hcju/isabelleps huggingface dataset:
fds_truncate_integral: fixes m :: "nat"   and c :: "'b"   and f :: "'b fds" assumes "0 < m \<or> c = (0::'b)" shows "fds_truncate m (fds_integral c f) = fds_integral c (fds_truncate m f)"
coeff_map_poly_hom: fixes p :: "'q mod_ring poly"   and i :: "nat" shows "poly.coeff (map_poly rebase p) i = @poly.coeff p i"
one_closed: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "subgroup H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
t_ins_form_all_def: fixes X :: "'b t_type" shows "t_ins_form_all X \<equiv> t_ins_form X \<or> subtrees X = []"
assms_3: shows "alphaAbs_all_distinct_qAFresh xs x X xs' x' X'"
RA1: fixes P :: "('d, 'f) urel"   and Q :: "('f, 'g) urel"   and R :: "('g, 'e) urel" shows "P ;; Q ;; R = (P ;; Q) ;; R"
cases: fixes base :: "nat"   and a1 :: "nat"   and a2 :: "hmultiset" assumes "2 \<le> base"   and "aligned\<^sub>d base a1 a2" obtains M :: "hmultiset" and e :: "nat" where "a1 = e" and "a2 = M" and "\<forall>m\<in>#hmsetmset M. e \<le> decode base 0 m"
ImpR_inv: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta>" shows "F, \<Gamma> \<Rightarrow> G, \<Delta>"
circline_set_x_axis_I: fixes z :: "complex" assumes "is_real z" shows "of_complex z \<in> circline_set x_axis"
leI: fixes x :: "'e word"   and y :: "'e word" assumes "\<not> x <s y" shows "y \<le>s x"
one_time_pad': fixes f :: "nat \<Rightarrow> nat"   and q :: "nat" assumes "inj_on f ({..<q} - {0})"   and "f ` ({..<q} - {0}) = {..<q} - {0}" shows "map_spmf f (sample_uniform_units q) = sample_uniform_units q"
lcoset_rel_def: fixes A :: "'a set" shows "lcoset_rel A \<equiv> {(x, y). - x + y \<in> A}"
WTrt_binopI_2: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty" shows "P \<turnstile> T1\<guillemotleft>NotEq\<guillemotright>T2 : Boolean"
non_interference_induct_case: fixes n :: "nat"   and s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "((\<forall>i t. i \<le> n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<longrightarrow> (\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2))) \<and> (\<forall>i t. i \<le> Suc n \<and> SEQ i s1 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}) \<and> (\<forall>i t. i \<le> Suc n \<and> SEQ i s2 = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {})" shows "\<exists>t1. Some t1 = (case SEQ n s1 of None \<Rightarrow> None | Some x \<Rightarrow> NEXT x) \<and> (\<exists>t2. Some t2 = (case SEQ n s2 of None \<Rightarrow> None | Some x \<Rightarrow> NEXT x) \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2)"
fVectorSpace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FGModule G smult V" shows "VectorSpace (aezfun_scalar_mult.fsmult smult) V"
sum_distl: fixes Y :: "'b set"   and x :: "'b" assumes "finite Y" shows "x \<cdot> \<Sum> Y = \<Sum> {x \<cdot> y |y. y \<in> Y}"
Step: shows "cfg \<turnstile> execMsg n \<mapsto> execConf (n + 1)"
right_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "b \<in> S"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (\<tau>'\<tau>.map\<^sub>0 a) = F.map\<^sub>0 a"
select_result_eq: fixes h :: "'b"   and f :: "('b, 'c, 'a) prog"   and h' :: "'b" assumes "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x" shows "|h \<turnstile> f|\<^sub>r = |h' \<turnstile> f|\<^sub>r"
aux5: fixes a :: "'b"   and b :: "'b"   and p :: "('b, 'c) Combinators list" assumes "DenyAllFromTo a b \<in> set p" shows "a \<in> set (net_list p)"
thmD13: fixes k :: "nat"   and u :: "nat" assumes "items_le k (\<J> k 0) = Gen (paths_le k (\<P> k 0))"   and "k \<le> length Doc" shows "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u)) \<and> \<T> k u = \<Z> k u \<and> items_le k (\<I> k) = Gen (paths_le k (\<Q> k))"
bernoulli_discounted_integrable: fixes N :: "bool stream measure"   and q :: "real"   and der :: "bool stream \<Rightarrow> real"   and n :: "nat"   and r :: "real"   and m :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "der \<in> borel_measurable (nat_filtration n)"   and "- 1 < r" shows "integrable N (discounted_value r (\<lambda>m. der) m)"
map_le_trans: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and m3 :: "'a \<Rightarrow> 'b option" assumes "m1 \<subseteq>\<^sub>m m2"   and "m2 \<subseteq>\<^sub>m m3" shows "m1 \<subseteq>\<^sub>m m3"
inv_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "local.iso f"   and "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>local.inv f : b \<Rightarrow> a\<guillemotright>"
properfactor_nz: fixes y :: "'a"   and x :: "'a" assumes "y \<noteq> (0::'a)" shows "comm_semiring_1_class.properfactor x y = (x dvd y \<and> \<not> y dvd x)"
pow_eq_div2: fixes x :: "'c"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier (H\<lparr>carrier := h ` carrier G\<rparr>)"   and "x [^]\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub> m = x [^]\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub> n" shows "x [^]\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub> (m - n) = \<one>\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub>"
induct: fixes P :: "'b formula \<Rightarrow> bool"   and a0 :: "'b formula" assumes "\<And>\<phi>. P (FNot \<phi>)"   and "\<And>\<phi>1 \<phi>2. \<lbrakk>P \<phi>1; P \<phi>2\<rbrakk> \<Longrightarrow> P (FAnd \<phi>1 \<phi>2)"   and "\<And>\<phi>1 \<phi>2. \<lbrakk>P \<phi>1; P \<phi>2\<rbrakk> \<Longrightarrow> P (FOr \<phi>1 \<phi>2)"   and "\<And>v va. P (FQ v va)"   and "\<And>v va. P (FLess v va)"   and "\<And>v va. P (FIn v va)"   and "\<And>v. P (FExists v)"   and "\<And>v. P (FEXISTS v)" shows "P a0"
WhileT\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and c :: "nat exp"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and e\<^sub>3 :: "nat exp"   and s\<^sub>3 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>2\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>"
ub2: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
proportionality_equiv: shows "equiv vec.non_zero_vectors vec.proportionality"
insert_text: fixes x :: "'a"   and b :: "'a textbuffer" shows "get_text (Challenge1.insert x b) = take (get_pos b) (get_text b) @ x # drop (get_pos b) (get_text b)"
independent_span_bound: fixes T :: "'b set"   and S :: "'b set" assumes "finite T"   and "local.independent S"   and "S \<subseteq> local.span T" shows "finite S \<and> card S \<le> card T"
real_mult: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and real_of :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_embedding plus zero minus uminus less_eq less sgn abs times one real_of" shows "real_of (times x y) = real_of x * real_of y"
sorted_keys: fixes t :: "('d, 'e) RBT.rbt" shows "sorted (RBT.keys t)"
no_Trace_implies_no_Failure: fixes s :: "'a event list"   and P :: "'a process" assumes "s \<notin> T P" shows "(s, {}) \<notin> F P"
in_quotient_imp_non_empty: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set" assumes "equiv A r"   and "X \<in> A // r" shows "X \<noteq> {}"
pat_cases: fixes t :: "term" obtains s :: "name" where "t = Free s"   | name :: "name" and args :: "term list" where "linears args" and "t = name $$ args"   | "\<not> linear t"
uninst_trivial_phis_def: fixes p :: "('a, 'b, 'c, 'd) gen_ssa_cfg_wf \<Rightarrow> ('a \<times> 'd, 'd list) mapping" shows "uninst_trivial_phis p \<equiv> CFG_SSA_wf_base_code.trivial_phis p"
wlog_real_zero: fixes u :: "complex_homo"   and P :: "complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "is_real (to_complex u)"   and "\<And>a u. \<lbrakk>u \<in> unit_disc; is_real a; cmod a < 1; P (moebius_pt (blaschke a) u)\<rbrakk> \<Longrightarrow> P u"   and "P 0\<^sub>h" shows "P u"
lnull_imp_lfinite: fixes xs :: "'a llist" assumes "lnull xs" shows "lfinite xs"
invar_remove1: fixes q :: "('c, 'd) SkewBinomialTree list"   and t :: "('c, 'd) SkewBinomialTree" assumes "queue_invar q" shows "queue_invar (remove1 t q)"
psubstEnv_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "rho' &[rho] \<equiv> \<lambda>xs x. case rho' xs x of None \<Rightarrow> rho xs x | Some X \<Rightarrow> Some X #[rho]"
simps_2: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_minus (Suc i) e = h_minus i e + (\<Squnion>j. h_minus_aux (\<lambda>e'. h_plus (Suc i) e' - h_minus i e') j e)"
pos_bounded: shows "\<exists>K>0. \<forall>x. norm (g x) \<le> norm x * K"
rep_eq: fixes x :: "('d, 'e) \<I>" shows "Rep_\<I> (e\<I> x) = (\<lambda>x. Rep_\<I> x x \<times> UNIV)"
hyps: fixes P' :: "(('a, 'rows) vec, 'rows) vec"   and B' :: "(('a, 'cols) vec, 'rows) vec"   and Q' :: "(('a, 'cols) vec, 'cols) vec"   and P :: "(('a, 'rows) vec, 'rows) vec"   and B :: "(('a, 'cols) vec, 'rows) vec"   and Q :: "(('a, 'cols) vec, 'cols) vec" assumes "k < min (nrows A) (ncols A)"   and "(P', B', Q') = diagonal_to_Smith_aux_PQ [0..<k] bezout (P, B, Q)"   and "invertible P"   and "invertible Q"   and "P ** A ** Q = B"   and "isDiagonal B" shows "B' = P' ** A ** Q' \<and> invertible P' \<and> invertible Q' \<and> B' = diagonal_to_Smith_aux B [0..<k] bezout"
simps_7: fixes y :: "nat" shows "pratt1_max [y]\<^sub>1 \<infinity>\<^sub>1 = \<infinity>\<^sub>1"
MGU_exists: fixes t :: "'b trm"   and s :: "'b trm" shows "\<forall>\<sigma>. t \<lhd> \<sigma> = s \<lhd> \<sigma> \<longrightarrow> unify t s \<noteq> None"
mem_lists_non_Nil: fixes xss :: "'a list list" shows "(xss \<in> lists (- {[]})) = (\<forall>x\<in>list.set xss. x \<noteq> [])"
cod_eqI: fixes b :: "'c"   and f :: "'c" assumes "J'.ide b"   and "J'.seq b f" shows "J'.cod f = b"
CCexp_def: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and \<Gamma> :: "exp" shows "CoCallArityAnalysis.CCexp cccExp \<Gamma> = (\<Lambda> a. snd (cccExp \<Gamma>\<cdot>a))"
add_spvec_commute: fixes a :: "(nat \<times> 'a) list"   and b :: "(nat \<times> 'a) list" shows "add_spvec a b = add_spvec b a"
intros_5: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> \<sigma>" shows "Collection \<tau> \<sqsubset> Collection \<sigma>"
trd_def: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.trd fs p = punit.trd_aux fs p 0"
set_status_nth: fixes xs :: "'b list"   and n :: "nat"   and i :: "nat"   and \<sigma> :: "'c status"   and f :: "'c" assumes "length xs = n"   and "i \<in> set (status \<sigma> (f, n))" shows "i < length xs \<and> xs ! i \<in> set xs"
hseq_leg\<^sub>0: shows "hseq s\<^sub>0 w"
cblinfun_eq_mat_of_cblinfunI: fixes a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "mat_of_cblinfun a = mat_of_cblinfun b" shows "a = b"
prefix_conc_fst: fixes j :: "nat"   and w :: "'b list"   and w' :: "nat \<Rightarrow> 'b" assumes "j \<le> length w" shows "prefix j (w \<frown> w') = take j w"
countable_UN: fixes A :: "'b \<Rightarrow> 'a set"   and X :: "'b set"   and M :: "'a measure" assumes "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
set_intros_12: fixes yj :: "'b"   and x111 :: "'b ltlc"   and x112 :: "'b ltlc" assumes "yj \<in> atoms_ltlc x111" shows "yj \<in> atoms_ltlc (x111 U\<^sub>c x112)"
weakCongPar2: fixes Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<Longrightarrow>\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P \<parallel> Q'"
nfoldli_while: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c"   and I :: "'d list \<times> 'c \<Rightarrow> bool" shows "nfoldli l c f \<sigma> \<le> WHILE\<^sub>T\<^bsup>I\<^esup> (FOREACH_cond c) (\<lambda>x. ASSERT (FOREACH_cond c x) \<bind> (\<lambda>_. FOREACH_body f x)) (l, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)"
fun_inter_def: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" shows "f \<sqinter> g \<equiv> \<lambda>p. f p \<inter> g p"
eq_on_sym: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b"   and X :: "'a set" assumes "s = t on X" shows "t = s on X"
perfect_sec_P1_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd \<times> 'e) spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and Out1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> ('d \<times> 'e) spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and m1 :: "'a"   and m2 :: "'b" shows "Semi_Honest_Def.sim_non_det_def.perfect_sec_P1 R1 S1 Out1 funct m1 m2 \<equiv> R1 m1 m2 = funct m1 m2 \<bind> (\<lambda>(s1, s2). Semi_Honest_Def.sim_non_det_def.Ideal1 S1 Out1 m1 m2 s1)"
assms: shows "\<lless>\<pi>, e\<ggreater> V\<rightarrow> \<lless>\<pi>' @ X, e'\<ggreater>"
2: fixes \<U> :: "'a set set" assumes "\<U> \<subseteq> Collect P" shows "\<exists>\<U>'\<subseteq>{S. \<exists>\<U>\<subseteq>Collect P. \<Union> \<U> = S}. \<Union> \<U>' = \<Union> \<U>"
assms: shows "Src t = Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
not_matches_add_match_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>add_match m rs, Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[], Undecided\<rangle> \<Rightarrow> t"
rev_nth: fixes n :: "nat"   and xs :: "'b list" assumes "n < \<^bold>|xs\<^bold>|" shows "rev xs ! n = xs ! (\<^bold>|xs\<^bold>| - Suc n)"
f_def: shows "f \<equiv> \<lambda>(l, R). (l, SOME u. u \<in> R)"
gen_keys_simps_3: fixes kts :: "('c \<times> ('c, 'd) rbt) list"   and c :: "color"   and l :: "('c, 'd) rbt"   and k :: "'c"   and v :: "'d"   and r :: "('c, 'd) rbt" shows "RBT_Impl.gen_keys kts (Branch c l k v r) = RBT_Impl.gen_keys ((k, r) # kts) l"
swapidseq_ext_finite: fixes S :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext S n p" shows "finite S"
is_span: shows "C.span HH\<nu>\<pi>\<rho>.leg0 HH\<nu>\<pi>\<rho>.leg1"
gs_powr_0_eq_one: shows "\<forall>\<^sub>F x in F. (\<Prod>g\<leftarrow>gs. get_fun g x powr 0) = 1"
dims_product: shows "st2.d0 = st2.d1 * st2.d2"
split_gallery: fixes C :: "'a set"   and D :: "'a set"   and Cs :: "'a set list" assumes "C \<in> f \<turnstile> folding_g.\<C>"   and "D \<in> folding_g.\<C> - f \<turnstile> folding_g.\<C>"   and "folding_g.gallery (C # Cs @ [D])" shows "\<exists>As A B Bs. A \<in> f \<turnstile> folding_g.\<C> \<and> B \<in> folding_g.\<C> - f \<turnstile> folding_g.\<C> \<and> C # Cs @ [D] = As @ A # B # Bs"
ih_2: assumes "getSATFlag state' = UNDEF"   and "InvariantConsistent (getM (solve_loop_body state' Vbl))"   and "InvariantUniq (getM (solve_loop_body state' Vbl))"   and "InvariantWatchesEl (getF (solve_loop_body state' Vbl)) (getWatch1 (solve_loop_body state' Vbl)) (getWatch2 (solve_loop_body state' Vbl))"   and "InvariantWatchesDiffer (getF (solve_loop_body state' Vbl)) (getWatch1 (solve_loop_body state' Vbl)) (getWatch2 (solve_loop_body state' Vbl))"   and "InvariantWatchCharacterization (getF (solve_loop_body state' Vbl)) (getWatch1 (solve_loop_body state' Vbl)) (getWatch2 (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl))"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList (solve_loop_body state' Vbl)) (getF (solve_loop_body state' Vbl))"   and "InvariantWatchListsUniq (getWatchList (solve_loop_body state' Vbl))"   and "InvariantWatchListsCharacterization (getWatchList (solve_loop_body state' Vbl)) (getWatch1 (solve_loop_body state' Vbl)) (getWatch2 (solve_loop_body state' Vbl))"   and "InvariantUniqQ (getQ (solve_loop_body state' Vbl))"   and "InvariantQCharacterization (getConflictFlag (solve_loop_body state' Vbl)) (getQ (solve_loop_body state' Vbl)) (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl))"   and "InvariantConflictFlagCharacterization (getConflictFlag (solve_loop_body state' Vbl)) (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl))"   and "InvariantNoDecisionsWhenConflict (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl)) (currentLevel (getM (solve_loop_body state' Vbl)))"   and "InvariantNoDecisionsWhenUnit (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl)) (currentLevel (getM (solve_loop_body state' Vbl)))"   and "InvariantGetReasonIsReason (getReason (solve_loop_body state' Vbl)) (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl)) (set (getQ (solve_loop_body state' Vbl)))"   and "getSATFlag (solve_loop_body state' Vbl) = UNDEF \<longrightarrow> InvariantEquivalentZL (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl)) F0'"   and "InvariantConflictClauseCharacterization (getConflictFlag (solve_loop_body state' Vbl)) (getConflictClause (solve_loop_body state' Vbl)) (getF (solve_loop_body state' Vbl)) (getM (solve_loop_body state' Vbl))"   and "getSATFlag (solve_loop_body state' Vbl) = FALSE \<longrightarrow> \<not> satisfiable F0'"   and "getSATFlag (solve_loop_body state' Vbl) = TRUE \<longrightarrow> satisfiable F0'"   and "finite Vbl"   and "vars F0' \<subseteq> vars F0"   and "vars F0 \<subseteq> Vbl"   and "InvariantVarsM (getM (solve_loop_body state' Vbl)) F0 Vbl"   and "InvariantVarsF (getF (solve_loop_body state' Vbl)) F0 Vbl"   and "InvariantVarsQ (getQ (solve_loop_body state' Vbl)) F0 Vbl" shows "let state' = solve_loop (solve_loop_body state' Vbl) Vbl in getSATFlag state' = FALSE \<and> \<not> satisfiable F0' \<or> getSATFlag state' = TRUE \<and> satisfiable F0'"
correctCompositionKS_exprChannel_k_Pex: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and key :: "Keys"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "kKS key \<notin> LocalSecrets PQ"   and "ch \<in> ins P"   and "exprChannel ch (kE key)"   and "kKS key \<notin> specKeysSecrets PQ"   and "correctCompositionIn PQ" shows "\<exists>ch. ch \<in> ins PQ \<and> exprChannel ch (kE key)"
omega_vplus_commutative: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<omega>"   and "b \<in>\<^sub>\<circ> \<omega>" shows "a + b = b + a"
imbal_size: fixes t :: "'a tree" assumes "\<not> bal_i (size t) (height t)"   and "bal_i (size (hchild t)) (height (hchild t))"   and "t \<noteq> \<langle>\<rangle>" shows "real (size1 t) \<le> e * real (imbal t + 1)"
init_satisfies: fixes t :: "(nat \<times> linear_poly) list" shows "\<Turnstile> (init t)"
IH: fixes i :: "nat" assumes "xs ! i = y"   and "i < length xs - 1"   and "1 < length xs" shows "y \<in> set (butlast xs)"
insort_wrt_sort_key: fixes xs :: "'b list"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "set xs \<subseteq> A" shows "insort_wrt (linorder_from_keys A f) xs = sort_key f xs"
fmmap_keys_hom: fixes m\<^sub>1 :: "('c, 'd) fmap"   and m\<^sub>2 :: "('c, 'd) fmap"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "fmdom' m\<^sub>1 \<inter> fmdom' m\<^sub>2 = {}" shows "fmmap_keys f (m\<^sub>1 ++\<^sub>f m\<^sub>2) = fmmap_keys f m\<^sub>1 ++\<^sub>f fmmap_keys f m\<^sub>2"
funpow_diff: fixes p :: "'b \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and a :: "'b" assumes "inj p"   and "i \<le> j"   and "(p ^^ i) a = (p ^^ j) a" shows "(p ^^ (j - i)) a = a"
lossless_funct_14_OT: fixes funct_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "lossless_spmf (funct_14_OT M C)"
WTBinOp: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "expr"   and T\<^sub>2 :: "ty"   and T :: "ty"   and bop :: "bop" assumes "P,E \<turnstile> e\<^sub>1 :: T\<^sub>1"   and "P,E \<turnstile> e\<^sub>2 :: T\<^sub>2"   and "case bop of Eq \<Rightarrow> T\<^sub>1 = T\<^sub>2 \<and> T = Boolean | Add \<Rightarrow> T\<^sub>1 = Integer \<and> T\<^sub>2 = Integer \<and> T = Integer" shows "P,E \<turnstile> e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2 :: T"
g_def: shows "g = (\<lambda>F i. let a_n = isolate_variable_sparse p var i; exp = if i mod 2 = 0 then Const 1 else Const (- 1) in or (fm.Atom (Less (exp * a_n))) (and (fm.Atom (Eq a_n)) F))"
distinct_3: fixes x3 :: "nat" shows "DBZero \<noteq> DBInd x3"
SC_sym': fixes X :: "'p set"   and Y :: "'p set" shows "SC X Y = SC Y X"
adjacent_V_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.adjacent G v v'" shows "v' \<in> nodes G"
rep_eq: fixes x :: "('a, 'b) ffun"   and xa :: "('a, 'b) ffun" shows "pfun_of (x + xa) = pfun_of x + pfun_of xa"
rOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "\<And>C. Prop C \<Psi> P M (\<lparr>\<nu>*(xvec @ yvec)\<rparr>N \<prec>' P')"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* yvec"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "yvec \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M (\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>N \<prec>' P')"
prod_split: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y \<oplus> ad x \<cdot> d y = ad x"
cont: shows "continuous_on (g ` {0..1}) (\<lambda>xa. (1 - xa) ^ n / (1 - (1 - x * y) * xa))"