hcju/isabelleps huggingface dataset:
order_consistentI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a a'. \<lbrakk>r a a'; s a' a\<rbrakk> \<Longrightarrow> a = a'" shows "order_consistent r s"
simps_17: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (SCOPE c, s) = Some (c;; Assign_Locals s, <<>|s>)"
invar_r: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" assumes "invar (Node k v l r)" shows "invar r"
le_cases3: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" obtains "x \<le>s y" and "y \<le>s z"   | "y \<le>s x" and "x \<le>s z"   | "x \<le>s z" and "z \<le>s y"   | "z \<le>s y" and "y \<le>s x"   | "y \<le>s z" and "z \<le>s x"   | "z \<le>s x" and "x \<le>s y"
sinh_real_nonpos_iff: fixes x :: "real" shows "(sinh x \<le> 0) = (x \<le> 0)"
normality_factorization: fixes N :: "'a set"   and H :: "'a set" assumes "N \<lhd> G"   and "N \<subseteq> H"   and "H \<lhd> G" shows "rcosets\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> N \<lhd> G Mod N"
Refl: fixes P :: "pi" shows "P \<equiv>\<^sub>s P"
RedBinOp: fixes bop :: "bop"   and v\<^sub>1 :: "val"   and v\<^sub>2 :: "val"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> Val v\<^sub>2,s\<rangle> \<rightarrow> \<langle>Val v,s\<rangle>"
assms_3: shows "\<pp>\<^sub>0[b, c] \<cdot> f = f0"
pol_expr_edeg: fixes p :: "'a"   and d :: "nat" assumes "p \<in> carrier R"   and "deg R S X p \<le> an d" shows "\<exists>f. pol_coeff S f \<and> fst f = d \<and> p = polyn_expr R X d f"
assms_2: shows "G \<simeq>\<^sub>F H"
mult_one_div_unit_factor: fixes a :: "'a"   and b :: "'a" shows "a * ((1::'a) div unit_factor b) = a div unit_factor b"
absorb2: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "max a b = b"
ap_gpv_comp: fixes u :: "('j \<Rightarrow> 'f, 'g, 'h) gpv"   and v :: "('i \<Rightarrow> 'j, 'g, 'h) gpv"   and w :: "('i, 'g, 'h) gpv" shows "Generative_Probabilistic_Value.Done (\<circ>) \<diamondop> u \<diamondop> v \<diamondop> w = u \<diamondop> (v \<diamondop> w)"
ord_eq_le_trans: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a = b"   and "b \<subseteq># c" shows "a \<subseteq># c"
\<alpha>': shows "\<beta> = \<alpha>' @ \<delta> \<and> LeftDerives1 \<alpha> i r \<alpha>' \<and> i + length (snd r) \<le> length \<alpha>'"
mergesort_by_rel_simps_2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "mergesort_by_rel R [x] = [x]"
space_lim: shows "space local.lim = space (Pi\<^sub>M I M)"
le_infE: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream"   and v\<^sub>1 :: "'a list" assumes "traces ind"   and "traces.le_inf ind w\<^sub>1 w\<^sub>2"   and "v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1" obtains "traces.le_fininf ind v\<^sub>1 w\<^sub>2"
lub_spmf_least: fixes Y :: "'c spmf set"   and z :: "'c spmf" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "\<And>x. x \<in> Y \<Longrightarrow> ord_spmf (=) x z" shows "ord_spmf (=) (lub_spmf Y) z"
when_mult: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and P :: "bool"   and b :: "'a" assumes "class.mult_zero times zero" shows "times (zero.when zero a P) b = zero.when zero (times a b) P"
right_neutral: fixes a :: "'a" shows "a \<squnion> \<bottom> = a"
Map_dom: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Map (local.dom f) = Dom f"
inj_on_inv_into: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "B \<subseteq> f ` A" shows "inj_on (inv_into A f) B"
diff_iff: fixes f :: "'a \<Rightarrow> 'e" shows "diff k M1 M3 f = (\<forall>x\<in>f.src.carrier. \<exists>c1\<in>f.src.atlas. \<exists>c2\<in>g.dest.atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1))"
power_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a"   and i :: "nat" assumes "z \<squnion> y \<cdot> x \<le> y" shows "z \<cdot> x ^ i \<le> y"
simps_3: fixes bs :: "nat \<Rightarrow> 'a"   and a :: "'a poly" shows "insertion bs (Neg a) = - insertion bs a"
sterm_continuous: fixes I :: "('d, 'e, 'f) interp"   and \<theta> :: "('d, 'f) trm" assumes "is_interp I"   and "dfree \<theta>" shows "continuous_on UNIV (sterm_sem I \<theta>)"
Rep_perm_uminus: fixes p :: "perm" shows "Rep_perm (- p) = inv (Rep_perm p)"
symm_f: fixes x :: "'a"   and y :: "'a" shows "symm_f x y = symm_f y x"
B: shows "ofs \<in> offsets_is (list_of_array ins)"
IH: fixes xs :: "('a \<times> 'b \<times> bool) list" assumes "xsa__ prefix of i"   and "apply_operations xsa__ = Some xs" shows "set (map fst xs) = set (indices xsa__)"
numbers_of_marks_sieve: fixes n :: "nat"   and bs :: "bool list" shows "numbers_of_marks (Suc n) (sieve n bs) = {q \<in> numbers_of_marks (Suc n) bs. \<forall>m\<in>numbers_of_marks (Suc n) bs. \<not> m dvd_strict q}"
not_msb_from_less: fixes v :: "'b word" assumes "v < 2 ^ (LENGTH('b) - 1)" shows "\<not> msb v"
dim_hyp2: shows "matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) \<in> carrier_mat (length signs) (length signs)"
before_in_setD2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x < y in xs" shows "y \<in> set xs"
homotopic_paths_def: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" shows "homotopic_paths s p q \<equiv> homotopic_with_canon (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
mon_w_alt: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and w :: "('h, 'k) label list" shows "mon_w fg w \<equiv> \<Union> (mon_e fg ` set w)"
Suc_1: fixes i :: "nat"   and h :: "heap" assumes "k = j - i"   and "i \<le> j"   and "\<And>k h. \<lbrakk>I k h; i \<le> k; k < j\<rbrakk> \<Longrightarrow> success (f k) h"   and "I i h"   and "\<And>k h h' r. \<lbrakk>I k h; i \<le> k; k < j; effect (f k) h h' r\<rbrakk> \<Longrightarrow> I (Suc k) h'" shows "\<exists>h'. effect (for [i..<j] f) h h' () \<and> I j h'"
relAPP_def: fixes f :: "('b \<times> 'c) set \<Rightarrow> 'a"   and x :: "('b \<times> 'c) set" shows "\<langle>x\<rangle>f \<equiv> f x"
validV2: shows "isViewOn \<V>2 E\<^bsub>ES2\<^esub>"
map_mono_prefix: fixes xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "prefix xs ys" shows "prefix (map f xs) (map f ys)"
lunit_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<l>[f] : trg f \<star> f \<Rightarrow> f\<guillemotright>"
Pareto_iff: fixes x :: "'alt pmf"   and y :: "'alt pmf" shows "Pareto (SD \<circ> R) x y = (\<forall>i\<in>agents. (SD \<circ> R) i x y)"
varsC_def: fixes c :: "('c, 'd) lit list" shows "varsC c = \<Union> (set (map varsL c))"
valuate_zero: fixes v :: "nat \<Rightarrow> 'b" shows "0 \<lbrace> v \<rbrace> = (0::'b)"
cSup_abs_le: fixes S :: "'b set"   and a :: "'b" assumes "S \<noteq> {}"   and "\<And>x. x \<in> S \<Longrightarrow> \<bar>x\<bar> \<le> a" shows "\<bar>Sup S\<bar> \<le> a"
fpxs_shift_compose_power: fixes r :: "rat"   and f :: "'b fpxs"   and s :: "rat" shows "fpxs_shift r (fpxs_compose_power f s) = fpxs_compose_power (fpxs_shift (r / s) f) s"
Quotient_rep_abs: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'b" assumes "Quotient R Abs Rep T"   and "R r r" shows "R (Rep (Abs r)) r"
set_take: fixes n :: "nat"   and s :: "'b list" shows "set (take n s) = {s ! i |i. i < n \<and> i < length s}"
\<gamma>_lt_\<delta>': shows "zhmset_of \<gamma> < zhmset_of \<delta>"
unwinding_theorem_BSI: fixes ur :: "('s \<times> 's) set" assumes "lrb ur"   and "osc ur" shows "BSI \<V> Tr\<^bsub>(induceES SES)\<^esub>"
le_Suc: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less a b" shows "less_eq (plus a one) b"
Basic: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "(Some (Basic f), s) -c\<rightarrow> (None, f s)"
uses'_code_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and n :: "'node" shows "uninst_code.uses'_code u p g next n \<equiv> uninst_code.substNext_code p g next ` (usesOf \<circ> u) g n"
precise_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> assn" shows "precise R \<equiv> \<forall>a a' h p F F'. h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F' \<longrightarrow> a = a'"
keys_rep_list_subset: fixes t :: "'a"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "t \<in> keys (rep_list r)" obtains v :: "'t" and s :: "'a" where "v \<in> keys r" and "s \<in> Keys (set fs)" and "t = pp_of_term v + s"
submatrix_def: fixes A :: "'b mat"   and I :: "nat set"   and J :: "nat set" shows "submatrix A I J = mat (card {i. i < dim_row A \<and> i \<in> I}) (card {j. j < dim_col A \<and> j \<in> J}) (\<lambda>(i, j). A $$ (pick I i, pick J j))"
prefix_slength_strengthen: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "prefix\<cdot>xs\<cdot>ys = (le\<cdot>(slength\<cdot>xs)\<cdot>(slength\<cdot>ys) andalso prefix\<cdot>xs\<cdot>ys)"
fin_inf_append_def: fixes us :: "'a list"   and s :: "nat \<Rightarrow> 'a" shows "fin_inf_append us s \<equiv> \<lambda>i. if i < length us then us ! i else s (i - length us)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F (\<Phi>'.Y a) \<Psi>.map"
seq_compact_eq_countably_compact: fixes U :: "'a set" shows "seq_compact U = countably_compact U"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case B of R \<Rightarrow> f1 | B \<Rightarrow> f2) = f2"
openin_open_trans: fixes T :: "'b set"   and S :: "'b set" assumes "openin (top_of_set T) S"   and "open T" shows "open S"
ThrowRedsNull: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>null,s',b'\<rangle>" shows "P \<turnstile> \<langle>throw e,s,b\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s',b'\<rangle>"
nn_integral_diff: fixes f :: "'k \<Rightarrow> ennreal"   and M :: "'k measure"   and g :: "'k \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integral\<^sup>N M g \<noteq> \<infinity>"   and "AE x in M. g x \<le> f x" shows "\<integral>\<^sup>+ x. f x - g x \<partial>M = integral\<^sup>N M f - integral\<^sup>N M g"
CondF: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and e' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
dma_control_var\<^sub>C_def: fixes v :: "nat" shows "dma_control_var\<^sub>C v \<equiv> if v = 0 then Low else High"
mem_in_image2: fixes a :: "'h"   and A :: "'h set"   and f :: "'h \<Rightarrow> 'i" assumes "a \<in> A" shows "f a \<in> f ` A"
suc_greatest_not_card: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"
endpoint_in_S_def: fixes a :: "proj2"   and b :: "proj2" shows "endpoint_in_S a b \<equiv> SOME p. p \<in> S \<and> B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt p"
fmdrop_fset_fmmap: fixes A :: "'a fset"   and f :: "'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmdrop_fset A (fmmap f m) = fmmap f (fmdrop_fset A m)"
mem_aforms_err_Cons_iff_Ex_conv: fixes x :: "real list"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and XS :: "((real \<times> real pdevs) \<times> real) list" shows "(x \<in> aforms_err e (X # XS)) = (\<exists>y ys. x = y # ys \<and> y \<in> aform_err e X \<and> ys \<in> aforms_err e XS)"
conf_f_def: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap" shows "conf_f P h sh \<equiv> \<lambda>(ST, LT) is (stk, loc, C, M, pc, ics). P,h \<turnstile> stk [:\<le>] ST \<and> P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT \<and> pc < length is \<and> P,h,sh \<turnstile>\<^sub>i (C, M, pc, ics)"
continuous: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "normal F" shows "continuous F"
rapprox_posrat_less1: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "0 \<le> x"   and "0 < y"   and "2 * x < y" shows "real_of_float (rapprox_posrat n x y) < 1"
get_if_const_insertion: fixes p :: "real mpoly"   and r :: "real"   and f :: "nat \<Rightarrow> real" assumes "get_if_const p = Some r" shows "insertion f p = r"
games_lfp_def: shows "games_lfp \<equiv> lfp fixgames"
Sup_upper: fixes x :: "real"   and X :: "real set" assumes "x \<in> X"   and "bdd_above X" shows "x \<le> Sup X"
int_encode_def: fixes i :: "int" shows "int_encode i = sum_encode (if 0 \<le> i then Inl (nat i) else Inr (nat (- i - 1)))"
eucl_sup: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "sup x y = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (sup (inner x i) (inner y i)) i) Basis"
assms: fixes Gxy :: "'a set"   and x :: "'a"   and y :: "'a"   and Gxz :: "'a set"   and z :: "'a"   and Gyz :: "'a set" assumes "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gxz x z"   and "geodesic_segment_between Gyz y z" shows "\<exists>w. infdist w Gxy \<le> delta \<and> infdist w Gxz \<le> delta \<and> infdist w Gyz \<le> delta"
rel_set_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "rel_set (BNF_Def.Grp A f) = BNF_Def.Grp {B. B \<subseteq> A} ((`) f)"
xs: shows "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"
rv_less_rv_measurable: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "{w. f w < g w} \<in> measurable_sets M"
distinct_67: fixes x41 :: "('a, 'b, 'c) com"   and x42 :: "('a, 'b, 'c) com"   and x111 :: "('a, 'b, 'c) com"   and x112 :: "('a, 'b, 'c) com" shows "Seq x41 x42 \<noteq> Catch x111 x112"
set_map_3: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and f3 :: "'h \<Rightarrow> 'c"   and v :: "('f, 'g, 'h) F1" shows "F1set3 (F1map f1 f2 f3 v) = f3 ` F1set3 v"
lower_def: fixes interval :: "'a interval" shows "interval.lower interval = (case interval of Interval x1 x2 \<Rightarrow> x1)"
star_unfold_left_Nil: fixes A :: "'b list set" shows "star A = (A - {[]}) @@ star A \<union> {[]}"
rel_refl: fixes R1a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R3a :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "('d, 'e, 'f) generat" assumes "\<And>x. R1a x x"   and "\<And>x. R2a x x"   and "\<And>x. R3a x x" shows "rel_generat R1a R2a R3a x x"
takeWhile_nth: fixes j :: "nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "j < length (takeWhile P xs)" shows "takeWhile P xs ! j = xs ! j"
D_GET_H: fixes x :: "Var"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and y :: "Var"   and f :: "Field" assumes "CONTEXT x = high" shows "(G, Get x y f, HighSec) \<in> Deriv"
reds: shows "P \<turnstile> \<langle>e,(h, l(Vs [\<mapsto>] vs), sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>"
if_doesnt_read: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and e :: "'c"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and annos :: "'b ModeUpd list" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "x \<notin> bexp_vars e" shows "sifum_security.doesnt_read (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) (sifum_modes.annotate (Stmt.If e c\<^sub>1 c\<^sub>2) annos) x"
assms: shows "\<Psi> \<rhd> P \<sim> Q"
\<Sigma>_protocol_def: shows "\<Sigma>0.\<Sigma>_protocol \<equiv> \<Sigma>0.completeness \<and> \<Sigma>0.special_soundness \<and> \<Sigma>0.HVZK"
simps_8: fixes \<alpha> :: "('b, 'c, 'a) hp" shows "BVP \<alpha>** = BVP \<alpha>"
simps_9: fixes Vs :: "char list list"   and e\<^sub>1 :: "char list exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (e\<^sub>1\<bullet>F{D} := e\<^sub>2) = compE\<^sub>1 Vs e\<^sub>1\<bullet>F{D} := compE\<^sub>1 Vs e\<^sub>2"
all_local_vars_def: fixes n :: "('form, 'rule) graph_node" shows "all_local_vars n = \<Union> (local_vars n ` fset (inPorts n))"
subst_cls_list_comp_subst: fixes Cs :: "'a literal multiset list"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "Cs \<cdot>cl (\<tau> \<odot> \<sigma>) = Cs \<cdot>cl \<tau> \<cdot>cl \<sigma>"
assms_3: fixes t2 :: "'a trm"   and t1 :: "'a trm" assumes "is_Const t2" shows "P (Eq t1 t2)"
set_diff_eq: fixes A :: "'c set"   and B :: "'c set" shows "A\<setminus>B = {x \<in> A. x \<notin> B}"
borel_eq_sigmaI2: fixes G :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and B :: "('d \<times> 'e) set"   and F :: "'f \<Rightarrow> 'g \<Rightarrow> 'c set"   and A :: "('f \<times> 'g) set" assumes "borel = sigma UNIV ((\<lambda>(i, j). G i j) ` B)"   and "\<And>i j. (i, j) \<in> B \<Longrightarrow> G i j \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"   and "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel" shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
submodule_pOp_closed: fixes H :: "'a set"   and h :: "'a"   and k :: "'a" assumes "submodule R M H"   and "h \<in> H"   and "k \<in> H" shows "h \<plusminus> k \<in> H"
embedding_functors_compose: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c" assumes "embedding_functor A B F"   and "embedding_functor B C G" shows "embedding_functor A C (G \<circ> F)"
real_vector_space_on_axioms: fixes k :: "enat"   and p :: "'c" shows "real_vector_space_on (manifold_eucl.tangent_space k p)"
plus_mat_def: fixes A :: "'a mat"   and B :: "'a mat" shows "A + B \<equiv> mat (dim_row B) (dim_col B) (\<lambda>ij. A $$ ij + B $$ ij)"
nat_less_as_int: shows "(<) = (\<lambda>a b. int a < int b)"
norm_cross: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "(norm (x \<times> y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
simps_2: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F' :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> = (F' = [] \<and> (\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. \<exists>a. \<Gamma> (Var x) = Var a) \<or> (\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> pair ` set F' \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X)))"
QCongA_def: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "QCongA a \<equiv> \<exists>A B C. A \<noteq> B \<and> C \<noteq> B \<and> (\<forall>X Y Z. (A B C CongA X Y Z) = a X Y Z)"
subtensor_combine_def: fixes ds :: "nat list"   and As :: "'b tensor list" shows "subtensor_combine ds As = tensor_from_vec (length As # ds) (concat (map vec As))"
\<pi>_def: shows "\<pi> \<equiv> \<lambda>(m, n). (m, Suc (m + n))"
Sigma_mset_Un_distrib1: fixes I :: "'b multiset"   and J :: "'b multiset"   and C :: "'b \<Rightarrow> 'c multiset" shows "Sigma_mset (I \<union># J) C = Sigma_mset I C \<union># Sigma_mset J C"
m_D: shows "P \<turnstile> D sees M', b :  Ts\<rightarrow>T = (mxs', mxl', ins', xt') in D'"
functor_at_ide_is_functor: fixes F :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "'a" assumes "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F"   and "A.ide a" shows "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at a F)"
F1set3_image_Lev2: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xb :: "'e"   and xd :: "bd_type_F"   and xa :: "'d" assumes "xe \<in> Lev2 s1 s2 n xb"   and "xe @ [Inr xd] \<in> Lev2 s1 s2 (Suc n) xb"   and "rv2 s1 s2 xe xb = Inl xa" shows "xd \<in> tobd_F13 s1 xa ` F1set3 (s1 xa)"
top_on_opt_narrow: fixes o1 :: "'a st option"   and X :: "char list set"   and o2 :: "'a st option" assumes "top_on\<^sub>o o1 X"   and "top_on\<^sub>o o2 X" shows "top_on\<^sub>o (o1 \<triangle> o2) X"
inj_on_image_eqpoll_2: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and A :: "'c set" assumes "inj_on f B" shows "(A \<approx> f ` B) = (A \<approx> B)"
FL_bisimilar_reflp: fixes F :: "'effect fs_set" shows "reflp (FL_bisimilar F)"