hcju/isabelleps huggingface dataset:
is_sup_join: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "dpo.is_sup le x y (dlat.join le x y)"
matrix_scaleR_vector_ac: fixes A :: "((real, 'b) vec, 'a) vec"   and k :: "real"   and v :: "(real, 'b) vec" shows "A *v k *\<^sub>R v = k *\<^sub>R A *v v"
square_free_factorization_int_main: fixes f :: "int poly"   and fs :: "(int poly \<times> nat) list" assumes "square_free_factorization_int_main f = fs"   and "content f = 1"   and "0 < lead_coeff f"   and "degree f \<noteq> 0" shows "square_free_factorization f (1, fs) \<and> (\<forall>fi i. (fi, i) \<in> set fs \<longrightarrow> content fi = 1 \<and> 0 < lead_coeff fi) \<and> distinct (map snd fs)"
multrow_mat_def: fixes n :: "nat"   and k :: "nat"   and a :: "'c" shows "multrow_mat n k a = mat n n (\<lambda>(i, j). if k = i \<and> k = j then a else if i = j then 1::'c else (0::'c))"
sums_minus: fixes a :: "'a" assumes "f sums a" shows "(\<lambda>n. - f n) sums - a"
CH_range: fixes A :: "'x set" shows "CH A \<subseteq> A"
supp_perm_eq_test: fixes x :: "'b"   and p :: "perm" assumes "supp x \<sharp>* p" shows "p \<bullet> x = x"
SBHReadBuffered: fixes sb :: "'b memref list"   and a :: "nat"   and v :: "nat"   and volatile :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "buffered_val sb a = Some v" shows "(Read volatile a t # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>(t \<mapsto> v), sb @ [Read\<^sub>s\<^sub>b volatile a t v], m, \<D>, \<O>, \<R>, \<S>)"
set_iterator_genord_foldri_correct: fixes xs :: "'d list"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "distinct xs"   and "sorted_wrt R (rev xs)" shows "set_iterator_genord (foldri xs) (set xs) R"
length_chop_part_le: fixes ys :: "'b list"   and n :: "nat"   and xs :: "'b list" assumes "ys \<in> set (chop n xs)" shows "length ys \<le> n"
fresh_1: fixes a :: "name"   and x2 :: "'a"   and x1 :: "('a, 'b, 'c) psi" shows "a \<sharp> \<rparr>x2.x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
pfd_homomorphism_def: fixes lift :: "'a \<Rightarrow> 'b" shows "pfd_homomorphism lift \<equiv> ((\<forall>a b. lift (a + b) = lift a + lift b) \<and> (\<forall>a b. lift (a * b) = lift a * lift b)) \<and> lift (0::'a) = (0::'b) \<and> lift (1::'a) = (1::'b)"
gen_succ: fixes rel :: "('a \<times> 'a) list"   and as :: "'a list" shows "set (gen_succ rel as) = {b. \<exists>a\<in>set as. (a, b) \<in> set rel}"
unsynlr_sup: fixes x :: "('a, 'b) synlr"   and y :: "('a, 'b) synlr" shows "unsynlr (sup x y) = unsynlr x \<union> unsynlr y"
discrete_top_def: fixes X :: "'b set" shows "discrete_top X = Pow X"
valid_reads: fixes m :: "nat \<Rightarrow> nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "valid_reads m ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "reads_consistent False \<O> m sb"
IChoose_def: fixes s :: "Istate"   and s' :: "Istate"   and p :: "Proc" shows "IChoose s s' p = (ioutput s p = NotAnInput \<and> (if ichosen s = NotAnInput then \<exists>ip\<in>iallInput s. ichosen s' = ip \<and> ioutput s' = (ioutput s)(p := ip) else ioutput s' = (ioutput s)(p := ichosen s) \<and> ichosen s' = ichosen s) \<and> iinput s' = iinput s \<and> iallInput s' = iallInput s)"
mask_zero: fixes x :: "'a word"   and a :: "nat" assumes "is_aligned x a" shows "x && mask a = 0"
comm_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and a :: "'a" assumes "comm_monoid f z" shows "f a z = a"
nn_integral_set_ennreal: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "\<integral>\<^sup>+ x. ennreal (f x) * indicator A x \<partial>M = \<integral>\<^sup>+ x. ennreal (f x * indicat_real A x) \<partial>M"
Lr_def: fixes F'' :: "'a filter"   and g'' :: "'a \<Rightarrow> real" shows "Lr F'' g'' = {f. \<exists>c>0. \<forall>\<^sub>F x in F''. R (norm (f x)) (c * norm (g'' x))}"
ord_iso_ord_embed: fixes f :: "'c \<Rightarrow> 'd" assumes "ord_iso f" shows "ord_embed f"
finsum_addf: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> carrier R"   and "g \<in> A \<rightarrow> carrier R" shows "(\<Oplus>x\<in>A. f x \<oplus> g x) = finsum R f A \<oplus> finsum R g A"
plus_conv: fixes x :: "'a" shows "(x\<^sup>+)\<^sup>T = x\<^sup>T\<^sup>+"
cod_dom: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.cod (VoV.dom f) = VoV.dom f"
assms_7: shows "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)"
nrows_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "nrows A = CARD('d)"
set_mtf2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "set (mtf2 n x xs) = set xs"
inverse_le_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse a) (inverse b) = ((less zero (times a b) \<longrightarrow> less_eq b a) \<and> (less_eq (times a b) zero \<longrightarrow> less_eq a b))"
arctan_less_pi4_neg: fixes x :: "real" assumes "- 1 < x" shows "- (pi / 4) < arctan x"
poly_roots_degree: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "card {x. poly p x = (0::'b)} \<le> degree p"
continuous_on_inverse: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "\<forall>x\<in>s. f x \<noteq> (0::'c)" shows "continuous_on s (\<lambda>x. inverse (f x))"
graph_nodes_it_def: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) graph"   and invar :: "'a \<Rightarrow> bool"   and nodes_list_it :: "'a \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list" shows "graph_nodes_it \<alpha> invar nodes_list_it \<equiv> graph \<alpha> invar \<and> graph_nodes_it_axioms \<alpha> invar nodes_list_it"
wf_idecl_mhead: fixes G :: "prog"   and I :: "qtname"   and i :: "iface"   and sig :: "sig"   and mh :: "mhead" assumes "wf_idecl G (I, i)"   and "(sig, mh) \<in> set (imethods i)" shows "wf_mhead G (pid I) sig mh \<and> \<not> is_static mh \<and> accmodi mh = Public"
stable_no_blocking_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "PdXXX.stable_no_blocking_on ds X = (\<forall>h X'. \<not> PdXXX.blocking_on ds X h X')"
cont_case_prodI: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and lubb :: "'e set \<Rightarrow> 'e"   and luba :: "'d set \<Rightarrow> 'd" assumes "monotone (rel_prod orda ordb) leq (\<lambda>(x, y). f x y)"   and "\<And>x. cont lubb ordb lub leq (f x)"   and "\<And>y. cont luba orda lub leq (\<lambda>x. f x y)"   and "class.preorder orda (mk_less orda)"   and "class.preorder ordb (mk_less ordb)" shows "cont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (\<lambda>(x, y). f x y)"
finite_in_span: fixes S :: "'c set"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "v \<in> span S" shows "\<exists>a. a \<in> S \<rightarrow> carrier K \<and> lincomb a S = v"
abs_convergent_prod_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'e"   and m :: "nat" assumes "abs_convergent_prod f" shows "abs_convergent_prod (\<lambda>n. f (n + m))"
postdominate_implies_path: fixes n' :: "'node"   and n :: "'node" assumes "n' postdominates n" obtains as :: "'edge list" where "n -as\<rightarrow>* n'"
renaming: fixes xvec :: "name list"   and Tvec :: "'a list"   and p :: "(name \<times> name) list"   and T :: "'a" assumes "length xvec = length Tvec"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "(p \<bullet> xvec) \<sharp>* T" shows "substTerm T xvec Tvec = substTerm (p \<bullet> T) (p \<bullet> xvec) Tvec"
map_cong: fixes x :: "'b stream"   and ya :: "'b stream"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> sset ya \<Longrightarrow> f z = g z" shows "smap f x = smap g ya"
inf_sup_distrib2: fixes 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 y :: "'d"   and z :: "'d"   and x :: "'d" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
subset_dual: fixes X :: "'b set"   and Y :: "'b set" shows "(X = \<partial> ` Y) = (\<partial> ` X = Y)"
sel: fixes x :: "('a, 'b, ('a, 'b) sp\<^sub>\<nu>) sp\<^sub>\<mu>" shows "out (In x) = x"
resE_positive: fixes e :: "nat \<times> nat" assumes "e \<in> l'.cf.E" shows "(0::'a) < l'.cf e"
older_seniors_tower'': fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "\<not> sink (token_run x n)"   and "\<not> sink (token_run y n)"   and "older_seniors x n = older_seniors x (Suc n)"   and "older_seniors y n \<subseteq> older_seniors x n" shows "older_seniors y n = older_seniors y (Suc n)"
failures_traces: fixes xs :: "'c list"   and X :: "'c set"   and P :: "'c process" assumes "(xs, X) \<in> failures P" shows "xs \<in> traces P"
zero_matrix_def: shows "0 \<equiv> Matrix (\<lambda>i j. 0::'c)"
vint_mult_closed: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x *\<^sub>\<int> y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
sminus_zero: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.sminus pair_of_term term_of_pair v (0::'b) = v"
borel_measurable_restrict_space_iff_ennreal: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(f \<in> borel_measurable (restrict_space M \<Omega>)) = ((\<lambda>x. f x * indicator \<Omega> x) \<in> borel_measurable M)"
h2_def: shows "h2 = lead_coeff G2 ^ (degree G1 - degree G2)"
CgE: shows "c\<^sup>2 + 3 * d\<^sup>2 = (gcd c d)\<^sup>2 * (e\<^sup>2 + 3 * f\<^sup>2)"
Ball_set: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
assms_2: shows "g : a \<mapsto>\<^bsub>dg_prod (I -\<^sub>\<circ> J) \<AA>\<^esub> b"
analytic_on_subset: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and T :: "complex set" assumes "f analytic_on S"   and "T \<subseteq> S" shows "f analytic_on T"
zip_insertl: fixes xs :: "'a list"   and ys :: "'b list"   and n :: "nat"   and x :: "'a"   and y :: "'b" assumes "length xs = length ys" shows "zip (insertl n x xs) (insertl n y ys) = insertl n (x, y) (zip xs ys)"
row_carrier: fixes A :: "'a mat"   and i :: "nat" shows "Matrix.row A i \<in> carrier_vec (dim_col A)"
ord_eq_less_trans: fixes a :: "'c"   and b :: "'c"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c" assumes "a = b"   and "less b c" shows "less a c"
term_to_nterm_vars: fixes t :: "Term_Class.term"   and F :: "name fset" assumes "wellformed t" shows "frees (fresh_frun (term_to_nterm [] t) F) |\<subseteq>| frees t"
simulation_silent2: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "r2.silent_move t s2 s2'" shows "\<exists>s1'. r1.silent_moves t s1 s1' \<and> t \<turnstile> s1' \<approx> s2'"
isCont_iff: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0::'a\<rightarrow> f x"
get_owner_document_tups_def: shows "get_owner_document_tups \<equiv> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document_tups get_root_node get_disconnected_nodes"
red_wf_finite: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "wfP (punit.red F)\<inverse>\<inverse>"
wf_no_infinite_down_chainI: fixes r :: "('c \<times> 'c) set" assumes "\<And>f. (\<And>i. (f (Suc i), f i) \<in> r) \<Longrightarrow> False" shows "wf r"
hoare_conseq_left: fixes P' :: "'e set"   and R :: "('e \<times> 'e) set"   and Q :: "'e set"   and P :: "'e set" assumes "{P'} R {> Q}"   and "P \<subseteq> P'" shows "{P} R {> Q}"
assms_5: fixes x' :: "'a option list" assumes "wf_tuple n A' x'"   and "P x'" shows "P (restrict A x')"
rank_skew_rank_invar: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_skew_invar (t # bq)" shows "SkewBinomialHeapStruc.rank_invar bq"
cEnum_finite_1_def: shows "CENUM(Enum.finite_1) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
assms_2: fixes k :: "nat"   and lholed :: "Lholed" shows "\<not> Lfilled k lholed [$Return] cs_es"
ord_spmf_bind_reflI: fixes p :: "'c spmf"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd spmf"   and g :: "'c \<Rightarrow> 'd spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) (g x)" shows "ord_spmf R (p \<bind> f) (p \<bind> g)"
inject_3: fixes x41 :: "stmt"   and x42 :: "stmt"   and y41 :: "stmt"   and y42 :: "stmt" shows "(Seq x41 x42 = Seq y41 y42) = (x41 = y41 \<and> x42 = y42)"
w\<^sub>h_is_map: shows "is_left_adjoint w\<^sub>h"
upper_density_good_direction_invertible: fixes d :: "real" assumes "I.invertible_qmpt"   and "0 < d"   and "d \<le> 1" shows "\<exists>delta. (\<forall>l. 0 < delta l) \<and> delta \<longlonglongrightarrow> 0 \<and> ennreal (1 - d) \<le> emeasure MI {x \<in> space MI. 1 - d \<le> upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - delta l * real l < vI n x - vI (n - l) ((TI ^^ l) x)}}"
zero_flow: shows "flow \<Delta>' (\<lambda>_. 0)"
star_right_induct: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<squnion> x * y \<le> x \<longrightarrow> z * y\<^sup>\<star> \<le> x"
sum_if_distrib_add: fixes A :: "'c set"   and b :: "'c"   and X :: "'c \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'd" assumes "finite A"   and "b \<in> A" shows "(\<Sum>a\<in>A. if a = b then X b + Y a else X a) = sum X A + Y b"
notTerminalNforms: fixes f :: "'a \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and n :: "'a" assumes "\<not> terminal subs (f n)" shows "nforms (f n) \<noteq> []"
a_coset_join2: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier R"   and "subgroup H (add_monoid R)"   and "x \<in> H" shows "H +> x = H"
uniformly_Cauchy_onI': fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>x\<in>X. \<forall>m\<ge>M. \<forall>n>m. dist (f m x) (f n x) < e" shows "uniformly_Cauchy_on X f"
Quotient_alt_def3: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (\<exists>z. T x z \<and> T y z)))"
cod_comp: fixes g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.seq g f" shows "J_C.cod (g \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] f) = J_C.cod g"
mod_mult_self1_is_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times b a) b = zero"
LeastI2: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and Q :: "'d \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (Least P)"
f': shows "\<guillemotleft>f' : a \<rightarrow>\<^sub>B b\<guillemotright> \<and> B.is_left_adjoint f' \<and> F = \<lbrakk>f'\<rbrakk>\<^sub>B"
nth_prime_bigtheta: shows "(\<lambda>x. real (nth_prime x)) \<in> \<Theta>(\<lambda>x. real x * ln (real x))"
size_list: fixes t :: "'b tree" shows "length (list t) = size t"
have_inbetween_point_les: fixes les :: "(real \<times> real \<times> real) list"   and r :: "real" assumes "\<forall>(d, e, f)\<in>set les. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0" shows "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
expectation_gpv2_def: shows "expectation_gpv2 \<equiv> expectation_gpv (expectation_gpv fail \<I> f gpv') \<I> f"
get_nat_sqrt_def: fixes n :: "nat" shows "get_nat_sqrt n = (if is_square n then Some (Discrete.sqrt n) else None)"
mirror_d_\<beta>: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs) \<and> d_\<beta> p 1" shows "iszlfm (mirror p) (a # bs) \<and> d_\<beta> (mirror p) 1"
GreenThm_typeI_divisible_region_boundary: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set"   and \<gamma> :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "green_typeI_chain i j F two_chain s"   and "gen_division s (cubeImage ` two_chain)"   and "two_chain_horizontal_boundary two_chain \<subseteq> \<gamma>"   and "\<gamma> \<subseteq> two_chain_boundary two_chain" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
assms_1: shows "e = (\<lambda>\<sigma>. \<sigma> x - \<sigma> y)"
mproj1_def: fixes R :: "('f, 'g) Ring_scheme"   and M1 :: "('e, 'f, 'h) Module_scheme"   and M2 :: "('e, 'f, 'h) Module_scheme"   and M :: "('e, 'f, 'h) Module_scheme" shows "mproj1 R M1 M2 M = (\<lambda>x\<in>carrier M. THE x1. x1 \<in> carrier M1 \<and> x \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> x1 \<in> carrier M2)"
simps: fixes a1 :: "nat"   and a2 :: "nat"   and xs :: "'a list" shows "((a1, a2) \<in> inversions xs) = (\<exists>i j. a1 = i \<and> a2 = j \<and> i < j \<and> j < length xs \<and> xs ! j < xs ! i)"
segment_a'_a_in: shows "segment G.face_cycle_succ a' a \<subseteq> arcs H"
permute_swap_cancel2: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "atom"   and b :: "atom"   and x :: "'a" assumes "class.pt permute" shows "permute (a \<rightleftharpoons> b) (permute (b \<rightleftharpoons> a) x) = x"
in_outs_\<I>_iff_responses_\<I>: fixes out :: "'h"   and \<I> :: "('h, 'i) \<I>" shows "(out \<in> outs_\<I> \<I>) = (responses_\<I> \<I> out \<noteq> {})"
iwlsFSw_termFSwImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" assumes "iwlsFSw MOD" shows "termFSwImorph h hA MOD = ipresCons h hA MOD"
distinct_mset_add_mset: fixes a :: "'b"   and L :: "'b multiset" shows "distinct_mset (add_mset a L) = (a \<notin># L \<and> distinct_mset L)"
assms_2: shows "a \<in>\<^sub>\<circ> A"
simps_3: fixes uw :: "nat" shows "tvs (Bv uw) = {}"
image_smult_cbox: fixes m :: "real"   and a :: "'c"   and b :: "'c" shows "(*\<^sub>R) m ` cbox a b = (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
precondition_after_cswitch: shows "\<forall>s d n a. AS_precondition s d a \<longrightarrow> AS_precondition (cswitch n s) d a"
bind_game_eq_dis_log: fixes \<A> :: "'a \<Rightarrow> ('a \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "ped_commit.bind_game \<A> = discrete_log.dis_log (dis_log_\<A> \<A>)"
a_remove_child_def: fixes ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" shows "a_remove_child ptr child = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. if child \<notin> set children then error NotFoundError else Heap_Error_Monad.bind (get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)) (\<lambda>owner_document. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes owner_document (child # disc_nodes)) (\<lambda>_. set_child_nodes ptr (remove1 child children)))))"
comp: fixes Z :: "'b set set"   and g :: "'a \<Rightarrow> 'b" assumes "ChamberComplexMorphism X Z g" shows "ChamberComplexMorphism X Z (g \<circ> f)"
infinite_chain_mono: fixes S :: "('a \<times> 'a) set"   and T :: "('a \<times> 'a) set" assumes "S \<subseteq> T"   and "\<exists>f. \<forall>i. (f i, f (Suc i)) \<in> S" shows "\<exists>f. \<forall>i. (f i, f (Suc i)) \<in> T"
ide_char: fixes f :: "'a" shows "ide f = (f \<in> Obj)"
circline_set_moebius_circline: fixes M :: "moebius"   and H :: "circline" shows "circline_set (moebius_circline M H) = moebius_pt M ` circline_set H"
above: fixes y :: "'a" assumes "y \<in> Lower L A" shows "y \<sqsubseteq> i"
seq_loc_reach: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle>" shows "(\<exists>c''. c' = c'' ;; c\<^sub>2 \<and> \<langle>c'', mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>1, mds, mem\<rangle>) \<or> (\<exists>c'' mds'' mem''. \<langle>Stop, mds'', mem''\<rangle> \<in> loc_reach \<langle>c\<^sub>1, mds, mem\<rangle> \<and> \<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>c\<^sub>2, mds'', mem''\<rangle>)"
LeftDerivationFix_is_sentence: fixes a :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and b :: "symbol list" assumes "LeftDerivationFix a i D j b" shows "is_sentence a \<and> is_sentence b"
foundation9'': fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> not x" shows "\<tau> \<Turnstile> \<delta> x"
t_def: shows "t = (\<lambda>n. s (n + k))"
eint_pow_int_is_pos: fixes n :: "int" assumes "0 < n" shows "0 < eint n"
map_ident: fixes t :: "('a, 'b) fmap" shows "fmmap (\<lambda>x. x) t = t"
intensional_simps_19: fixes P :: "'b \<Rightarrow> bool" shows "#False \<and> P \<equiv> #False"
below_refl: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.po below" shows "below x x"