hcju/isabelleps huggingface dataset:
bijective_open_imp_closed_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "open_map X X' f"   and "f ` topspace X = topspace X'"   and "inj_on f (topspace X)" shows "closed_map X X' f"
interaction_bounded_by'_exec_gpv_count: fixes "consider" :: "'b \<Rightarrow> bool"   and gpv :: "('c, 'b, bool list) gpv"   and n :: "nat"   and x :: "'c"   and s' :: "'b \<Rightarrow> bool list option"   and s :: "'b \<Rightarrow> bool list option"   and count :: "('b \<Rightarrow> bool list option) \<Rightarrow> nat" assumes "interaction_bounded_by' consider gpv n"   and "(x, s') \<in> set_spmf (exec_gpv oracle gpv s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (oracle s x); dom \<sigma> \<subseteq> dom s; consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (oracle s x); dom \<sigma> \<subseteq> dom s; \<not> consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "dom \<sigma> \<subseteq> dom s" shows "count s' \<le> n + count s"
simple_path_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" shows "simple_path (part_circlepath z r s t) = (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2 * pi)"
add_del_block_inv: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl :: "'b set" assumes "incidence_system point_set block_collection"   and "bl \<subseteq> point_set" shows "incidence_system.del_block (incidence_system.add_block block_collection bl) bl = block_collection"
cases: fixes x :: "'b sort_key" obtains xa :: "'b \<times> sort_dir" and y :: "'b sort_key" where "x = NonFinal xa y"   | i :: "'b" where "x = Final i"
lossless_S1_12: fixes m :: "bool \<times> bool"   and out1 :: "unit" shows "lossless_spmf (S1_OT12 m out1)"
goBrk_impl_def: shows "goBrk_impl \<equiv> fst"
prime_gt_Suc_0_nat: fixes p :: "nat" assumes "prime p" shows "Suc 0 < p"
jmpl_instr_low_equal_sub2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "'c \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1))))))) PC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1))))))))))) \<and> t2 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2))))))) PC (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))))))))))" shows "low_equal t1 t2"
ipresFreshAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'c, 'd, 'i) Iteration.model_scheme" shows "ipresFreshAll h hA MOD \<equiv> ipresFresh h MOD \<and> ipresFreshAbs hA MOD"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_empty_case_1: fixes F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D = []" shows "D = []"
zmset_of_plus: fixes M :: "'a multiset"   and N :: "'a multiset" shows "zmset_of (M + N) = zmset_of M + zmset_of N"
aasc_seqn1: fixes a :: "ant"   and A :: "ant set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "aasc_seq A a n + 1 \<le> aasc_seq A a (Suc n)"
UP_simps_2: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (UP \<mu>) = UP\<^sub>0 (src\<^sub>B \<mu>)"
runit_naturality: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "V \<mu> (bicategory.runit V H \<a> \<i> src (partial_magma.dom V \<mu>)) = V (bicategory.runit V H \<a> \<i> src (partial_magma.cod V \<mu>)) (H \<mu> (src \<mu>))"
rotate_Suc: fixes n :: "nat"   and xs :: "'g list" shows "rotate (Suc n) xs = rotate1 (rotate n xs)"
less_funE: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "less_fun f g" obtains k :: "'a" where "f k < g k" and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'"
abc_comp_commute: fixes A :: "abc_inst list"   and B :: "abc_inst list"   and C :: "abc_inst list" shows "A [+] B [+] C = A [+] (B [+] C)"
knows2knowPQ_s: fixes P :: "specID"   and m :: "Secrets"   and Q :: "specID" assumes "knows P [sE m] \<or> knows Q [sE m]" shows "know P (sKS m) \<or> know Q (sKS m)"
simps: fixes i :: "nat"   and j :: "nat" shows "local.neville_aitken_main i j = (if i < j then smult (inverse (x j - x i)) (local.X i * local.neville_aitken_main (i + 1) j - local.X j * local.neville_aitken_main i (j - 1)) else [:f i:])"
exe_wf_theory_translate_imp_wf_theory: fixes a :: "exetheory" assumes "exe_wf_theory a" shows "wf_theory (translate_theory a)"
swap_6: fixes ii :: "nat" assumes "ii \<le> m" shows "rat_of_int (d fs' ii) = (if ii = k then \<parallel>fs'.gs.gso (k - 1)\<parallel>\<^sup>2 / \<parallel>fs.gs.gso (k - 1)\<parallel>\<^sup>2 * rat_of_int (d fs k) else rat_of_int (d fs ii))"
Quotient_uint64: shows "Quotient (=) Abs_uint64 Rep_uint64 cr_uint64"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_5: fixes t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (delete\<langle>t,s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>) \<union> fv (s \<cdot> \<theta>)"
ideal_eq_zero_iff: fixes F :: "'c set" shows "(ideal F = {0::'c}) = (F \<subseteq> {0::'c})"
fls_base_factor_to_fps_mult_conv_unit_factor: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor_to_fps (f * g) = unit_factor (fls_base_factor_to_fps f * fls_base_factor_to_fps g)"
residue_plus_zero_r: fixes k :: "nat"   and y :: "int" shows "0 \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = y mod p ^ k"
learn_fin_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" shows "learn_fin \<psi> U s \<equiv> environment \<psi> U s \<and> (\<forall>f\<in>U. \<exists>i n\<^sub>0. \<psi> i = f \<and> (\<forall>n<n\<^sub>0. s (f \<triangleright> n) \<down>= 0) \<and> (\<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= Suc i))"
lemma_interval_lt: fixes a :: "real"   and x :: "real"   and b :: "real" assumes "a < x"   and "x < b" shows "\<exists>d>0. \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a < y \<and> y < b"
uncurry_simp: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'a \<times> 'b"   and \<tau> :: "'a \<Rightarrow> ('b \<Rightarrow> 'c, 'b \<Rightarrow> 'c) concrete_category.arr" assumes "currying A1 A2 B"   and "partial_magma.arr (product_category.comp A1 A2) f" shows "currying.uncurry A1 A2 B \<tau> f = evaluation_functor.map A2 B (\<tau> (fst f), snd f)"
WT_gpv_iff_outs_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" shows "\<I> \<turnstile>g gpv \<surd> = (outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>)"
adal_w: shows "(ad, al) \<in> action_loc P E (ws i)"
permutations_of_set_aux_correct: fixes A :: "'a set" shows "permutations_of_set_aux [] A = permutations_of_set A"
psize_eq_0_iff: fixes p :: "'a poly" shows "(psize p = 0) = (p = 0)"
smult_assoc_simp: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier K"   and "b \<in> carrier K"   and "x \<in> carrier V" shows "a \<odot>\<^bsub>V\<^esub> (b \<odot>\<^bsub>V\<^esub> x) = a \<otimes>\<^bsub>K\<^esub> b \<odot>\<^bsub>V\<^esub> x"
infinite_image_imp_infinite: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "infinite (f ` A)" shows "infinite A"
unit_divides: fixes u :: "'a"   and a :: "'a" assumes "u \<in> Units G"   and "a \<in> carrier G" shows "u divides a"
pred_prod_inject: fixes P1 :: "'c \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" shows "pred_prod P1 P2 (a, b) = (P1 a \<and> P2 b)"
wp1_alternativesep: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (separate p)"
resolution_class_blocks_constant_size: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset"   and u_block_size :: "int"   and S :: "'b set multiset"   and bl :: "'b set" assumes "r_block_design point_set block_collection partition u_block_size"   and "S \<in># partition"   and "bl \<in># S" shows "int (card bl) = u_block_size"
vars0_def: shows "st1.vars0 = vars1 \<union> vars2"
A_def: shows "A = UNIV - (if a \<in> \<int> then {0, 1} else {0})"
bisimilar_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a" assumes "nominal_ts satisfies transition" shows "nominal_ts.bisimilar satisfies transition P Q \<equiv> \<exists>R. nominal_ts.is_bisimulation satisfies transition R \<and> R P Q"
\<alpha>edges_aux_def: fixes g :: "'m" shows "\<alpha>edges_aux g \<equiv> {(u, v). \<exists>xs d. M_lookup g u = Some xs \<and> (v, d) \<in> set xs}"
agree_on_ext: fixes G :: "('b, 'c) labeled_graph"   and f\<^sub>1 :: "('c \<times> 'd) set"   and f\<^sub>2 :: "('c \<times> 'd) set"   and g :: "('d \<times> 'e) set" assumes "agree_on G f\<^sub>1 f\<^sub>2" shows "agree_on G (f\<^sub>1 O g) (f\<^sub>2 O g)"
simps_9: fixes v :: "char list"   and va :: "aexp"   and vb :: "'b"   and vc :: "'b"   and vd :: "bexp"   and ve :: "'b acom"   and vf :: "'b" shows "(v ::= va {vb} \<sqsubseteq> {vc} WHILE vd DO ve {vf}) = False"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c heap list \<Rightarrow> 'b" shows "(case heap.Empty of heap.Empty \<Rightarrow> f1 | Hp x xa \<Rightarrow> f2 x xa) = f1"
subset_image_inj: fixes S :: "'e set"   and f :: "'f \<Rightarrow> 'e"   and T :: "'f set" shows "(S \<subseteq> f ` T) = (\<exists>U\<subseteq>T. inj_on f U \<and> S = f ` U)"
ex_imp_not_empty: fixes A :: "'a set" assumes "\<exists>x. x \<in> A" shows "A \<noteq> {}"
extremum_strict: fixes a :: "nat" shows "\<not> a < 0"
inf_complement_intro: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> - y) \<sqinter> - - x = - y \<sqinter> - - x"
sum_wt_ParT_sub_WtFT_pickFT_0: fixes cl :: "('test, 'atom, 'choice) cmd list"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "WtFT cl = 1"   and "I \<subseteq> {..<brn (cl ! pickFT cl)}"   and "0 \<in> I" shows "sum (wt (ParT cl) s) ((+) (brnL cl (pickFT cl)) ` I) = 1"
diff_var_axiom_def: shows "diff_var_axiom \<equiv> Equals (Differential (Var vid1)) ($' vid1)"
UP_smult_r_distr: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P"   and "q \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
Val\<^sub>1: fixes P :: "nat exp prog"   and v :: "val"   and s :: "heap \<times> val list \<times> sheap" shows "P \<turnstile>\<^sub>1 \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
sum40E: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "sum4 a b c d = 0" obtains "a = 0" and "b = 0" and "c = 0" and "d = 0"
rep_list_subset_ideal: fixes B :: "('t \<Rightarrow>\<^sub>0 'b) set" shows "rep_list ` B \<subseteq> ideal (set fs)"
sets_Collect_single: fixes i :: "'b"   and I :: "'b set"   and A :: "'c set"   and M :: "'b \<Rightarrow> 'c measure" assumes "i \<in> I"   and "A \<in> sets (M i)" shows "{x \<in> space (Pi\<^sub>M I M). x i \<in> A} \<in> sets (Pi\<^sub>M I M)"
Pi_UNIV: fixes A :: "'d set" shows "A \<rightarrow> UNIV = UNIV"
lunit_ide_eq: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f" shows "bicategory.lunit (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) f = \<lparr>Chn = prj0 (partial_magma.cod C (Leg1 (Dom f))) (Leg1 (Dom f)), Dom = \<lparr>Leg0 = C (Leg0 (Dom f)) (prj0 (partial_magma.cod C (Leg1 (Dom f))) (Leg1 (Dom f))), Leg1 = prj1 (partial_magma.cod C (Leg1 (Dom f))) (Leg1 (Dom f))\<rparr>, Cod = Cod f\<rparr>"
assms_2: shows "subgraph G\<^sub>1 G\<^sub>2"
gt_sub_arg: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity.wary arity_sym arity_var (App s t)" shows "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym (App s t) t"
not_ev: fixes \<phi> :: "'b stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> ev \<phi> xs) = alw (\<lambda>xs. \<not> \<phi> xs)"
that: fixes f' :: "'a fps" assumes "f = fls_shift n (fps_to_fls f')" shows "thesis"
zcount_union: fixes M :: "'a zmultiset"   and N :: "'a zmultiset"   and a :: "'a" shows "zcount (M + N) a = zcount M a + zcount N a"
rel_envT_def: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and M :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "Monomorphic_Monad.rel_envT R M = BNF_Def.vimage2p run_env run_env (R ===> M)"
nondecF: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "f x \<le> f y"
map_upd_map_add_left: fixes m :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and b :: "'b"   and m' :: "'a \<Rightarrow> 'b option" shows "m(a \<mapsto> b) ++ m' = m ++ (if a \<notin> dom m' then m'(a \<mapsto> b) else m')"
cond_bind_spmf: fixes p :: "'i spmf"   and f :: "'i \<Rightarrow> 'h spmf"   and A :: "'h set" shows "cond_spmf (p \<bind> f) A = cond_bind_spmf p f A \<bind> (\<lambda>x. cond_spmf (f x) A)"
map_comp_assoc: fixes f :: "'c \<Rightarrow> 'b option"   and g :: "'d \<Rightarrow> 'c option"   and h :: "'a \<Rightarrow> 'd option" shows "f \<circ>\<^sub>m (g \<circ>\<^sub>m h) = f \<circ>\<^sub>m g \<circ>\<^sub>m h"
sets_measure_of_st_vec: fixes x :: "'b st_vec" shows "sets (measure_of_st_vec x) = UNIV"
value_zero_nonzero: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x = 0" shows "x \<noteq> \<zero>"
points_subset: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "sub_set_system \<U> \<A> \<V> \<B>" shows "\<U> \<subseteq> \<V>"
inner_Basis_sup_left: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (sup x y) i = sup (inner x i) (inner y i)"
case_split_right_equal: fixes w :: "'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "w \<squnion> z = top"   and "x \<sqinter> w = y \<sqinter> w"   and "x \<sqinter> z = y \<sqinter> z" shows "x = y"
prod: shows "(\<Prod>j = 1..n. real_of_int (a j)) * (1 / (\<Prod>j = 1..s - 1. real_of_int (a j))) = (\<Prod>j = s..n. real_of_int (a j))"
ord_option_simps_2: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option" shows "ord_option ord x None = (x = None)"
limproc_is_thelub_fun: fixes S :: "nat \<Rightarrow> 'a \<Rightarrow> 'b process"   and c :: "'a" assumes "chain S" shows "Lub S c = lim_proc (range (\<lambda>x. S x c))"
poly_of_pm_def: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_of_pm x p = Abs_poly (\<lambda>d. lookup p (monomial d x))"
subcls1: shows "G\<turnstile>C\<prec>\<^sub>C1D"
h_meas: fixes n :: "nat" shows "h n \<in> borel_measurable lebesgue"
rank_nullity_theorem_matrices: fixes A :: "(('b, 'c) vec, 'd) vec" shows "ncols A = vec.dim (null_space A) + vec.dim (col_space A)"
bounded_antilinear_cinner_left: fixes b :: "'b" shows "bounded_antilinear (\<lambda>a. \<langle>a, b\<rangle>)"
S_def: fixes c :: "'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b ref"   and r :: "'b \<Rightarrow> 'b ref" shows "S c l r = (\<lambda>x. if c x then r x else l x)"
sum_fun_distr: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "finite X" shows "\<Sum> (f ` X) \<cdot> y = \<Sum> {f x \<cdot> y |x. x \<in> X}"
Inr_dec_finite: fixes n :: "nat"   and s :: "('a \<times> bool list) stream"   and P :: "nat set"   and FO :: "nat set" assumes "\<forall>i<n. finite {p. snd (s !! p) ! i}"   and "Inr P \<in> set (stream_dec n FO s)" shows "finite P"
powr_bigo_iff: fixes g :: "'b \<Rightarrow> real"   and F :: "'b filter"   and p :: "real"   and q :: "real" assumes "filterlim g at_top F"   and "F \<noteq> bot" shows "((\<lambda>x. g x powr p) \<in> O[F](\<lambda>x. g x powr q)) = (p \<le> q)"
inp13: fixes s :: "real" assumes "0 \<le> s"   and "s \<le> t" shows "mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid3 UNIV"
simps_13: fixes p :: "fm"   and q :: "fm" shows "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
assms: fixes m :: "nat"   and n :: "nat" assumes "N0 < m"   and "N0 < n" shows "u (m + n) \<le> u m + u n"
nxt_nontotal_hyp: fixes b :: "nat"   and i :: "nat" assumes "nxt b \<up>"   and "s b \<down>= i" shows "\<exists>x. \<phi> i x \<up>"
simps_5: fixes b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "merge_guards (Cond b c\<^sub>1 c\<^sub>2) = Cond b (merge_guards c\<^sub>1) (merge_guards c\<^sub>2)"
not_af: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<not> af A P" shows "(\<exists>x y. x \<in> A \<and> y \<in> A \<and> \<not> P x y) \<and> (\<exists>x\<in>A. \<not> af A (\<lambda>y z. P y z \<or> P x y))"
meet_join_absorb: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "dlat.meet le x (dlat.join le x y) = x"
mono_generate: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H" shows "generate (add_monoid R) K \<subseteq> generate (add_monoid R) H"
derivableI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set"   and x :: "'b" assumes "fixed_point_proof A f"   and "fixed_point_proof.derivation A less_eq f X"   and "x \<in> X" shows "fixed_point_proof.derivable A less_eq f x"
coeff_Var1: fixes x :: "nat" shows "coeff (Var x) x = 1"
inj_on_iff_eq_card: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "finite A" shows "inj_on f A = (card (f ` A) = card A)"
s2: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) [y, x] = type1 init x y"
n_inj_polymap_findlinear: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and p :: "'a poly" assumes "VectorSpaceEnd smult V T"   and "\<And>p. 0 < degree p \<Longrightarrow> \<exists>c. poly p c = (0::'a)"   and "p \<noteq> 0"   and "\<not> inj_on (VectorSpaceEnd.polymap smult V T p) V" shows "\<exists>c. \<not> inj_on (VectorSpaceEnd.polymap smult V T [:- c, 1::'a:]) V"
subprob_cexpr_def: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" shows "subprob_cexpr V V' \<Gamma> e \<equiv> \<forall>\<rho>\<in>space (state_measure V' \<Gamma>). \<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge V V' (x, \<rho>)) e)) \<partial>state_measure V \<Gamma> \<le> 1"
hom_Max_commute: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'a"   and N :: "'a set" assumes "class.linorder less_eq less"   and "\<And>x y. h (ord.max less_eq x y) = ord.max less_eq (h x) (h y)"   and "finite N"   and "N \<noteq> {}" shows "h (linorder.Max less_eq N) = linorder.Max less_eq (h ` N)"
Ivl_simps_2: fixes a :: "'a"   and b :: "'a" shows "upper (Ivl a b) = b"
subsetI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x::'b. x \<in> (A::'b set) \<Longrightarrow> x \<in> (B::'b set)" shows "(A::'b set) \<subseteq> (B::'b set)"
simps_26: fixes x1 :: "'a"   and x101 :: "'a local.term"   and x102 :: "'a local.term"   and x103 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[x101, x102, x103\<^bold>]"
rel_compp_Grp: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_H R = (Grp {x. set_H x \<subseteq> {(x, y). R x y}} (map_H fst))\<inverse>\<inverse> OO Grp {x. set_H x \<subseteq> {(x, y). R x y}} (map_H snd)"
quasi_measure_preserving_absolutely_continuous: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and u :: "'b \<Rightarrow> ennreal" assumes "f \<in> quasi_measure_preserving M N"   and "u \<in> borel_measurable M" shows "absolutely_continuous N (distr (density M u) N f)"
compatible_cas: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "compatible G H" shows "pre_digraph.cas G = pre_digraph.cas H"
f_Exec_Stream_State_Idle_replicate_NoMsg_output: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState c)" shows "map output_fun (f_Exec_Comp_Stream trans_fun NoMsg\<^bsup>n\<^esup> c) = NoMsg\<^bsup>n\<^esup>"
poly_diff: fixes p :: "'c poly"   and q :: "'c poly"   and x :: "'c" shows "poly (p - q) x = poly p x - poly q x"
cnj_zero_iff: fixes z :: "quat" shows "(Quaternions.cnj z = 0) = (z = 0)"