hcju/isabelleps huggingface dataset:
le_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq a (divide b c) = (if less zero c then less_eq (times a c) b else if less c zero then less_eq b (times a c) else less_eq a zero)"
logically_equivalent_eqvt: fixes P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "P =\<cdot> Q" shows "p \<bullet> P =\<cdot> p \<bullet> Q"
axioms_1: fixes G :: "('d, 'e) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
prime_divides: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and p :: "'c" assumes "comm_monoid_cancel G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "prime G p"   and "p divides\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> b" shows "p divides\<^bsub>G\<^esub> a \<or> p divides\<^bsub>G\<^esub> b"
sup_top_left: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and top :: "'c"   and x :: "'c" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup top x = top"
lfp_exp_unfold: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "mono_trans t"   and "t R \<tturnstile> R"   and "sound R" shows "lfp_exp t = t (lfp_exp t)"
All2I: fixes A :: "fm"   and j :: "name"   and i :: "name" assumes "ss_fm A"   and "atom j \<sharp> (i, A)" shows "ss_fm (All2 i (Var j) A)"
e'_2: fixes d' :: "nat" assumes "d' \<in> {d..<d + length X}" shows "pdevs_val e (filter_pdevs (- (\<lambda>i x. I i (pdevs_applys X i))) (snd (zip [d..<d + length X] X ! (d' - d)))) = e' d' * tdev' p (filter_pdevs (- (\<lambda>i x. I i (pdevs_applys X i))) (snd (zip [d..<d + length X] X ! (d' - d))))"
assms: shows "valid_mmuaux' args cur dt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) auxlist"
weakly_preferred_by_men_def: fixes \<mu> :: "(M \<times> W) set"   and \<mu>' :: "(M \<times> W) set" shows "Non_Strict.weakly_preferred_by_men \<mu> \<mu>' = (\<forall>m. \<forall>w\<in>{w. (m, w) \<in> \<mu>}. \<exists>w'\<in>{w. (m, w) \<in> \<mu>'}. w' \<in> Non_Strict.m_weakly_prefers m \<mu>)"
le_minus_one_simps_1: 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" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus one) zero"
is_word_is_terminal: fixes i :: "nat"   and u :: "symbol list" assumes "i < length u"   and "is_word u" shows "is_terminal (u ! i)"
simplex_ns_sat: fixes solve_exec_ns :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a list + (nat, 'b) mapping"   and cs :: "('a \<times> 'b ns_constraint) list"   and v :: "(nat, 'b) mapping" assumes "Solve_exec_ns solve_exec_ns"   and "solve_exec_ns cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  (snd ` set cs)"
assms_1: shows "1 / p + 1 / q = 1"
bind_spmf_cong: fixes p :: "'a spmf"   and q :: "'a spmf"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "bind_spmf p f = bind_spmf q g"
gb_red_aux_spoly_reducible: fixes p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "(p, q) \<in> set ps" shows "(red (fst ` set bs \<union> set (gb_red_aux bs ps)))\<^sup>*\<^sup>* (spoly (fst p) (fst q)) 0"
less_num_simps_2: fixes n :: "num" shows "(num.One < num.Bit0 n) = True"
\<mu>: shows "\<not> \<mu> \<le> h \<mu> \<and> \<mu> \<in> D"
af_trans_extension_imp_wf: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "\<And>x y. P x y \<Longrightarrow> Q x y"   and "almost_full_on P A"   and "transp_on Q A" shows "wfp_on (strict Q) A"
succss_def: fixes xs :: "'a set" shows "succss xs = (\<Union>x\<in>xs. set (succs x))"
convergent_eq_Cauchy: fixes S :: "nat \<Rightarrow> 'a" shows "(\<exists>l. S \<longlonglongrightarrow> l) = Cauchy S"
squarefree_imp_coprime_pderiv: fixes p :: "'a poly" assumes "squarefree p"   and "content p = (1::'a)" shows "algebraic_semidom_class.coprime p (pderiv p)"
intro: fixes hom :: "'c \<Rightarrow> 'b" assumes "hom (0::'c) = (0::'b)" shows "zero_hom hom"
Max_by_ge: fixes A :: "'c set"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<in> A" shows "f x \<le> f (Max_by f A)"
complex_roots_1: fixes p :: "complex poly" shows "Polynomial.smult (lead_coeff p) (\<Prod>a\<leftarrow>complex_roots_complex p. [:- a, 1:]) = p"
simps_1: fixes g1 :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'a \<times> 'b)"   and g2 :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'a \<times> 'c)"   and s1 :: "'b"   and s2 :: "'c" shows "append_next g1 g2 (Inl (s1, s2)) = (if fst g1 s1 then let (x, s1') = snd g1 s1 in (x, Inl (s1', s2)) else append_next g1 g2 (Inr s2))"
ring_simprules_19: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = \<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
anonymous_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set" shows "anonymous scf A Is \<equiv> \<forall>P f x y. profile A Is P \<and> bij_betw f Is Is \<and> x \<in> A \<and> y \<in> A \<longrightarrow> (x \<^bsub>(scf P)\<^esub>\<preceq> y) = (x \<^bsub>(scf (P \<circ> f))\<^esub>\<preceq> y)"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and p :: "'psym"   and Tl :: "'fsym trm list" shows "I.satA \<xi> (Pr p Tl) = I_intP p (map (I.int \<xi>) Tl)"
lexlist_le_def: fixes xs :: "'b lexlist"   and ys :: "'b lexlist" shows "(xs \<le> ys) = list_le (unlex xs) (unlex ys)"
implies_not_sub: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> c\<^sub>1 implies not c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "r \<in> \<lbrakk> c\<^sub>1 implies not c\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
preserves_cod: fixes f :: "'b" assumes "D.arr f" shows "D.cod (src\<^sub>D f) = src\<^sub>D (D.cod f)"
fst_cond: shows "m \<noteq> m' \<and> valid_msg m \<and> valid_msg m'"
infsetsum_prod_PiE: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"   and "\<And>x. x \<in> A \<Longrightarrow> f x abs_summable_on B x" shows "(\<Sum>\<^sub>ag\<in>Pi\<^sub>E A B. \<Prod>x\<in>A. f x (g x)) = (\<Prod>x\<in>A. infsetsum (f x) (B x))"
power_inductl: fixes z :: "'a ppath set"   and x :: "'a ppath set"   and y :: "'a ppath set"   and n :: "nat" assumes "z \<union> pp_prod x y \<subseteq> y" shows "pp_prod (ppath_dioid.power x n) z \<subseteq> y"
minim_least: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "b \<in> B" shows "(local.minim B, b) \<in> r"
12: shows "infs even (smap f (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p))))"
fst_eq_Domain: fixes R :: "('e \<times> 'f) set" shows "fst ` R = Domain R"
wfrec: fixes R :: "('a \<times> 'a) set"   and F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a" assumes "wf R" shows "wfrec R F a = F (cut (wfrec R F) R a) a"
complete_delete: fixes t :: "'b tree23"   and x :: "'b" assumes "complete t" shows "complete (delete x t)"
"1_2": fixes k :: "nat" shows "prime (4 * k + 3) \<longrightarrow> even (multiplicity (4 * k + 3) n)"
intro: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b set"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "inj_on f P"   and "OrderingSetMap less_eq' less' less_eq less (f ` P) (the_inv_into P f)" shows "OrderingSetIso_axioms less_eq less less_eq' less' P f"
conv_galois_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "less_eq (composition x y) z = less_eq y (uminus (composition (converse x) (uminus z)))"
dghm_const_is_dghm: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "digraph \<alpha> \<DD>"   and "f : a \<mapsto>\<^bsub>\<DD>\<^esub> a" shows "dghm_const \<CC> \<DD> a f : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
1: shows "avl \<langle>l, (a, n), r\<rangle>"
simps_2: fixes step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s :: "'c"   and xs :: "'a list"   and x :: "'a" shows "c_tr step out s (xs @ [x]) = c_tr step out s xs @ [(x, out (foldl step s xs) x)]"
ccProd_empty': fixes S :: "Vars.var set" shows "S G\<times> {} = \<bottom>"
set_of_algs: shows "subfield {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x} L"
closure_minus_Collect: fixes A :: "'b set"   and B :: "'b set" shows "{x - y |x y. x \<in> closure A \<and> y \<in> closure B} \<subseteq> closure {x - y |x y. x \<in> A \<and> y \<in> B}"
seqI: fixes g1 :: "'a"   and f1 :: "'a"   and g2 :: "'a"   and f2 :: "'a" assumes "C.seq g1 f1 \<and> C.seq g2 f2" shows "CC.seq (g1, g2) (f1, f2)"
ereal_inverse_mult: fixes a :: "ereal"   and b :: "ereal" assumes "a \<noteq> 0"   and "b \<noteq> 0" shows "inverse (a * b) = inverse a * inverse b"
a_rcos_sum: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "I +> a <+> (I +> b) = I +> a \<oplus> b"
set_le_two: fixes a :: "'a"   and b :: "'a" shows "card {a, b} \<le> 2"
prems_2: fixes x' :: "'a node \<times> 'a node set" assumes "case x' of (n, ns) \<Rightarrow> expand_inv \<phi> (n, ns)"   and "(x', x) \<in> expand_ord \<phi>" shows "f x' \<le> SPEC (\<lambda>a. case a of (uu_, nds) \<Rightarrow> old_next_pair ` snd x' \<subseteq> old_next_pair ` nds \<and> expand_inv_result \<phi> nds)"
bij_betw_onto: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "f ` A = B"
le_step_down_nat: fixes i :: "nat"   and n :: "nat"   and P :: "bool" assumes "i \<le> n"   and "i = n \<longrightarrow> P"   and "i \<le> n - 1 \<longrightarrow> P" shows "P"
surj_image_vimage_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'c set" assumes "surj f" shows "f ` f -` A = A"
wt0_ge_min_ground_head: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and \<zeta> :: "('s, 'v) hd" assumes "legal_zpassign A" shows "zhmset_of (wt_sym (min_ground_head \<zeta>)) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt0 \<zeta>))"
add_diff_cancel_left': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
mult_coprime: fixes f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat" assumes "multiplicative_function f"   and "coprime a b" shows "f (a * b) = f a * f b"
absn_cong: fixes s :: "dB"   and t :: "dB"   and n :: "nat" assumes "s \<leftrightarrow> t" shows "(Abs ^^ n) s \<leftrightarrow> (Abs ^^ n) t"
suminf_pos2: fixes i :: "nat" assumes "summable f"   and "\<And>n. (0::'a) \<le> f n"   and "(0::'a) < f i" shows "(0::'a) < suminf f"
f_Exec_Stream_Acc_Output_append: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and ys :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (xs @ ys) c = f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c @ f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun ys (f_Exec_Comp trans_fun (xs \<odot> k) c)"
enum_ex_D2_def: fixes P :: "D2 \<Rightarrow> bool" shows "enum_class.enum_ex P = (P D1 \<or> P D2)"
measure_restrict_spmf: fixes p :: "'h spmf"   and A :: "'h set"   and X :: "'h set" shows "Sigma_Algebra.measure (measure_spmf (p \<upharpoonleft> A)) X = Sigma_Algebra.measure (measure_spmf p) (X \<inter> A)"
fps_nth_cases: fixes y :: "nat \<Rightarrow> 'a" assumes "y \<in> {f. True}" obtains x :: "'a fps" where "y = fps_nth x"
insert_seq_last_ref: fixes start :: "'a option"   and xs :: "('a \<times> 'a option) list"   and xi :: "'a"   and xr :: "'a option"   and yi :: "'a"   and yr :: "'a option" assumes "insert_seq start (xs @ [(xi, xr), (yi, yr)])" shows "yr = Some xi"
Arg2pi_ge_0: fixes z :: "complex" shows "0 \<le> Arg2pi z"
that: fixes y :: "'a" assumes "\<alpha>_limit_set x = {y}"   and "f y = (0::'a)" shows "thesis"
semClosed_Nil: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "'b dfmla"   and env :: "pvar \<Rightarrow> nat" assumes "Shallow S s0 \<delta> AP L AP'"   and "wff \<phi>"   and "FV \<phi> = {}" shows "Shallow.semClosed S s0 \<delta> AP L AP' \<phi> = Shallow.sem S s0 \<delta> AP L AP' \<phi> env []"
distinct_9: fixes x2 :: "name"   and x41 :: "dbtm"   and x42 :: "dbtm" shows "DBVar x2 \<noteq> DBEats x41 x42"
ts_ok_def: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and ts :: "'f \<Rightarrow> ('g \<times> 'i \<Rightarrow>f nat) option"   and m :: "'h" shows "ts_ok P ts m \<equiv> \<forall>t. case ts t of None \<Rightarrow> True | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> P t x m"
cos_gt_zero: fixes x :: "real" assumes "0 < x"   and "x < pi / 2" shows "0 < cos x"
swap_def: fixes p :: "'e \<times> 'd" shows "prod.swap p = (snd p, fst p)"
is_word_Derivation_derivation_ge: fixes w :: "symbol list"   and a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "is_word w"   and "Derivation (w @ a) D b" shows "derivation_ge D (length w)"
if\<^sub>1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "(\<Gamma>, scrut ? e1 : e2, S) \<Rightarrow> (\<Gamma>, scrut, Alts e1 e2 # S)"
"4_2": shows "fv\<^sub>s\<^sub>s\<^sub>t T1 = wfvarsoccs\<^sub>s\<^sub>s\<^sub>t T1"
assms_4: shows "op \<in> set (map (strips_op_to_sasp \<Psi>) ops')"
ax_sound2: fixes G :: "prog"   and A :: "'b triple set"   and ts :: "'b triple set" assumes "wf_prog G"   and "G,A|\<turnstile>ts" shows "G,A|\<Turnstile>\<Colon>ts"
p_zero: shows "ego_other.p 0 = s\<^sub>o - s\<^sub>e"
permutes_invI: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and p' :: "'c \<Rightarrow> 'c" assumes "p permutes S"   and "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"   and "\<And>x. x \<notin> S \<Longrightarrow> p' x = x" shows "inv p = p'"
arev_arev: fixes a :: "'b" shows "rev_H' (rev_H' a) = a"
Weierstrass_m_test: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> real" assumes "\<And>n x. x \<in> A \<Longrightarrow> norm (f n x) \<le> M n"   and "summable M" shows "uniform_limit A (\<lambda>n x. \<Sum>i<n. f i x) (\<lambda>x. \<Sum>i. f i x) sequentially"
C: shows "C = four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"
inj_map_strong: fixes x :: "'e set['f]"   and xa :: "'e set['f]"   and f :: "'e \<Rightarrow> 'g"   and fa :: "'e \<Rightarrow> 'g" assumes "\<And>z za. \<lbrakk>z \<in> set_bset x; za \<in> set_bset xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_bset f x = map_bset fa xa" shows "x = xa"
iia_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set"   and a :: "'a" assumes "iia_on A f"   and "B \<subseteq> A"   and "C \<subseteq> B"   and "a \<in> C"   and "a \<in> f B" shows "a \<in> f C"
get_root_node_no_parent: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> Shadow_DOM.get_parent node_ptr \<rightarrow>\<^sub>r None" shows "h \<turnstile> Shadow_DOM.get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr"
independent_substdbasis: fixes d :: "'a set" assumes "d \<subseteq> Basis" shows "independent d"
ordinal_div_self: fixes x :: "ordinal" assumes "0 < x" shows "x div x = 1"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "bdd_map f (Leaf x1) = Leaf (f x1)"
P1D1Q1: shows "(P1, D1, Q1) = Smith_mxn A2"
hd_filter_app: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list"   and zs :: "'d list"   and ys :: "'d list" assumes "filter P xs \<noteq> []"   and "zs = xs @ ys" shows "hd (filter P zs) = hd (filter P xs)"
iface_name_is_wildcard_alt': fixes eth :: "char list" shows "iface_name_is_wildcard eth = (eth \<noteq> [] \<and> hd (rev eth) = CHR ''+'')"
rsd_winners_permute_profile: fixes \<sigma> :: "'alt \<Rightarrow> 'alt"   and agents' :: "'agent list" assumes "\<sigma> permutes alts"   and "set agents' \<subseteq> agents" shows "rsd_winners (permute_profile \<sigma> R) alts agents' = \<sigma> ` rsd_winners R alts agents'"
lb_exp: fixes x :: "float"   and y :: "real"   and prec :: "nat" assumes "exp (real_of_float x) \<le> y" shows "real_of_float (lb_exp prec x) \<le> y"
adj_mat_1: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "adj_mat A \<in> carrier_mat n n"
fermat_liar_witness_of_composition_2: shows "local.fermat_witness \<noteq> local.fermat_liar"
word_sint_msb_eq: fixes x :: "'a Word.word" shows "sint x = uint x - (if msb x then 2 ^ size x else 0)"
less_eq_object_ptr_def: fixes x :: "(_) object_ptr"   and y :: "(_) object_ptr" shows "x \<le> y \<equiv> case x of Ext i \<Rightarrow> case y of Ext j \<Rightarrow> i \<le> j"
a4: shows "all_security_requirements_fulfilled (get_ACS M) G"
simps: fixes a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat list"   and a4 :: "nat list"   and a5 :: "nat list" shows "Form_Body a1 a2 a3 a4 a5 = (\<exists>xs ys a as b bs ka kb c d zs. a1 = ka \<and> a2 = kb \<and> a3 = xs \<and> a4 = ys \<and> a5 = zs \<and> length xs < length ys \<and> xs = concat (a # as) \<and> ys = concat (b # bs) \<and> a # as \<in> lists (- {[]}) \<and> b # bs \<in> lists (- {[]}) \<and> length (a # as) = ka \<and> length (b # bs) = kb \<and> c = acc_lengths 0 (a # as) \<and> d = acc_lengths 0 (b # bs) \<and> zs = concat [c, a, d, b] @ interact as bs \<and> strict_sorted zs)"
list_update_code_2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" shows "(x # xs)[0 := y] = y # xs"
connected_component_in_topspace: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" assumes "connected_component_of X x y" shows "x \<in> topspace X \<and> y \<in> topspace X"
in_arcs_H: shows "{uw, rev_H uw, vw, rev_H vw} \<subseteq> arcs H"
g_le_G: fixes t :: "real" assumes "t \<in> {a..b}" shows "g t \<le> G t"
discr_MtransC: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.discr tval aval indis c"   and "PL.MtransC tval aval (c, s) (c', s')" shows "PL_Indis.discr tval aval indis c'"
upred_bottom: shows "\<bottom>\<^bsub>\<P>\<^esub> = true"
snd_swap: fixes x :: "'b \<times> 'c" shows "snd (prod.swap x) = fst x"
triple_cross_mult_hmset: fixes An :: "hmultiset"   and Bn :: "hmultiset"   and Cn :: "hmultiset"   and Bp :: "hmultiset"   and Cp :: "hmultiset"   and Ap :: "hmultiset" shows "An * (Bn * Cn + Bp * Cp - (Bn * Cp + Cn * Bp)) + (Cn * (An * Bp + Bn * Ap - (An * Bn + Ap * Bp)) + (Ap * (Bn * Cp + Cn * Bp - (Bn * Cn + Bp * Cp)) + Cp * (An * Bn + Ap * Bp - (An * Bp + Bn * Ap)))) = An * (Bn * Cp + Cn * Bp - (Bn * Cn + Bp * Cp)) + (Cn * (An * Bn + Ap * Bp - (An * Bp + Bn * Ap)) + (Ap * (Bn * Cn + Bp * Cp - (Bn * Cp + Cn * Bp)) + Cp * (An * Bp + Bn * Ap - (An * Bn + Ap * Bp))))"
charpoly_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "charpoly A = det (mat (monom (1::'b) (Suc 0)) - mat2matofpoly A)"
contr_graph_slim_eq: fixes G :: "'b pair_pre_digraph" assumes "pair_sym_digraph G" shows "gen_contr_graph (with_proj (pair_pre_digraph.slim G)) (verts3 (with_proj G)) = gen_contr_graph (with_proj G) (verts3 (with_proj G))"
wp3Skip: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "wp\<^sub>3\<^sub>' SKIP Q = (Q \<and>* $ 1)"
distrib_imp2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
map_dom_ran_finite: fixes M :: "'a \<Rightarrow> 'b option" assumes "finite (dom M)" shows "finite (ran M)"
vd: fixes x :: "real" assumes "x \<in> {0..1}"   and "x \<notin> (\<lambda>t. (v - u) *\<^sub>R t + u) -` S" shows "vector_derivative (\<lambda>x. g ((v - u) * x + u)) (at x) = (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u))"
intersect_segment_xline_eq_empty: fixes p0 :: "real \<times> real"   and p1 :: "real \<times> real"   and x :: "real"   and prec :: "nat" assumes "closed_segment p0 p1 \<inter> {p. fst p = x} = {}" shows "intersect_segment_xline prec (p0, p1) x = None"
eq_refl: fixes a :: "Arity"   and v :: "Value" shows "eq a v v"
typeI_edges_are_valid_paths: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "typeI_twoCube twoC"   and "(k, \<gamma>) \<in> boundary twoC" shows "valid_path \<gamma>"
dghm_ArrMap_is_arr: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
sum_two_elements: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "a \<noteq> b" shows "sum f {a, b} = f a + f b"