hcju/isabelleps huggingface dataset:
ordertype_eqI: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and s :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "wf r"   and "total_on A r"   and "small A"   and "wf s"   and "bij_betw f A B"   and "\<forall>x\<in>A. \<forall>y\<in>A. ((f x, f y) \<in> s) = ((x, y) \<in> r)" shows "ordertype A r = ordertype B s"
divides_zero: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "ring R"   and "a \<in> carrier R" shows "a divides\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub>"
pivot_vars: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "tvars (\<T> (pivot x\<^sub>i x\<^sub>j s)) = tvars (\<T> s)"
TSL: shows "SIDE_GEN_ALGO (is_set_to_sorted_list R Rk Rs tsl)"
assms: shows "tfr\<^sub>s\<^sub>s\<^sub>t (A @ B)"
ide_eval_Ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'b \<Rightarrow> 'd"   and t :: "'b local.term" assumes "evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Ide C t" shows "partial_magma.ide D (evaluation_map.eval D T \<alpha> \<iota> V t)"
idtm2: shows "map (\<lambda>k. v ! k * a + scalar_prod (row m k) cc) [0..<length r] = map (\<lambda>k. v ! k * a + scalar_prod (tm ! k) cc) [0..<length r]"
reduce_simple_not_trap: fixes es :: "e list"   and es' :: "e list" assumes "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>" shows "es \<noteq> [Trap]"
col_def: fixes A :: "'b mat"   and j :: "nat" shows "col A j = vec (dim_row A) (\<lambda>i. A $$ (i, j))"
has_integral_negligible: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> f x = (0::'b)" shows "(f has_integral (0::'b)) T"
intros_10: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<equiv>\<^sub>s P"
measurable_on_cmul: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and c :: "real" assumes "f measurable_on S" shows "(\<lambda>x. c * f x) measurable_on S"
expands_to_zero: fixes basis :: "(real \<Rightarrow> real) list" assumes "basis_wf basis"   and "length basis = expansion_level TYPE('a)" shows "((\<lambda>_. 0) expands_to zero_expansion) basis"
oalist_ntm_eqI: fixes xs :: "('a, 'b) oalist_ntm"   and ys :: "('a, 'b) oalist_ntm" assumes "list_of_oalist_ntm xs = list_of_oalist_ntm ys" shows "xs = ys"
characteristic_formula_def: fixes P :: "'state" shows "characteristic_formula P \<equiv> Conj (Abs_bset {distinguishing_formula P Q |Q. \<not> P =\<cdot> Q})"
prime_def: fixes p :: "'b" shows "prime p = (prime_elem p \<and> normalize p = p)"
card_less_nth_prime: fixes n :: "nat" shows "card {q. prime q \<and> q < nth_prime n} = n"
open_segment_eq_sing: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "open_segment a b \<noteq> {c}"
nth_list_of: fixes xs :: "'b llist" assumes "lfinite xs" shows "(!) (list_of xs) = ($) xs"
IH2: fixes s :: "'a net_state" assumes "s \<in> init (closed (pnet np p2))"   and "i \<in> net_tree_ips p2" shows "the (netmap s i) \<in> init (np i)"
conc_fun_chain: fixes R :: "('e \<times> 'f) set"   and S :: "('f \<times> 'g) set"   and M :: "'g nres" shows "\<Down> R (\<Down> S M) = \<Down> (R O S) M"
lm126: fixes x :: "'a"   and n :: "nat"   and l :: "'a list" shows "set (rotateLeft (length (x # rotateLeft n l) - n mod length (x # rotateLeft n l)) (x # rotateLeft n l)) = {x} \<union> set l"
conjugate_cancel_iff: fixes conjugate :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.conjugate conjugate" shows "(conjugate a = conjugate b) = (a = b)"
countable_wtFsym: shows "countable {f. wtFsym f}"
has_absolute_integral_change_of_variables_linear: fixes g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "linear g" shows "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \<and> integral (g ` S) f = b)"
apply_not_equal: fixes x :: "'b"   and y :: "'b" shows "apply_commute Uprod.not_equal x y = (x \<noteq> y)"
simps_6: fixes \<alpha> :: "('e, 'f, 'd) hp"   and p :: "('e, 'f, 'd) formula" shows "FVF (\<langle> \<alpha> \<rangle> p) = FVP \<alpha> \<union> (FVF p - MBV \<alpha>)"
ln_exp_cf6_lower_bound_pos: fixes x :: "real" assumes "0 \<le> x" shows "ln (exp_cf6 x) \<le> x"
singleton_insert_inj_eq': fixes a :: "'b"   and A :: "'b set"   and b :: "'b" shows "(a \<triangleright> A = {b}) = (a = b \<and> A \<subseteq> {b})"
distinct_Cons_remove1: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "distinct (x # remove1 x xs) = distinct xs"
\<QQ>_def: fixes \<Sigma> :: "'a list" shows "\<QQ> \<Sigma> \<equiv> embed2.samequot_exec lookup (\<epsilon> \<Sigma>) (\<lambda>(x, y). Singleton x y)"
iterative_HSem: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list"   and ESem :: "'b \<Rightarrow> ('a \<Rightarrow> 'c) \<rightarrow> 'c"   and e :: "'b"   and \<rho> :: "'a \<Rightarrow> 'c" assumes "x \<notin> domA \<Gamma>" shows "has_ESem.HSem ESem ((x, e) # \<Gamma>)\<cdot>\<rho> = (\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> has_ESem.HSem ESem \<Gamma>\<cdot>xa)(x := ESem e\<cdot>xa))"
imp_split_smeq_axioms: shows "imp_split_smeq split_fun"
DERIV_isCont: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b" assumes "(f has_field_derivative D) (at x)" shows "isCont f x"
dbind_mono_1: fixes M :: "'b dres"   and M' :: "'b dres"   and f :: "'b \<Rightarrow> 'c dres"   and f' :: "'b \<Rightarrow> 'c dres" assumes "M \<le> M'"   and "\<And>x. dRETURN x \<le> M \<Longrightarrow> f x \<le> f' x" shows "M \<bind> f \<le> M' \<bind> f'"
less_than_char_def: fixes x :: "char"   and y :: "char" shows "((x, y) \<in> less_than_char) = less_char x y"
elims_2: fixes x :: "'c formula" assumes "is_lit_plus x" obtains "x = \<bottom>"   | "x = \<^bold>\<not> \<bottom>"   | uu_ :: "'c" where "x = Atom uu_"   | uv_ :: "'c" where "x = \<^bold>\<not> (Atom uv_)"
g_ipurge_tr: fixes L :: "'a \<Rightarrow> g_level"   and xs :: "'a list" shows "ipurge_tr g_I L High xs = filter (\<lambda>x. L x = Low) xs"
path_image_mirror: fixes g :: "real \<Rightarrow> 'a" shows "path_image (- g) = uminus ` path_image g"
rep_eq: fixes x :: "'c regular"   and xa :: "'c regular" shows "Rep_regular (x \<squnion> xa) = Rep_regular x \<squnion> Rep_regular xa"
simps_12: fixes b :: "bool"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and objRef :: "expr"   and mn :: "mname"   and pTs :: "ty list"   and args :: "expr list" shows "assigns_if b ({accC,statT,mode}objRef\<cdot>mn( {pTs}args)) = assignsE ({accC,statT,mode}objRef\<cdot>mn( {pTs}args))"
acc_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "acc A r = wf {(y, x). x \<in> A \<and> y \<in> A \<and> x \<sqsubset>\<^bsub>r\<^esub> y}"
upset_set_to_downset_set: shows "\<Up> = (`) \<partial> \<circ> \<Down> \<circ> (`) \<partial>"
fBexI: fixes P :: "'h \<Rightarrow> bool"   and x :: "'h"   and A :: "'h fset" assumes "P x"   and "x |\<in>| A" shows "fBex A P"
norms_1: shows "gauss_int_norm (of_nat x + \<i>\<^sub>\<int> * of_nat y) = p"
return_bind: fixes x :: "'aa"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog" shows "Heap_Error_Monad.bind (return x) g = g x"
llist_of_tllist_simps_1: fixes x :: "'g" shows "llist_of_tllist (TNil x) = LNil"
R1: shows "R \<V>1 Tr\<^bsub>ES1\<^esub>"
simps: fixes ctf :: "String.literal \<Rightarrow> typ option"   and arf :: "String.literal \<Rightarrow> nat option"   and oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" shows "wf_sig (ctf, arf, oss) = (wf_osig oss \<and> dom (tcsigs oss) = dom arf \<and> (\<forall>type\<in>dom (tcsigs oss). \<forall>ars\<in>ran (the (tcsigs oss type)). the (arf type) = length ars) \<and> Ball (ran ctf) (wf_type (ctf, arf, oss)))"
is_Sup_upper: fixes A :: "'b set"   and sup :: "'b"   and x :: "'b" assumes "is_Sup A sup"   and "x \<in> A" shows "x \<sqsubseteq> sup"
as: shows "as \<equiv> THE ds. length ds = length vs \<and> v = ds \<bullet>\<cdot> vs"
int_pow_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> i \<in> carrier G"
xevw: shows "xevw = \<lbrakk>\<guillemotleft>Ae\<guillemotright>\<rbrakk>e"
that: fixes d :: "'a" assumes "a = fst (normalize_quot (a, b)) * d"   and "b = snd (normalize_quot (a, b)) * d"   and "d dvd a"   and "d dvd b"   and "d \<noteq> (0::'a)" shows "thesis"
ran_m_fmdrop: fixes C :: "'c"   and N :: "('c, 'd) fmap" assumes "C \<in># dom_m N" shows "ran_m (fmdrop C N) = remove1_mset (the (fmlookup N C)) (ran_m N)"
split: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
assms_1: shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T"
Quotient3_rel_abs: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and Abs :: "'k \<Rightarrow> 'l"   and Rep :: "'l \<Rightarrow> 'k"   and r :: "'k"   and s :: "'k" assumes "Quotient3 R Abs Rep"   and "R r s" shows "Abs r = Abs s"
hoare_valid_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "(\<Turnstile> {P} c {Q}) = (\<forall>s t. P s \<and> (c, s) \<Rightarrow> t \<longrightarrow> Q t)"
u_simps_5: shows "cod r\<^sub>0s\<^sub>1.p\<^sub>0 = r\<^sub>0s\<^sub>1.p\<^sub>0"
count: fixes x :: "'k multiset" shows "count x \<in> multiset"
prefix_Cons: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))"
m: shows "max_Var_floatariths fas \<le> DIM('a) + length params"
in_set_suffixes: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<in> set (suffixes ys)) = suffix xs ys"
s: shows "n - Suc (length w) + 1 = n - length w"
fmdom_drop_fset: fixes A :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmdom (fmdrop_fset A m) = fmdom m |-| A"
expandTrans_2: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q" shows "P \<parallel> Q \<longmapsto> a[b] \<prec> P' = R \<longmapsto> a[b] \<prec> P'"
\<eta>0'_def: shows "\<eta>0' = inv_renaming \<rho> \<odot> \<eta>0"
finite_has_minimal2: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. m \<le> a \<and> (\<forall>b\<in>A. b \<le> m \<longrightarrow> m = b)"
assms: fixes i :: "'b" assumes "i \<in> Basis" shows "((\<lambda>x. blinfun_apply (f x) i) has_derivative blinfun_apply (f' i)) (at x)"
codomain_restrict_chart_if: fixes S :: "'d set"   and c :: "('d, 'c) chart" shows "codomain (restrict_chart S c) = (if open S then codomain c \<inter> inv_chart c -` S else {})"
oFix_least: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and a :: "ordinal" assumes "mono F"   and "F x = x"   and "a \<le> x" shows "oFix F a \<le> x"
Sup_upper: fixes x :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set" assumes "s2p A x" shows "less_eq_prod x (Sup_prod A)"
lspace_1: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "S \<in> sets lebesgue" shows "(f \<in> lspace (lebesgue_on S) 1) = (f absolutely_integrable_on S)"
validD_L: fixes Cn :: "'b set \<Rightarrow> 'b set"   and \<phi> :: "'b"   and A :: "'b set" assumes "Tarskian_logic Cn"   and "\<phi> \<in> Cn {}" shows "\<phi> \<in> Cn A"
prover_def: fixes N :: "((nat, nat) Term.term literal list \<times> nat) list" shows "prover N = (case deterministic_RP (St0 N 0) of None \<Rightarrow> True | Some R \<Rightarrow> [] \<notin> set R)"
uint_or: fixes x :: "'a word"   and y :: "'a word" shows "uint (x OR y) = uint x OR uint y"
t3_space: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and y :: "'c" assumes "class.t3_space open"   and "topological_space.closed open S"   and "y \<notin> S" shows "\<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}"
R1_14_interm1_def: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and choice :: "bool \<times> bool" shows "R1_14_interm1 msgs choice = (let (m00, m01, m10, m11) = msgs; (c0, c1) = choice in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. S1_OT12 (S0, S1) () \<bind> (\<lambda>a. R1_OT12 (S2, S3) c1 \<bind> (\<lambda>b. R1_OT12 (S4, S5) c1 \<bind> (\<lambda>c. return_spmf (msgs, (S0, S1, S2, S3, S4, S5), a, b, c)))))))))))"
iprev_singleton_cut_less_empty_iff: fixes t0 :: "nat"   and I :: "nat set" shows "({iprev t0 I} \<down>< t0 = {}) = (I \<down>< t0 = {} \<or> t0 \<notin> I)"
update_by_fun_pair_ko_def: fixes ko :: "'a key_order" shows "update_by_fun_pair_ko ko \<equiv> comparator.update_by_fun_pair (key_compare ko)"
SourcesA7_L0: shows "Sources level0 sA7 = {sA6}"
max_input_def: fixes g :: "vname aexp" shows "max_input g = (let inputs = enumerate_aexp_inputs g in if inputs = {} then None else Some (Max inputs))"
s: shows "(x * y + (q - (x * ((y + (q - b)) mod q) + (q - r)) mod q)) mod q = ((x + a) mod q * b + (q - (a * b + (q - r)) mod q)) mod q"
map_ticket_addsimps_4: fixes f :: "nat \<Rightarrow> nat"   and t :: "thread"   and tk :: "nat" shows "(map_ticket f t = REL tk) = (\<exists>tk'. tk = f tk' \<and> t = REL tk')"
bisim_composeI: fixes bisim12 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'h"   and bisim23 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and s3 :: "'i" assumes "bisim12 s1 s2"   and "bisim23 s2 s3" shows "(bisim12 \<circ>\<^sub>B bisim23) s1 s3"
plus_pp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "rep_nat_pp (plus x y) = rep_nat_pp x + rep_nat_pp y"
dghm_HomDom: shows "\<FF>\<lparr>HomDom\<rparr> = \<AA>"
rep_eq: fixes x :: "'a ltln\<^sub>Q"   and xa :: "'a ltln\<^sub>Q" shows "local.ltln\<^sub>Q_eq_test x xa = rep_ltln\<^sub>Q x \<sim>\<^sub>Q rep_ltln\<^sub>Q xa"
symm_closure_of_indRelR_2: shows "indRelEQ = (symcl (indRelR\<^sup>=))\<^sup>+"
cong_mersenne_number_int: fixes k :: "int"   and n :: "nat" shows "[k mod 2 ^ n + k div 2 ^ n = k] (mod 2 ^ n - 1)"
option_of_exception_exception_of_option: fixes x :: "'h option" shows "option_of_exception (exception_of_option x) = x"
subset_insertI2: fixes A :: "'c set"   and B :: "'c set"   and b :: "'c" assumes "A \<subseteq> B" shows "A \<subseteq> insert b B"
minimal: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.pcpo below" shows "below (pcpo.bottom below) x"
DC: shows "\<forall>ts'. ts \<^bold>\<Rightarrow> ts' \<longrightarrow> (\<forall>x xa. x \<noteq> xa \<longrightarrow> (\<nexists>va u. move ts ts' v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts' x = ext v \<and> restrict v (res ts') x = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' xa = ext v \<and> restrict v (res ts') xa = lan v \<and> |lan v| = 1) \<and> True)) \<and> True)) \<longrightarrow> (\<forall>ts'a. ts' \<^bold>\<leadsto> ts'a \<longrightarrow> (\<nexists>va u. move ts' ts'a (move ts ts' v)=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts'a x = ext v \<and> restrict v (res ts'a) x = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts'a xa = ext v \<and> restrict v (res ts'a) xa = lan v \<and> |lan v| = 1) \<and> True)) \<and> True))))"
t1: assumes "Q\<^sup>T *\<^sub>v v - v = 0\<^sub>v (dim_vec v)" shows "(Q\<^sup>T - 1\<^sub>m (dim_row Q)) *\<^sub>v v = 0\<^sub>v (dim_vec v)"
simps_3: fixes OP :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b"   and r :: "'b"   and v :: "'b"   and va :: "'b list" shows "rexp_of_list OP N (r # v # va) = OP r (rexp_of_list OP N (v # va))"
GR: fixes q :: "('a, 'b) fm"   and p :: "('a, 'b) fm"   and a :: "'a" assumes "\<turnstile> q \<^bold>\<longrightarrow> p\<langle>\<^bold>\<dagger> a []/0\<rangle>"   and "a \<notin> params {p, q}" shows "\<turnstile> q \<^bold>\<longrightarrow> \<^bold>\<forall> p"
nn_integral_cong_AE: fixes u :: "'c \<Rightarrow> ennreal"   and v :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "AE x in M. u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
isomorphic_density: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2" shows "Ugraph_Lemmas.density G\<^sub>1 = Ugraph_Lemmas.density G\<^sub>2"
The_Min: fixes P :: "'b \<Rightarrow> bool" assumes "Ex1 P" shows "The P = Min {x. P x}"
p'_def: shows "convert_poly var p (xs' @ x # xs) = Some p'"
nvlr_length: fixes old :: "nat"   and v :: "nat"   and f :: "face"   and ls :: "nat list" shows "|natToVertexListRec old v f ls| = |ls|"
refl: fixes g :: "'g"   and \<phi> :: "'val" assumes "unnecessaryPhi g \<phi>" shows "\<phi> \<in> reachable g \<phi>"
path_first_edge': fixes v :: "'a"   and xs :: "'a list" assumes "H.path (v # xs)"   and "xs \<noteq> []" shows "v \<rightarrow>\<^bsub>H\<^esub> hd xs"
join_eq_bind_pmf: fixes M :: "'c pmf pmf" shows "join_pmf M = M \<bind> id"
bprv_\<phi>G_eqv: shows "bprv (eqv \<phi>G (neg (PP \<langle>\<phi>G\<rangle>)))"
jumpF_polyL_mod: fixes q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jumpF_polyL q p x = jumpF_polyL (q mod p) p x"
simps: fixes a :: "real" shows "(a \<in> radical_sqrt) = ((\<exists>x. a = x \<and> x \<in> \<rat>) \<or> (\<exists>x. a = - x \<and> x \<in> radical_sqrt) \<or> (\<exists>x. a = 1 / x \<and> x \<in> radical_sqrt \<and> x \<noteq> 0) \<or> (\<exists>x y. a = x + y \<and> x \<in> radical_sqrt \<and> y \<in> radical_sqrt) \<or> (\<exists>x y. a = x * y \<and> x \<in> radical_sqrt \<and> y \<in> radical_sqrt) \<or> (\<exists>x. a = sqrt x \<and> x \<in> radical_sqrt \<and> 0 \<le> x))"
exhaustive_nat_def: fixes f :: "nat \<Rightarrow> (bool \<times> term list) option"   and d :: "natural" shows "exhaustive_class.exhaustive f d = exhaustive_class.exhaustive (\<lambda>x. f (nat_of_natural x)) d"
kh2: fixes y :: "real" assumes "y \<in> {0..1}" shows "k (y, 0) = h2 0"
takedropWhile_queue_fst: fixes f :: "'a \<Rightarrow> bool"   and q :: "'a queue" shows "fst (takedropWhile_queue f q) = dropWhile_queue f q"
distinct_3: fixes x31 :: "'b"   and x32 :: "'b"   and x33 :: "'c" shows "DenyAll \<noteq> AllowPortFromTo x31 x32 x33"
elims: fixes x :: "'d decision decision"   and y :: "'d decision" assumes "flat_orD x = y" obtains ya :: "'d" where "x = allow (allow ya__)" and "y = allow ya__"   | ya :: "'d" where "x = allow (deny yaa__)" and "y = deny yaa__"   | ya :: "'d" where "x = deny (allow yab__)" and "y = deny yab__"   | ya :: "'d" where "x = deny (deny ya)" and "y = deny ya"
assms_2: shows "{0..} \<subseteq> existence_ivl0 x"
Rep_formula_inverse: fixes x :: "('b, 'c, 'd) formula" shows "Abs_formula (Rep_formula x) = x"
simps: fixes a :: "'a"   and p :: "'b \<times> 'a" shows "(a \<in> Basic_BNFs.snds p) = (a = snd p)"
free_thread_id_def: shows "free_thread_id \<equiv> \<lambda>ts t. lfp (\<lambda>p. ts t = None)"
elims_1: fixes x :: "'b \<times> 'b"   and y :: "bool" assumes "is_interval x = y" obtains l :: "'b" and u :: "'b" where "x = (l, u)" and "y = (l < u)"
simps_3: fixes x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (ELam x e) \<rho> = {VFun f |f. \<forall>v1 v2. (v1, v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)}"
hyps_1: shows "\<Gamma>,\<Theta>_\<Turnstile>\<^sub>t\<^bsub>/F_\<^esub> P_ c_ Q_,A_"
IH: fixes z :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term"   and yb :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" assumes "z \<in> set T"   and "yb \<in> set S"   and "equal_mod_timpls TI z yb"   and "wf\<^sub>t\<^sub>r\<^sub>m z"   and "\<forall>f\<in>funs_term z. is_Abs f \<longrightarrow> the_Abs f \<in> A" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) z yb"
sound_modes_no_read: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem :: "'a \<Rightarrow> 'b"   and x :: "'a"   and i :: "nat" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.sound_mode_use dma \<C>_vars eval (cms, mem)"   and "x \<in> (map snd cms ! i) GuarNoReadOrWrite"   and "i < length cms" shows "sifum_security_init.doesnt_read_or_modify \<C>_vars eval (fst (cms ! i)) x"
Length_SNOC: fixes x :: "bool list"   and \<AA> :: "interp" shows "Length (SNOC x \<AA>) = Suc (Length \<AA>)"
simps_95: fixes x61 :: "'b formula"   and x62 :: "'b formula" shows "atoms (x61 \<^bold>\<rightarrow> x62) = atoms x61 \<union> atoms x62"
update_convs_3: fixes Dom' :: "'d \<Rightarrow> 'd"   and Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e" shows "Dom_update Dom' \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = \<lparr>Pol = Pol, Map = Map, Dom = Dom' Dom, In = In, Out = Out, \<dots> = more\<rparr>"
exec_Normal_elim_cases_4: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>Skip,Normal s\<rangle> \<Rightarrow> t" obtains "t = Normal s"
assms_4: shows "martingale N G (discounted_value r (prices Mkt stk))"
mpath'_correct: fixes wl :: "'V \<Rightarrow> 'W infty option"   and res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and v :: "'V" assumes "dinvarm (wl, res)" shows "mpath' (res v) = map_option rev (\<alpha>r res v)"
that: fixes io :: "('a \<times> 'b) list" assumes "sequence_to_failure M1 M2 io" shows "thesis"
lem1: shows "\<forall>z. infnorm (negatex z) = infnorm z"
focus_eq_zero_iff: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "(focus X p = 0) = (p = 0)"
TS_non_containment_causes_final_suc: fixes vs :: "'d list"   and xs :: "'d list"   and M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<notin> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "set xs \<subseteq> inputs M2"   and "final_iteration M2 M1 \<Omega> V m i"   and "OFSM M2" obtains xr :: "'d list" and j :: "nat" where "xr \<noteq> xs" and "prefix xr xs" and "Suc j \<le> i" and "vs @ xr \<in> RM M2 M1 \<Omega> V m (Suc j)"
card_verts: fixes G :: "graph" shows "card (verts G) = Dijkstra.size G"
comp_null_2: fixes f :: "'a" shows "L.comp f L.null = L.null"