hcju/isabelleps huggingface dataset:
inf_sup_aci_6: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
semi_mojmir: fixes \<psi> :: "'a ltl" shows "semi_mojmir \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w"
right_left: shows "right (\<lambda>x y. y < x) \<and> left (<) \<or> right (<) \<and> left (\<lambda>x y. y < x)"
scaleC: fixes f :: "'d \<Rightarrow> 'e"   and r :: "complex"   and x :: "'d" assumes "antilinear f" shows "f (r *\<^sub>C x) = cnj r *\<^sub>C f x"
sum_pos: fixes M :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'b" assumes "finite M"   and "\<forall>x\<in>M. (0::'c) \<le> f x"   and "y \<in> M"   and "(0::'c) < f y" shows "(0::'c) < sum f M"
rtc_appL: fixes s :: "trm"   and u :: "trm"   and t :: "trm" assumes "s \<longlonglongrightarrow>\<^sup>* u" shows "s \<degree> t \<longlonglongrightarrow>\<^sup>* u \<degree> t"
set_union: fixes xs :: "'a list"   and ys :: "'a list" shows "set (List.union xs ys) = set xs \<union> set ys"
dickson_grading_adds_imp_le: fixes d :: "'e \<Rightarrow> nat"   and s :: "'e"   and t :: "'e" assumes "dickson_grading d"   and "s adds t" shows "d s \<le> d t"
rational_algebraic_int_is_int: fixes x :: "'b" assumes "algebraic_int x"   and "x \<in> \<rat>" shows "x \<in> \<int>"
weakly_preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (local.map (src\<^sub>C \<mu>)) (src\<^sub>D (local.map \<mu>))"
mset_iff_set_tree: fixes x :: "'b"   and t :: "'b tree" shows "(x \<in># mset_tree t) = (x \<in> set_tree t)"
suffixI: fixes ys :: "'a list"   and zs :: "'a list"   and xs :: "'a list" assumes "ys = zs @ xs" shows "suffix xs ys"
fmpred_empty: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "fmpred P fmempty"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>)" shows "\<CC>\<lparr>Cod\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>)"
of_complex_ii: shows "of_complex \<i> = ii\<^sub>h"
heap_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "Leftist_Heap.linorder_class.heap t" shows "Leftist_Heap.linorder_class.heap (insert x t)"
assms_1: shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r"
us: shows "us = getFrN (map snd txs) (map fst txs) [all y \<phi>] (length txs)"
simps_2: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and L :: "lock_action"   and Ls :: "lock_action list" shows "lock_actions_ok' l t (L # Ls) = (L = Lock \<and> \<not> may_lock l t \<or> lock_action_ok l t L \<and> lock_actions_ok' (upd_lock l t L) t Ls)"
is_OK_returns_heap_E: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog" assumes "h \<turnstile> ok f" obtains h' :: "'ap" where "h \<turnstile> f \<rightarrow>\<^sub>h h'"
sfilter_const_TT: fixes xs :: "[:'r:]" shows "sfilter\<cdot>(\<Lambda> x. TT)\<cdot>xs = xs"
integrate_insert_commute_insert_axioms: shows "integrate_insert_commute_insert M a s m"
assms_2: shows "n = \<Prod>A2 * s2\<^sup>2"
chamber_retraction2: fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>" shows "f ` C = C"
USigma_iff: fixes \<I> :: "'a set set"   and J :: "'a set \<Rightarrow> 'a set"   and x :: "'a set" assumes "\<And>I j. I \<in> \<I> \<Longrightarrow> I \<lless> J I \<and> finite I" shows "(x \<in> USigma \<I> J) = usplit (\<lambda>I j. I \<in> \<I> \<and> j \<in> J I \<and> x = insert j I) x"
INF_apply: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'b" shows "Inf (f ` A) x = (INF y\<in>A. f y x)"
simps_1: fixes p :: "('a, 'b) form" shows "put_unis 0 p = p"
assms: shows "infer \<Gamma> (Fst P) = Some \<tau>"
adverse_Suc: fixes s :: "nat \<Rightarrow> nat option"   and x :: "nat"   and z :: "nat"   and i :: "nat"   and j :: "nat" assumes "s \<in> \<R>"   and "0 < x" shows "adverse s z i j (Suc x) \<down>= (if s (e_snoc (adverse s z i j \<triangleright> x) 0) \<noteq> s (adverse s z i j \<triangleright> x) then 0 else if s (e_snoc (adverse s z i j \<triangleright> x) 1) \<noteq> s (adverse s z i j \<triangleright> x) then 1 else z)"
simps_1: fixes x1 :: "'d"   and x2 :: "'e"   and y1 :: "'d"   and y2 :: "'e" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
coprime_heuristic: fixes f :: "int poly"   and g :: "int poly" assumes "coprime_heuristic f g"   and "content f = 1 \<or> content g = 1" shows "comm_monoid_mult_class.coprime f g"
expand_new_name_step: fixes n :: "'b node" shows "name n < expand_new_name (name n)"
update_by_fun_gr_pair_eq_update_by_fun_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and k :: "'a"   and f :: "'d \<Rightarrow> 'd" assumes "oalist_inv_raw x xs" shows "update_by_fun_gr_pair x k f xs = update_by_fun_pair x k f xs"
sum_mset_sum_count: fixes M :: "nat multiset" shows "\<Sum>\<^sub># M = (\<Sum>i\<in>set_mset M. count M i * i)"
set_bind_spmf: fixes M :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "set_spmf (M \<bind> f) = set_spmf M \<bind> set_spmf \<circ> f"
isomorphic_transitive: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "R.isomorphic f g"   and "R.isomorphic g h" shows "R.isomorphic f h"
distinct_9: fixes trm1 :: "trm"   and trm2 :: "trm"   and name' :: "name"   and trm' :: "trm" shows "App trm1 trm2 \<noteq> \<Lambda> name' . trm'"
d_1: fixes p :: "int poly" shows "complex_roots_of_int_poly_all p = (let n = degree p in if 3 \<le> n then complex_roots_of_int_poly3 p else if n = 1 then [roots1 (complex_of_int_poly p)] else if n = 2 then croots2 (complex_of_int_poly p) else [])"
a: shows "pp_of_term ` (Keys (set (fst z)) \<union> set (fst (snd (snd z)))) \<subseteq> dgrad_set d m"
case_prodE: fixes c :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and p :: "'x \<times> 'y" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'x" and y :: "'y" where "p = (x, y)" and "c x y"
x: shows "LeftDerives1 (u @ v) (fst d) (snd d) x \<and> LeftDerivation x D w"
simps: fixes a :: "'a"   and T :: "('a, 'b) TS_scheme" shows "(a \<in> reach T) = ((\<exists>s. a = s \<and> s \<in> init T) \<or> (\<exists>s t. a = t \<and> (s, t) \<in> TS.trans T \<and> s \<in> reach T))"
ObjectC_def: shows "ObjectC \<equiv> (Object, undefined, [], [])"
phi_top: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "phi top = Filters.filter.Abs_filter UNIV"
LIMSEQ_def: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
GetHigh: 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 \<rhd> Get x y f : HighSec"
Un_upper1: fixes A :: "'a set"   and B :: "'a set" shows "A \<subseteq> A \<union> B"
in_sorted_domain_iff_lookup: fixes k :: "'a"   and ko :: "'o"   and xs :: "'x" shows "(k \<in> set (sorted_domain ko xs)) = (lookup xs k \<noteq> (0::'b))"
cr_zmultiset_def: shows "cr_zmultiset \<equiv> \<lambda>x. (=) (abs_zmultiset x)"
Rep_cfun_inject: fixes x :: "'e \<rightarrow> 'f"   and y :: "'e \<rightarrow> 'f" shows "(Rep_cfun x = Rep_cfun y) = (x = y)"
asm: shows "(as, x) \<approx>lst (=) supp p' (as', x')"
DJ_def: fixes f :: "fm \<Rightarrow> fm"   and p :: "fm" shows "DJ f p = evaldjf f (disjuncts p)"
v_2: shows "C0 = insert v (C0 \<inter> D0)"
\<pi>_defined: fixes u :: "'v" assumes "Q u \<noteq> \<infinity>" shows "\<pi> u \<noteq> None"
ltl_lmap: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "ltl (lmap f xs) = lmap f (ltl xs)"
fix_ind: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b \<rightarrow> 'b" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (F\<cdot>x)" shows "P (fix\<cdot>F)"
subst_apply_term_empty: fixes t :: "('d, 'e) Term.term" shows "t \<cdot> Var = t"
nonneg1_imp_zdiv_pos_iff: fixes a :: "int"   and b :: "int" assumes "0 \<le> a" shows "(0 < a div b) = (b \<le> a \<and> 0 < b)"
even_or_iff: fixes a :: "'a"   and b :: "'a" shows "even (a OR b) = (even a \<and> even b)"
intro: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd" assumes "category A1"   and "category A2"   and "category B"   and "product_category A1 A2"   and "functor (product_category.comp A1 A2) B F" shows "binary_functor A1 A2 B F"
gext_gupd: fixes s :: "st"   and r :: "oref"   and x :: "obj" assumes "globs s r = None" shows "s\<le>|gupd(r\<mapsto>x) s"
spoly_swap: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.spoly p q = - punit.spoly q p"
gtrp_loc: fixes s :: "'c"   and c :: "'c multiset"   and e :: "'d"   and s' :: "'c"   and c' :: "'c multiset"   and S :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set" assumes "((s, c), e, s', c') \<in> S" shows "((s, c), LOC e, s', c') \<in> gtrp S"
rename_rename_guarantees_eq: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and X :: "'c program set"   and Y :: "'c program set" assumes "bij h" shows "(rename h F \<in> rename h ` X guarantees rename h ` Y) = (F \<in> X guarantees Y)"
ae: shows "AE x in M. prob {x} \<noteq> 0"
llength_lappend_endE: fixes n :: "enat"   and w :: "'b llist" assumes "eSuc n \<le> llength w" obtains a :: "'b" and v :: "'b llist" where "w = v $ <a>" and "n \<le> llength v"
span_mono: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "A \<subseteq> B" shows "local.punit.pmdl A \<subseteq> local.punit.pmdl B"
not_ordLeq_ordLess: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r \<le>o r'" shows "(r', r) \<notin> ordLess"
linear_independent_injective_image: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and s :: "'e set" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f"   and "\<not> module.dependent s1 s"   and "inj_on f (module.span s1 s)" shows "\<not> module.dependent s2 (f ` s)"
type_wf_defs: fixes h :: "(_) heap" shows "ShadowRootClass.type_wf h = (DocumentClass.type_wf h \<and> (\<forall>shadow_root_ptr\<in>fset (shadow_root_ptr_kinds h). get\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h \<noteq> None))"
c_f_list_at_Suc: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and y :: "nat"   and x :: "nat" shows "c_f_list f (Suc y) x = c_cons (f (Suc y) x) (c_f_list f y x)"
wp_g_odei: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> bool"   and U :: "'a \<Rightarrow> real set"   and S :: "'a set"   and t\<^sub>0 :: "real"   and Q :: "'a \<Rightarrow> bool" assumes "\<lceil>P\<rceil> \<subseteq> \<lceil>I\<rceil>"   and "\<lceil>I\<rceil> \<subseteq> wp x\<acute>=f & G on U S @ t\<^sub>0 \<lceil>I\<rceil>"   and "\<lceil>\<lambda>s. I s \<and> G s\<rceil> \<subseteq> \<lceil>Q\<rceil>" shows "\<lceil>P\<rceil> \<subseteq> wp x\<acute>=f & G on U S @ t\<^sub>0 DINV I  \<lceil>Q\<rceil>"
scast_ucast_norm_2: fixes b :: "'b signed word"   and a :: "'b word" shows "(b = UCAST('b \<rightarrow> 'b signed) a) = (a = SCAST('b signed \<rightarrow> 'b) b)"
less_def: fixes x :: "int"   and y :: "int" shows "int.less x y = (x \<le> y \<and> x \<noteq> y)"
vector_one_nth: fixes x :: "('a, 1) vec" shows "vec (x $ 1) = x"
deg_comp_pp_def: fixes cmp :: "('a, 'b) pp \<Rightarrow> ('a, 'b) pp \<Rightarrow> order" shows "deg_comp cmp = (\<lambda>x y. case comparator_of (deg_pp x) (deg_pp y) of Eq \<Rightarrow> cmp x y | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
rd_exists: fixes x :: "char list"   and p :: "assertion" shows "rd_ass (Exists x p) = rd_ass p"
arr_dom_iff_arr: fixes f :: "'c" shows "Cop.arr (Cop.dom f) = Cop.arr f"
map_mat_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c mat" shows "map_mat f A \<equiv> Matrix.mat (dim_row A) (dim_col A) (\<lambda>ij. f (A $$ ij))"
source_nonempty: shows "\<exists>s. s \<in> \<S>"
comp_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "dual_category C" shows "dual_category.comp C g f \<equiv> C f g"
3: shows "bprv (imp (PP \<langle>\<phi>L \<phi>\<rangle>) (PP \<langle>PP \<langle>\<phi>L \<phi>\<rangle>\<rangle>))"
representation_raw_def: shows "representation \<equiv> module.representation (*\<^sub>R)"
ide_pbdom: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "ide (f \<down>\<down> g)"
approx_slp_lengthD: fixes p :: "nat"   and slp :: "floatarith list"   and a :: "((real \<times> real pdevs) \<times> real) list"   and xs :: "((real \<times> real pdevs) \<times> real) list" assumes "approx_slp p slp a = Some xs" shows "length xs = length slp + length a"
exec_step_ind_Putfield_Null: fixes stk :: "val list"   and F :: "char list"   and C :: "char list"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "hd (tl stk) = Null" shows "exec_step_ind (StepI (Putfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
simple_functionD_2: fixes M :: "'b measure"   and g :: "'b \<Rightarrow> 'c"   and X :: "'c set" assumes "simple_function M g" shows "g -` X \<inter> space M \<in> sets M"
IH: shows "le_trans (iterates body G i) (iterates body G (Suc i))"
inverts_mat_length_1: fixes A :: "'b mat"   and B :: "'b mat" assumes "square_mat A"   and "inverts_mat A B"   and "inverts_mat B A" shows "dim_row B = dim_row A"
add_del_arc_collapse: fixes a :: "'b" shows "pre_digraph.add_arc (del_arc a) a = add_arc a"
HSem_below: fixes h :: "('d \<times> 'e) list"   and \<rho> :: "'d \<Rightarrow> 'f"   and r :: "'d \<Rightarrow> 'f"   and ESem :: "'e \<Rightarrow> ('d \<Rightarrow> 'f) \<rightarrow> 'f" assumes "\<And>x. x \<notin> domA h \<Longrightarrow> \<rho> x \<sqsubseteq> r x"   and "\<And>x. x \<in> domA h \<Longrightarrow> ESem (the (map_of h x))\<cdot>r \<sqsubseteq> r x" shows "has_ESem.HSem ESem h\<cdot>\<rho> \<sqsubseteq> r"
assms_1: shows "internal_equivalence f g \<eta> \<epsilon>"
add_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus c a) (plus c b)"
const_in: fixes x :: "'b" assumes "x \<in> Field s" shows "rs.const x = zero r"
span_explicit: fixes b :: "'b set" assumes "b \<subseteq> U\<^sub>V\<^sub>S" shows "local.span b = {x \<in> U\<^sub>V\<^sub>S. \<exists>y\<subseteq>U\<^sub>V\<^sub>S. \<exists>f. (finite y \<and> y \<subseteq> b) \<and> x = (\<Sum>a\<in>y. f a *s a)}"
associated_polynomials_imp_same_roots: fixes p :: "'a list"   and q :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "q \<in> carrier (poly_ring R)"   and "p \<sim>\<^bsub>poly_ring R\<^esub> q" shows "roots p = roots q"
create_expunge_commute: fixes i1 :: "'d"   and e1 :: "'c"   and e2 :: "'c"   and mo :: "'d"   and i2 :: "'d" shows "\<langle>Create i1 e1\<rangle> \<rhd> \<langle>Expunge e2 mo i2\<rangle> = \<langle>Expunge e2 mo i2\<rangle> \<rhd> \<langle>Create i1 e1\<rangle>"
negHorbit_list_indV: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" assumes "set gs \<subseteq> H"   and "T ` set as \<subseteq> indV" shows "set (concat (negHorbit_list gs T as)) \<subseteq> indV"
zero_def: 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" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "zero = Abs 0"
replace_f: shows "\<forall>pt. matches \<gamma> (replace_f pt) a p = matches \<gamma> (negation_type_to_match_expr_f C pt) a p"
simply_connected_iff_simple: fixes S :: "complex set" assumes "open S"   and "bounded S" shows "simply_connected S = (connected S \<and> connected (- S))"
sumIdemp: fixes P :: "pi" shows "P \<oplus> P \<sim> P"
remove_hd: fixes z :: "'a"   and xs :: "'a list" shows "remove1 z (z # xs) = xs"
gtrp_loc: fixes s :: "'a"   and c :: "'a multiset"   and e :: "'b"   and s' :: "'a"   and c' :: "'a multiset"   and S :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((s, c), e, s', c') \<in> S" shows "((s, c), LOC e, s', c') \<in> gtrp S"
OV_inv3D: fixes x :: "'c obsv_state_scheme"   and r :: "nat"   and p :: "process"   and v :: "val" assumes "x \<in> OV_inv3"   and "obsv_state.obs x r p = Some v" shows "obs_safe r x v"
obj_Op: fixes C :: "('d, 'e, 'f) Category_scheme" shows "obj\<^bsub>Op C\<^esub> = obj\<^bsub>C\<^esub>"
abSubgroup_inner_dirsum_el_decomp_set: fixes G :: "'a set"   and Hs :: "'a set list"   and x :: "'a" assumes "AbGroup G"   and "\<forall>H\<in>set Hs. Group H \<and> H \<subseteq> G"   and "add_independentS Hs"   and "x \<in> (\<Oplus>H\<leftarrow>Hs. H)" shows "set (\<Oplus>Hs\<leftarrow> x) \<subseteq> G"
inj_single: fixes k :: "'c" shows "inj (Poly_Mapping.single k)"
ssorted_iff_mono: fixes s :: "'b stream" shows "ssorted s = (\<forall>i j. i \<le> j \<longrightarrow> s !! i \<le> s !! j)"
ws_update_foreach_def: fixes E :: "('d \<times> 'c) set"   and u :: "'d"   and p :: "'d list"   and V :: "'c set"   and ws :: "'c \<Rightarrow> 'd list option" shows "ws_update_foreach E u p V ws \<equiv> FOREACH (LIST_SET_REV_TAG (E `` {u})) (\<lambda>v (V, ws). if v \<in> V then RETURN (V, ws) else ASSERT (v \<notin> dom ws) \<bind> (\<lambda>_. RETURN (insert v V, ws(v \<mapsto> u # p)))) (V, ws)"
prems_1: shows "enc (w, I) = enc (w', I')"
uminus_2_2: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "invariant_2 (uminus_2 x)"
compact_imp_seq_compact: fixes U :: "'a set" assumes "compact U" shows "seq_compact U"
nth_rotate1: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "rotate1 xs ! n = xs ! (Suc n mod length xs)"
lin_indpt_list_def: fixes fs :: "real vec list" shows "gso.lin_indpt_list fs = (set fs \<subseteq> carrier_vec n \<and> distinct fs \<and> gso.lin_indpt (set fs))"
frag_extend_def: fixes b :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and x :: "'d \<Rightarrow>\<^sub>0 int" shows "frag_extend b x \<equiv> \<Sum>i\<in>keys x. frag_cmul (lookup x i) (b i)"
cong_imp_coprime: fixes a :: "'b"   and b :: "'b"   and m :: "'b" assumes "[a = b] (mod m)"   and "coprime a m" shows "coprime b m"
s1': shows "s1' \<in> pmdl.syzygy_module (set bs)"
sdsj: fixes F :: "'fmla set" assumes "F \<subseteq> fmla"   and "finite F" shows "sdsj F \<in> fmla"
LIMSEQ_lessThan_iff_atMost: fixes f :: "nat set \<Rightarrow> 'b"   and x :: "'b" shows "(\<lambda>n. f {..<n}) \<longlonglongrightarrow> x = (\<lambda>n. f {..n}) \<longlonglongrightarrow> x"
assms_2: shows "set (concat mss) \<subseteq> M"
basis_card_eq_dim: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set" assumes "B \<subseteq> V"   and "V \<subseteq> vec.span B"   and "vec.independent B" shows "card B = vec.dim V"
univ_poly_a_inv_consistent: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> p = \<ominus>\<^bsub>carrier R [X]\<^esub> p"
setdist_gt_0_compact_closed: fixes S :: "'c set"   and T :: "'c set" assumes "compact S"   and "closed T" shows "(0 < setdist S T) = (S \<noteq> {} \<and> T \<noteq> {} \<and> S \<inter> T = {})"
maximalideal_axioms_def: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" shows "maximalideal_axioms I R \<equiv> carrier R \<noteq> I \<and> (\<forall>J. ideal J R \<longrightarrow> I \<subseteq> J \<longrightarrow> J \<subseteq> carrier R \<longrightarrow> J = I \<or> J = carrier R)"
cases: fixes x :: "'a list list \<times> 'a list list" obtains p1 :: "'a list list" and p2 :: "'a list list" where "x = (p1, p2)"
aform_of_point_def: fixes x :: "'a" shows "aform_of_point x = aform_of_ivl x x"
set_def: fixes l :: "'b" shows "local.set l = set_mset (multiset l)"
lens_indep_right_comp: fixes y :: "'d \<Longrightarrow> 'e"   and z :: "'d \<Longrightarrow> 'e"   and x :: "'f \<Longrightarrow> 'd" assumes "y \<bowtie> z" shows "x ;\<^sub>L y \<bowtie> x ;\<^sub>L z"
B: shows "Esigma (h * M) = h * M + h * N"
deriv_c2: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q,A"
bisim2c_bad: fixes cipher :: "bool list \<times> bool list"   and \<sigma> :: "'a"   and s :: "bool list \<Rightarrow> bool list option"   and x :: "bool list"   and r :: "bool list" shows "rel_spmf (\<lambda>(x, uu_) (y, uu_). x = y) (exec_gpv rf_encrypt (\<A>2 cipher \<sigma>) (s(x \<mapsto> r))) (exec_gpv (rf_encrypt_bad x) (\<A>2 cipher \<sigma>) (s(x \<mapsto> r), False))"
qSkel_qSwapAll: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A :: "('a, 'b, 'c, 'd, 'e) qAbs" shows "qSkel (X #[[x \<and> y]]_zs) = qSkel X \<and> qSkelAbs (A $[[x \<and> y]]_zs) = qSkelAbs A"
mult_eSuc': fixes m :: "eint"   and n :: "eint" shows "(m + 1) * n = n + m * n"
ssuf_ext: fixes r :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "r <s u" shows "r <s v \<cdot> u"
absorb2: fixes b :: "'l"   and a :: "'l" assumes "b \<le> a" shows "min a b = b"
iso_sym: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme" assumes "Group.group G"   and "G \<cong> H" shows "H \<cong> G"
equalityD1: fixes A :: "'b set"   and B :: "'b set" assumes "A = B" shows "order.greater_eq B A"
consis_take: shows "weak_sharing_consistent \<O> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
master_theorem_rtl: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set" assumes "X \<subseteq> subformulas\<^sub>\<mu> \<phi>"   and "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>"   and "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu>"   and "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)"   and "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)" shows "w \<Turnstile>\<^sub>n \<phi>"
antisymp_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "antisymp r = (\<forall>x y. r x y \<longrightarrow> r y x \<longrightarrow> x = y)"
that: fixes T :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'a \<times> 'b" assumes "closed T"   and "homeomorphism S T f fst" shows "thesis"