hcju/isabelleps huggingface dataset:
fundfoldpair_unique_half_chamber_system_f: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs"   and "(f', g') \<in> fundfoldpairs"   and "Abs_induced_automorph f' g' = Abs_induced_automorph f g" shows "f' \<turnstile> \<C> = f \<turnstile> \<C>"
widen_ArrayPrimT: fixes G :: "prog"   and t :: "prim_ty"   and T :: "ty" assumes "G\<turnstile>PrimT t.[]\<preceq>T" shows "T = Class Object \<or> T = PrimT t.[]"
degree_eqI: fixes x :: "'b pdevs"   and d :: "nat" assumes "pdevs_apply x d \<noteq> (0::'b)"   and "\<And>j. d < j \<Longrightarrow> pdevs_apply x j = (0::'b)" shows "degree x = Suc d"
card_UNIV_def: shows "card_UNIV TYPE('a) = card UNIV"
1: shows "i \<in> liset A w \<inter> {..<k}"
a_rcos_module_minus: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme"   and x :: "'a"   and x' :: "'a" assumes "abelian_subgroup H G"   and "ring G"   and "x \<in> carrier G"   and "x' \<in> carrier G" shows "(x' \<in> H +>\<^bsub>G\<^esub> x) = (x' \<ominus>\<^bsub>G\<^esub> x \<in> H)"
initial_call_no_ret: fixes p :: "'b"   and l :: "('b, 'c) edgeAnnot"   and v :: "'a" assumes "initialproc fg p"   and "(entry fg p, l, v) \<in> edges fg" shows "\<exists>p'. l = Call p' \<and> isolated_ret fg p'"
assms_2: shows "uverts H\<^sub>2 \<subseteq> T"
word_lsb_numeral: fixes bin :: "num" shows "lsb (numeral bin) = odd (numeral bin)"
direct_decomp_ideal_normal_form: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "direct_decomp P[X] [ideal F \<inter> P[X], normal_form F ` P[X]]"
B: fixes i :: "'n"   and j :: "'n" shows "A $h i $h j \<le> B $h i $h j"
hyps_1: shows "(ts, m, \<S>) \<Rightarrow>\<^sub>s\<^sub>b (ts'', m'', \<S>'')"
periodic_set_tan_linear: fixes a :: "real"   and c :: "real"   and b :: "real" assumes "a \<noteq> 0"   and "c \<noteq> 0" shows "periodic_set (roots (\<lambda>x. a * tan (x / c) + b)) (c * pi)"
Am_def: fixes t :: "'b tree" shows "Am t = real (T_splay_max t) + local.\<Phi> (splay_max t) - local.\<Phi> t"
finite_blocks: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>D" shows "finite b"
hom_zero: fixes c :: "'a" shows "c * (0::'a) = (0::'a)"
MGT_CALL1: shows "\<forall>p. {} |\<turnstile>\<^sub>t {MGT\<^sub>t (CALL p)}"
monoE: fixes g :: "'a"   and f :: "'a"   and f' :: "'a" assumes "S.mono g"   and "S.seq g f"   and "S.seq g f'"   and "g \<cdot>\<^sub>S f = g \<cdot>\<^sub>S f'" shows "f' = f"
Red_F_def: fixes N :: "'f set" shows "Red_F N = (\<Inter>q\<in>Q. Red_F_q q N)"
Hausdorff_space_topspace_empty: fixes X :: "'c topology" assumes "topspace X = {}" shows "Hausdorff_space X"
simps: fixes aa :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" shows "(aa \<in> segment f a b) = (aa = f a \<and> f a \<noteq> b \<or> (\<exists>x. aa = f x \<and> x \<in> segment f a b \<and> f x \<noteq> b))"
game_equiv_def: fixes \<alpha> :: "game"   and \<beta> :: "game" shows "game_equiv \<alpha> \<beta> \<equiv> \<forall>I X. game_sem I \<alpha> X = game_sem I \<beta> X"
widen_PrimT_strong: fixes G :: "prog"   and x :: "prim_ty"   and T :: "ty" assumes "G\<turnstile>PrimT x\<preceq>T" shows "T = PrimT x"
measure_countable_Union_le: fixes S :: "nat \<Rightarrow> 'c set"   and M :: "'c measure"   and B :: "real" assumes "\<And>i. S i \<in> fmeasurable M"   and "\<And>n. Sigma_Algebra.measure M (\<Union> (S ` {..n})) \<le> B" shows "Sigma_Algebra.measure M (\<Union> (range S)) \<le> B"
param_out_slice2: fixes n'' :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p:V\<rightarrow>\<^bsub>out\<^esub> n'"   and "sum_SDG_slice2p n n'" shows "sum_SDG_slice2p n n''"
remove_Max_eq: fixes I :: "'a set"   and n :: "'a" assumes "finite I"   and "I \<noteq> {}"   and "n \<noteq> Max I" shows "Max (I - {n}) = Max I"
simps_1: shows "tmf_polys [] = (0\<^sub>p, 0\<^sub>p)"
list_slice_0: fixes xs :: "'b list" shows "list_slice xs 0 = []"
int_pow_inv: fixes x :: "'c"   and i :: "int" assumes "x \<in> carrier H" shows "inv\<^bsub>H\<^esub> x [^]\<^bsub>H\<^esub> i = inv\<^bsub>H\<^esub> (x [^]\<^bsub>H\<^esub> i)"
inject: fixes int :: "int"   and int' :: "int" shows "(eint int = eint int') = (int = int')"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "x \<in>\<^sub>\<circ> B"
linepath_half_finite_inter: fixes a :: "'a"   and c :: "'a"   and d :: "real"   and b :: "'a" assumes "a \<bullet> c \<noteq> d \<or> b \<bullet> c \<noteq> d" shows "finite {t. linepath a b t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
Iff_MP2_same: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IFF B"   and "H \<turnstile> B" shows "H \<turnstile> A"
distinct_131: shows "GreaterThan \<noteq> BinOr"
oth_class_taut_5_i: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> \<psi> \<^bold>\<equiv> \<phi> \<^bold>& \<psi> \<^bold>\<or> \<^bold>\<not>\<phi> \<^bold>& \<^bold>\<not>\<psi> in v]"
phl_skip: fixes x :: "'a" shows "\<lbrace>x\<rbrace>1::'a\<lbrace>x\<rbrace>"
meld_correct_1: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and q' :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.invar q"   and "BinomialHeapStruc.invar q'" shows "BinomialHeapStruc.invar (BinomialHeapStruc.meld q q')"
preorder_on_restrict: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "Order_Predicates.preorder_on carrier le" shows "Order_Predicates.preorder_on (carrier \<inter> A) (restrict_relation A le)"
translate_collision_range: assumes "s\<^sub>o \<le> u_max"   and "u_max < other.s_stop" shows "delayed_safe_distance.collision {0..} = collision_react {\<delta>..}"
inv_rev: shows "queue_invar (rev (filter (\<lambda>t. 0 < rank t) (children (getMinTree q))))"
remove_all_def: fixes x :: "'d"   and xs :: "'d list" shows "remove_all x xs = (if x \<in> set xs then tl (remove_cycles xs x []) else xs)"
rtrancl_into_rtrancl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
real_lim_then_eventually_real: fixes u :: "'b \<Rightarrow> ereal"   and l :: "real"   and F :: "'b filter" assumes "(u \<longlongrightarrow> ereal l) F" shows "\<forall>\<^sub>F n in F. u n = ereal (real_of_ereal (u n))"
hs_get_roots_locs_def: shows "hs_get_roots_locs = prefixed ''hs_get_roots_''"
prj_simps_12: shows "C.seq gf.prj\<^sub>0 \<p>\<^sub>0[f.leg0, g.leg1 \<cdot> gf.prj\<^sub>1]"
ring_l_one: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d" assumes "Ring R"   and "x \<in> carrier R" shows "1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>r\<^bsub>R\<^esub> x = x"
matrix_mul_lid: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "mat (1::'a) ** A = A"
closedD: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "closed A f"   and "x \<in> A"   and "y \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<in> A"
SeqWRP_LstSeqP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqWRP s k t} \<turnstile> LstSeqP s k t"
AbGroup_subset_sum_list: fixes Gs :: "'b set list"   and H :: "'b set" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "H \<in> set Gs" shows "H \<subseteq> (\<Sum>G\<leftarrow>Gs. G)"
ex_new_if_finite: fixes A :: "'c set" assumes "infinite UNIV"   and "finite A" shows "\<exists>a. a \<notin> A"
vi: fixes x :: "real" assumes "x \<in> {0..1} - s \<inter> {0..1}" shows "(\<phi> has_real_derivative D x) (at x within {0..1})"
card_sccs_verts: shows "card H.sccs_verts = card H.sccs"
prob_Collect_eq_1: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "prob_space M"   and "{x \<in> space M. P x} \<in> sets M" shows "(\<P>(x in M. P x) = 1) = almost_everywhere M P"
app\<^sub>i_Invoke: fixes M :: "String.literal"   and n :: "nat"   and P :: "'d prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Invoke M n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length ST \<and> (ST ! n \<noteq> NT \<longrightarrow> (\<exists>C D Ts T m. class_type_of' (ST ! n) = \<lfloor>C\<rfloor> \<and> P \<turnstile> C sees M: Ts\<rightarrow>T = m in D \<and> P \<turnstile> rev (take n ST) [\<le>] Ts)))"
tendsto_mult_ereal: fixes f :: "'b \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal"   and m :: "ereal" assumes "(f \<longlongrightarrow> l) F"   and "(g \<longlongrightarrow> m) F"   and "\<not> (l = 0 \<and> \<bar>m\<bar> = \<infinity> \<or> m = 0 \<and> \<bar>l\<bar> = \<infinity>)" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
pp_bijective_conv_mapping: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_bijective x = matrix_stone_relation_algebra.pp_mapping (x\<^sup>t)"
QF_Suc: fixes k :: "nat"   and j :: "nat" shows "QF k (Suc j, 0) = enum (nxt N (Suc (Max (QF k (j, m - Suc 0))))) ` {..<del k (if Suc j = k then m - 1 else 0) (Suc j)}"
filter_distinct_at6: fixes xs :: "'a list"   and as :: "'a list"   and u :: "'a"   and bs :: "'a list"   and us :: "'a list"   and zs :: "'a list" assumes "distinct xs"   and "xs = as @ u # bs"   and "filter (\<lambda>v. v = u \<or> v \<in> set us) xs = u # us"   and "set zs \<inter> set xs \<subseteq> {u} \<union> set as" shows "filter (\<lambda>v. v \<in> set us) (zs @ bs) = us \<and> filter (\<lambda>v. v \<in> set us) bs = us"
null_sets_completion: fixes N' :: "'b set"   and M :: "'b measure"   and N :: "'b set" assumes "N' \<in> null_sets M"   and "N \<subseteq> N'" shows "N \<in> sets (completion M)"
less_bot_Bot: fixes x :: "'a bot" shows "(x < Bot) = False"
lconcat_LCons: fixes xs :: "'c llist"   and xss :: "'c llist llist" shows "lconcat (LCons xs xss) = lappend xs (lconcat xss)"
bernpoly_0: fixes n :: "nat" shows "bernpoly n (0::'a) = of_real (bernoulli n)"
IH_2: fixes d :: "nat"   and t' :: "'a tree"   and n' :: "nat"   and h' :: "nat" assumes "ins3 n d x r = Unbal2 t' n' h'"   and "bal_i n (height r + d)" shows "\<Phi> t' - \<Phi> r \<le> 6 * e * real (height r)"
strict_coboundedI1: fixes a :: "'a Word.word"   and c :: "'a Word.word"   and b :: "'a Word.word" assumes "a <s c" shows "signed.min a b <s c"
Range_Collect_case_prod: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
loop: fixes st :: "'s \<times> ('c \<Rightarrow> 't)" shows "return_pmf st \<in> K st"
rep_eq: fixes x :: "'a ltln\<^sub>P"   and xa :: "'a ltln\<^sub>P" shows "local.ltln\<^sub>P_eq_test x xa = rep_ltln\<^sub>P x \<sim>\<^sub>P rep_ltln\<^sub>P xa"
invoke_ptr_in_heap: fixes h :: "(_) heap"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'aa \<Rightarrow> ((_) heap, exception, 'z) prog)) list"   and ptr :: "(_) object_ptr"   and args :: "'aa" assumes "h \<turnstile> ok invoke xs ptr args" shows "ptr |\<in>| object_ptr_kinds h"
C: shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
nta_bisim_simps_sym_2: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta :: "('a, 'b, 'c) new_thread_action"   and t :: "'a"   and b :: "bool" shows "nta_bisim bisim ta (ThreadExists t b) = (ta = ThreadExists t b)"
not_eta_reducible_imp_eta_norm_no_change: fixes t :: "Core.term" assumes "\<not> eta_reducible t" shows "eta_norm t = t"
summable: fixes f :: "'a \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'a" assumes "bounded_linear f"   and "summable X" shows "summable (\<lambda>n. f (X n))"
assms_4: shows "f \<in> carrier L \<rightarrow> carrier L"
confluence_var: fixes y :: "'a"   and x :: "'a" shows "(y ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>) = (y\<^sup>\<star> ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>)"
cons_to_append: fixes a :: "'c"   and b :: "'c list" shows "a # b = [a] @ b"
comp_null_1: fixes f :: "'a \<times> 'a" shows "VoV.comp VoV.null f = VoV.null"
lpd: shows "\<pi> l pd\<rightarrow> \<pi> j"
mod_ring_finite_field_ops_int: shows "mod_ring_gen (finite_field_ops_int p) mod_ring_rel p"
imp: shows "as \<Turnstile>\<^sub>i \<B>\<I> s"
bst_mrbst_union: fixes t' :: "'a tree"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_union t1 t2)"   and "bst t1"   and "bst t2" shows "bst t'"
simps_2: fixes l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "rlh \<langle>l, uu, r\<rangle> = 1 - rh l r + rlh r"
fresh_var_not_free: fixes e :: "exp" shows "fresh_var e \<notin> fv e"
starfun_o2: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "(\<lambda>x. (*f* f) ((*f* g) x)) = *f* (\<lambda>x. f (g x))"
assms_4: shows "v \<in> set (\<Pi>\<^sub>\<V>)"
fn_norm_ge_zero: fixes f :: "'a \<Rightarrow> real" assumes "Function_Norm.continuous V f norm" shows "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (vc C\<^sub>1 Q \<and> vc C\<^sub>2 Q)"
lexn_conv: fixes r :: "('g \<times> 'g) set"   and n :: "nat" shows "lexn r n = {(xs, ys). length xs = n \<and> length ys = n \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y) \<in> r)}"
simps: fixes uu :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" shows "eval (MS uu f) = f"
case_cong_weak: fixes generat :: "('a, 'b, 'c) generat"   and generat' :: "('a, 'b, 'c) generat"   and f1 :: "'a \<Rightarrow> 'd"   and f2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "generat = generat'" shows "(case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (case generat' of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa)"
set_times_mono3_b: fixes a :: "'c"   and C :: "'c set"   and x :: "'c"   and D :: "'c set" assumes "a \<in> C"   and "x \<in> a *o D" shows "x \<in> C * D"
cInf_singleton: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Inf (\<eta> x) = x"
prems: shows "s \<in> deriv c (Star r) \<rightarrow> v"
abs_eq_iff: fixes x :: "freemsg"   and y :: "freemsg" shows "(abs_msg x = abs_msg y) = (x \<sim> y)"
subset_le_dim: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'c) vec set" assumes "S \<subseteq> vec.span T" shows "vec.dim S \<le> vec.dim T"
lunit_simps_2: fixes f :: "'a" assumes "ide f" shows "src \<l>[f] = src f"
Dup_def: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set" shows "Dup p i branch xs \<equiv> \<forall>(v, v')\<in>xs. is_at p i branch v v' \<and> is_elsewhere p i branch xs"
exec_call: shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =n\<Rightarrow> t"
dismantle_red: fixes m :: "trm"   and m' :: "trm"   and k :: "stack" assumes "m \<mapsto> m'" shows "m \<star> k \<mapsto> m' \<star> k"
has_integral_restrict_closed_subintervals_eq: fixes c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b" assumes "cbox c d \<subseteq> cbox a b" shows "((\<lambda>x. if x \<in> cbox c d then f x else (0::'b)) has_integral i) (cbox a b) = (f has_integral i) (cbox c d)"
absc_distr_self: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg" shows "MDP.MC.T (absc cfg) = distr (MDP.MC.T cfg) (stream_space (count_space UNIV)) (smap absc)"
simps_8: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> G \<phi> = (\<forall>k. suffix k w \<Turnstile> \<phi>)"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F H \<tau>\<sigma>.map"
rev_compose_def: fixes R :: "('c, 'd) ring_scheme" shows "rev_compose R = eval R (UP R) (to_polynomial R)"
abs_eq: fixes x :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "L_transform_Tree\<^sub>\<alpha> (FL_Formula.abs_Tree\<^sub>\<alpha> x) = L_transform_Tree x"
p_l_r_def_3: shows "r \<equiv> rai_ub y"
GroupEnd_axioms_def: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'a" shows "GroupEnd_axioms G T \<equiv> T ` G \<subseteq> G"
statically_complete_calculus_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "statically_complete_calculus Bot Inf entails Red_I Red_F \<equiv> calculus Bot Inf entails Red_I Red_F \<and> statically_complete_calculus_axioms Bot Inf entails Red_I"
oSuc_not_0: fixes x :: "ordinal" shows "oSuc x \<noteq> 0"
weak_SN_both_mono_ordered_semiring_1_def: fixes weak_gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and default :: "'a"   and arc_pos :: "'a \<Rightarrow> bool" shows "weak_SN_both_mono_ordered_semiring_1 weak_gt default arc_pos \<equiv> \<forall>xys. (\<forall>x y. (x, y) \<in> set xys \<longrightarrow> weak_gt x y) \<longrightarrow> (\<exists>gt. SN_both_mono_ordered_semiring_1 default gt arc_pos \<and> (\<forall>x y. (x, y) \<in> set xys \<longrightarrow> gt x y))"
P_f: fixes C :: "'a acom" assumes "P C" shows "P (f C)"
closed_subsets_zero: shows "\<V> {\<zero>} = Spec"
inj_add: fixes x :: "nat"   and q :: "nat"   and x' :: "nat"   and y :: "nat" assumes "x < q"   and "x' < q"   and "(y + x) mod q = (y + x') mod q" shows "x = x'"
has_integral_stretch: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and m :: "'a \<Rightarrow> real" assumes "(f has_integral i) (cbox a b)"   and "\<forall>k\<in>Basis. m k \<noteq> 0" shows "((\<lambda>x. f (\<Sum>k\<in>Basis. (m k * (x \<bullet> k)) *\<^sub>R k)) has_integral (1 / \<bar>prod m Basis\<bar>) *\<^sub>R i) ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x \<bullet> k)) *\<^sub>R k) ` cbox a b)"
msetext_dersh_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "msetext_dersh gt (xs @ [x]) xs"
add_mono_neutl: fixes b :: "'a"   and a :: "'a" assumes "\<one> \<le> b" shows "a \<le> b + a"
single_valued_subset: fixes r :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "r \<subseteq> s"   and "single_valued s" shows "single_valued r"
A_lb_finite: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and x :: "nat" assumes "LoadBalancing m"   and "LoadBalancing.lb t m T A j"   and "x \<in> {1..m}" shows "finite (A x)"
that: fixes Y1 :: "'a multiset"   and Y2 :: "'a multiset" assumes "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "(X2, Y2) \<in> multpw ns" shows "thesis"
a_zero: shows "a \<longlonglongrightarrow> 0"
simps_10: fixes sh :: "sheap"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "'a exp" shows "iconf sh (C\<bullet>\<^sub>sF{D} := e\<^sub>2) = iconf sh e\<^sub>2"
iT_Plus_Int: fixes A :: "nat set"   and B :: "nat set"   and k :: "nat" shows "A \<inter> B \<oplus> k = (A \<oplus> k) \<inter> (B \<oplus> k)"
B: fixes y :: "'b process" assumes "cont f"   and "cont g" shows "cont (\<lambda>x. f x |-| y)"
perm_refl: fixes l :: "'c list" shows "l <~~> l"
elims_1: fixes x :: "'a \<times> nat"   and xa :: "('a, 'b) event \<Rightarrow> bool"   and y :: "bool" assumes "event_pred x xa = y" obtains i :: "'a" and j :: "nat" and p :: "('a, 'b) event \<Rightarrow> bool" where "x = (i, j)" and "xa = p" and "y = (is_valid_event_id (i, j) \<and> p (events i ! j))"
ordinal_numberTr5: fixes x :: "'c Order set"   and D :: "'c Order" assumes "x \<in> ODnums"   and "D \<in> x" shows "x = ordinal_number D"
X2: fixes a :: "'b" assumes "(x, w) = arc_to_ends G a" shows "a \<notin> arcs G"
sameDom_def: fixes inp1 :: "'c \<Rightarrow> 'd option"   and inp2 :: "'c \<Rightarrow> 'e option" shows "sameDom inp1 inp2 \<equiv> \<forall>i. (inp1 i = None) = (inp2 i = None)"
lookup_map_values: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and m :: "('b, 'c) mapping"   and k :: "'b" shows "Mapping.lookup (Mapping.map_values f m) k = map_option (f k) (Mapping.lookup m k)"
roolz: fixes f :: "'a" assumes "f \<in> Hom A B" shows "set_dom (Hom(A,_) \<^bsub>\<a>\<^esub> f) = Hom A A"
all_vars: shows "set (opaque x) \<union> set (opaque y) \<subseteq> {0..<n}"
thetaWhileW_WretrT: shows "thetaWhileW \<subseteq> WretrT (thetaWhileW \<union> WbisT)"
leg_simps_4: shows "C.dom HfHgh.cod.leg1 = HfHgh.cod.apex"
Transset_SUP: fixes A :: "'c set"   and B :: "'c \<Rightarrow> V" assumes "\<And>x. x \<in> A \<Longrightarrow> Transset (B x)" shows "Transset (\<Squnion> (B ` A))"
sum_SDG_edge_valid_SDG_node_1: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and b :: "bool"   and n' :: "'node SDG_node" assumes "sum_SDG_edge n Vopt popt b n'" shows "valid_SDG_node n"
cong: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" shows "comm_monoid_add.sum plus zero \<equiv> comm_monoid_add.sum plus zero"
nat2Nat_Nat2nat: fixes n :: "ZF" assumes "Elem n HOLZF.Nat" shows "nat2Nat (Nat2nat n) = n"
wf_digraph_axioms: shows "wf_digraph (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
path_component_of_maximal: fixes X :: "'a topology"   and s :: "'a set"   and x :: "'a" assumes "path_connectedin X s"   and "x \<in> s" shows "s \<subseteq> path_component_of_set X x"
subcringE_3: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subcring H R" shows "\<one>\<^bsub>R\<^esub> \<in> H"
Healthy_Sequence: fixes P :: "'a hrel"   and Q :: "'a hrel" assumes "P is \<H>"   and "Q is \<H>" shows "P ;; Q is \<H>"
punit_adds_pp: shows "MPoly_Type_Class.punit.adds_pp = (adds)"
fci_def: fixes bdd :: "bddi" shows "fci bdd \<equiv> return (0, bdd)"
min_union_comm: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<union>\<^sub>m B = B \<union>\<^sub>m A"
comm_monoid_add_ow_axioms_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "comm_monoid_add_ow_axioms S pls z \<equiv> (\<forall>a. a \<in> S \<longrightarrow> pls z a = a) \<and> z \<in> S"
f0acmcile: fixes X :: "complex mat"   and Y :: "complex mat" assumes "X \<le>\<^sub>L Y"   and "X \<in> carrier_mat d d"   and "Y \<in> carrier_mat d d" shows "f 0 + 1 / c \<cdot>\<^sub>m X \<le>\<^sub>L f 0 + 1 / c \<cdot>\<^sub>m Y"
measure_measure_spmf_def: fixes p :: "'a spmf" shows "measure_measure_spmf p = Sigma_Algebra.measure (measure_spmf p)"
assms: shows "st \<in> reachable (closed (pnet np (p1 \<parallel> p2))) TT"
ltake_lfind_conv_1: fixes P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" assumes "lfilter P lst \<noteq> LNil" shows "ltake (enat (lfind P lst)) lst = ltakeWhile (Not \<circ> P) lst"
sim_sim: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<turnstile> H"   and "G \<triangleright> \<Gamma> \<turnstile> F" shows "G \<triangleright> \<Gamma> \<turnstile> H"
higher_differentiable_on_const: fixes S :: "'b set"   and c :: "'c"   and n :: "nat" shows "higher_differentiable_on S (\<lambda>x. c) n"
block_closed: fixes a :: "'a set \<times> 'b"   and A :: "('a set \<times> 'b) set" assumes "a \<in> A"   and "A \<in> rel.Partition" shows "a \<in> Sigma I \<FF>"
prems_7: shows "card (insert x H <#> J) = card (insert x H) * card J"
relpowp_code_def: shows "Transitive_Closure.relpowp = compow"
condition1_index_le_zero_row: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k a (Suc k) A" shows "mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<le> a"