hcju/isabelleps huggingface dataset:
modifies_trans: fixes vs :: "char list set"   and a :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "char list \<Rightarrow> int \<Rightarrow> int"   and c :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "modifies vs a b"   and "modifies vs b c" shows "modifies vs a c"
prems_4: fixes i :: "nat" assumes "Suc i < length (a # p)" shows "head G ((a # p) ! i) = tail G ((a # p) ! Suc i)"
Rep_io_inverse: fixes x :: "'c io" shows "Abs_io (Rep_io x) = x"
OUTfromVCorrect1_data16: shows "OUTfromVCorrect1 data16"
ladder_shift_j_cons: fixes d :: "nat"   and x :: "nat \<times> nat \<times> nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_shift_j d (x # L) = (fst x, fst (snd x) - d, snd (snd x)) # ladder_shift_j d L"
e_B_def: shows "e_B \<equiv> cons_exec e_B' (a, t)"
setcong: fixes I :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> I \<Longrightarrow> g x = h x" shows "{g x |x. x \<in> I} = {h x |x. x \<in> I}"
irs_SDG_path_realizable: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n irs-ns\<rightarrow>\<^sub>d* n'"   and "n \<noteq> n'" obtains ns' :: "'node SDG_node list" where "realizable (CFG_node (_Entry_)) ns' n'" and "n \<in> set ns'"
XY: shows "\<forall>x\<in>X. \<forall>y\<in>Y. \<exists>z\<in>X \<union> Y. x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
cycleE: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list" assumes "Graph G"   and "Graph.cycle G xs" shows "(Graph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> 2 < length xs \<and> last xs \<rightarrow>\<^bsub>G\<^esub> hd xs"
kerD: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "a \<in> ker f" shows "f a = (0::'b)"
AUX: fixes it :: "('k \<times> 'v) set"   and k :: "'k"   and v :: "'v"   and \<sigma> :: "'s" assumes "it \<subseteq> {(k, v). \<alpha> m k = Some v}"   and "\<alpha> m k = Some v"   and "(k, v) \<notin> it"   and "{(k, v). \<alpha> \<sigma> k = Some v} = it \<inter> Collect P" shows "k \<notin> dom (\<alpha> \<sigma>)"
funs_term_subterms_eq_2: fixes M :: "('a, 'b) Term.term set" shows "\<Union> (funs_term ` subterms\<^sub>s\<^sub>e\<^sub>t M) = \<Union> (funs_term ` M)"
list_Previous_Suc: fixes xs :: "'d list"   and init :: "'d"   and n :: "nat" shows "xs\<^bsup>\<leftarrow>' init\<^esup> Suc n = xs ! n"
blank_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "blank P C \<equiv> (C, Collect (init_obj P C))"
cnj_transpose_is_dagger: fixes M :: "complex Matrix.mat" shows "M\<^sup>\<star>\<^sup>t = M\<^sup>\<dagger>"
dim_col_of_iter_tensor: fixes n :: "nat"   and A :: "complex Matrix.mat" assumes "1 \<le> n" shows "dim_col (A \<otimes>\<^bsup>n\<^esup>) = dim_col A ^ n"
right_edge_explicit: shows "(\<lambda>y. twoC (y, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
ennreal_real_conv_ennreal_of_enat: fixes n :: "nat" shows "ennreal (real n) = ennreal_of_enat (enat n)"
ring_simprules_16: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
simps: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "kbo s t = (if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s then if weight t < weight s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict (f, length ss) (g, length ts) then (True, True) else if pr_weak (f, length ss) (g, length ts) then lex_ext_unbounded kbo ss ts else (False, False) else (False, False))"
homotopic_paths_trans: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and r :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q"   and "homotopic_paths s q r" shows "homotopic_paths s p r"
norm1_def: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec" shows "norm1 v = (\<Sum>i\<in>UNIV. norm (v $h i))"
dvdm_imp_degree_le: fixes p :: "int poly"   and q :: "int poly" assumes "p dvdm q"   and "monic p"   and "Mp q \<noteq> 0"   and "1 < p" shows "degree p \<le> degree q"
Quotient_rep_abs: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a" assumes "Quotient R Abs Rep T"   and "R r r" shows "R (Rep (Abs r)) r"
prognosis_env_cong: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and u :: "Arity"   and e :: "exp"   and S :: "stack_elem list" assumes "ae f|` domA \<Gamma> = ae' f|` domA \<Gamma>" shows "prognosis ae as u (\<Gamma>, e, S) = prognosis ae' as u (\<Gamma>, e, S)"
refl_on_def: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
vinsert_obtain_ne: fixes A :: "V" assumes "A \<noteq> []\<^sub>\<circ>" obtains a :: "V" and A' :: "V" where "A = vinsert a A'" and "a \<notin>\<^sub>\<circ> A'"
getOneIp_elem: fixes W :: "'b wordinterval" assumes "\<not> wordinterval_empty W" shows "wordinterval_element (getOneIp W) W"
card_even: fixes n :: "nat" shows "card {a \<in> Collect even. a < 2 * n} = n"
isCont_borel_pred: fixes f :: "'c \<Rightarrow> 'd" shows "Measurable.pred borel (isCont f)"
runit_simps_5: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "cod \<r>[f] = f"
NegChecks: shows "s = \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>"
gfp_trans_least: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t. \<lbrakk>le_utrans t (T t); \<And>P. unitary P \<Longrightarrow> unitary (t P)\<rbrakk> \<Longrightarrow> le_utrans t u"   and "\<And>P. unitary P \<Longrightarrow> unitary (u P)" shows "le_utrans (gfp_trans T) u"
l: shows "sigma_sets (space M) (\<Union>i\<in>{m. m \<le> n}. {X i -` A \<inter> space M |A. A \<in> sets borel}) \<subseteq> sigma_sets (space M) {proj_stoch_proc X n -` B \<inter> space M |B. B \<in> sets (stream_space borel)}"
Nil_is_map_conv: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "([] = list.map f xs) = (xs = [])"
Entry_target: fixes a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" assumes "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a"   and "trg a = NewEntry" shows "False"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b + (a + c) = a + (b + c)"
in_intra_SDG_path_in_HRB_slice: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "n i-ns\<rightarrow>\<^sub>d* n'"   and "n'' \<in> set ns"   and "n' \<in> S" shows "n'' \<in> HRB_slice S"
assms_4: shows "Z = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n"
red_Ret: fixes s :: "trm"   and t :: "trm" assumes "[s] \<mapsto> t" shows "\<exists>s'. t = [s'] \<and> s \<mapsto> s'"
mult_1_left: fixes a :: "'b" shows "(0::'b) + a = a"
set_cases: fixes e :: "'b"   and a :: "'b stream" assumes "e \<in> sset a" obtains z2 :: "'b stream" where "a = e ## z2__"   | z1 :: "'b" and z2 :: "'b stream" where "a = z1 ## z2" and "e \<in> sset z2"
assms_1: shows "\<forall>t2<t. i2 t2 = []"
subcls1I: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "C \<noteq> Object" shows "G\<turnstile>C\<prec>\<^sub>C1super c"
assms_2: shows "Q \<Longrightarrow>\<^sup>^coAction \<alpha> \<prec> Q'"
hyps: shows "(1::'a) + of_nat n * x \<le> ((1::'a) + x) ^ n"
col_fst_is_col: fixes M :: "'a Matrix.mat" shows "col_fst M = Matrix.col M 0"
minus_Mp_1: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.Mp m (poly_mod.Mp m f - g) = poly_mod.Mp m (f - g)"
simps_2: fixes x :: "'c" shows "the_default x None = x"
phiUses_exI: fixes m :: "'node"   and g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "m \<in> set (old.predecessors g n)"   and "phis g (n, v) = Some vs"   and "n \<in> set (\<alpha>n g)" obtains v' :: "'val" where "v' \<in> phiUses g m" and "v' \<in> set vs"
abs_summable_on_nat_iff': fixes f :: "nat \<Rightarrow> 'a" shows "(f abs_summable_on UNIV) = summable (\<lambda>n. norm (f n))"
simps_5: fixes a :: "tm"   and b :: "tm" shows "tmbound0 (tm.Add a b) = (tmbound0 a \<and> tmbound0 b)"
sl: shows "take (length bs) (snd (TSdet init h ((as @ x # bs) @ [x]) (Suc (length as + length bs)))) = rev bs"
distinct_10: fixes x41 :: "dbtm"   and x42 :: "dbtm"   and x2 :: "name" shows "DBEats x41 x42 \<noteq> DBVar x2"
fps_lr_inverse_mult_ring1_2: fixes x :: "'a"   and f :: "'a fps"   and y :: "'a"   and g :: "'a fps" assumes "x * fps_nth f 0 = (1::'a)"   and "fps_nth f 0 * x = (1::'a)"   and "y * fps_nth g 0 = (1::'a)"   and "fps_nth g 0 * y = (1::'a)" shows "fps_right_inverse (f * g) (y * x) = fps_right_inverse g y * fps_right_inverse f x"
star_sub_one: fixes x :: "'a" assumes "coreflexive x" shows "x\<^sup>\<star> = (1::'a)"
small: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>}"
in_carrierp_def: shows "in_carrierp = (\<lambda>(x, y, z). x \<in> carrier R \<and> y \<in> carrier R \<and> z \<in> carrier R)"
synth_inversion_6: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> synth H" obtains "\<langle>X, Y\<rangle> \<in> H"   | "X \<in> synth H" and "Y \<in> synth H"
set1_FGco_def: fixes x :: "('p, 'q, 'r, ('q, 's, 't, 'u, 'v, 'w, 'x) G, 't, 'y, 'v, 'z, 'aa, 'ab) F" shows "set1_FGco x = set1_F x"
assms: shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
Vagree_union: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and W :: "variable set" assumes "Vagree \<nu> \<nu>' V"   and "Vagree \<nu> \<nu>' W" shows "Vagree \<nu> \<nu>' (V \<union> W)"
u_facts_4: shows "u \<notin> snd ` set txs"
cases: fixes \<I> :: "('c, 'd) \<I>"   and a :: "('c, 'd) resource" assumes "\<I> \<turnstile>res a \<surd>" obtains res :: "('c, 'd) resource" where "a = res" and "\<And>q r res'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, res') \<in> set_spmf (run_resource res q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>"
is_down_def: shows "is_down \<equiv> map_fun (map_fun Word.Word Word.rep) id (\<lambda>_. LENGTH('c) \<le> LENGTH('b))"
ANR_connected_component_ANR: fixes S :: "'a set"   and x :: "'a" assumes "ANR S" shows "ANR (connected_component_set S x)"
res: shows "local.select_correct_factor_main init polys [] l' r' 0 = ((q, ri), l, r)"
F1: shows "\<forall>x\<^sub>1. (0::'a) \<le> \<bar>x\<^sub>1\<bar>"
subresultant_hom: fixes J :: "nat"   and f :: "'b poly"   and g :: "'b poly" shows "map_poly to_fract (subresultant J f g) = subresultant J (map_poly to_fract f) (map_poly to_fract g)"
ostep_invariantD: fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e \<Rightarrow> bool"   and \<sigma> :: "'d"   and s :: "'e"   and a :: "'f"   and \<sigma>' :: "'d"   and s' :: "'e" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P ((\<sigma>, s), a, \<sigma>', s')"
P1: shows "A B C B C A SumA G H I \<and> G H I C A B SumA D E F"
lec_of_constraint: fixes c :: "constraint"   and v :: "nat \<Rightarrow> rat" assumes "is_le c" shows "v \<Turnstile>\<^sub>l\<^sub>e lec_of_constraint c = v \<Turnstile>\<^sub>c c"
NEUT_def: fixes t :: "trm" shows "NEUT t \<equiv> (\<exists>a. t = Var a) \<or> (\<exists>t1 t2. t = App t1 t2)"
bind_assoc_return_val: fixes M :: "val measure"   and t :: "pdf_type"   and f :: "val \<Rightarrow> val"   and t' :: "pdf_type"   and g :: "val \<Rightarrow> val"   and t'' :: "pdf_type" assumes "sets M = sets (stock_measure t)"   and "f \<in> stock_measure t \<rightarrow>\<^sub>M stock_measure t'"   and "g \<in> stock_measure t' \<rightarrow>\<^sub>M stock_measure t''" shows "M \<bind> (\<lambda>x. return_val (f x)) \<bind> (\<lambda>x. return_val (g x)) = M \<bind> (\<lambda>x. return_val (g (f x)))"
T_BIT_eq: shows "T_BIT (length qs) = T_on_rand' BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs"
distinct_69: fixes x9 :: "('a, String.literal) phantom" shows "String \<noteq> Enum x9"
shows_prec_prod_def: fixes d :: "nat"   and p :: "'a \<times> 'b" shows "shows_prec d p = shows_paren (shows (fst p) \<circ> ('','' +#+ shows (snd p)))"
Spy_spies_bad_shrK: fixes A :: "agent"   and evs :: "event list" assumes "A \<in> bad" shows "Key (shrK A) \<in> knows Spy evs"
equiv_out_deselect: fixes o1 :: "'prog_out"   and o2 :: "'prog_out"   and P :: "'prog"   and P' :: "'prog" assumes "equiv_out o1 o2"   and "deselect P o1 P'" shows "deselect P o2 P'"
assms_1: fixes m :: "nat" shows "f m \<in> {0..<p ^ m}"
inj_on_convol_ident: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set" shows "inj_on (\<lambda>x. (x, f x)) X"
member_le_sum_list: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "x \<le> sum_list xs"
finite_Di': fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and y :: "nat list" shows "finite (hlde_ops.Di' a b i y)"
simps_1: fixes f1 :: "'j \<Rightarrow> 'h"   and f2 :: "'k \<Rightarrow> 'i"   and a :: "'j" shows "map_sum f1 f2 (Inl a) = Inl (f1 a)"
cases: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" assumes "Axp a1 a2" obtains i :: "nat" where "a1 = []" and "a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)"   | "a1 = []" and "a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)"
P_fs_def: fixes x :: "'a" shows "P_fs x \<equiv> if x \<in> P then fs else []"
pfinite_converter_2: shows "a \<in> outs_\<I> \<I>"
fmupd_fmdrop: fixes xm :: "'f \<rightharpoonup>\<^sub>f 'e"   and k :: "'f"   and x :: "'e" assumes "fmlookup xm k = Some x" shows "xm = fmdrop k xm(k \<mapsto>\<^sub>f x)"
assms_2: shows "arcs H = {e \<in> arcs G. tail G e \<in> verts H \<and> head G e \<in> verts H}"
B: shows "\<turnstile>\<^sub>2 {\<lambda>s. (if bval b s then wp\<^sub>2 c (wp\<^sub>2 (WHILE b DO c) Q) s else Q s) + \<up> (bval b s)} c {\<lambda>t. (if bval b t then wp\<^sub>2 c (wp\<^sub>2 (WHILE b DO c) Q) t else Q t) + 1}"
many_strongly_connected_iff_6: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>T ; x\<^sup>\<star> \<le> x\<^sup>+)"
simps_6: fixes \<alpha> :: "('a, 'b, 'c) hp"   and \<beta> :: "('a, 'b, 'c) hp"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Psubst (\<alpha> \<union>\<union> \<beta>) \<sigma> = (Psubst \<alpha> \<sigma> \<union>\<union> Psubst \<beta> \<sigma>)"
cases: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and x :: "'b literal" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" obtains t :: "'b trm" and s :: "'b trm" where "x = Pos (Eq t__ s__)"   | t :: "'b trm" and s :: "'b trm" where "x = Neg (Eq t s)"
seq_comp_prop_2: fixes P :: "'c option process"   and xs :: "'c option list"   and x :: "'c option"   and X :: "'c option set"   and Q :: "'c option process" assumes "weakly_sequential P"   and "(xs @ [x], X) \<in> seq_comp_failures P Q" shows "(xs, {}) \<in> seq_comp_failures P Q"
first_entrance_rec: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "first_entrance_set A (Suc n) = T --` first_entrance_set A n - A"
degree_def: fixes f :: "nat \<Rightarrow>\<^sub>0 'b" shows "degree f = Max (insert 0 (Suc ` keys f))"
connected: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "v \<rightarrow>\<^sup>* w"
divideC_field_splits_simps_1_1: fixes a :: "'b"   and c :: "complex"   and b :: "'b" shows "(a = b /\<^sub>C c) = (if c = 0 then a = (0::'b) else c *\<^sub>C a = b)"
star_n_add: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" shows "star_n X + star_n Y = star_n (\<lambda>n. X n + Y n)"
in_rel: shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
conjugate_dist_mul: fixes conjugate :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (times a b) = times (conjugate a) (conjugate b)"
le_approx_lemma_T: fixes P :: "'c process"   and Q :: "'c process" assumes "P \<sqsubseteq> Q" shows "T Q \<subseteq> T P"
erfc_minus: fixes z :: "'a" shows "erfc (- z) = (2::'a) - erfc z"
\<sigma>_\<sigma>1: fixes v :: "'a" assumes "v \<in> attractor p K - K" shows "\<sigma> v = \<sigma>1 v"
red: shows "M \<longrightarrow>\<^sub>a* M'"
reachableNodes_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat" shows "Graph.reachableNodes c u \<equiv> {v. Graph.connected c u v}"
assms_1: shows "\<forall>x. f x \<in> {l..u}"
bind_bind: fixes A :: "'g set"   and B :: "'g \<Rightarrow> 'f set"   and C :: "'f \<Rightarrow> 'e set" shows "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
emeasure_point_measure_finite: fixes A :: "'c set"   and X :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "finite A"   and "X \<subseteq> A" shows "emeasure (point_measure A f) X = sum f X"
that: assumes "nop A (2\<^sub>\<nat>) f" shows "thesis"
SET'_def: shows "SET' \<equiv> \<lparr>Obj = {x. True}, Mor = {f. isZFfun f}, Dom = ZFfunDom, Cod = ZFfunCod, Id = \<lambda>x. ZFfun x x (\<lambda>x. x), Comp = (|o|)\<rparr>"
vsv_vunion_app_left: fixes r :: "V"   and s :: "V"   and x :: "V" assumes "vsv r"   and "vsv s"   and "\<D>\<^sub>\<circ> r \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> s = []\<^sub>\<circ>"   and "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(r \<union>\<^sub>\<circ> s)\<lparr>x\<rparr> = r\<lparr>x\<rparr>"
assms_6: shows "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"
mut_m_not_idle_no_fM_write: fixes s :: "'f process_name \<Rightarrow> ('e, 'f, 'g, 'h) Model.local_state"   and m :: "'f"   and p :: "'f process_name"   and fl :: "bool"   and ws :: "('e, 'g, 'h) mem_store_action list" assumes "ghost_hs_phase (s (mutator m)) \<noteq> hp_Idle"   and "fM_rel_inv s"   and "handshake_phase_inv s"   and "tso_store_inv s"   and "p \<noteq> sys" shows "sys_mem_store_buffers p s \<noteq> mw_fM fl # ws"
pmf_map_outside: fixes x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf" assumes "x \<notin> f ` set_pmf M" shows "pmf (map_pmf f M) x = 0"
assms_1: fixes x :: "'a" shows "0 \<le> f x"
represents_of_rat: fixes p :: "int poly"   and x :: "rat" shows "p represents of_rat x = p represents x"
guarantees_imp_extend_guarantees: fixes F :: "'a program"   and X :: "'a program set"   and Y :: "'a program set" assumes "F \<in> X guarantees Y" shows "extend h F \<in> extend h ` X guarantees extend h ` Y"
simps_8: fixes f :: "'b \<Rightarrow> 'a" shows "map_option f None = None"
\<G>_semantics: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<G> \<phi> w \<equiv> {\<psi> \<in> subformulas\<^sub>\<nu> \<phi>. w \<Turnstile>\<^sub>n G\<^sub>n \<psi>}"
ord_option_mono': fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "\<And>x y. A x y \<longrightarrow> B x y" shows "ord_option A x y \<longrightarrow> ord_option B x y"
select_convs_1: fixes arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b" shows "arrays \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = arrays"
signed_take_bit_numeral_bit0: fixes l :: "num"   and k :: "num" shows "signed_take_bit (numeral l) (numeral (num.Bit0 k)) = signed_take_bit (pred_numeral l) (numeral k) * 2"
uminus_not_id': fixes x :: "'a literal" shows "- x \<noteq> x"
simps_11: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Cos a) vs = approx_floatarith p a vs \<bind> cos_aform_err p"
order_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
dbm_le_def: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "a \<preceq> b \<equiv> a \<prec> b \<or> a = b"
tp_sin: fixes n :: "real" assumes "n \<in> \<nat>" shows "trigpoly (\<lambda>x. sin (n * x))"
sms_def: shows "sms = - (changes_alt_itv_smods 0 1 (pR1 div gc1) (pI1 div gc1) + changes_alt_itv_smods 0 1 (pR2 div gc2) (pI2 div gc2) + changes_alt_itv_smods 0 1 (pR3 div gc3) (pI3 div gc3) + changes_alt_itv_smods 0 1 (pR4 div gc4) (pI4 div gc4)) div 4"
steps_non_empty': fixes xs :: "'a list" assumes "steps xs" shows "xs \<noteq> []"
historyI: fixes I :: "state \<Rightarrow> bool"   and sigma :: "behavior"   and N :: "state \<times> state \<Rightarrow> bool"   and vs :: "state \<Rightarrow> 'a"   and ha :: "state \<Rightarrow> 'b"   and HI :: "(state \<Rightarrow> 'b) \<Rightarrow> state \<Rightarrow> bool"   and hb :: "'b \<Rightarrow> state \<times> state \<Rightarrow> 'b"   and HN :: "(state \<Rightarrow> 'b) \<Rightarrow> state \<times> state \<Rightarrow> bool" assumes "sigma \<Turnstile> Init I"   and "sigma \<Turnstile> \<box>N"   and "basevars vs"   and "\<And>h. basevars (h, vs) \<Longrightarrow> \<turnstile> I \<and> h = ha \<longrightarrow> HI h"   and "\<And>h s t. \<lbrakk>basevars (h, vs); N (s, t); h t = hb (h s) (s, t)\<rbrakk> \<Longrightarrow> HN h (s, t)" shows "(\<exists>\<exists> h. Init HI h \<and> \<box>HN h) sigma"
r10': fixes x :: "name"   and y :: "name"   and u :: "trm"   and s :: "trm"   and t :: "trm" assumes "x \<sharp> y"   and "x \<sharp> u" shows "(s to x in t) to y in u \<mapsto> s to x in t to y in u"
i_dvd_j: shows "\<forall>j\<in>J. i dvd j"
integral_iff: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "domain R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>) = (a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
maxU34: shows "wmin (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2) \<equiv>\<^sub>L min (rl1 * ru2) (ru1 * ru2)"
TRIV: shows "makespan (T(min\<^sub>k T m := T (min\<^sub>k T m) + t (Suc j))) = Max\<^sub>0 (t ` {1..Suc j})"
fbrelationD1: fixes r :: "V"   and x :: "V" assumes "fbrelation r"   and "x \<in>\<^sub>\<circ> r" shows "\<exists>a b. x = [a, b]\<^sub>\<circ>"
maj: fixes t :: "state" assumes "F (s, t)" shows "G (s, t)"
cod_dom: fixes f :: "'b" assumes "B.arr f" shows "B.cod (B.dom f) = B.dom f"
simps_2: fixes a :: "'b set"   and b :: "'b set"   and n :: "nat" shows "trivial_series2 a b (Suc n) = (if n = 0 then b else {})"
adj_reachable_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow>\<^bsub>del_vert u\<^esub> b"   and "b \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> c" shows "a \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> c"
norm_Arccos_bounded: fixes w :: "complex" shows "cmod (Arccos w) \<le> pi + cmod w"