hcju/isabelleps huggingface dataset:
I: shows "dist x m \<le> Gromov_product_at x p y + 2 * deltaG TYPE('a)"
reachable_modes_subset: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')" shows "reachable_mode_states (cms', mem') \<subseteq> reachable_mode_states (cms, mem)"
prems_5: shows "{l..u} \<subseteq> Csafe"
sets_SUP: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure" assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"   and "I \<noteq> {}" shows "sets (\<Squnion> (M ` I)) = sets N"
lrt: shows "Dag ((rep \<propto> low) q) (rep \<propto> low) (rep \<propto> high) lrt"
typ_of1_arg_typ: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and R :: "typ"   and U :: "typ" assumes "typ_of1 Ts (t $ u) = Some R"   and "typ_of1 Ts u = Some U" shows "typ_of1 Ts t = Some (U \<rightarrow> R)"
set_iterator_genord_image_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and g :: "'e \<Rightarrow> 'f"   and R' :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_genord it S R"   and "inj_on g S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; R x y\<rbrakk> \<Longrightarrow> R' (g x) (g y)" shows "set_iterator_genord (set_iterator_image g it) (g ` S) R'"
filt_equiv_sets: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure" assumes "filt_equiv F M N" shows "sets M = sets N"
f1: fixes h :: "hmultiset"   and ha :: "hmultiset" shows "h - (ha + h) = 0"
rational_root_test_1: fixes p :: "rat poly"   and x :: "rat" assumes "rational_root_test p = Some x" shows "poly p x = 0"
Mp_smult_m_0: fixes m :: "int"   and f :: "int poly" shows "poly_mod.Mp m (smult m f) = 0"
multiplicity_decompose: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "\<not> dvd.dvd times p (divide x (power.power one times p (factorial_semiring.multiplicity times one p x)))"
ring_hom_lucas_lehmer_hom: fixes m :: "nat" assumes "1 < m" shows "lucas_lehmer_hom m \<in> ring_hom lucas_lehmer_ring (lucas_lehmer_ring_mod m)"
is_tiny_dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB> \<and> tiny_digraph \<alpha> \<AA> \<and> tiny_digraph \<alpha> \<BB>"
X: fixes i :: "'a" assumes "i \<in> I" shows "X i \<in> N \<rightarrow>\<^sub>M M i"
Ord_vsubset_closed: fixes \<alpha> :: "V"   and \<gamma> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<gamma>"   and "\<alpha> \<subseteq>\<^sub>\<circ> \<beta>"   and "\<beta> \<in>\<^sub>\<circ> \<gamma>" shows "\<alpha> \<in>\<^sub>\<circ> \<gamma>"
isnpoly_def: fixes p :: "poly" shows "isnpoly p = isnpolyh p 0"
Ide_in_Hom_2: fixes t :: "'a local.term" assumes "Ide t" shows "t \<in> VHom t t"
const_on4: shows "const_on (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (domA \<Gamma>) TTree.empty"
has_contour_integral_unique: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex"   and j :: "complex" assumes "(f has_contour_integral i) g"   and "(f has_contour_integral j) g" shows "i = j"
P: fixes i :: "nat" shows "(f i, f (Suc i)) \<in> rwof_rel init cond step"
measurable_alw_stream: fixes P :: "'a stream \<Rightarrow> bool" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "Measurable.pred (stream_space (count_space UNIV)) (alw P)"
Conf_InitStatus_InitConf: fixes A :: "('a, 'b, 'c) hierauto" shows "Conf (InitStatus A) = InitConf A"
\<delta>: shows "\<gamma> = \<omega> \<up> \<delta>"
ListRed1: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e # es,s,b\<rangle> [\<rightarrow>] \<langle>e' # es,s',b'\<rangle>"
polyn_Suc_split: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (Suc n, f)" shows "polyn_expr R X (Suc n) (Suc n, f) = polyn_expr R X n (n, f) \<plusminus> f (Suc n) \<cdot>\<^sub>r X^\<^bsup>R Suc n\<^esup>"
summable_comparison_test_bigo: fixes g :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "summable (\<lambda>n. norm (g n))"   and "f \<in> O(g)" shows "summable f"
131: shows "set b\<^sub>1 \<subseteq> set b"
left_idem: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.min a (linordered_monoid.min a b) = linordered_monoid.min a b"
cases: fixes a1 :: "'a list"   and a2 :: "nat"   and a3 :: "'a" assumes "nth_el1 a1 a2 a3" obtains x :: "'a" and xs :: "'a list" where "a1 = x__ # xs__" and "a2 = 0" and "a3 = x__"   | xs :: "'a list" and i :: "nat" and y :: "'a" and x :: "'a" where "a1 = x # xs" and "a2 = Suc i" and "a3 = y" and "nth_el1 xs i y"
arith_special_11: shows "- (1::'b) + (1::'b) = (0::'b)"
scale_zero_left: fixes x :: "'a" shows "(0::'a) \<cdot> x = (0::'a)"
rep_eq: fixes x :: "'a fpxs"   and xa :: "'a fpxs" shows "fpxs_nth (x * xa) = (\<lambda>xb. \<Sum>(y, z)\<in>{(y, z). y \<in> supp (fpxs_nth x) \<and> z \<in> supp (fpxs_nth xa) \<and> xb = y + z}. fpxs_nth x y * fpxs_nth xa z)"
eSuc_minus_eSuc: fixes n :: "enat"   and m :: "enat" shows "eSuc n - eSuc m = n - m"
paa0_sub: shows "poss AA0 \<subseteq># remove1_mset (Pos A0) C"
fbox_seq_var: fixes p :: "'a"   and x :: "'a"   and p' :: "'a"   and y :: "'a"   and q :: "'a" assumes "p \<le>  |x] p'"   and "p' \<le>  |y] q" shows "p \<le>  |x \<cdot> y] q"
ipresIGWls_def: fixes h :: "'u \<Rightarrow> 'v"   and MOD :: "('w, 'x, 'y, 'z, 'aa, 'ab, 'u, 'ac, 'ad) model_scheme"   and MOD' :: "('ae, 'af, 'ag, 'z, 'ah, 'ai, 'v, 'aj, 'ak) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
assms_2: shows "insertion (nth_default 0 (xs' @ x # xs)) b = B"
abs_eq: fixes xa :: "nat \<Rightarrow>\<^sub>0 nat"   and x :: "'b" shows "monom xa x = MPoly (Poly_Mapping.single xa x)"
wf_permute_agents: fixes \<sigma> :: "'b \<Rightarrow> 'b" assumes "\<sigma> permutes agents" shows "pref_profile_wf agents alts (R2 \<circ> \<sigma>)"
poly_dvd_1: fixes p :: "'a poly" shows "(p dvd 1) = (degree p = 0 \<and> coeff p 0 dvd (1::'a))"
r_nand_def: shows "r_nand \<equiv> Cn 2 r_not [r_add]"
prob_space: fixes s :: "'s option" shows "K2.T.prob s (space (K2.T s)) = 1"
minus_iarray_component: fixes i :: "nat"   and A :: "'a iarray"   and B :: "'a iarray" assumes "i < IArray.length A"   and "i < IArray.length B" shows "(A - B) !! i = A !! i - B !! i"
param_takeWhile: fixes a :: "('a \<times> 'b) set" shows "(takeWhile, takeWhile) \<in> (a \<rightarrow> bool_rel) \<rightarrow> \<langle>a\<rangle>list_rel \<rightarrow> \<langle>a\<rangle>list_rel"
preserves_reflects_arr: fixes f :: "'a" shows "arr (\<ll>.map f) = arr f"
transformation_by_components_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow> 'c" shows "transformation_by_components A B F G t \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> transformation_by_components_axioms A B F G t"
empty_not_insert: fixes a :: "'k"   and A :: "'k set" shows "{} \<noteq> insert a A"
map_le_substate_conv: shows "(\<subseteq>\<^sub>m) = (\<preceq>)"
proj_undefined: fixes i :: "'d"   and P :: "'d \<Rightarrow>\<^sub>F 'e" assumes "i \<notin> domain P" shows "(P)\<^sub>F i = undefined"
weak_control_dependence_def: fixes n :: "'node"   and n' :: "'node" shows "n weakly controls n' = (\<exists>a a' as. n' \<notin> set (sourcenodes (a # as)) \<and> n -a # as\<rightarrow>* n' \<and> n' strongly-postdominates targetnode a \<and> valid_edge a' \<and> sourcenode a' = n \<and> \<not> n' strongly-postdominates targetnode a')"
unit_div: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one"   and "dvd.dvd times b one" shows "dvd.dvd times (divide a b) one"
simps_2: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b Trace"   and x :: "'b" shows "tMap f (xs \<leadsto> x) = tMap f xs \<leadsto> f x"
IH: shows "lprob_comp p (xs @ [x]) = lprob_comp p xs * (if x then p else 1 - p)"
integrable_on_lebesgue: fixes f :: "'c \<Rightarrow> 'd" assumes "integrable lebesgue f" shows "f integrable_on UNIV"
id: shows "charpoly (cis (arg \<alpha>) *k cA) = charpoly cA"
assms_2: shows "x1 1 \<in> carrier_mat d d"
WT: shows "WT_rest \<I>_adv \<I>_usr I rest"
sup_least: fixes y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" assumes "y \<preceq> x"   and "z \<preceq> x" shows "y \<oplus> z \<preceq> x"
oexp_def: shows "rs.oexp = {(f, g). f \<in> rs.FINFUNC \<and> g \<in> rs.FINFUNC \<and> ((let m = wo_rel.max_fun_diff s f g in (f m, g m) \<in> r) \<or> f = g)}"
cnj_commute: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" assumes "H \<turnstile> B AND A" shows "H \<turnstile> A AND B"
octo_mult_cnj_conv_norm: fixes x :: "octo" shows "x * Octonions.cnj x = (octo_of_real (norm x))\<^sup>2"
wellformed: fixes x :: "nat"   and A :: "'b set" assumes "x \<in> {i. enat i < esize A}" shows "nth_least A x \<in> A"
eq: fixes a :: "'a"   and b :: "'b" assumes "a \<sim> b" shows "b = f a"
analz_synth: fixes H :: "msg set" shows "analz (synth H) = analz H \<union> synth H"
continuous: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set"   and m :: "'c set" assumes "continuous f S T"   and "m open\<^bsub>T\<^esub>" shows "carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>"
irrefl_trancl_rD: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "\<forall>x. (x, x) \<notin> r\<^sup>+"   and "(x, y) \<in> r" shows "x \<noteq> y"
at_eq: fixes e :: "cars"   and c :: "'a"   and d :: "'a" shows "\<forall>ts v. (\<forall>v'. v = e > v' \<longrightarrow> c = d) = (c = d)"
dom_map_of_conv_image_fst: fixes xys :: "('c \<times> 'd) list" shows "dom (map_of xys) = fst ` set xys"
isCont_inverse_function: fixes d :: "real"   and x :: "real"   and g :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "0 < d"   and "\<And>z. \<bar>z - x\<bar> \<le> d \<Longrightarrow> g (f z) = z"   and "\<And>z. \<bar>z - x\<bar> \<le> d \<Longrightarrow> isCont f z" shows "isCont g (f x)"
eq_upto_BAD_non_empty: fixes i :: "nat" shows "eq_upto (BAD P) lb i \<noteq> {}"
simps_4: fixes x21 :: "'a"   and x22 :: "'a"   and x1 :: "'a" shows "Edge x21 x22 \<noteq> Vertex x1"
degen_acc_run_complete_aux1': fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat"   and i :: "nat"   and q :: "'Q"   and n :: "nat" assumes "num_acc \<noteq> 0"   and "graph_defs.is_run (degeneralize_ext m) r"   and "\<forall>n<num_acc. \<exists>\<^sub>\<infinity>i. n \<in> acc (fst (r i))"   and "r i = (q, n)" shows "\<exists>j\<ge>i. \<exists>q'. r j = (q', n) \<and> n \<in> acc q'"
to_nat_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 x :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep x + 1 < CARD('b)" shows "mod_type.to_nat Rep (plus x one) = mod_type.to_nat Rep x + 1"
qAbs_alphaAbs_iff: fixes xs :: "'c"   and x :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A' :: "('a, 'b, 'c, 'd, 'e) qAbs" shows "(qAbs xs x X $= A') = (\<exists>x' y X'. A' = qAbs xs x' X' \<and> y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "GE.satC \<xi> \<equiv> list_ex (GE.satL \<xi>)"
HMA_M_mult: shows "(Mod_Type_Connect.HMA_M ===> Mod_Type_Connect.HMA_M ===> Mod_Type_Connect.HMA_M) (*) (**)"
blinding_of_T_root_hash: fixes bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rh :: "'e \<Rightarrow> 'f" assumes "bo \<le> vimage2p rh rh (=)" shows "blinding_of_T rh bo \<le> vimage2p (root_hash_T rh) (root_hash_T rh) (=)"
fempty_not_finsert: fixes a :: "'b"   and A :: "'b fset" shows "{||} \<noteq> finsert a A"
Framed_Tangle_Diagram_Equivalence_def: fixes T1 :: "Tangle_Diagram"   and T2 :: "Tangle_Diagram" shows "T1 ~T T2 \<equiv> Rep_Tangle_Diagram T1 ~ Rep_Tangle_Diagram T2"
DynPDG_path_Append: fixes n'' :: "'node"   and as' :: "'edge list"   and n' :: "'node"   and n :: "'node"   and as :: "'edge list" assumes "n'' -as'\<rightarrow>\<^sub>d* n'"   and "n -as\<rightarrow>\<^sub>d* n''" shows "n -as @ as'\<rightarrow>\<^sub>d* n'"
measurable_measure_of: fixes N :: "'d set set"   and \<Omega> :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and M :: "'e measure"   and \<mu> :: "'d set \<Rightarrow> ennreal" assumes "N \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>"   and "\<And>y. y \<in> N \<Longrightarrow> f -` y \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M measure_of \<Omega> N \<mu>"
FW_3: shows "fw m n n n n k j = D m k j n"
obtain_blocks_intersect_num: fixes n :: "int" assumes "n \<in> source.intersection_numbers" shows "\<exists>b1 b2. b1 \<in># \<B> \<and> b2 \<in># remove1_mset b1 \<B> \<and> b1 |\<inter>| b2 = n"
top_sorted_abs_mem: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'d"   and l :: "'d list"   and x :: "'d" assumes "top_sorted_abs R (h # l)"   and "ListMem x l" shows "\<not> R x h"
function_ring_defs_6: fixes R :: "('a, 'b) ring_scheme"   and S :: "'c set" assumes "ring_functions R" shows "(\<odot>\<^bsub>function_ring S R\<^esub>) = function_scalar_mult S R"
head_arev: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "head G (arev a) = tail G a"
check_del_l_check_del: fixes A :: "(nat, (char list list \<times> int) list) fmap"   and B :: "(nat, int mpoly) fmap"   and x3 :: "nat"   and x3a :: "nat"   and spec :: "'a" assumes "(A, B) \<in> \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(x3, x3a) \<in> nat_rel" shows "check_del_l spec A (pac_src1 (Del x3)) \<le> \<Down> {(st, b). (\<not> is_cfailed st) = b \<and> (b \<longrightarrow> st = CSUCCESS)} (check_del B (pac_src1 (Del x3a)))"
ord_adds: fixes s :: "'a"   and t :: "'a" assumes "s adds t" shows "s \<preceq> t"
mono: fixes P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i a. a <\<^bsub>P i\<^esub> f P \<longrightarrow> a <\<^bsub>P' i\<^esub> f P" shows "f P' = f P"
l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs \<equiv> type_wf = DocumentClass.type_wf \<and> set_attribute = l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_attribute \<and> set_attribute_locs = l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_attribute_locs"
nodes_graph: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "finite V"   and "finite E" shows "nodes (graph V E) = V \<union> fst ` E \<union> snd ` E"
scale_cancel_right: fixes a :: "complex"   and x :: "'a"   and b :: "complex" shows "(a *\<^sub>C x = b *\<^sub>C x) = (a = b \<or> x = (0::'a))"
TS_compet': assumes "List_Factoring.pairwise (Partial_Cost_Model.embed (rTS []))" shows "\<forall>s0\<in>{init. distinct init \<and> init \<noteq> []}. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS [])) (fst (Partial_Cost_Model.embed (rTS [])) s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs \<le> 2 * real (T\<^sub>p_opt s0 qs) + b"
section_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "section f \<equiv> \<exists>g. ide (local.comp g f)"
cSup_least: fixes X :: "'c multiset set"   and z :: "'c multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X \<subseteq># z"
BooleanLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "bool"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (BooleanLiteral c) : \<tau>" obtains "\<tau> = Boolean[1]"
IH: shows "geom_rand_walk u d v n w = lgeom_proc u d v (stake n w)"
assms_1: shows "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>"
l_set_val_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs get_tag_name get_tag_name_locs \<equiv> l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs \<and> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
comp_fun_commute_axioms: shows "comp_fun_commute (\<lambda>x. Mapping.update x p)"
eq_linorder_on_axioms: shows "eq_linorder_on D cmp"
disj_conj_dsitr: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool" shows "(P \<or> Q \<and> R) = ((P \<or> Q) \<and> (P \<or> R))"
Hermite_of_upt_row_i_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Hermite_of_upt_row_i A i ass res = foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<i])"
assms_2: assumes "A \<lesssim>\<^sub>\<circ> B"   and "\<not> A \<approx>\<^sub>\<circ> B" shows "P"
vertices_cases: fixes v :: "'form \<times> nat list" assumes "v |\<in>| vertices" obtains c :: "'form" where "c__ \<in> set conclusions" and "v = (c__, [])"   | c :: "'form" and "is" :: "nat list" where "c \<in> set conclusions" and "is \<in> it_paths (it' c)" and "v = (c, 0 # is)"
eventually_at_filter: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and s :: "'c set" shows "eventually P (at a within s) = (\<forall>\<^sub>F x in nhds a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
sum_decomp_f: shows "(\<Sum>i = 0..<length t. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0) = (\<Sum>i = 0..<i. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0) + (\<Sum>i = i..<j + 1. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0) + (\<Sum>i = j + 1..<length t. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0)"
contraposition_counterexample: shows "\<not> valid_in {1, 2} (Eql' (Imp' (Pro ''p'') (Pro ''q'')) (Imp' (Neg' (Pro ''q'')) (Neg' (Pro ''p''))))"
idx_spec: fixes xs :: "'a list" shows "incr xs = (\<forall>i<length xs - 1. xs ! i < xs ! Suc i)"
white_heap_def: shows "white_heap = (\<lambda>s. \<forall>x. obj_at (\<lambda>s. True) x s \<longrightarrow> white x s)"
flat_codom: fixes f :: "'c \<rightarrow> 'b"   and x :: "'c"   and c :: "'b" assumes "f\<cdot>x = c" shows "f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
matrix_inv_Gauss_Jordan_iarrays_PA: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "invertible A" shows "matrix_to_iarray (matrix_inv A) = fst (Gauss_Jordan_iarrays_PA (matrix_to_iarray A))"
enres_lift_def: fixes m :: "'h nres" shows "enres_lift m \<equiv> m \<bind> (\<lambda>x. RETURN (Inr x))"
hyps_1: fixes c :: "('a, 'b, 'c, 'd) com"   and ictxt :: "('a, 'b, 'c, 'd) com \<Rightarrow> ('a, 'b, 'c, 'd) com"   and fctxt :: "('a, 'b, 'c, 'd) com \<Rightarrow> ('a, 'b, 'c, 'd) com list"   and cs :: "('a, 'b, 'c, 'd) com list" assumes "(c, ictxt, fctxt) \<in> decompose_com c01" shows "(c, loc_compC c (fctxt c @ cs)) \<in> fragments c01 (atCs cs)"
l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Shadow_DOM.remove"
finite_lin_spanTr3: fixes A :: "'b set"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "ideal R A" shows "h \<in> {j. j \<le> n} \<rightarrow> carrier M \<longrightarrow> (\<forall>naa. \<forall>s\<in>{j. j \<le> naa} \<rightarrow> A. \<forall>f\<in>{j. j \<le> naa} \<rightarrow> h ` {j. j \<le> n}. \<exists>t\<in>{j. j \<le> n} \<rightarrow> A. l_comb R M naa s f = l_comb R M n t h)"
trans: fixes b :: "'g"   and a :: "'g"   and c :: "'g" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
elims: fixes x :: "nat"   and xa :: "rat vec"   and y :: "constraint list" assumes "from_index_geq0_vector x xa = y" obtains i :: "nat" and v :: "rat vec" where "x = i" and "xa = v" and "y = map (\<lambda>j. GEQ (lp_monom 1 (i + j)) (v $ j)) [0..<dim_vec v]"