hcju/isabelleps huggingface dataset:
comm2Aux: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and R :: "('a, 'b, 'c) psi"   and K :: "'a"   and N :: "'a"   and R' :: "('a, 'b, 'c) psi"   and A\<^sub>R :: "name list"   and \<Psi>\<^sub>R :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> R \<longmapsto> K\<lparr>N\<rparr> \<prec> R'"   and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<Psi> \<otimes> \<Psi>\<^sub>Q \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K"   and "\<langle>A\<^sub>Q, (\<Psi> \<otimes> \<Psi>\<^sub>Q) \<otimes> \<Psi>\<^sub>R\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>P, (\<Psi> \<otimes> \<Psi>\<^sub>P) \<otimes> \<Psi>\<^sub>R\<rangle>"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>P"   and "distinct A\<^sub>R"   and "A\<^sub>R \<sharp>* A\<^sub>P"   and "A\<^sub>R \<sharp>* A\<^sub>Q"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q"   and "A\<^sub>R \<sharp>* R"   and "A\<^sub>R \<sharp>* K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* R"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>Q \<sharp>* R"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>R \<sharp>* xvec"   and "xvec \<sharp>* M" obtains K' :: "'a" where "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> R \<longmapsto> K'\<lparr>N\<rparr> \<prec> R'" and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K'" and "A\<^sub>R \<sharp>* K'"
not01: fixes z :: "complex" assumes "z \<in> cball 0 1" shows "\<not> (f z = 0 \<or> f z = 1)"
leg1_simps_3: shows "src f\<^sub>0gh\<^sub>1.p\<^sub>1 = src f\<^sub>0gh\<^sub>1.p\<^sub>0"
graph_pred_it_axioms_def: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) graph"   and invar :: "'a \<Rightarrow> bool"   and pred_list_it :: "'a \<Rightarrow> 'b \<Rightarrow> (('b \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list) \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list" shows "graph_pred_it_axioms \<alpha> invar pred_list_it \<equiv> \<forall>g v. invar g \<longrightarrow> set_iterator (pred_list_it g v) (pred (\<alpha> g) v)"
subst_cls_empty: fixes \<sigma> :: "'s" shows "{#} \<cdot> \<sigma> = {#}"
mempty_right: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and liftemb :: "'d\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'd\<^sub>\<bottom>"   and liftdefl :: "'d itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'d \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'd"   and defl :: "'d itself \<Rightarrow> udom defl"   and mempty :: "'d"   and mappend :: "'d \<rightarrow> 'd \<rightarrow> 'd"   and xs :: "'d" assumes "class.monoid below liftemb liftprj liftdefl emb prj defl mempty mappend" shows "mappend\<cdot>xs\<cdot>mempty = xs"
Add: shows "\<forall>x\<in>P. \<forall>y\<in>P. x + y \<in> P"
arctan_inverse_pos: fixes x :: "real" assumes "0 < x" shows "arctan x = pi / 2 - arctan (inverse x)"
redundant_infer_def: fixes N :: "'a literal multiset set"   and \<iota> :: "'a literal multiset inference" shows "G.redundant_infer N \<iota> = (\<exists>DD\<subseteq>N. DD \<union> set (side_prems_of \<iota>) \<TTurnstile>e {concl_of \<iota>} \<and> (\<forall>D\<in>DD. D < main_prem_of \<iota>))"
csum_cinfinite_bound: fixes p :: "('c \<times> 'c) set"   and r :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p \<le>o r"   and "q \<le>o r"   and "Card_order p"   and "Card_order q"   and "Cinfinite r" shows "p +c q \<le>o r"
h5: shows "invertible_mat (get_matrix_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
not_fic_subst2: fixes M :: "trm"   and a :: "coname"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "\<not> fic M a" shows "\<not> fic (M{c:=y.P}) a"
rq_triv: fixes z :: "'b list"   and u :: "'b list" shows "(z \<cdot> u)\<^sup><\<inverse>u  = z"
prime_elem_multiplicity_mult_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "factorial_semiring.multiplicity times one p (times x y) = factorial_semiring.multiplicity times one p x + factorial_semiring.multiplicity times one p y"
a_join_d: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion> d_aux a b c = a \<squnion> b \<sqinter> c"
w\<^sub>g'_in_hhom: shows "\<guillemotleft>w\<^sub>g' : src u\<^sub>g \<rightarrow> src (tab\<^sub>0 g)\<guillemotright>"
inject: fixes x1 :: "real interval"   and x2 :: "real interval"   and y1 :: "real interval"   and y2 :: "real interval" shows "(Complex_Interval x1 x2 = Complex_Interval y1 y2) = (x1 = y1 \<and> x2 = y2)"
simps_2: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "maxindex (CN c n p) = max (n + 1) (max (maxindex c) (maxindex p))"
iT_Plus_neg_empty: fixes k :: "nat" shows "{} \<oplus>- k = {}"
h3: shows "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = [] \<and> vc t = [vc_com] \<and> st_out t = y"
not_eint_eq: fixes x :: "eint" shows "(\<forall>y. x \<noteq> eint y) = (x = \<infinity>)"
nn_integral_multc: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * c \<partial>M = integral\<^sup>N M f * c"
divides_prod: fixes A :: "'b mod_ring set"   and g :: "'b mod_ring \<Rightarrow> 'b mod_ring poly"   and f :: "'b mod_ring poly" assumes "\<forall>c1 c2. c1 \<in> A \<and> c2 \<in> A \<and> c1 \<noteq> c2 \<longrightarrow> comm_monoid_mult_class.coprime (g c1) (g c2)"   and "\<forall>c\<in>A. g c dvd f" shows "prod g A dvd f"
stable_on_def: fixes ds :: "D2 set"   and X :: "X5 set" shows "BSI.stable_on ds X = (BSI.individually_rational_on ds X \<and> BSI.stable_no_blocking_on ds X)"
enabledPrefix: fixes SES :: "('c, 'd) SES_rec"   and s :: "'c"   and t1 :: "'d list"   and t2 :: "'d list" assumes "enabled SES s (t1 @ t2)" shows "enabled SES s t1"
collapse: fixes nondetT :: "('d, 'e) nondetT" shows "NondetT (run_nondet nondetT) = nondetT"
soundnessForSAT: fixes F0 :: "Literal list list"   and decisionVars :: "nat set"   and state0 :: "State"   and state :: "State" assumes "vars F0 \<subseteq> decisionVars"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation decisionVars"   and "\<not> formulaFalse (getF state) (elements (getM state))"   and "decisionVars \<subseteq> vars (elements (getM state))" shows "model (elements (getM state)) F0"
T_Read: fixes p :: "local_expr"   and \<tau> :: "thread_state"   and l' :: "V + bool \<Rightarrow> nat"   and v :: "V"   and e :: "local_expr"   and R' :: "nat set"   and a :: "abs_level" assumes "eval_bool p \<tau>"   and "l' = (l \<tau>)(Inl v := sh \<tau> (eval_word e \<tau>))"   and "R' = R \<tau> \<union> {eval_word e \<tau>}"   and "a \<in> {No_Abst, Eq_Abst}" shows "step_t a (\<tau>, Read (Var v) e, p) (\<tau>\<lparr>l := l', R := R'\<rparr>)"
sm_enum_DF: fixes k :: "nat"   and i :: "nat" shows "strict_mono_on (enum (DF k i)) {..k}"
is_pseudonatural_equivalence: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" assumes "converse_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G F \<Phi>\<^sub>F (converse_pseudonatural_equivalence.map\<^sub>0 V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<tau>\<^sub>0) (converse_pseudonatural_equivalence.map\<^sub>1 src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F G \<tau>\<^sub>0 \<tau>\<^sub>1)"
degree_fps_poly_swap2_le: fixes p :: "'b poly fps"   and d :: "nat" assumes "\<And>n. degree (p $ n) \<le> d" shows "degree (fps_poly_swap2 p) \<le> d"
r1: shows "R1 = T1 OO T1\<inverse>\<inverse>"
DBM_reset'_empty: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and cs :: "'c list"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n" shows "([M]\<^bsub>v,n\<^esub> = {}) = ([reset' M n cs v d]\<^bsub>v,n\<^esub> = {})"
simps_1: fixes xs :: "'varSort"   and x :: "'var" shows "qGood (qVar xs x) = True"
lift_Suc_mono_le: fixes f :: "nat \<Rightarrow> 'c list"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. prefix (f n) (f (Suc n))"   and "n \<le> n'" shows "prefix (f n) (f n')"
mult_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"
transfer_rule_numeral: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R (0::'b) (0::'c)"   and "R (1::'b) (1::'c)"   and "(R ===> R ===> R) (+) (+)" shows "((=) ===> R) numeral numeral"
bcomp_split: fixes b :: "bexp"   and f :: "bool"   and i :: "int"   and P' :: "instr list"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and n :: "nat"   and j :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "bcomp b f i @ P' \<turnstile> (0, s, stk) \<rightarrow>^n (j, s', stk')"   and "j \<notin> {0..<size (bcomp b f i)}"   and "0 \<le> i" shows "\<exists>s'' stk'' i' k m. bcomp b f i \<turnstile> (0, s, stk) \<rightarrow>^k (i', s'', stk'') \<and> (i' = size (bcomp b f i) \<or> i' = i + size (bcomp b f i)) \<and> bcomp b f i @ P' \<turnstile> (i', s'', stk'') \<rightarrow>^m (j, s', stk') \<and> n = k + m"
map_1: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "map_poly f (x1)\<^sub>p = (f x1)\<^sub>p"
substitution_def: fixes g :: "'g" shows "step.substitution g \<equiv> THE v'. step.isTrivialPhi g (step.chooseNext g) v'"
sum_diff: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "B \<subseteq> A" shows "sum f (A - B) = sum f A - sum f B"
p1_1: fixes i :: "nat" shows "local.coeff p1 i \<in> carrier R"
rep_eq: fixes x :: "String.literal"   and xa :: "String.literal" shows "(x \<le> xa) = ord.lexordp_eq (\<lambda>c d. of_char c < of_char d) (literal.explode x) (literal.explode xa)"
check_eqv_def: fixes r :: "nat rexp"   and s :: "nat rexp" shows "Equivalence_Checking.check_eqv r s = (let nr = norm r; ns = norm s; as = add_atoms nr (add_atoms ns []) in case Equivalence_Checking.closure as (nr, ns) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False)"
w_simps_3: shows "src Tfg_Hfg.chine = src (tab\<^sub>0 g \<star> Tfg.p\<^sub>0)"
x2gt: shows "\<forall>x2>x1. sign_vec qs x1 \<noteq> sign_vec qs x2 \<longrightarrow> (\<exists>r>x1. r \<le> x2 \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
invT_eq: fixes g :: "'m"   and h :: "'n" assumes "g \<in> M"   and "T g = h" shows "invT h = g"
ty_decr: shows "typ_of (decr 0 t) = Some (\<tau> \<rightarrow> \<tau>')"
icard_def: fixes A :: "'a set" shows "icard A \<equiv> if finite A then enat (card A) else \<infinity>"
assms_4: shows "case_sum f2 g2 \<in> rst.FINFUNC"
g_def: shows "g = Cn 1 r_ifeq_else_diverg [f, r_const 1, r_const 1]"
ord_spmf_return_spmf1: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and p :: "'c spmf" shows "ord_spmf R (return_spmf x) p = (lossless_spmf p \<and> (\<forall>y\<in>set_spmf p. R x y))"
fun_eq_in_ivl: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<forall>\<^sub>F \<xi> in at x. f \<xi> = f x)" shows "f a = f b"
flowpipe_def: fixes X0 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and hl :: "real"   and hu :: "real"   and CX :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'a \<Rightarrow>\<^sub>L 'a) set" shows "flowpipe X0 hl hu CX X1 = (0 \<le> hl \<and> hl \<le> hu \<and> fst ` X0 \<subseteq> X \<and> fst ` CX \<subseteq> X \<and> fst ` X1 \<subseteq> X \<and> (\<forall>(x0, d0)\<in>X0. \<forall>h\<in>{hl..hu}. h \<in> existence_ivl0 x0 \<and> (flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0..h}. (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX)))"
smap_stoch_proc_nth: fixes p :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and k :: "nat"   and w :: "'c" assumes "Suc p \<le> Suc n" shows "smap_stoch_proc (Suc n) f k w ! p = f (k + p) w"
r_delay_aux: fixes n :: "nat"   and x :: "nat"   and i :: "nat" assumes "n \<le> e_length x" shows "eval r_delay_aux [n, i, x] \<down>= (if \<exists>j<n. 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x]) then Suc (GREATEST j. j < n \<and> 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x])) else 0)"
distinct_29: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi1' :: "pi"   and pi2' :: "pi" shows "name1{name2}.pi \<noteq> pi1' \<parallel> pi2'"
preserves_seq: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "D.seq (G g) (G f)"
braun2_of_size_braun: fixes x :: "'b"   and n :: "nat"   and s :: "'b tree"   and t :: "'b tree" assumes "braun2_of x n = (s, t)" shows "size s = n \<and> size t = n + 1 \<and> braun s \<and> braun t"
root_unity: fixes i :: "'a" assumes "i \<in> I" shows "\<exists>d. d \<noteq> 0 \<and> x i ^ d = 1"
Minr_strip_Suc: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" assumes "Minr Rr xs w \<le> w" shows "Minr Rr xs (Suc w) = Minr Rr xs w"
iso_runit': fixes f :: "'b" assumes "D.ide f" shows "D.iso \<r>\<^sub>D\<^sup>-\<^sup>1[f]"
BoxWFI: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> WF(A)_v \<longrightarrow> \<box>WF(A)_v"
Z_def: fixes n :: "nat"   and m :: "nat" shows "Z n m = {(f, \<pi>) |f \<pi>. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> \<pi> permutes {0..<n}}"
THE_defaultI': fixes P :: "'b \<Rightarrow> bool"   and d :: "'b" assumes "\<exists>!x. P x" shows "P (THE_default d P)"
common_range_limit: fixes x :: "nat \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> 'c" assumes "finite (range x)"   and "finite (range y)" obtains i :: "nat" where "limit x = range (suffix i x)" and "limit y = range (suffix i y)"
Extend_def: fixes A :: "'d set"   and S :: "(('c, 'd) fm list \<times> 'd) set"   and f :: "nat \<Rightarrow> ('c, 'd) fm list \<times> 'd" shows "Extend A S f \<equiv> \<Union> (range (extend A S f))"
graph_plan_lemma_1: fixes s :: "('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list"   and vs :: "'d set" assumes "sat_precond_as s as" shows "exec_plan (fmrestrict_set vs s) (as_proj as vs) = fmrestrict_set vs (exec_plan s as)"
LiberalConseq: fixes P :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'e set"   and c :: "('d, 'f, 'e) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<forall>t. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> ((s \<in> P' \<longrightarrow> t \<in> Normal ` Q' \<union> Abrupt ` A') \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
assms: shows "\<not> is_character_data_ptr_kind node_ptr"
assms: shows "Pascal_Property.inter l m \<noteq> Pascal_Property.inter l' m"
ldeg_p_zero: fixes p :: "'a"   and d :: "nat" assumes "p = \<zero>" shows "ldeg_p R S X d p = \<zero>"
eq2: shows "Imon (drop (Suc 0) l) M \<otimes> head l [^] j \<otimes> (Ipol l (snd (mfactor P (mkMinj (Suc 0) M))) \<otimes> head l [^] (i - j)) = Imon (drop (Suc 0) l) M \<otimes> Ipol l (snd (mfactor P (mkMinj (Suc 0) M))) \<otimes> (head l [^] (i - j) \<otimes> head l [^] j)"
WT_obsf_distinguisher: fixes \<I> :: "('m, 'n) \<I>"   and \<A> :: "('m, 'n) distinguisher" assumes "\<I> \<turnstile>g \<A> \<surd>" shows "exception_\<I> \<I> \<turnstile>g obsf_distinguisher \<A> \<surd>"
elims: fixes x :: "(val \<times> val) list"   and y :: "funty" assumes "Tf x = y" obtains "x = []" and "y = \<top>"   | v1 :: "val" and v2 :: "val" and t :: "(val \<times> val) list" where "x = (v1, v2) # t" and "y = (T v1 \<rightarrow> T v2) \<sqinter> Tf t"
c: shows "[d2 = q + q + e2 * b + q * b - (s2 + (q - r)) mod q + (q - r)] (mod q)"
res_eq_to_cong: fixes m :: "int"   and a :: "int"   and b :: "int" assumes "residues m" shows "(a mod m = b mod m) = [a = b] (mod m)"
functor_axioms: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.R"
exp_Ln: fixes z :: "complex" assumes "z \<noteq> 0" shows "exp (Ln z) = z"
power_Suc0_right: fixes a :: "'a" shows "a\<^bsup>Suc 0\<^esup> = a"
Lp_complete: fixes p :: "ennreal"   and M :: "'a measure" shows "complete\<^sub>N (\<LL> p M)"
char_poly_0_column: fixes n :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "\<And>j. j < n \<Longrightarrow> A $$ (j, i) = (0::'b)"   and "A \<in> carrier_mat n n"   and "i < n" shows "char_poly A = monom (1::'b) 1 * char_poly (mat_delete A i i)"
multpw_emptyL: fixes X :: "'a multiset"   and ns :: "('a \<times> 'a) set" shows "(({#}, X) \<in> multpw ns) = (X = {#})"
subst_dterm: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<theta> :: "('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "Tadmit \<sigma> \<theta>"   and "dsafe \<theta>"   and "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'"   and "\<And>f f'. SPredicates \<sigma> f = Some f' \<Longrightarrow> fsafe f'" shows "dterm_sem I (Tsubst \<theta> \<sigma>) \<nu> = dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> \<nu>"
lemma_2_10_32: fixes b :: "'a"   and a :: "'a" shows "(((b l\<rightarrow> a) r\<rightarrow> a) l\<rightarrow> b) l\<rightarrow> b l\<rightarrow> a = b l\<rightarrow> a"
rev_image_eqI: fixes x :: "'e"   and A :: "'e set"   and b :: "'f"   and f :: "'e \<Rightarrow> 'f" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
ann_hoare_induct: fixes P2 :: "'b ann_com \<Rightarrow> 'b set \<Rightarrow> bool"   and P1 :: "'b set \<Rightarrow> 'b com \<Rightarrow> 'b set \<Rightarrow> bool"   and x4 :: "'b ann_com"   and x5 :: "'b set" assumes "\<And>r f q. r \<subseteq> {s. f s \<in> q} \<Longrightarrow> P2 (AnnBasic r f) q"   and "\<And>c0 c1 q. \<lbrakk>\<turnstile> c0  pre c1; P2 c0 (pre c1); \<turnstile> c1  q; P2 c1 q\<rbrakk> \<Longrightarrow> P2 (AnnSeq c0 c1) q"   and "\<And>r b c1 q c2. \<lbrakk>r \<inter> b \<subseteq> pre c1; \<turnstile> c1  q; P2 c1 q; r \<inter> - b \<subseteq> pre c2; \<turnstile> c2  q; P2 c2 q\<rbrakk> \<Longrightarrow> P2 (AnnCond1 r b c1 c2) q"   and "\<And>r b c q. \<lbrakk>r \<inter> b \<subseteq> pre c; \<turnstile> c  q; P2 c q; r \<inter> - b \<subseteq> q\<rbrakk> \<Longrightarrow> P2 (AnnCond2 r b c) q"   and "\<And>r i b c q. \<lbrakk>r \<subseteq> i; i \<inter> b \<subseteq> pre c; \<turnstile> c  i; P2 c i; i \<inter> - b \<subseteq> q\<rbrakk> \<Longrightarrow> P2 (AnnWhile r b i c) q"   and "\<And>c r b q. \<lbrakk>atom_com c; \<parallel>- (r \<inter> b) c q; P1 (r \<inter> b) c q\<rbrakk> \<Longrightarrow> P2 (AnnAwait r b c) q"   and "\<And>c q q'. \<lbrakk>\<turnstile> c  q; P2 c q; q \<subseteq> q'\<rbrakk> \<Longrightarrow> P2 c q'"   and "\<And>Ts. \<lbrakk>\<forall>i<length Ts. \<exists>c q. Ts ! i = (Some c, q) \<and> \<turnstile> c  q \<and> P2 c q; interfree Ts\<rbrakk> \<Longrightarrow> P1 (\<Inter>i\<in>{i. i < length Ts}. pre (the (OG_Hoare.com (Ts ! i)))) (Parallel Ts) (\<Inter>i\<in>{i. i < length Ts}. OG_Hoare.post (Ts ! i))"   and "\<And>f q. P1 {s. f s \<in> q} (Basic f) q"   and "\<And>p c1 r c2 q. \<lbrakk>\<parallel>- p c1 r; P1 p c1 r; \<parallel>- r c2 q; P1 r c2 q\<rbrakk> \<Longrightarrow> P1 p (Seq c1 c2) q"   and "\<And>p b c1 q c2. \<lbrakk>\<parallel>- (p \<inter> b) c1 q; P1 (p \<inter> b) c1 q; \<parallel>- (p \<inter> - b) c2 q; P1 (p \<inter> - b) c2 q\<rbrakk> \<Longrightarrow> P1 p (Cond b c1 c2) q"   and "\<And>p b c i. \<lbrakk>\<parallel>- (p \<inter> b) c p; P1 (p \<inter> b) c p\<rbrakk> \<Longrightarrow> P1 p (While b i c) (p \<inter> - b)"   and "\<And>p' p c q q'. \<lbrakk>p' \<subseteq> p; \<parallel>- p c q; P1 p c q; q \<subseteq> q'\<rbrakk> \<Longrightarrow> P1 p' c q'" shows "\<turnstile> x4  x5 \<longrightarrow> P2 x4 x5"
S_base: fixes u :: "'a"   and a :: "'c"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and M :: "'d set"   and P :: "'b multiset"   and k :: "nat" assumes "(u, Base a, v) \<in> edges fg"   and "(u, M, P) \<in> S_cs fg k" shows "(v, M, P) \<in> S_cs fg k"
strongdlHPPB_NDCIDCaux: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pp :: "'a \<Rightarrow> nat"   and DA :: "'b \<Rightarrow> 'f"   and lH :: "('f \<times> 'e \<times> nat) set"   and d :: "'f"   and PP :: "nat set"   and R :: "('a list \<times> 'a list) set"   and V :: "'a list"   and V' :: "'a list"   and i :: "nat" assumes "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP R"   and "(V, V') \<in> R"   and "i < length V" shows "WHATWHERE.NDC SR DA d (V ! i) \<or> WHATWHERE.IDC SR E DA d (V ! i) (WHATWHERE.htchLoc lH (pp (V ! i)))"
block_size_lt_v: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size" shows "u_block_size \<le> int (card point_set)"
numeral_code_1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"