hcju/isabelleps huggingface dataset:
cts_wp_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "bd_cts (wp a)"   and "bd_cts (wp b)"   and "healthy (wp a)"   and "healthy (wp b)" shows "bd_cts (wp (a \<Sqinter> b))"
dynmethd_Ext_foo: shows "dynmethd tprg Base Ext \<lparr>name = foo, parTs = [Class Base]\<rparr> = Some (Ext, snd Ext_foo)"
simps_8: fixes x31 :: "com"   and x32 :: "com" shows "x31;; x32 \<noteq> SKIP"
equivp_implies_part_equivp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "equivp R" shows "part_equivp R"
simulation1: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl1 :: "'tl1"   and s1' :: "'s1" assumes "s1 \<approx> s2"   and "s1 -1-tl1\<rightarrow> s1'" shows "\<exists>s2' tl2. s2 -2-tl2\<rightarrow> s2' \<and> s1' \<approx> s2' \<and> tl1 \<sim> tl2"
mix_pmf_def: fixes \<alpha> :: "real"   and p :: "'b pmf"   and q :: "'b pmf" shows "mix_pmf \<alpha> p q = bernoulli_pmf \<alpha> \<bind> (\<lambda>X. if X then p else q)"
nth_default_map2: fixes xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and d1 :: "'a"   and d2 :: "'b"   and d :: "'c"   and n :: "nat" assumes "length xs = length ys"   and "f d1 d2 = d" shows "nth_default d (map2 f xs ys) n = f (nth_default d1 xs n) (nth_default d2 ys n)"
ics_SDG_path_valid_SDG_node_2: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n'" shows "valid_SDG_node n'"
p: shows "p = hd_prog p\<^sub>s\<^sub>b sb"
well_related_extend: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "well_related_set A r"   and "well_related_set B r"   and "\<forall>a\<in>A. \<forall>b\<in>B. r a b" shows "well_related_set (A \<union> B) r"
intros: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and F1 :: "'a"   and F2 :: "'f"   and H :: "'i"   and st :: "'j list" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"   and "std_inca_simulation.rel_fundefs \<DD>\<ee>\<II>\<nn>\<ll> (Fstd_get F1) (Finca_get F2)" shows "std_inca_simulation.match Fstd_get Finca_get \<DD>\<ee>\<II>\<nn>\<ll> (State F1 H st) (State F2 H st)"
FP_Orig_weakest: fixes F :: "'b program"   and A :: "'b set" assumes "\<And>B. F \<in> stable (A \<inter> B)" shows "A \<subseteq> FP_Orig F"
safe_config: fixes k :: "nat"   and c :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and l :: "nat" assumes "k \<le> n"   and "trace c 0 k"   and "c 0 = c\<^sub>0"   and "l \<le> k" shows "safe_delayed (c l)"
accept\<^sub>R_def: fixes R :: "('d \<Rightarrow> 'e \<Rightarrow> 'd) \<times> 'd \<times> (('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set) set"   and w :: "nat \<Rightarrow> 'e" shows "accept\<^sub>R R w \<equiv> \<exists>P\<in>snd (snd R). accepting_pair\<^sub>R (fst R) (fst (snd R)) P w"
DERIV_diff: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'b"   and E :: "'b" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at x within s)" shows "((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
closed_INT: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'b set" assumes "class.topological_space open"   and "\<forall>x\<in>A. topological_space.closed open (B x)" shows "topological_space.closed open (\<Inter> (B ` A))"
even_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (power.power one times a n) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<and> 0 < n)"
mult_less_cancel_left_pos: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "(0::'a) < c" shows "(c * a < c * b) = (a < b)"
assms: fixes e :: "real" assumes "0 < e" shows "AE x in M. F x \<le> G x + ennreal e"
stable_regularCard: fixes r :: "('a \<times> 'a) set" assumes "Card_order r"   and "infinite (Field r)"   and "stable r" shows "regularCard r"
extend_maximal: fixes f :: "nat \<Rightarrow> (nat, 'a) form"   and C :: "(nat, 'a) form set set"   and S :: "(nat, 'a) form set" assumes "\<forall>y. \<exists>n. y = f n"   and "finite_char C" shows "maximal (Extend S C f) C"
abs_valid_operator_6': fixes name :: "char list"   and pre :: "(nat \<times> nat) list"   and eff :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pre, eff, cost)" shows "list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') (map (\<lambda>(uu_, v, uu_, a). (v, a)) eff)) (map (\<lambda>(uu_, v, uu_, a). (v, a)) eff)"
filterlim_at_infinity: fixes c :: "real"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "0 \<le> c" shows "filterlim f at_infinity F = (\<forall>r>c. \<forall>\<^sub>F x in F. r \<le> norm (f x))"
max_coeff_bound'_2: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions" shows "\<forall>j<length y. y ! j \<le> Max (set a)"
fps_binomial_p_compose: fixes p :: "nat"   and n :: "nat" assumes "p \<noteq> 0" shows "(1 + fps_X ^ p) ^ n = (1 + fps_X) ^ n oo fps_X ^ p"
decr: fixes p :: "fm"   and bbs :: "bool list"   and x :: "int"   and bs :: "int list" assumes "bound0 p" shows "Ifm bbs (x # bs) p = Ifm bbs bs (decr p)"
atomize_vectorize_poly: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.atomize_poly pair_of_term (term_powerprod.vectorize_poly term_of_pair p) = p"
max_subgraph_cong: fixes H :: "('a, 'b) pre_digraph"   and H' :: "('a, 'b) pre_digraph"   and P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and P' :: "('a, 'b) pre_digraph \<Rightarrow> bool" assumes "H = H'"   and "\<And>H''. \<lbrakk>subgraph H' H''; subgraph H'' G\<rbrakk> \<Longrightarrow> P H'' = P' H''" shows "max_subgraph P H = max_subgraph P' H'"
n_star_sim_right_var: fixes y :: "'a"   and x :: "'a" assumes "!y \<cdot> x = x \<cdot> !y" shows "x\<^sup>\<star> \<cdot> !y = !y \<cdot> (x \<cdot> !y)\<^sup>\<star>"
and_mask: fixes w :: "'a word"   and n :: "nat" shows "w AND mask n = w << size w - n >> size w - n"
lookup_val_cons: fixes a :: "'f"   and x :: "'h"   and xs :: "'h list" shows "lookup (Val a) (x # xs) = None"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "semilattice f" shows "semilattice_set f"
type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap" assumes "l_type_wf\<^sub>N\<^sub>o\<^sub>d\<^sub>e type_wf"   and "type_wf h" shows "NodeClass.type_wf h"
assms_2: shows "i < j \<and> j < k \<and> (finite X \<longrightarrow> k < card X)"
exec_step_ind_Invoke_Static: fixes r :: "val"   and stk :: "val list"   and n :: "nat"   and C :: "char list"   and h :: "heap"   and D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and M :: "char list"   and m :: "jvm_method"   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 "r = stk ! n"   and "C = cname_of h (the_Addr r)"   and "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "r \<noteq> Null"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D" shows "exec_step_ind (StepI (Invoke M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt IncompatibleClassChangeError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
size1_height: fixes t :: "'c tree" shows "size1 t \<le> 2 ^ height t"
npesub_correct: fixes xs :: "'a list"   and e1 :: "pexpr"   and e2 :: "pexpr" assumes "in_carrier xs" shows "peval xs (npesub e1 e2) = peval xs (PExpr1 (PSub e1 e2))"
dot_square_norm: fixes x :: "'c" shows "x \<bullet> x = (norm x)\<^sup>2"
Cons_7: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
exp_base_def: fixes x :: "'a list"   and n :: "nat" shows "UP.exp_base x n = map (([^]\<^bsub>K [X]\<^esub>) x) (rev [0..<n])"
PxPy_nonneg: shows "AE x in S \<Otimes>\<^sub>M T. 0 \<le> Px (fst x) * Py (snd x)"
cinner_real: fixes x :: "'c" shows "x \<bullet>\<^sub>C x \<in> \<real>"
add_less_cancel_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus c a) (plus c b) = less a b"
complex_Im_of_int: fixes z :: "int" shows "Im (of_int z) = 0"
bot_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.finite_lattice_complete Inf Sup inf less_eq less sup bot top" shows "bot = semilattice_inf.Inf_fin inf UNIV"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (f, cod \<iota>))"
gs: shows "gs \<subseteq> {q. irreducible q \<and> monic q}"
Re_divide_of_real: fixes z :: "complex"   and r :: "real" shows "Re (z / cor r) = Re z / r"
is_density_exprD_1: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and t :: "pdf_type"   and e :: "cexpr" assumes "is_density_expr (vs, vs', \<Gamma>, \<delta>) t e" shows "case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL"
gcd_coprime_poly: fixes p :: "'a poly"   and q :: "'a poly"   and p' :: "'a poly"   and q' :: "'a poly" assumes "p \<noteq> 0 \<or> q \<noteq> 0"   and "p = p' * gcd p q"   and "q = q' * gcd p q" shows "coprime p' q'"
"1_1": shows "v\<^sub>2 \<le>\<^sub>F\<^sub>I u @- v"
assms_4: shows "\<forall>l s. P l s \<longrightarrow> pre C Q l s"
elims_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.strict_sorted less x = y" obtains "x = []" and "y"   | xa :: "'c" and ys :: "'c list" where "x = xa # ys" and "y = (Ball (set ys) (less xa) \<and> linorder.strict_sorted less ys)"
reachable_dom_acyclic2: fixes G :: "nat graph_rec"   and n2 :: "nat"   and n1 :: "nat" assumes "cfg_doms G"   and "cfg_doms.reachable G n2"   and "cfg_doms.strict_dominate G n1 n2" shows "\<not> cfg_doms.dominate G n2 n1"
qmsg_send_receive_or_tau: shows "qmsg \<TTurnstile>\<^sub>A (\<lambda>(uu_, a, uu_). \<exists>m. a = send m \<or> a = receive m \<or> a = \<tau>)"
tuples_def: shows "tuples = (\<lambda>Y. {f \<in> (X \<rightarrow>\<^sub>E Roots) \<rightarrow>\<^sub>E X. image_mset f (mset_set (X \<rightarrow>\<^sub>E Roots)) = Y})"
vconn_implies_hpar_3: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C g = src\<^sub>C f"
fmrel_on_fset_alt_def: fixes S :: "'c fset"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'e) fmap" shows "fmrel_on_fset S P m n = fBall S (\<lambda>x. rel_option P (fmlookup m x) (fmlookup n x))"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (C'.T (C.cod \<iota>\<^sub>C, f))"
image_paired_Times: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'f set" shows "(\<lambda>(x, y). (f x, g y)) ` (A \<times> B) = f ` A \<times> g ` B"
weakBisimInputPres: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a" assumes "\<And>Tvec. length xvec = length Tvec \<Longrightarrow> \<Psi> \<rhd> P[xvec::=Tvec] \<approx> Q[xvec::=Tvec]" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<approx> M\<lparr>\<lambda>*xvec N\<rparr>.Q"
mult_cancel_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times a c = times b c) = (c = zero \<or> a = b)"
sum_zero_eq_neg: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "x \<oplus> y = \<zero>" shows "x = \<ominus> y"
new_privs_preserved': fixes c :: "'Com\<^sub>C"   and mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c' :: "'Com\<^sub>C"   and mds' :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val"   and m :: "Mode" assumes "\<langle>c, mds, mem\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "mds m \<inter> - range var\<^sub>C_of = mds' m \<inter> - range var\<^sub>C_of"
subspace_UNIV: shows "??.Real_Vector_Spaces.subspace S"
assms: fixes j2 :: "'a"   and j1 :: "'a" assumes "j2 \<in> Field r"   and "(j1, j2) \<in> r" shows "R j1 initial_segment_of R j2"
Chernoff_c: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "Chernoff_on A f = (\<forall>B C. B \<subseteq> A \<and> C \<subseteq> A \<longrightarrow> f (B \<union> C) \<subseteq> f (f B \<union> C))"
the_\<nu>_props_1: shows "\<guillemotleft>\<mu>.the_\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> \<mu>.chine\<guillemotright>"
intro: fixes term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c" assumes "\<And>v. term_of_pair (pair_of_term v) = v"   and "\<And>p. pair_of_term (term_of_pair p) = p" shows "term_powerprod pair_of_term term_of_pair"
valid_call_list_valid_edges: fixes cs :: "'edge list"   and n :: "'node" assumes "valid_call_list cs n" shows "\<forall>c\<in>set cs. valid_edge c"
is_a_proper_intersec_is_a_intersec: fixes P :: "Points"   and A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" assumes "is_a_proper_intersec P A B C D" shows "is_a_intersec P A B C D"
submodule_inc_0: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "\<zero>\<^bsub>M\<^esub> \<in> H"
word_ao_absorbs_2: fixes x :: "'b word"   and y :: "'b word" shows "x OR y AND x = x"
leaves_def: shows "leaves = fold_tree (\<lambda>_. 1) (+)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> String.literal \<Rightarrow> 'b val list \<Rightarrow> 'b val \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> htype \<Rightarrow> 'a"   and f5 :: "'c \<Rightarrow> 'a"   and f6 :: "'c \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'b \<Rightarrow> 'a"   and f9 :: "'c \<Rightarrow> 'a"   and f10 :: "'c \<Rightarrow> 'a"   and obs_event :: "('b, 'c) obs_event" shows "P (case obs_event of ExternalCall x xa xb xc \<Rightarrow> f1 x xa xb xc | ReadMem x xa xb \<Rightarrow> f2 x xa xb | WriteMem x xa xb \<Rightarrow> f3 x xa xb | NewHeapElem x xa \<Rightarrow> f4 x xa | ThreadStart x \<Rightarrow> f5 x | ThreadJoin x \<Rightarrow> f6 x | SyncLock x \<Rightarrow> f7 x | SyncUnlock x \<Rightarrow> f8 x | ObsInterrupt x \<Rightarrow> f9 x | ObsInterrupted x \<Rightarrow> f10 x) = (\<not> ((\<exists>x11 x12 x13 x14. obs_event = ExternalCall x11 x12 x13 x14 \<and> \<not> P (f1 x11 x12 x13 x14)) \<or> (\<exists>x21 x22 x23. obs_event = ReadMem x21 x22 x23 \<and> \<not> P (f2 x21 x22 x23)) \<or> (\<exists>x31 x32 x33. obs_event = WriteMem x31 x32 x33 \<and> \<not> P (f3 x31 x32 x33)) \<or> (\<exists>x41 x42. obs_event = NewHeapElem x41 x42 \<and> \<not> P (f4 x41 x42)) \<or> (\<exists>x5. obs_event = ThreadStart x5 \<and> \<not> P (f5 x5)) \<or> (\<exists>x6. obs_event = ThreadJoin x6 \<and> \<not> P (f6 x6)) \<or> (\<exists>x7. obs_event = SyncLock x7 \<and> \<not> P (f7 x7)) \<or> (\<exists>x8. obs_event = SyncUnlock x8 \<and> \<not> P (f8 x8)) \<or> (\<exists>x9. obs_event = ObsInterrupt x9 \<and> \<not> P (f9 x9)) \<or> (\<exists>x10. obs_event = ObsInterrupted x10 \<and> \<not> P (f10 x10))))"
pivot_tableau: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and v :: "nat \<Rightarrow> 'b" assumes "Pivot eq_idx_for_lvar pivot"   and "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "let s' = pivot x\<^sub>i x\<^sub>j s in v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s' \<and> \<triangle> (\<T> s')"
tuple_since'_def: shows "tuple_since' \<equiv> Mapping.filter (\<lambda>as _. proj_tuple_in_join pos maskL as X) tuple_since"
induct: fixes x :: "('a, 'b) atomic"   and P :: "('a, 'b) atomic \<Rightarrow> bool" assumes "lformula0 x"   and "\<And>m. P (Fo m)"   and "\<And>m n. P (Eq_Const None m n)"   and "\<And>m1 m2. P (Less None m1 m2)"   and "\<And>m1 m2 n. P (Plus_FO None m1 m2 n)"   and "\<And>m1 m2. P (Eq_FO False m1 m2)"   and "\<And>M1 M2. P (Eq_SO M1 M2)"   and "\<And>bl M1 M2. P (Suc_SO False bl M1 M2)"   and "\<And>M. P (Empty M)"   and "\<And>M. P (Singleton M)"   and "\<And>M1 M2. P (Subset M1 M2)"   and "\<And>m M. P (In False m M)"   and "\<And>m M. P (Eq_Max False m M)"   and "\<And>m M. P (Eq_Min False m M)"   and "\<And>M1 M2 M3. P (Eq_Union M1 M2 M3)"   and "\<And>M1 M2 M3. P (Eq_Inter M1 M2 M3)"   and "\<And>M1 M2 M3. P (Eq_Diff M1 M2 M3)"   and "\<And>M1 M2. P (Disjoint M1 M2)"   and "\<And>M n. P (Eq_Presb None M n)" shows "P x"
simps: fixes i :: "nat"   and t :: "nat"   and e :: "nat"   and s :: "bdd" shows "mi.param_opt_impl i t e s = destrmi' i s \<bind> (\<lambda>ii. destrmi' t s \<bind> (\<lambda>ti. destrmi' e s \<bind> (\<lambda>ei. tmi' s \<bind> (\<lambda>(tn, s). fmi' s \<bind> (\<lambda>(fn, s). Some (if ii = TD then Some t else if ii = FD then Some e else if ti = TD \<and> ei = FD then Some i else if t = e then Some t else if ei = TD \<and> i = t then Some tn else if ti = FD \<and> i = e then Some fn else None, s))))))"
in_hom_char: fixes f :: "'a \<times> 'a \<times> 'a"   and a :: "'a \<times> 'a \<times> 'a"   and b :: "'a \<times> 'a \<times> 'a" shows "VVV.in_hom f a b = (VVV.arr a \<and> VVV.arr b \<and> VVV.arr f \<and> VxVxV.in_hom f a b)"
inj_onI: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
map_FGl_def: fixes l1 :: "'n \<Rightarrow> 'aa"   and l2 :: "'o \<Rightarrow> 'ab"   and l3 :: "'u \<Rightarrow> 'ag"   and co1 :: "'p \<Rightarrow> 'ac"   and co2 :: "'q \<Rightarrow> 'ad"   and co3 :: "'v \<Rightarrow> 'ah"   and co4 :: "'w \<Rightarrow> 'ai"   and contra1 :: "'ae \<Rightarrow> 'r"   and contra2 :: "'af \<Rightarrow> 's"   and contra3 :: "'aj \<Rightarrow> 'x"   and contra4 :: "'ak \<Rightarrow> 'y" shows "map_FGl l1 l2 l3 co1 co2 co3 co4 contra1 contra2 contra3 contra4 = map_F (map_G l1 l2 co1 co2 contra1 contra2) l1 l3 co1 co3 co4 contra1 contra3 contra4"
suminf_split_initial_segment: fixes f :: "nat \<Rightarrow> 'e"   and k :: "nat" assumes "summable f" shows "suminf f = (\<Sum>n. f (n + k)) + sum f {..<k}"
d_2: fixes p :: "('a \<times> 'a set) set" assumes "p tagged_division_of cbox a b"   and "\<gamma> fine p" shows "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e"
less_eq_odlist_def: fixes xs :: "'a odlist"   and ys :: "'a odlist" shows "xs \<le> ys \<equiv> local.less_eq_list (toList xs) (toList ys)"
inconsistentPrefix: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" assumes "isPrefix valuation1 valuation2"   and "inconsistent valuation1" shows "inconsistent valuation2"
add_strict_increasing2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'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_cancel_comm_monoid_add plus minus zero less_eq less"   and "less_eq zero a"   and "less b c" shows "less b (plus a c)"
sep_disj_commuteI: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "sep_disj y x"
simps_4: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "Unf (\<phi> R\<^sub>n \<psi>) = (\<phi> R\<^sub>n \<psi>) or\<^sub>n Unf \<phi> and\<^sub>n Unf \<psi>"
semiring_norm_76: fixes n :: "num" shows "(num.One < num.Bit0 n) = True"
vimage_subset_eq: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and A :: "'c set" assumes "bij f" shows "(f -` B \<subseteq> A) = (B \<subseteq> f ` A)"
le: fixes y :: "'b"   and x :: "real" assumes "y \<in> A"   and "a'' \<le> x" shows "norm (f y x) \<le> g x"
Cons_1: fixes y :: "'a" assumes "|filter P xs| = |xs|"   and "y \<in> set xs" shows "P y"
2: shows "(\<Union>i. a \<inter> A i) \<in> smallest_ccdi_sets \<Omega> M"
dvd_mult_unit_iff: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b dvd (1::'b)" shows "(a dvd c * b) = (a dvd c)"