hcju/isabelleps huggingface dataset:
update_convs_3: fixes c_msg' :: "('c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list) \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_msg_update c_msg' \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg' c_msg, c_glob = c_glob, \<dots> = more\<rparr>"
assms_1: shows "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"
RecvMarker_given_channel: fixes ev :: "('a, 'b, 'c) event"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   and c :: "('a, 'b, 'c) configuration" assumes "isRecvMarker ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "can_occur ev c" shows "ev = RecvMarker cid q p"
e_cons_elementary: fixes e :: "nat"   and es :: "nat" shows "e_cons e es = (if es = 0 then Suc (prod_encode (0, e)) else Suc (prod_encode (e_length es, prod_encode (e, pdec2 (es - 1)))))"
bv_to_int_def: fixes w :: "bit list" shows "bv_to_int w = (case bv_msb w of \<zero> \<Rightarrow> int (bv_to_nat w) | \<one> \<Rightarrow> - int (bv_to_nat (bv_not w) + 1))"
emeasure_lborel_UNIV: shows "emeasure lborel UNIV = \<infinity>"
base_1: shows "r = (\<Sum>i\<in>S. a i * Measure.measure M (A i))"
matrix_A_def: fixes sign_list :: "rat list list"   and subset_list :: "nat list list" shows "M_mat sign_list subset_list = mat_of_rows_list (length sign_list) (map (mtx_row sign_list) subset_list)"
coprime: fixes p :: "rat poly"   and q :: "rat poly" assumes "p \<in> set fs"   and "q \<in> set fs"   and "p \<noteq> q" shows "comm_monoid_mult_class.coprime p q"
Node_3: assumes "avl r" shows "avl (AVL_Set_Code.insert x r)"
inf_fres: fixes y :: "'c \<times> 'd"   and x :: "'c \<times> 'd" shows "times_prod' y x = Inf_prod {z. less_eq_prod y (prod.fres z x)}"
eq_refl: fixes x :: "'b"   and y :: "'b" assumes "x = y" shows "ordb x y"
lower_asymptotic_density_infinite_interval_1: fixes N :: "nat" shows "lower_asymptotic_density {N..} = 1"
open_vimage_ProjI: fixes T :: "'b proj_space set" assumes "open T" shows "open (Proj -` T)"
orthogonal_gso: shows "orthogonal (map gso [0..<length fs])"
elims: fixes x :: "('b, 'c) result"   and y :: "('b list, 'c) result" assumes "list_result x = y" obtains v2 :: "'b" where "x = Rval v2" and "y = Rval [v2]"   | e :: "'c error_result" where "x = Rerr e" and "y = Rerr e"
Irr_generates_ker: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "a_kernel (K [X]) R (\<lambda>p. local.eval p x) = PIdl\<^bsub>K [X]\<^esub> Irr K x"
induct: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'b"   and a2 :: "('a \<times> 'b) list" assumes "\<And>k v. P k v []"   and "\<And>k v l w ls. (k \<noteq> l \<Longrightarrow> P k v ls) \<Longrightarrow> P k v ((l, w) # ls)" shows "P a0 a1 a2"
scale_prod: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'d"   and a :: "'e"   and b :: "'f" assumes "module_prod s1 s2" shows "module_prod.scale s1 s2 x (a, b) = (s1 x a, s2 x b)"
remainder_theorem: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "carrier R \<noteq> {\<zero>}" shows "\<exists>q r. q \<in> carrier P \<and> r \<in> carrier P \<and> f = (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0"
max_nat_def: fixes a :: "nat"   and b :: "nat" shows "max_nat a b = (if a \<le> b then b else a)"
amor_del_min: fixes t :: "'b tree" shows "real (T_dm t) + \<Phi> (del_min t) - \<Phi> t \<le> 2 * \<phi> t + 1"
Fun: fixes ss :: "('d, 'e) Term.term list"   and ts :: "('d, 'f) Term.term list"   and P :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset"   and f :: "'d"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term option" assumes "length ss = length ts" shows "MATCH1 (P + {#(Fun f ss, Fun f ts)#}, \<sigma>) (P + mset (zip ss ts), \<sigma>)"
rotate14: fixes A14 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and A9 :: "fm"   and A10 :: "fm"   and A11 :: "fm"   and A12 :: "fm"   and A13 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A14 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H))))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H))))))))))))) \<turnstile> B"
nonInitBlks_def: fixes s :: "state"   and p :: "Proc" shows "nonInitBlks s p = {bs \<in> blocksSeen s p. inp bs \<in> Inputs}"
norm_ge_zero: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "0 \<le> norm x"
of_int_less_1_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) one = (z < 1)"
semiring_axioms_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.semiring_axioms plus times \<equiv> (\<forall>a b c. times (plus a b) c = plus (times a c) (times b c)) \<and> (\<forall>a b c. times a (plus b c) = plus (times a b) (times a c))"
lconjseq_snoc: fixes xs :: "'a list"   and x :: "'a" shows "lconjseq (xs @ [x]) = lconjseq xs @ [lconjby (sum_list xs) x]"
PlusE: fixes u :: "'c + 'd"   and A :: "'c set"   and B :: "'d set" assumes "u \<in> A <+> B" obtains x :: "'c" where "x \<in> A" and "u = Inl x"   | y :: "'d" where "y \<in> B" and "u = Inr y"
derived_S5_rules_2_b: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
min_add_distrib_right: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "x + subset_mset.min y z = subset_mset.min (x + y) (x + z)"
trancl_to_fmrel: fixes f :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'f"   and ym :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "(fmrel f)\<^sup>+\<^sup>+ xm ym" shows "fmrel f\<^sup>+\<^sup>+ xm ym"
bounded_iff: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "semilattice_order f less_eq less" shows "less_eq a (f b c) = (less_eq a b \<and> less_eq a c)"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> VV.arr f" shows "fst f \<star> snd f = null"
inject: fixes a :: "'f"   and b :: "'g"   and a' :: "'f"   and b' :: "'g" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
open_codomain: fixes c :: "('d, 'c) chart" shows "open (codomain c)"
H_2: shows "\<forall>n. u n \<in> space\<^sub>N bcontfun\<^sub>N"
\<alpha>'_nonneg: shows "0 \<le> \<alpha>'"
in_genby_S_reduced_letter_set: fixes A :: "'w set"   and a :: "'w" assumes "order.greater_eq S A"   and "a \<in> \<langle>A\<rangle>" shows "a \<in> \<langle>reduced_letter_set A a\<rangle>"
inject: fixes a :: "'c"   and b :: "'d"   and a' :: "'c"   and b' :: "'d" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
Bseq_add: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "Bseq f" shows "Bseq (\<lambda>x. f x + c)"
InvariantGetReasonIsReasonAfterApplyLearn: fixes state :: "State" assumes "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))" shows "let state' = applyLearn state in InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
zero_aezfun_def: shows "0 \<equiv> Abs_aezfun 0"
BuildSeq_trunc: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf"   and l :: "hf" assumes "BuildSeq B C s k y"   and "l \<^bold>\<in> k" shows "BuildSeq B C s l (app s l)"
simps_6: fixes va :: "iface" shows "compress_extra (MatchNot (Match (OIface va))) = MatchNot (compress_extra (Match (OIface va)))"
inj: shows "inj_on (\<lambda>b. (y + (q - b)) mod q) {..<q}"
rec_2: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
extremum_uniqueI: fixes a :: "'a" assumes "a \<le> (zero_class.zero::'a)" shows "a = (zero_class.zero::'a)"
Mul_interfree_Mutator_Mutator: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n"   and "i \<noteq> j" shows "interfree_aux (Some (Mul_Mutator i n), {}, Some (Mul_Mutator j n))"
viewnres_split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "('d \<times> 'e) option" shows "P (case x of None \<Rightarrow> f1 | Some (xa, xb) \<Rightarrow> f2 xa xb) = ((x = None \<longrightarrow> P f1) \<and> (\<forall>a b. x = Some (a, b) \<longrightarrow> P (f2 a b)))"
assms: shows "\<forall>s\<in>RED \<tau>. t[x::=s] \<in> RED \<sigma>"
distinct_5: fixes x2 :: "nat"   and x3 :: "nat" shows "Regions.intv.Intv x2 \<noteq> Regions.intv.Greater x3"
simps_6: fixes x7 :: "'a OCL_Types.type"   and y7 :: "'a OCL_Types.type" shows "(Bag x7 = Bag y7) = (x7 = y7)"
19: shows "infs (accepting A) (smap snd (fromN 1 ||| r))"
object_ptr_kinds_eq_h: shows "object_ptr_kinds h = object_ptr_kinds h2"
hyps: fixes v :: "'v" assumes "finite E"   and "is_trail v xs v'" shows "(v \<noteq> v') = (odd (degree v' (rem_unPath xs G)) = even (degree v' G))"
BinOp2Reds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and bop :: "bop" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>Val v \<guillemotleft>bop\<guillemotright> e,s\<rangle> \<rightarrow>* \<langle>Val v \<guillemotleft>bop\<guillemotright> e',s'\<rangle>"
induct: fixes P :: "'a list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "nat" assumes "\<And>n. P [] n"   and "\<And>x r n. (\<not> n \<le> 0 \<Longrightarrow> P r (n - 1)) \<Longrightarrow> P (x # r) n" shows "P a0 a1"
lincomb_union: fixes A :: "'c set"   and B :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "finite (A \<union> B)"   and "A \<union> B \<subseteq> carrier V"   and "A \<inter> B = {}"   and "a \<in> A \<union> B \<rightarrow> carrier K" shows "lincomb a (A \<union> B) = lincomb a A \<oplus>\<^bsub>V\<^esub> lincomb a B"
ccard_diff_lgth: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "dbc n' - dbc n \<le> card {i. n < i \<and> i \<le> n' \<and> dmining i}"
inorder_joinL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "bheight l \<le> bheight r" shows "Tree2.inorder (joinL l x r) = Tree2.inorder l @ x # Tree2.inorder r"
ind: fixes s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps0 (s, l, r) tp stp = (s', l', r')"
plus_Nat_eval: fixes x :: "nat discr"   and y :: "nat discr" shows "box\<cdot>x + box\<cdot>y = box\<cdot>(x + y)"
As_def: shows "As = lmap (active_subset \<circ> snd) TNs"
con_rews_4: fixes a :: "'a"   and slist :: "[:'a:]" shows "(a :# slist = \<bottom>) = (a = \<bottom> \<or> slist = \<bottom>)"
perm_fresh_fresh_aux: fixes pi :: "(W.tvar \<times> W.tvar) list"   and z :: "'a" assumes "\<forall>(x, y)\<in>set pi. x \<sharp> z \<and> y \<sharp> z" shows "pi \<bullet> z = z"
r_e2ls_prim: shows "prim_recfn 1 r_e2ls"
Lyndon_Schutzenberger': fixes x :: "'b list"   and a :: "nat"   and y :: "'b list"   and b :: "nat"   and z :: "'b list"   and c :: "nat" assumes "x \<^sup>@ a \<cdot> y \<^sup>@ b = z \<^sup>@ c"   and "2 \<le> a"   and "2 \<le> b"   and "2 \<le> c" shows "x \<cdot> y = y \<cdot> x"
RG_el_decomp_aezdeltafun: fixes c :: "('a, 'g) aezfun" assumes "c \<in> group_ring" shows "\<exists>ras. set (map snd ras) \<subseteq> G \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"
rsp: shows "fractrel (1::'b, 1::'b) (1::'b, 1::'b)"
collapse_2: fixes sum :: "'g + 'h" assumes "\<not> isl sum" shows "Inr (projr sum) = sum"
simps_2: fixes x :: "'g"   and xs :: "'g list"   and i :: "nat"   and v :: "'g" shows "(x # xs)[i := v] = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # xs[j := v])"
iso_block_img_alt_rep: shows "image_mset ((`) \<pi>) \<B> = \<B>'"
assms_2: shows "(g has_vector_derivative g') (at x within s)"
SD_singleton: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "SD le (return_pmf x) (return_pmf y) = le x y"
mat_erase_carrier: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat"   and nr :: "nat"   and nc :: "nat" shows "(mat_erase A i j \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
name_list_set_fresh: fixes xvec :: "name list"   and x :: "'e" shows "set xvec \<sharp>* x = xvec \<sharp>* x"
ACI_nullable: fixes r :: "'a rexp"   and s :: "'a rexp" assumes "ACI r s" shows "nullable r = nullable s"
frameResChainEqvt: fixes perm :: "(name \<times> name) list"   and xvec :: "name list"   and F :: "'d frame" shows "perm \<bullet> \<lparr>\<nu>*xvec\<rparr>F = \<lparr>\<nu>*perm \<bullet> xvec\<rparr>perm \<bullet> F"
multipl_div_mono: fixes f :: "nat \<Rightarrow> complex"   and d :: "nat"   and k :: "nat" assumes "completely_multiplicative_function f"   and "d dvd k"   and "0 < d"   and "\<And>p. prime p \<Longrightarrow> f p \<noteq> 0" shows "f (k div d) = f k / f d"
0: shows "Poly_Mapping.single x ` {d. lookup p (monomial d x) \<noteq> (0::'a)} \<subseteq> {d. lookup p d \<noteq> (0::'a)}"
trss_bot_proc_const: fixes s :: "'a list"   and u :: "'a"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and u' :: "'a"   and c' :: "'a list multiset" assumes "((s @ [u], c), w, s' @ [u'], c') \<in> trcl (trss fg)" shows "proc_of fg u = proc_of fg u'"
ok_lctxt_swap: fixes \<phi>s :: "Formula.formula list"   and \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and mr :: "mregex" assumes "ok_lctxt \<phi>s \<kappa> \<kappa>'" shows "from_mregex (\<kappa> mr) \<phi>s = \<kappa>' (from_mregex mr \<phi>s)"
left_unit: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c" assumes "monoid M composition unit"   and "a \<in> M" shows "composition unit a = a"
max_ipv4_addr_max: shows "\<forall>a. a \<le> max_ipv4_addr"
envsOf_preserves_good: fixes P :: "('a, 'b, 'varSort, 'var, 'c) param"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodPar P"   and "rho \<in> envsOf P" shows "goodEnv rho"
HStartBallot_HInv4a_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv4a s q"   and "p \<noteq> q" shows "HInv4a s' q"
TR4_sub_carr: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR4 R M N P \<subseteq> carrier P"
iter_def: fixes x :: "'a" shows "iter x \<equiv> \<Sqinter>{y. \<exists>i. y = x ^ i}"
Abs_perm_inverse: fixes y :: "atom \<Rightarrow> atom" assumes "y \<in> perm" shows "Rep_perm (Abs_perm y) = y"
assms_2: shows "theta\<inverse> \<subseteq> ZOretr (theta\<inverse>)"
elims_3: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "('d, 'e, 'f) FSM" assumes "\<not> x \<preceq> xa" obtains M1 :: "('d, 'e, 'f) FSM" and M2 :: "('d, 'e, 'f) FSM" where "x = M1" and "xa = M2" and "\<not> L M1 \<subseteq> L M2"
f3: fixes z :: "'a" shows "ad (x \<cdot> y) \<cdot> (x \<cdot> z) = ad (x \<cdot> y) \<cdot> (x \<cdot> (ad (ad (ad y) \<cdot> y) \<cdot> z))"
process_rule_2_failures: fixes xs :: "'c list"   and xs' :: "'c list"   and X :: "'c set"   and P :: "'c process" assumes "(xs @ xs', X) \<in> failures P" shows "(xs, {}) \<in> failures P"
is_BoolVal_def: fixes val :: "val" shows "is_BoolVal val = (\<exists>x2. val = BoolVal x2)"
distinct_23: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a"   and x41 :: "nat"   and x42 :: "nat" shows "x31\<not>x32:arrive(x33) \<noteq> connect(x41, x42)"
heap_is_wellformed_disconnected_nodes_distinct: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "distinct disc_nodes"
FSP: shows "(([u], {#}), LOC (LCall p # wsl), [v, u'], ch) \<in> ntrp fg"
finiteSends: fixes p :: "'p"   and s :: "'s"   and m :: "'v messageValue" shows "finite {v. v \<in># sends p s m}"
hom_graph_imp_finite: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b" assumes "hom_graph f S x a" shows "finite S"
n_def: shows "n = n_of_wstate (llast Sts)"
verit_eq_simplify_2: fixes n :: "num" shows "(numeral n = (1::'a)) = (n = num.One)"
D_in_C: shows "D \<in> \<G>_F C"
fps_compose_0_right: fixes a :: "'b fps" shows "a oo 0 = fps_const (a $ 0)"
set_global: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and v :: "v"   and s' :: "s"   and vs :: "v list" assumes "supdate_glob s i j v = s'" shows "\<lparr>s;vs;[$C v, $Set_global j]\<rparr> \<leadsto>_ i \<lparr>s';vs;[]\<rparr>"
restrict_current_IN_not_RF: fixes \<Gamma> :: "('k, 'l) web_scheme"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> g"   and "x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_IN (g \<upharpoonleft> \<Gamma> / f) x = d_IN g x"
\<eta>_leg_simps_2: shows "\<eta>.dom.leg1 = f.dsrc"
WHILEIT_refine_WHILEI: fixes V :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and I' :: "'b \<Rightarrow> bool"   and x' :: "'b"   and x :: "'a"   and R :: "('a \<times> 'b) set"   and b' :: "'b \<Rightarrow> bool"   and f' :: "'b \<Rightarrow> 'b nres" assumes "wf V"   and "\<And>s. \<lbrakk>I s; b s; f s \<le> SPEC I\<rbrakk> \<Longrightarrow> f s \<le> (spec s'. (s', s) \<in> V)"   and "I' x' \<Longrightarrow> (x, x') \<in> R"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I' x'\<rbrakk> \<Longrightarrow> I x"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I x; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "while\<^sub>T\<^bsup>I\<^esup> b f x \<le> \<Down> R (while\<^bsup>I'\<^esup> b' f' x')"
2: fixes x :: "real" assumes "- (pi / 2) < x"   and "x * 2 < pi" shows "isCont (\<lambda>x. 1 + (tan x)\<^sup>2) x"
2: shows "rv \<in> fringe prb \<and> rv \<noteq> src re \<or> rv = tgt re"
finished_step: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and final :: "'a \<Rightarrow> bool"   and s :: "'a"   and s' :: "'a" assumes "semantics step final"   and "step s s'" shows "\<not> finished step s"
coprime_mult_left_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (times a b) c = (algebraic_semidom.coprime times one a c \<and> algebraic_semidom.coprime times one b c)"
inorder_delete: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (RBT_Set.delete x t) = del_list x (Tree2.inorder t)"
cprank_max_upper_bound: fixes A :: "'b tensor" shows "cprank_max (prod_list (dims A)) A"
node\<^sub>i_bal: fixes ts :: "('a btree \<times> 'a) list"   and t :: "'a btree"   and k :: "nat" assumes "bal (Node ts t)" shows "bal_up\<^sub>i (node\<^sub>i k ts t)"
intros_1: fixes a :: "'c"   and ars :: "('c \<times> 'd \<times> 'c) set" shows "(a, []) \<in> seq ars"
ide_map\<^sub>0_obj: fixes a :: "'c" assumes "C.obj a" shows "D.ide (\<sigma>\<^sub>0 a)"
pdens: shows "has_density M (stock_measure INTEG) \<delta>"
comp_arr_inv: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c" assumes "C.VV.inverse_arrows f g" shows "C.VV.comp f g = C.VV.dom g"
mult_on_union': fixes M :: "'b multiset"   and N :: "'b multiset"   and r :: "('b \<times> 'b) set"   and K :: "'b multiset" assumes "(M, N) \<in> mult r" shows "(M + K, N + K) \<in> mult r"
accepts_def: fixes A :: "'c \<times> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> bool)" shows "accepts A = (\<lambda>w. fin A (delta A w (start A)))"
pw_RES_bind_choose_1: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'b nres" shows "nofail (RES X \<bind> f) = (\<forall>x\<in>X. nofail (f x))"
abs_sgn: fixes k :: "'d" shows "\<bar>k\<bar> = k * sgn k"
pred_spmf_return: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "pred_spmf P (return_spmf x) = P x"
simps: fixes x :: "'b"   and xl :: "'b bintree"   and xr :: "'b bintree" shows "t_val (Branch x xl xr) = x"
that: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b" shows "thesis"
efficient_power_code: fixes m :: "nat"   and y :: "nat"   and x :: "nat"   and n :: "nat" shows "mod_exp_nat.efficient_power m y x n = (if n = 0 then y else if n = 1 then x * y mod m else if even n then mod_exp_nat.efficient_power m y (x * x mod m) (n div 2) else mod_exp_nat.efficient_power m (x * y mod m) (x * x mod m) (n div 2))"
catch_option_def: fixes return :: "'e option \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('e option \<Rightarrow> 'f) \<Rightarrow> 'f"   and m :: "('e, 'f) optionT"   and h :: "('e, 'f) optionT" shows "catch_option return bind m h = OptionT (bind (run_option m) (\<lambda>x. if x = None then run_option h else return x))"