hcju/isabelleps huggingface dataset:
list_update_code_3: fixes x :: "'c"   and xs :: "'c list"   and i :: "nat"   and y :: "'c" shows "(x # xs)[Suc i := y] = x # xs[i := y]"
abs_eq: fixes xa :: "'b ltl"   and x :: "'b set list" shows "af_G_abs.f_foldl_abs (Abs xa) x = Abs (af\<^sub>G xa x)"
subgroupI: fixes H :: "nat set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> inv\<^bsub>G\<^esub> a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>G\<^esub> b \<in> H" shows "subgroup H G"
p_id_assoc2: fixes R :: "('a \<times> 'a set) set"   and T :: "('a \<times> 'a set) set" shows "R \<cdot> 1\<^sub>\<pi> \<cdot> T = R \<cdot> (1\<^sub>\<pi> \<cdot> T)"
src_cod: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "src (cod \<mu>) = src \<mu>"
continuous_norm: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "continuous F f" shows "continuous F (\<lambda>x. norm (f x))"
Card_order_Plus_empty2: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "r =o |{} <+> Field r|"
\<phi>_ide_simp: fixes f :: "'a" assumes "\<guillemotleft>f : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>"   and "ide f" shows "d\<^sub>0e\<^sub>0.\<phi> f = (e\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[d\<^sub>1, f \<star> e\<^sub>0, d\<^sub>0]) \<cdot> \<a>[e\<^sub>1, d\<^sub>1, (f \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, e\<^sub>0, d\<^sub>0]) \<cdot> (local.inv \<epsilon>\<^sub>1 \<star> f \<star> local.inv \<epsilon>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src d\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f]"
pred_of_set_fold_sup: fixes A :: "'a set" assumes "finite A" shows "pred_of_set A = Finite_Set.fold (\<squnion>) \<bottom> (single ` A)"
AG_all_s: fixes x :: "'a"   and y :: "'a"   and s :: "'a set" assumes "x \<rightarrow>\<^sub>i* y"   and "x \<in> AG s" shows "y \<in> AG s"
Un_subset_iff: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "(A \<union> B \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
stateful_policy_compliance_axioms: shows "stateful_policy_compliance \<T> G M"
size_13: fixes x6 :: "'b ptrm" shows "size (PFst x6) = size x6 + Suc 0"
hd_take: fixes j :: "nat"   and xs :: "'c list" assumes "0 < j" shows "hd (take j xs) = hd xs"
known_set_def: fixes S :: "state" shows "known_set S = {..<length (est S)} - unknown_set S"
intros_30: fixes \<C> :: "t_context"   and n :: "nat"   and a :: "nat"   and tp :: "tp option"   and t :: "t"   and off :: "nat" assumes "memory \<C> = Some n"   and "load_store_t_bounds a tp t" shows "\<C> \<turnstile> [Store t tp a off] : [T_i32, t] _> []"
Q6''_a_def: shows "Q6''_a \<equiv> [: p, t, lnk, lbl, mrk \<leadsto> p', t', lnk', lbl', mrk' . p = nil \<and> p' = p \<and> t' = t \<and> lnk' = lnk \<and> lbl' = lbl \<and> mrk' = mrk :]"
vrrestriction_vlrestriction: fixes r :: "V"   and A :: "V" shows "(r \<restriction>\<^sup>r\<^sub>\<circ> A) \<restriction>\<^sup>l\<^sub>\<circ> A = r \<restriction>\<^sub>\<circ> A"
finite_measure_Diff: fixes M :: "'b measure"   and A :: "'b set"   and B :: "'b set" assumes "finite_measure M"   and "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
rev_drop: fixes i :: "nat"   and xs :: "'a list" shows "rev (drop i xs) = take (length xs - i) (rev xs)"
1: shows "isomorphic_rep r (tab\<^sub>0 r) (tab\<^sub>1 r)"
pw_ref_I: fixes S' :: "'a nres"   and S :: "'b nres"   and R :: "('b \<times> 'a) set" assumes "nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> (\<exists>s'. (x, s') \<in> R \<and> inres S' s'))" shows "S \<le> \<Down> R S'"
run_ipath: fixes G :: "('b, 'c) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" assumes "graph_defs.is_run G r" shows "ipath (g_E G) r"
invert_opposite_sides_of_square: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a"   and k :: "'a \<times> 'a" assumes "VoV.seq f g"   and "VoV.comp f g = VoV.comp h k"   and "VoV.iso f"   and "VoV.iso k" shows "VoV.seq g (VoV.inv k) \<and> VoV.seq (VoV.inv f) h \<and> VoV.comp g (VoV.inv k) = VoV.comp (VoV.inv f) h"
inj_on_fimage_set_diff: fixes f :: "'a \<Rightarrow> 'b"   and C :: "'a fset"   and A :: "'a fset"   and B :: "'a fset" assumes "inj_on f (fset C)"   and "A |\<subseteq>| C"   and "B |\<subseteq>| C" shows "f |`| (A |-| B) = f |`| A |-| f |`| B"
\<epsilon>_def: shows "\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) \<eta>\<epsilon>.FG.map \<epsilon>o"
decreasing_sum_le_integral_1: fixes f :: "real \<Rightarrow> real"   and I :: "real" assumes "\<And>x. 0 \<le> x \<Longrightarrow> 0 \<le> f x"   and "\<And>x y. \<lbrakk>0 \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> f y \<le> f x"   and "(f has_integral I) {0..}" shows "summable (\<lambda>i. f (real (Suc i)))"
mem_val_alt_10_unchanged0: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 (pa AND 68719476732) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 (pa AND 68719476732) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
rotate_awalkE: fixes u :: "'a"   and p :: "'b list"   and w :: "'a" assumes "awalk u p u"   and "w \<in> set (awalk_verts u p)" obtains q :: "'b list" and r :: "'b list" where "p = q @ r" and "awalk w (r @ q) w" and "set (awalk_verts w (r @ q)) = set (awalk_verts u p)"
list_emb_Cons_iff1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "P x y" shows "list_emb P (x # xs) (y # ys) = list_emb P xs ys"
start_remove_subs: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" shows "set (start_remove xs x ys) \<subseteq> set xs \<union> set ys"
M_def: shows "M = a * y - q / ((2::'a) * a)"
list_all2_refl: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<And>x. P x x" shows "list_all2 P xs xs"
simps_1: shows "P_D2_H2_d D1 = linord_of_list [(D1, H1), (D1, H2)]"
hn_refineD: fixes \<Gamma> :: "assn"   and c :: "'b Heap"   and \<Gamma>' :: "assn"   and R :: "'c \<Rightarrow> 'b \<Rightarrow> assn"   and m :: "'c nres" assumes "hn_refine \<Gamma> c \<Gamma>' R m"   and "nofail m" shows "<\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
that: fixes ns'' :: "'node SDG_node list" assumes "realizable n (ns @ ns'') n'" shows "thesis"
kind_Predicate_notin_slice_slice_kind_Predicate: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "valid_edge a"   and "kind a = (Q)\<^sub>\<surd>" obtains Q' :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" where "slice_kind S a = (Q')\<^sub>\<surd>" and "Q' = (\<lambda>s. False) \<or> Q' = (\<lambda>s. True)"
psize_def: fixes p :: "'a poly" shows "psize p = (if p = 0 then 0 else Suc (degree p))"
emeasure_eq_sum_singleton: fixes S :: "'a set"   and M :: "'a measure" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" shows "emeasure M S = (\<Sum>x\<in>S. emeasure M {x})"
assms_2: shows "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'"
mult_le_cancel_left1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq c (times c b) = ((less zero c \<longrightarrow> less_eq one b) \<and> (less c zero \<longrightarrow> less_eq b one))"
assms: shows "can_occur (Recv i p q s s' m) c"
smalloI_tendsto: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> (0::'c)) F"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "f \<in> o[F](g)"
adm: shows "ccpo.admissible (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. P (fst x) (snd x))"
dickson_gradingD1: fixes d :: "'c \<Rightarrow> nat"   and s :: "'c"   and t :: "'c" assumes "dickson_grading d" shows "d (s + t) = ord_class.max (d s) (d t)"
the_\<theta>_props_1: shows "\<guillemotleft>\<chi>.the_\<theta> : u\<^sub>0 \<star> \<chi>.chine \<Rightarrow> s\<^sub>0\<guillemotright>"
bounds_compare_contradictory_6: fixes c :: "'a"   and b :: "'a option" assumes "c \<ge>\<^sub>l\<^sub>b b"   and "c <\<^sub>l\<^sub>b b" shows "False"
onorm_le: fixes f :: "'d \<Rightarrow> 'e"   and b :: "real" assumes "\<And>x. norm (f x) \<le> b * norm x" shows "onorm f \<le> b"
appR: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "u $ s \<rightarrow>\<^sub>\<beta> u $ t"
simps_24: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case Integer of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f3"
tci_def: fixes bdd :: "bddi" shows "tci bdd \<equiv> return (1, bdd)"
galois_property: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d"   and y :: "'e" assumes "galois_connection G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "(\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y)"
span_superset: fixes S :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "S \<subseteq> phull S"
psubset_imp_ex_mem: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater B A" shows "\<exists>b. b \<in> B - A"
WeakI1: fixes Gamma :: "formula list"   and A :: "(formula list \<times> formula list set) set"   and Delta :: "formula list" assumes "Gamma \<in> deductions A"   and "Weaks \<subseteq> A" shows "Delta @ Gamma \<in> deductions A"
triangle_encompassing_polychain_of: fixes p :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "0 \<le> det3 p a b"   and "0 \<le> det3 p b c"   and "0 \<le> det3 p c a"   and "ccw' a b c" shows "p \<in> convex hull {a, b, c}"
hcomp_vcomp: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data"   and \<mu>' :: "'a arrow_of_spans_data"   and \<nu>' :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>"   and "arr \<mu>'"   and "arr \<nu>'"   and "src \<nu>' = trg \<mu>'"   and "seq \<mu>' \<mu>"   and "seq \<nu>' \<nu>" shows "\<nu>' \<bullet> \<nu> \<star> \<mu>' \<bullet> \<mu> = (\<nu>' \<star> \<mu>') \<bullet> (\<nu> \<star> \<mu>)"
process_rule_3: fixes xs :: "'c list"   and Y :: "'c set"   and P :: "'c process"   and X :: "'c set" assumes "(xs, Y) \<in> failures P"   and "X \<subseteq> Y" shows "(xs, X) \<in> failures P"
eq_numeral_iff_iszero_5: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(numeral.numeral one plus x = one) = ring_1.iszero zero (neg_numeral.sub minus plus one x num.One)"
func_pre: fixes f :: "nat \<Rightarrow> 'd"   and n :: "nat"   and A :: "'d set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> A" shows "f \<in> {j. j \<le> n} \<rightarrow> A"
assms: shows "fv\<^sub>s\<^sub>e\<^sub>t M \<subseteq> subst_domain \<theta>"
vars1'_def: shows "vars1' = ind_in_set vars0 ` vars1"
atUV_union: fixes U :: "'a set"   and V :: "'a set"   and c1 :: "'a list multiset"   and c2 :: "'a list multiset" shows "atUV U V (c1 + c2) = (atUV U V c1 \<or> atUV U V c2 \<or> atU U c1 \<and> atU V c2 \<or> atU V c1 \<and> atU U c2)"
coeff_finsum: fixes A :: "'c set"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "finite A"   and "p \<in> A \<rightarrow> carrier P" shows "coeff P (finsum P p A) k = (\<Oplus>i\<in>A. coeff P (p i) k)"
limsup_INF_SUP: fixes f :: "nat \<Rightarrow> 'a" shows "limsup f = (INF n. Sup (f ` {n..}))"
assms_2: shows "fmdom (xm(i \<mapsto>\<^sub>f x)) = fmdom ym"
RBT_set_empty: shows "RBT_set RBT_Set2.empty = {}"
lbk_option_def: fixes T :: "nat" shows "lbk_option T = (\<lambda>w. (MAX i\<in>{0..T}. prices Mkt stk i w) - prices Mkt stk T w)"
ct_list_eq_exists: fixes xs :: "ct list" shows "\<exists>ys. ct_list_eq xs (to_ct_list ys)"
mod_pure_star_dist: fixes h :: "pheap"   and P :: "assn"   and b :: "bool" shows "(h \<Turnstile> P * \<up> b) = (h \<Turnstile> P \<and> b)"
scale_zero_left: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e" assumes "module scale" shows "scale (0::'d) x = (0::'e)"
simps_1: fixes uu :: "'a"   and B :: "'a"   and uv :: "'b" shows "Cod (MkArr uu B uv) = B"
set_shadow_root_get_shadow_root_different_pointers: fixes ptr :: "(_) element_ptr"   and ptr' :: "(_) element_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr \<noteq> ptr'"   and "w \<in> set_shadow_root_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> get_shadow_root_locs ptr'" shows "r h h'"
liftAll_lift_comp: fixes phi :: "'d \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and inp :: "'e \<Rightarrow> 'f option" shows "liftAll phi (lift f inp) = liftAll (phi \<circ> f) inp"
build_first: fixes w :: "nat \<Rightarrow> 'b" shows "w 0 ## w [Suc 0..] = w"
Cons_1: fixes aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat" assumes "valid_mmuaux' args cur dt aux auxlist"   and "lin_ts_mmuaux aux = tss @ tss'"   and "\<And>ts. ts \<in> set (take (length tss) (lin_ts_mmuaux aux)) \<Longrightarrow> enat ts + right (args_ivl args) < enat dt" shows "valid_mmuaux' args cur dt (foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss) auxlist \<and> lin_ts_mmuaux (foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss) = tss'"
adjunction_in_bicategory_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" shows "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<equiv> adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<and> adjunction_in_bicategory_axioms V H \<a> \<i> src trg f g \<eta> \<epsilon>"
conjoin_non_aborting: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (c::'a::refinement_lattice)" shows "(d::'a::refinement_lattice) \<sqsubseteq> d \<iinter> (c::'a::refinement_lattice)"
dom_closed: fixes f :: "'c" assumes "Arr f" shows "Arr (C.dom f)"
hyps: fixes t1 :: "'a event list"   and t2 :: "'a event list"   and u :: "'a event list" assumes "s setinterleaves ((t1 @ t2, u), C)" shows "set t2 \<subseteq> set (drop (length t1) s)"
1: shows "B \<in> obj\<^bsub>NTCatDom (MakeNT (YMapInv' C X F x))\<^esub>"
Lambert_W'_eq_minus_exp_minus1: shows "Lambert_W' (- exp (- 1)) = - 1"
ge_eq_tpoly_trans: fixes r :: "('v pvar, hmultiset) tpoly"   and q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" assumes "r \<ge>\<^sub>p q"   and "q =\<^sub>p p" shows "r \<ge>\<^sub>p p"
substAbs_simp: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good Y"   and "good X"   and "xs \<noteq> ys \<or> x \<noteq> y"   and "fresh xs x Y" shows "Abs xs x X $[Y / y]_ys = Abs xs x (X #[Y / y]_ys)"
simps: fixes return :: "'d option \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d option \<Rightarrow> 'e) \<Rightarrow> 'e"   and x :: "'e"   and f :: "'d \<Rightarrow> ('d, 'e) optionT" shows "bind_option return bind (OptionT x) f = OptionT (bind x (\<lambda>x. case x of None \<Rightarrow> return None | Some y \<Rightarrow> run_option (f y)))"
assms_2: fixes i :: "nat" shows "s i \<in> borel_measurable M"
mkVarChannel_def: fixes v :: "varDecl"   and upd :: "((String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list) \<Rightarrow> 'b gState_scheme \<times> pState \<Rightarrow> 'b gState_scheme \<times> pState"   and g :: "'b gState_scheme"   and p :: "pState" shows "mkVarChannel v upd g p = (let (k, v, cs) = toVariable g p v; (g', p') = upd (lm.update k v) (g, p) in mkChannels g' p' cs)"
is_serial_solution_for_problem_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi> \<equiv> let I = \<Psi>\<^sub>I\<^sub>+; G = \<Psi>\<^sub>G\<^sub>+; ops = \<Psi>\<^sub>\<O>\<^sub>+ in G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I \<psi> \<and> list_all (\<lambda>op. ListMem op ops) \<psi>"
abs_summable_on_Sigma_project1': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B"   and "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)" shows "(\<lambda>x. infsetsum (f x) (B x)) abs_summable_on A"
p_def: fixes k :: "nat" shows "p k = (\<Sum>i<n. Root i ^ k)"
otherwith_def: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and P :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a"   and a :: "'c" shows "otherwith Q I P \<sigma> \<sigma>' a \<equiv> (\<forall>i. i \<notin> I \<longrightarrow> Q (\<sigma> i) (\<sigma>' i)) \<and> P \<sigma> a"
Aexp_Lam: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and y :: "Vars.var"   and e :: "exp"   and n :: "Arity" assumes "ArityAnalysisSafe Aexp" shows "env_delete y (Aexp e\<cdot>(pred\<cdot>n)) \<sqsubseteq> Aexp (Lam [y]. e)\<cdot>n"
field_lbound_gt_zero: fixes d1 :: "'e"   and d2 :: "'e" assumes "(0::'e) < d1"   and "(0::'e) < d2" shows "\<exists>e>0::'e. e < d1 \<and> e < d2"
minus_of_nat_in_nonpos_Ints: fixes n :: "nat" shows "- of_nat n \<in> \<int>\<^sub>\<le>\<^sub>0"
ctxt_subst_caseApp_1: fixes c :: "cmd" shows "\<forall>n E s. c[n=n liftM_ctxt E n]\<^sup>C[n=n \<diamond> \<^sup>\<bullet> liftM_trm s n]\<^sup>C = c[n=n liftM_ctxt E n \<^sup>\<bullet> liftM_trm s n]\<^sup>C"
MainCl_def: shows "MainCl = \<lparr>access = Public, cfields = [], methods = [], init = Skip, super = Object, superIfs = []\<rparr>"
alphaAbs_refl: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodAbs A" shows "A $= A"
icard_singleton: fixes a :: "'a" shows "icard {a} = eSuc 0"
simps: fixes s' :: "'a"   and s :: "'a"   and t' :: "'a"   and t :: "'a"   and r' :: "'a"   and r :: "'a" shows "euclid_ext_aux s' s t' t r' r = (if r = (0::'a) then let c = (1::'a) div unit_factor r' in ((s' * c, t' * c), normalize r') else let q = r' div r in euclid_ext_aux s (s' - q * s) t (t' - q * t) r (r' mod r))"
distinct_6: fixes x3 :: "num"   and x2 :: "num" shows "num.Bit1 x3 \<noteq> num.Bit0 x2"
intro: fixes M :: "'b \<Rightarrow> 'c measure" assumes "\<And>i. sigma_finite_measure (M i)" shows "product_sigma_finite M"
left_uniqueD: fixes A :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'b"   and z :: "'e"   and y :: "'b" assumes "left_unique A"   and "A x z"   and "A y z" shows "x = y"
gx: shows "\<forall>x\<in>M. g \<subseteq> x \<longrightarrow> g = x"
rel_inject_2: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x21 :: "'e"   and x22 :: "'e list"   and y21 :: "'f"   and y22 :: "'f list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
lhd_lunstream: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lhd (lunstream g s) = (case g s of Skip s' \<Rightarrow> lhd (lunstream g s') | Yield x xa \<Rightarrow> x)"
deriv_j_def: fixes d :: "nat \<times> nat \<times> nat" shows "deriv_j d = fst (snd d)"
basis_subset_carrier: fixes X :: "'a set" assumes "basis X" shows "X \<subseteq> carrier"
smap_onto_PosetComplex: shows "dual_order.poset_simplex_map \<P> ` \<P> = \<Sigma>"
ab: shows "int CARD('p) \<le> int CARD('pq)"
coalg_final: shows "coalg car_final1 car_final2 str_final1 str_final2"
lcm_dvdD2: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times (lcm a b) c" shows "dvd.dvd times b c"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)). \<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
hv_props_7: shows "\<forall>(k, \<gamma>)\<in>\<V>. valid_path \<gamma>"
cases: fixes a1 :: "'s"   and a2 :: "'s cfg" assumes "cfg_onp a1 a2" obtains cfg :: "'s cfg" and s :: "'s" where "a1 = s" and "a2 = cfg" and "state cfg = s" and "action cfg \<in> K s" and "\<And>t. t \<in> set_pmf (action cfg) \<Longrightarrow> cfg_onp t (cont cfg t)"
distinct_17: fixes x1 :: "char list"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "new x1 \<noteq> Call x101 x102 x103 x104"
wellformed_induce_subgraph: fixes G :: "('c, 'd) pre_digraph"   and vs :: "'c set" assumes "wf_digraph G" shows "wf_digraph (G \<restriction> vs)"
simps_2: fixes t' :: "'b"   and n :: "nat"   and t :: "'b" shows "has_locks \<lfloor>(t', n)\<rfloor> t = (if t = t' then Suc n else 0)"
is_arg_min_size_subprop: fixes m :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "is_arg_min m P x"   and "Q x"   and "\<And>y. Q y \<Longrightarrow> P y" shows "m (arg_min m Q) = m (arg_min m P)"
additive_sum: fixes f :: "'a set \<Rightarrow> ennreal"   and S :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "positive (lambda_system \<Omega> M f) f"   and "Measure_Space.additive (lambda_system \<Omega> M f) f"   and "finite S"   and "A ` S \<subseteq> lambda_system \<Omega> M f"   and "disjoint_family_on A S" shows "(\<Sum>i\<in>S. f (A i)) = f (\<Union> (A ` S))"
re_refine_aux2: shows "gen_dfs \<le> \<Down> (rel O \<langle>ES\<rangle>simple_state_rel) a.a.it_dfs"
mult_comm: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y = y \<cdot> x"
dist_float_def: shows "dist \<equiv> map_fun real_of_float (map_fun real_of_float id) dist"
1: shows "\<Gamma> (fresh_pgwt (\<Union> (funs_term ` T)) (\<Gamma> s)) = \<Gamma> s"