hcju/isabelleps huggingface dataset:
co_sd_id: fixes u :: "'b"   and w :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "(u, w) \<notin> set p"   and "(v, w) \<notin> set p" shows "co_path (u, v) w (sd_path (u, v) w p) = p"
rel_witness_option1: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "rel_option A x y" shows "rel_option (\<lambda>a (a', b). a = a' \<and> A a' b) x (rel_witness_option (x, y))"
while_def: fixes guard :: "'b \<Rightarrow> bool"   and body :: "'b \<Rightarrow> 'b spmf" shows "loop_spmf.while guard body \<equiv> spmf.fixp_fun (\<lambda>while s. if guard s then body s \<bind> while else return_spmf s)"
emv_subst_vars_term: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "emv s t"   and "s = t \<cdot> \<sigma>" shows "vars_term s = (the_Var \<circ> \<sigma>) ` vars_term t"
mem_equal_mod_cpu_reg: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and new_cpu1 :: "CPU_register \<Rightarrow> word32"   and cpu_reg2 :: "CPU_register \<Rightarrow> word32" assumes "mem_equal s1 s2 pa" shows "mem_equal (s1\<lparr>cpu_reg := new_cpu1\<rparr>) (s2\<lparr>cpu_reg := cpu_reg2\<rparr>) pa"
f_unchanged: shows "\<forall>a m. a \<notin> F \<longrightarrow> f m a = m a"
infinite_nat_greaterThan: fixes N :: "nat set"   and m :: "nat" assumes "infinite N" shows "infinite (N \<inter> {m<..})"
MStar_1: fixes ms :: "mregex" assumes "ok m mr"   and "ms \<in> RPD mr" shows "ok m ms"
from_nat_mono: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "i < j"   and "j < CARD('a)" shows "less (mod_type.from_nat Abs i) (mod_type.from_nat Abs j)"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', M\<lparr>\<lambda>*xvec N\<rparr>.P, M\<lparr>\<lambda>*xvec N\<rparr>.Q) \<in> Rel"
upper_asymptotic_density_bdd_interval_3: fixes N :: "nat" shows "upper_asymptotic_density {..<N} = 0"
hRe_hcomplex_of_hypreal: fixes z :: "real star" shows "hRe (hcomplex_of_hypreal z) = z"
False: shows "\<not> 0 \<le> additive_strength f xi"
f_non_negative: fixes e :: "nat \<times> nat" shows "(0::'a) \<le> f e"
inj_on_iff_Uniq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "inj_on f A = (\<forall>x\<in>A. \<exists>\<^sub>\<le>\<^sub>1 y. y \<in> A \<and> f x = f y)"
Mp_unique_factorization_m: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "unique_factorization_m (Mp f) cfs = unique_factorization_m f cfs"
entropy_rw: shows "\<H>(X) = - (\<Sum>i\<in>L. fi i * log b (fi i))"
assms_2: shows "M' \<longrightarrow>\<^sub>a* M''"
nvT_t: fixes \<sigma> :: "'tp"   and T :: "'fsym trm" assumes "\<not> unprot \<sigma>" shows "\<forall>x\<in>nvT (tT T). TE.tpOfV x \<noteq> \<sigma>"
vconverse_iff: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>) = (r\<lparr>a\<rparr> = b)"
S2: shows "(x + 1) * (x + 2) = x * (x + 1) + 2 * (x + 1)"
complete_net_reachable_is_subnet: fixes st :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree"   and I :: "'d node_action \<Rightarrow> bool" assumes "st \<in> reachable (closed (pnet np (p1 \<parallel> p2))) I" shows "\<exists>s t. st = SubnetS s t"
Lyndon_prim: fixes w :: "'a list" assumes "Lyndon w" shows "primitive w"
sturm_aux_gcd: fixes r :: "real poly"   and p :: "real poly"   and q :: "real poly" assumes "r \<in> set (sturm_aux p q)" shows "gcd p q dvd r"
lfp_eq_fixp: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "lfp f = ccpo_class.fixp f"
fmset_ee_aux: fixes cas :: "'a set list"   and cbs :: "'a set list"   and as :: "'a list"   and bs :: "'a list" assumes "cas <~~> cbs"   and "cas = map (assocs G) as"   and "cbs = map (assocs G) bs" shows "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = cbs"
dbl_simps_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus zero = zero"
zdvd_imp_le: fixes z :: "int"   and n :: "int" assumes "z dvd n"   and "0 < n" shows "z \<le> n"
simps: fixes a :: "('Var, 'AExp, 'BExp) Stmt" shows "no_await a = ((\<exists>x e. a = x \<leftarrow> e) \<or> (\<exists>c1 c2. a = c1 ;; c2 \<and> no_await c1 \<and> no_await c2) \<or> (\<exists>c1 c2 b. a = Stmt.If b c1 c2 \<and> no_await c1 \<and> no_await c2) \<or> (\<exists>c b. a = While b c \<and> no_await c) \<or> a = Skip \<or> a = Stop \<or> (\<exists>c m. a = c@[m] \<and> no_await c))"
mult_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"   and d :: "'a" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero b"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
length_0_conv: fixes xs :: "'g list" shows "(length xs = 0) = (xs = [])"
WHILEI_le_WHILEIT: fixes I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and s :: "'b" shows "WHILE\<^bsup>I\<^esup> b f s \<le> WHILE\<^sub>T\<^bsup>I\<^esup> b f s"
simps_3: fixes y :: "'c"   and ys :: "'c list" shows "foo [] (y # ys) = (1::'a)"
non_empty: fixes U :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U V" shows "U \<noteq> {}"
upper_density_good_direction: fixes d :: "real" assumes "0 < d"   and "d \<le> 1" shows "\<exists>delta. (\<forall>l. 0 < delta l) \<and> delta \<longlonglongrightarrow> 0 \<and> ennreal (1 - d) \<le> emeasure M {x \<in> space M. 1 - d \<le> upper_asymptotic_density {n. \<forall>l\<in>{1..n}. - delta l * real l < u n x - u (n - l) ((T ^^ l) x)}}"
order_trans: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term"   and z :: "('a, 'b) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
orient_lit_vars: fixes L :: "'a literal"   and t :: "'a trm"   and s :: "'a trm"   and p :: "sign" assumes "orient_lit L t s p" shows "vars_of t \<subseteq> vars_of_lit L \<and> vars_of s \<subseteq> vars_of_lit L"
get_proc_call: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "get_proc (targetnode a) = p"
lcf_of_X_minus: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<one> \<noteq> \<zero>" shows "X_poly_minus R a (deg R (X_poly_minus R a)) = \<one>"
poly_add_zero_1: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "ring R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p" shows "ring.poly_add R p [] = p"
urel_integer_mult: fixes x :: "integer"   and y :: "int"   and x' :: "integer"   and y' :: "int" assumes "urel_integer x y"   and "urel_integer x' y'" shows "urel_integer (mult_p_integer pp x x') (mult_p p y y')"
disjAll_def: fixes \<Delta>s :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool) set"   and s :: "'a"   and vl :: "'b"   and s1 :: "'c"   and vl1 :: "'d" shows "disjAll \<Delta>s s vl s1 vl1 \<equiv> \<exists>\<Delta>\<in>\<Delta>s. \<Delta> s vl s1 vl1"
triple_valid2_Suc: fixes G :: "prog"   and n :: "nat"   and t :: "'b triple" assumes "G\<Turnstile>Suc n\<Colon>t" shows "G\<Turnstile>n\<Colon>t"
unfold: shows "i_th = (\<Lambda> v. Fixrec.run\<cdot>(match_stcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(Nat_case\<cdot>x\<cdot>(i_th\<cdot>xs)))))"
CNV_def: fixes x :: "'a"   and y :: "'a" shows "CNV x y \<equiv> x = y"
lm044: fixes L1 :: "'c list list"   and L2 :: "'d list"   and f2 :: "'c set \<Rightarrow> 'e \<Rightarrow> 'd set"   and N :: "'e" assumes "\<forall>l\<in>set L1. set L2 = f2 (set l) N" shows "set (map (\<lambda>l. set L2) L1) = {f2 P N |P. P \<in> set (map set L1)}"
tr_s_blockI: fixes T :: "x \<Rightarrow> x option"   and s_s'_list :: "(s \<times> s) list" assumes "list_all (\<lambda>f. f) (map (\<lambda>(s_XXX, s_'). tr_s T s_XXX s_') s_s'_list)" shows "tr_s T (s_block (map (\<lambda>(s_XXX, s_'). s_XXX) s_s'_list)) (s_block (map (\<lambda>(s_XXX, s_'). s_') s_s'_list))"
Red1BinOp: fixes bop :: "bop"   and v1 :: "'addr val"   and v2 :: "'addr val"   and v :: "'addr val"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and s :: "'heap \<times> 'addr val list" assumes "binop bop v1 v2 = \<lfloor>Inl v\<rfloor>" shows "uf,P,t \<turnstile>1 \<langle>Val v1 \<guillemotleft>bop\<guillemotright> Val v2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val v,s\<rangle>"
zero_not_one: shows "\<zero>\<^bsub>K\<^esub> \<noteq> \<one>\<^bsub>K\<^esub>"
dom_llp2: fixes x :: "'a"   and y :: "'a" assumes "ad (ad x) \<le> ad (ad y)" shows "x \<le> ad (ad y) \<cdot> x"
0: shows "rawpsubst (rawpsubst (dsj \<phi>1 \<phi>2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = dsj (rawpsubst (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubst (rawpsubst \<phi>2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
insort_is_Cons: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "\<forall>x\<in>set xs. f a \<le> f x" shows "insort_key f a xs = a # xs"
norm_vec_def: fixes x :: "('c, 'd) vec" shows "norm x = L2_set (\<lambda>i. norm (x $ i)) UNIV"
l_get_parent_wf_axioms: shows "l_get_parent_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_parent"
l_2: shows "(flow0 x \<circ> s \<circ> r) \<longlonglongrightarrow> l"
assms: fixes k :: "'a"   and v :: "'b" assumes "(k, v) \<in> set (list_of_oalist_tc xs)" shows "f k v = g k v"
stable_on_allocation: fixes ds :: "D2 set"   and X :: "X5 set" assumes "BSI.stable_on ds X" shows "inj_on X5d X"
absorb1: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" shows "max a b = a"
DARS3: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "DenyAll \<notin> set p" shows "DenyAll \<notin> set (rm_MT_rules Cp p)"
map_def: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.map f = (if \<F>\<^sub>SC.arr f then f else \<F>\<^sub>SC.null)"
eq: fixes A :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "\<And>i. A i \<in> sets (M i)" shows "emeasure P (prod_emb UNIV M {..n} (Pi\<^sub>E {..n} A)) = emeasure Q (prod_emb UNIV M {..n} (Pi\<^sub>E {..n} A))"
is_volatile_Read\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_volatile_Read\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = volatile"
intros_1: fixes u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and c :: "int" assumes "u x - u y < real_of_int c" shows "intv'_elem x y u (Smaller' c)"
distinct_80: fixes pi1' :: "pi"   and pi2' :: "pi"   and pi1 :: "pi"   and pi2 :: "pi" shows "pi1' \<parallel> pi2' \<noteq> pi1 \<oplus> pi2"
assms_4: shows "y [^]\<^bsub>R\<^esub> CARD('a) ^ m' = y"
that: fixes g :: "complex \<Rightarrow> complex"   and r :: "nat \<Rightarrow> nat" assumes "g holomorphic_on S"   and "strict_mono r"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (r n) x) \<longlonglongrightarrow> g x"   and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) g sequentially" shows "thesis"
simps_2: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) extstrand_step list" shows "to_st (Step x # S) = x # to_st S"
Rep_CoCall_inverse: fixes x :: "CoCalls" shows "Abs_CoCall (Rep_CoCall x) = x"
l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs ShadowRootClass.known_ptr ShadowRootClass.type_wf get_host get_host_locs get_shadow_root get_shadow_root_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_tag_name Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
integrable_val_process: fixes pf :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and M :: "'d measure"   and Mkt :: "('d, 'c) discrete_market"   and n :: "nat" assumes "\<forall>asset\<in>support_set pf. integrable M (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)" shows "integrable M (val_process Mkt pf n)"
CAs0'_split: shows "\<forall>i<n. CAs0' ! i = Cs0' ! i + poss (AAs0' ! i)"
distinct_32: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm"   and x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PPair x51 x52 \<noteq> PFn x41 x42 x43"
A2: shows "Esigma p * Esigma q = Esigma r"
subdistl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<parallel> x \<le> z \<parallel> (x + y)"
OK_insert_iff: fixes i :: "'b"   and I :: "'b set"   and F :: "'b \<Rightarrow> 'c program" shows "OK (insert i I) F = (if i \<in> I then OK I F else OK I F \<and> F i ok JOIN I F)"
v_def: shows "v \<equiv> \<lambda>xs. int (sign_changes xs)"
Power: fixes y :: "ZF"   and x :: "ZF" shows "y |\<in>| Power x = y |\<subseteq>| x"
abs_ltln\<^sub>L_def: shows "abs_ltln\<^sub>L \<equiv> quot_type.abs (\<sim>\<^sub>L) Abs_ltln\<^sub>L"
rupture_surj_hom_1: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and p :: "('c multiset \<Rightarrow> 'a) list" assumes "subring K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "L.rupture_surj K p \<in> ring_hom (K [X]\<^bsub>L\<^esub>) (Rupt\<^bsub>L\<^esub> K p)"
Set_Cover_def: fixes w :: "nat \<Rightarrow> real"   and m :: "nat"   and S :: "nat \<Rightarrow> 'b set" shows "Set_Cover w m S \<equiv> (\<forall>i\<in>{1..m}. finite (S i)) \<and> (\<forall>i. 0 \<le> w i)"
num_sylow_normalizer: fixes P :: "'a set" assumes "P \<in> subgroups_of_size (p ^ a)" shows "card (rcosets\<^bsub>G\<lparr>carrier := conj.stabilizer P\<rparr>\<^esub> P) * p ^ a = card (conj.stabilizer P)"
interchange: fixes f :: "('e \<Rightarrow> 'd) af"   and x :: "'e" shows "f \<diamondop> pure x = pure (\<lambda>f. f x) \<diamondop> f"
not_gr_zero: fixes n :: "'b multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
in_Union_mset_iff: fixes x :: "'a"   and MM :: "'a multiset multiset" shows "(x \<in># \<Sum>\<^sub># MM) = (\<exists>M. M \<in># MM \<and> x \<in># M)"
rep_list_in_ideal: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list r \<in> ideal (set fs)"
word_gt_0: fixes y :: "'b word" shows "(0 < y) = (0 \<noteq> y)"
set_iterator_rev_linord_rule_P: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_rev_linord less_eq it S0"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<And>x'. x' \<in> S0 - S \<Longrightarrow> less_eq x x'; \<And>x'. x' \<in> S \<Longrightarrow> less_eq x' x\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<And>x x'. \<lbrakk>x \<in> S; x' \<in> S0 - S\<rbrakk> \<Longrightarrow> less_eq x x'; \<not> c \<sigma>; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
coeffs: shows "set (coeffs p) \<subseteq> \<real>"
unfold_llist_id: fixes xs :: "'c llist" shows "unfold_llist lnull lhd ltl xs = xs"
vsv_vlrestriction: fixes A :: "V" shows "vsv (T \<restriction>\<^sup>l\<^sub>\<circ> A)"
comp_3: fixes v :: "Vertex"   and w :: "Vertex" shows "((\<forall>v. s v \<noteq> v) \<longrightarrow> s v = w \<longrightarrow> s w = v) \<and> (s v = v \<longrightarrow> s w = w \<longrightarrow> v \<noteq> w \<longrightarrow> s = id)"
P1: shows "Bet A B B' \<and> Cong B B' A B"
ecarr: shows "somegcd G (c \<otimes> a) (c \<otimes> b) \<in> carrier G"
closure_of_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S \<subseteq> topspace X"
absurd_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "not_PL (not_PL p) \<in> Cn A" shows "p \<in> Cn A"
repeatable_def: fixes t :: "'a ttree" shows "repeatable t = (\<forall>x. possible t x \<longrightarrow> nxt t x = t)"
simps_2: fixes e :: "'c error_result" shows "list_result (Rerr e) = Rerr e"
negative_part_lemma: fixes C :: "'at Literal set" shows "negative_part C = validated_part {} C"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Left_a.in_hom f a b" obtains "Left_a.arr f" and "Left_a.dom f = a" and "Left_a.cod f = b"
adjoint_pair_antipar_1: fixes f :: "'a"   and g :: "'a" assumes "B.adjoint_pair f g" shows "B.ide f"
simps_9: fixes \<phi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_G_letter (F \<phi>) \<nu> = F \<phi> or af_G_letter \<phi> \<nu>"
Df: shows "dom\<^bsub>C\<^esub> f = X"
4: shows "trace (ioa.asig A) e_A = (if a \<in> externals (ioa.asig A) then Cons a (trace (ioa.asig A) e_A') else trace (ioa.asig A) e_A')"
bind_spmf_def: fixes x :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf" shows "x \<bind> f = bind_pmf x (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
injectivity_1: fixes P :: "nat set set"   and j :: "nat"   and P' :: "nat set set"   and j' :: "nat"   and B' :: "nat set"   and k :: "nat"   and B'' :: "nat set"   and k' :: "nat"   and Q :: "nat set set"   and Q' :: "nat set set"   and f :: "nat \<Rightarrow> nat set"   and g :: "nat \<Rightarrow> nat set"   and P'' :: "nat set set"   and P''' :: "nat set set" assumes "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "f = g"
nz: shows "\<not> (\<forall>z. z \<noteq> (0::'a) \<longrightarrow> poly p z = (0::'a))"
step_Normal_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (Throw, Normal s) \<rightarrow> u" shows "P"
assms_7: shows "Q_I_neg = filterQueryNeg I Qn"
lookup_chain: fixes r :: "'b ref"   and f :: "'a Heap" shows "!r \<bind> (\<lambda>_. f) = f"
Conj\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha> set['a]" assumes "finite (supp tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"
dom_simp: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.dom f = B.VxV.dom f"
mod_mult_cong: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and c :: "'d"   and a' :: "'d"   and b :: "'d"   and b' :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (times a b) c = modulo (times a' b') c"
GSMP_disjoint_snd_specific_not_fst_specific: fixes l :: "'b"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and l' :: "'b"   and Sec :: "('fun, 'var) Term.term set"   and m :: "('fun, 'var) Term.term" assumes "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"   and "proj_specific l' m \<A> Sec" shows "\<not> proj_specific l m \<A> Sec"
empty: fixes g :: "'c \<Rightarrow> 'a" shows "comm_monoid_add.sum (\<squnion>) bot g {} = bot"
resource_of_oracle_rprodl: fixes "oracle" :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c \<times> 'd \<times> 'e) spmf"   and s1 :: "'c"   and s2 :: "'d"   and s3 :: "'e" shows "RES ((rprodl ---> id ---> map_spmf (map_prod id lprodr)) oracle) ((s1, s2), s3) = RES oracle (s1, s2, s3)"
supp_cart_prod: fixes Xs :: "'a set"   and Ys :: "'c set" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "fs TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "finite Xs"   and "finite Ys"   and "Xs \<noteq> {}"   and "Ys \<noteq> {}" shows "supp (Xs \<times> Ys) = supp Xs \<union> supp Ys"
suc_nth_d_is: shows "lnth Ns n = (T2, N2)"
desargues_config_3D_coplanar_6_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A', B', C', \<alpha>, \<beta>, \<gamma>} = 3"
right_uniqueD: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and z :: "'d" assumes "right_unique A"   and "A x y"   and "A x z" shows "y = z"
comp_ide_self: fixes a :: "('o, 'a) arr" assumes "ide a" shows "COMP a a = a"
assms: shows "f : a \<mapsto>\<^bsub>dg_1 \<aa> \<ff>\<^esub> b"
Labels_Seq2: fixes c\<^sub>2 :: "cmd"   and l :: "nat"   and c :: "cmd"   and c\<^sub>1 :: "cmd" assumes "labels c\<^sub>2 l c" shows "labels (c\<^sub>1;; c\<^sub>2) (l + #:c\<^sub>1) c"
snd_monofun: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "x \<sqsubseteq> y" shows "snd x \<sqsubseteq> snd y"
E: shows "cfun_map\<cdot>(KM_map\<cdot>ValK_copy)\<cdot>(KM_map\<cdot>ValK_copy) \<sqsubseteq> snd (fix\<cdot>ValK_copy_rec)"
map_of_constant_assignments_defined_if: fixes l :: "('a \<times> 'b) list"   and v :: "'a"   and a :: "'b" assumes "\<forall>(v, a)\<in>set l. \<forall>(v', a')\<in>set l. v \<noteq> v' \<or> a = a'"   and "(v, a) \<in> set l" shows "map_of l v = Some a"