \<noteq> 1"   and "\<And>v. Mapping.lookup p (Entry g, v) \<in> {None, Some []}" shows "lookupDef g n ` set us = lookup_multimap u' n"
simps_2: shows "typeof Null = \<lfloor>NT\<rfloor>"
intro: fixes perm1 :: "'b perm \<Rightarrow> 'c \<Rightarrow> 'c"   and perm2 :: "'b perm \<Rightarrow> 'd \<Rightarrow> 'd" assumes "permute perm1"   and "permute perm2" shows "fun_permute perm1 perm2"
bal_tree_def: fixes n :: "nat"   and t :: "'b tree" shows "Balance.bal_tree n t = Balance.bal_list n (inorder t)"
transfer_forall_transfer_3: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A" shows "((A ===> (\<longrightarrow>)) ===> (\<longrightarrow>)) transfer_forall transfer_forall"
nonneg_Reals_minus_one_I: shows "- (1::'a) \<notin> \<real>\<^sub>\<ge>\<^sub>0"
gpv_coinduct_bind: fixes R :: "('d, 'e, 'f) gpv \<Rightarrow> ('d, 'e, 'f) gpv \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv' \<or> (\<exists>gpv2 gpv2' f f'. gpv = gpv2 \<bind> f \<and> gpv' = gpv2' \<bind> f' \<and> rel_gpv (\<lambda>x y. R (f x) (f' y)) (=) gpv2 gpv2')))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
simple_ruleset_append: fixes rs\<^sub>1 :: "'a rule list"   and rs\<^sub>2 :: "'a rule list" shows "simple_ruleset (rs\<^sub>1 @ rs\<^sub>2) = (simple_ruleset rs\<^sub>1 \<and> simple_ruleset rs\<^sub>2)"
elims_3: fixes x :: "'at Literal"   and xa :: "'at Literal" assumes "\<not> literal_ordering x xa" obtains L1 :: "'at Literal" and L2 :: "'at Literal" where "x = L1" and "xa = L2" and "(atom L1, atom L2) \<notin> atom_ordering"
hyps_5: fixes \<Gamma>' :: "(name \<times> ty) list"   and b :: "(name \<times> trm) list"   and ba :: "(name \<times> trm) list" assumes "\<Gamma>' \<turnstile> b is ba over (x, T\<^sub>1) # \<Gamma>"   and "valid \<Gamma>'" shows "\<Gamma>' \<turnstile> b<t\<^sub>2> is ba<t\<^sub>2> : T\<^sub>2"
dominatesE: fixes g :: "'g"   and n :: "'node"   and m :: "'node" assumes "dominates g n m" obtains "m \<in> set (\<alpha>n g)" and "\<And>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<Longrightarrow> n \<in> set ns"
card_gt_0_diff_singleton: fixes S :: "'a set"   and x :: "'a" assumes "0 < card S"   and "x \<in> S" shows "card (S - {x}) = card S - 1"
simps: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" shows "accessible_on P A a = (\<exists>x. a = x \<and> x \<in> A \<and> (\<forall>xa. xa \<in> A \<longrightarrow> P xa x \<longrightarrow> accessible_on P A xa))"
elem_is_gen_pow: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<exists>n. x = gen [^] n"
L_While_let: fixes X :: "char list set"   and b :: "bexp"   and c :: "com" assumes "finite X" shows "L (WHILE b DO c) X = (let f = \<lambda>Y. vars b \<union> X \<union> L c Y in while (\<lambda>Y. f Y \<noteq> Y) f {})"
mat_incr_def: fixes n :: "nat" shows "mat_incr n = mat n n (\<lambda>(i, j). if i = 0 then if j = n - 1 then 1 else 0 else if i = j + 1 then 1 else 0)"
foldli_foldl: fixes xs :: "'e list"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli xs (\<lambda>_. True) f \<sigma> = foldl (\<lambda>\<sigma> x. f x \<sigma>) \<sigma> xs"
Un_idem_collapse: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> (B \<union> A) = B \<union> A"
compact_interval_eq_Inf_Sup: fixes A :: "real set" assumes "is_interval A"   and "compact A"   and "A \<noteq> {}" shows "A = {Inf A..Sup A}"
boundedI: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> b \<sqinter> c"
"1_6": assumes "n \<noteq> 0"   and "n \<noteq> 1"   and "a \<noteq> 1"   and "\<not> n < 0"   and "odd n"   and "\<not> a < 0"   and "a \<noteq> 0"   and "\<not> n \<le> a"   and "odd a"   and "coprime a n" shows "Jacobi n a = jacobi_code n a"
All_lex: shows "All_lex = Algorithm.gen2 (Max (set b)) (Max (set a)) a b"
simps_2: fixes i :: "nat"   and "is" :: "nat list"   and lab_c :: "t list list" shows "same_lab (i # is) lab_c = (if length lab_c \<le> i then None else same_lab_h is lab_c (lab_c ! i))"
sees_field_fun: fixes P :: "'b prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal"   and T' :: "ty"   and fm' :: "fmod"   and D' :: "String.literal" assumes "P \<turnstile> C sees F:T (fm) in D"   and "P \<turnstile> C sees F:T' (fm') in D'" shows "T' = T \<and> D' = D \<and> fm = fm'"
soundness: fixes p :: "fol fm"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool" assumes "\<turnstile> p" shows "semantics e f g p"
d_def: fixes x :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d x = x \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma>"
P3: fixes j :: "nat" shows "foldl1 f2 ([One, Two] @ replicate j Zero) = Two"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) ; c = a ; c + b ; c"
rcis_cmod_arg: fixes z :: "complex" shows "rcis (cmod z) (arg z) = z"
Vagree_sym_rel: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set" assumes "Vagree \<nu> \<nu>' V" shows "Vagree \<nu>' \<nu> V"
le_less_TC_trans: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<sqsubseteq> y"   and "y \<sqsubset> z" shows "x \<sqsubset> z"
IH: assumes "are_all_operators_non_interfering ops"   and "are_operators_interfering op\<^sub>1 op\<^sub>2"   and "op\<^sub>1 \<noteq> op\<^sub>2" shows "op\<^sub>1 \<notin> set ops \<or> op\<^sub>2 \<notin> set ops"
mat_map_index: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat"   and j :: "nat"   and f :: "'c \<Rightarrow> 'c" assumes "mat nr nc m"   and "i < nc"   and "j < nr" shows "mat_map f m ! i ! j = f (m ! i ! j)"
set_offending_flows_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP = {F. F \<subseteq> edges G \<and> SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar F G nP}"
acquired_takeWhile_non_volatile_Write\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and A :: "nat set" shows "acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) A \<subseteq> A \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
noops_valid: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "is_valid_problem_sas_plus (prob_with_noop \<Psi>)"