hcju/isabelleps huggingface dataset:
mult_div_mod_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times b (divide a b)) (modulo a b) = a"
not_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" shows "(not_PL p \<in> Cn A) = (false_PL \<in> Cn (A \<union> {p}))"
SET_CR1: fixes evs1 :: "event list"   and C :: "agent"   and k :: "nat"   and NC1 :: "nat"   and i :: "nat" assumes "evs1 \<in> set_cr"   and "C = Cardholder k"   and "Nonce NC1 \<notin> used evs1" shows "Says C (CA i) \<lbrace>Agent C, Nonce NC1\<rbrace> # evs1 \<in> set_cr"
limproc_Sync_F: fixes Y :: "nat \<Rightarrow> 'c process"   and A :: "'c set"   and S :: "'c process" assumes "chain Y" shows "F (lim_proc (range Y) \<lbrakk>A\<rbrakk> S) = F (lim_proc (range (\<lambda>i. Y i \<lbrakk>A\<rbrakk> S)))"
expr_typing_unique: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile> e : t"   and "\<Gamma> \<turnstile> e : t'" shows "t = t'"
simps: fixes x :: "pexpr"   and y :: "pexpr" shows "npemul x y = (case x of PExpr1 (PCnst c) \<Rightarrow> if c = 0 then x else if c = 1 then y else case y of PExpr1 (PCnst d) \<Rightarrow> PExpr1 (PCnst (c * d)) | PExpr1 _ \<Rightarrow> PExpr2 (PMul x y) | PExpr2 pexpr2 \<Rightarrow> PExpr2 (PMul x y) | PExpr1 _ \<Rightarrow> case y of PExpr1 (PCnst d) \<Rightarrow> if d = 0 then y else if d = 1 then x else PExpr2 (PMul x y) | PExpr1 _ \<Rightarrow> PExpr2 (PMul x y) | PExpr2 pexpr2 \<Rightarrow> PExpr2 (PMul x y) | PExpr2 (PMul pexpr1 pexpr2a) \<Rightarrow> case y of PExpr1 (PCnst d) \<Rightarrow> if d = 0 then y else if d = 1 then x else PExpr2 (PMul x y) | PExpr1 _ \<Rightarrow> PExpr2 (PMul x y) | PExpr2 pexpr2 \<Rightarrow> PExpr2 (PMul x y) | PExpr2 (PPow e1 n) \<Rightarrow> case y of PExpr1 (PCnst d) \<Rightarrow> if d = 0 then y else if d = 1 then x else PExpr2 (PMul x y) | PExpr1 _ \<Rightarrow> PExpr2 (PMul x y) | PExpr2 (PMul pexpr1 pexpr2a) \<Rightarrow> PExpr2 (PMul x y) | PExpr2 (PPow e2 m) \<Rightarrow> if n = m then npepow (npemul e1 e2) n else PExpr2 (PMul x y))"
g_dom: fixes i :: "nat" shows "dom (g i) = reach A w i"
mono_inf: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "order.mono less_eq f" shows "f (inf A B) \<le> f A \<sqinter> f B"
C_restr_eqD: fixes r :: "C"   and f :: "C \<rightarrow> 'a"   and g :: "C \<rightarrow> 'a"   and r' :: "C" assumes "f|\<^bsub>r\<^esub> = g|\<^bsub>r\<^esub>"   and "r' \<sqsubseteq> r" shows "f\<cdot>r' = g\<cdot>r'"
gext_refl: fixes s :: "st" shows "s\<le>|s"
simps_2: fixes i :: "'a"   and l :: "'a factor_tree"   and r :: "'a factor_tree" shows "factors_of_factor_tree (Factor_Node i l r) = factors_of_factor_tree l + factors_of_factor_tree r"
bex_imageD: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and P :: "'d \<Rightarrow> bool" assumes "\<exists>x\<in>f ` A. P x" shows "\<exists>x\<in>A. P (f x)"
order_iff: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
simps_1: fixes x1 :: "'x"   and x2 :: "'y"   and y1 :: "'x"   and y2 :: "'y" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
replace_update_1: fixes k :: "'b"   and m :: "('b, 'c) mapping"   and v :: "'c" assumes "k \<notin> keys m" shows "replace k v m = m"
simple_integral_mono_AE: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "simple_function M f"   and "simple_function M g"   and "AE x in M. f x \<le> g x" shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
simps_103: fixes x8 :: "int"   and x7 :: "rat" shows "Pub_AgrK x8 \<noteq> Pri_AgrK x7"
uniform_snd_imp_proper_signed_list: fixes xs :: "('a \<times> bool) list"   and b :: "bool" assumes "order.greater_eq {b} (snd ` set xs)" shows "proper_signed_list xs"
timpl_closure'_timpls_trancl_eq': fixes c :: "('c set \<times> 'c set) set" shows "timpl_closure' {(a, b). (a, b) \<in> c\<^sup>+ \<and> a \<noteq> b} = timpl_closure' c"
linmap_im_lincomb: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i"   and H :: "'e set"   and s :: "nat \<Rightarrow> 'f"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e" assumes "R module M"   and "ideal R A"   and "R module N"   and "f \<in> mHom R M N"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> n} \<rightarrow> H" shows "f (l_comb R M n s g) = l_comb R N n s (cmp f g)"
simps_6: fixes \<alpha> :: "('d, 'e, 'f) hp"   and \<phi> :: "('d, 'e, 'f) formula"   and \<sigma> :: "('d, 'e, 'f) subst" shows "Fsubst (\<langle> \<alpha> \<rangle> \<phi>) \<sigma> = (\<langle> Psubst \<alpha> \<sigma> \<rangle> Fsubst \<phi> \<sigma>)"
restrict_apply': fixes x :: "'g"   and A :: "'g set"   and f :: "'g \<Rightarrow> 'h" assumes "x \<in> A" shows "restrict f A x = f x"
permutation_of_list_Cons: fixes x :: "'b"   and y :: "'b"   and xs :: "('b \<times> 'b) list"   and x' :: "'b" shows "permutation_of_list ((x, y) # xs) x' = (if x = x' then y else permutation_of_list xs x')"
smap_ctr: fixes f :: "'d \<Rightarrow> 'c"   and s :: "'d stream"   and x :: "'c"   and s' :: "'c stream" shows "(smap f s = x ## s') = (f (shd s) = x \<and> smap f (stl s) = s')"
weakBisimWeakCoinductAux: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<X> Q \<and> (Q, P) \<in> X" shows "P \<approx> Q"
UNIV: fixes A :: "'b set set" shows "UNIV \<in> \<sigma>_algebra A"
disjE_realizer: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and x :: "'d + 'e"   and R :: "'f \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q"   and "\<And>p. P p \<Longrightarrow> R (f p)"   and "\<And>q. Q q \<Longrightarrow> R (g q)" shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)"
Id_on_Gr: fixes A :: "'b set" shows "s2r A = BNF_Def.Gr A id"
abs_residual_def: shows "abs_residual \<equiv> quot_type.abs alpha_residual Abs_residual"
ordinal_le_def: fixes x :: "ordinal"   and y :: "ordinal" shows "(x \<le> y) = (\<forall>a\<in>Rep_ordinal x. \<forall>b\<in>Rep_ordinal y. a \<le> b)"
app_def: fixes i :: "instr"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and t :: "(ty list \<times> ty err list) option" shows "app i P mxs T\<^sub>r pc mpc xt t = (case t of None \<Rightarrow> True | \<lfloor>\<tau>\<rfloor> \<Rightarrow> app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt t). pc' < mpc))"
rind_hom_well_def: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R"   and "a \<in> carrier A" shows "f a = (f\<degree>\<^bsub>A,R\<^esub>) (a \<uplus>\<^bsub>A\<^esub> ker\<^bsub>A,R\<^esub> f)"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "drop (Suc n) (x # xs) = drop n xs"
nat_term_compD4'': fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and a :: "'a"   and v :: "'a"   and b :: "'a" assumes "comparator cmp"   and "nat_term_comp cmp"   and "fst (rep_nat_term u) = fst (rep_nat_term a)"   and "fst (rep_nat_term v) = fst (rep_nat_term b)"   and "snd (rep_nat_term u) \<le> snd (rep_nat_term v)"   and "snd (rep_nat_term a) = snd (rep_nat_term b)"   and "cmp a b \<noteq> Gt" shows "cmp u v \<noteq> Gt"
lastAssertedLiteralHasHighestElementLevel: fixes literal :: "Literal"   and clause :: "Literal list"   and M :: "(Literal \<times> bool) list" assumes "isLastAssertedLiteral literal clause (elements M)"   and "uniq (elements M)" shows "\<forall>l'. l' el clause \<and> l' el elements M \<longrightarrow> elementLevel l' M \<le> elementLevel literal M"
derivation_cases: fixes X :: "'a set"   and x :: "'a" assumes "derivation X"   and "x \<in> X" obtains Y :: "'a set" and y :: "'a" where "Ya__ = {y \<in> X. y \<sqsubset> x}" and "extreme Ya__ (\<sqsubseteq>) y" and "x = f y"   | Y :: "'a set" where "Y = {y \<in> X. y \<sqsubset> x}" and "f ` Y \<subseteq> Y" and "extreme_bound A (\<sqsubseteq>) Y x"
id_aux4: fixes p :: "('b set set, 'c) Combinators list"   and a :: "('b set set, 'd) Combinators"   and s :: "('b set set, 'c) Combinators" assumes "noDenyAll p"   and "\<And>s. s \<in> set p \<Longrightarrow> disjSD_2 a s"   and "s \<in> set (insertDenies p)"   and "\<not> member DenyAll a" shows "disjSD_2 a s"
intros_4: fixes c :: "f64"   and fop :: "unop_f" shows "\<lparr>[$C ConstFloat64 c, $Unop_f T_f64 fop]\<rparr> \<leadsto> \<lparr>[$C ConstFloat64 (app_unop_f fop c)]\<rparr>"
simps_1: fixes n :: "nat"   and m :: "nat"   and x :: "'b" shows "swap n m (x)\<^sub>p = (x)\<^sub>p"
vtimes_vsubset_VPowVPow: fixes A :: "V"   and B :: "V" shows "A \<times>\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> VPow (VPow (A \<union>\<^sub>\<circ> B))"
simps_2: fixes M :: "'d"   and xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P = ROut M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' P)"
hlit_of_flit_bij: shows "bij_betw hlit_of_flit {l. ground\<^sub>l l} UNIV"
PO_m1_refines_init_m1a: shows "init m1 \<subseteq> R1a1 `` init m1a"
divideC_field_splits_simps_2_1: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "0 < c" shows "(a \<le> b /\<^sub>C c) = (if 0 < c then c *\<^sub>C a \<le> b else if c < 0 then b \<le> c *\<^sub>C a else a \<le> (0::'b))"
hyps: fixes r :: "'a rexp" shows "slexer r s = lexer r s"
IH: fixes ts :: "'b list"   and u :: "'a \<Rightarrow> 'b" shows "\<forall>c. c \<notin> set cs_ \<longrightarrow> set_clocks cs_ ts u c = u c"
vector_component_5: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec"   and i :: "'c" shows "(x - y) $ i = x $ i - y $ i"
assms_3: fixes n :: "nat" shows "\<exists>h\<in>M \<rightarrow>\<^sub>M N. \<forall>x\<in>A n. f x = h x"
S: fixes s :: "'a" assumes "rwof m0 cond step s"   and "I s"   and "cond s" shows "step s \<le> SPEC I"
of_nat_ucast: fixes n :: "nat" assumes "is_down UCAST('b \<rightarrow> 'c)" shows "word_of_nat n = UCAST('b \<rightarrow> 'c) (word_of_nat n)"
pbij_lens_def: fixes x :: "'d \<Longrightarrow> 'e" shows "pbij_lens x \<equiv> weak_lens x \<and> pbij_lens_axioms x"
quasi_inverses_def: fixes f :: "'d"   and g :: "'d" shows "D.quasi_inverses f g \<equiv> \<exists>\<phi> \<psi>. D.internal_equivalence f g \<phi> \<psi>"
le_matrix_def: fixes A :: "'b matrix"   and B :: "'b matrix" shows "(A \<le> B) = (\<forall>j i. Rep_matrix A j i \<le> Rep_matrix B j i)"
cobounded1: fixes a :: "'d"   and b :: "'d" shows "a \<sqinter> b \<le> a"
ccomp_empty: fixes c :: "com"   and s :: "char list \<Rightarrow> int" assumes "ccomp c = []" shows "(c, s) \<Rightarrow> s"
snoc_2: shows "xs \<cdot> [x] = LynFac w"
strong_get_put: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and \<rho> :: "'c" assumes "bij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>"
open_B: fixes X :: "'b set"   and i :: "'a" assumes "X \<in> B i" shows "open X"
1: shows "suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)"
length_stream_dec: fixes \<Sigma> :: "'b list"   and n :: "nat"   and FO :: "nat set"   and x :: "('b \<times> bool list) stream" assumes "formula \<Sigma>" shows "length (formula.stream_dec n FO x) = n"
fset_of_list_remdups: fixes l :: "'a list" shows "fset_of_list (remdups l) = fset_of_list l"
lossless2: fixes \<sigma> :: "'a \<times> bool list"   and cipher :: "'b" shows "lossless_gpv \<I>_full (\<A>2 \<sigma> cipher)"
add_decreasing2: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<le> bot"   and "a \<le> b" shows "a \<squnion> c \<le> b"
insert_index_exclude: fixes i :: "nat"   and i' :: "nat" shows "insert_index i i' \<noteq> i"
plus_matrix_def: fixes A :: "'a matrix"   and B :: "'a matrix" shows "A + B = combine_matrix (+) A B"
assms_7: shows "z \<in> circline_set (poincare_line u v) \<inter> circline_set x_axis"
meet_top: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and P :: "'a hrel" assumes "utp_theory_continuous hcond"   and "P is hcond" shows "P \<sqinter> \<top>\<^bsub>utp_order hcond\<^esub> = P"
ws_class_induct': fixes G :: "prog"   and C :: "qtname"   and P :: "qtname \<Rightarrow> bool" assumes "is_class G C"   and "ws_prog G"   and "\<And>co. class G Object = Some co \<Longrightarrow> P Object"   and "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object; P (super c)\<rbrakk> \<Longrightarrow> P C" shows "P C"
reduce_system_matrix_match_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "M_mat_R (get_signs_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) (get_subsets_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) = get_matrix_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))"
keys_single: fixes k :: "'d"   and v :: "'e" shows "keys (monomial v k) = (if v = (0::'e) then {} else {k})"
cases: fixes a1 :: "'c"   and a2 :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a1, a2) \<in> r\<^sup>+" obtains a :: "'c" and b :: "'c" where "a1 = a__" and "a2 = b__" and "(a__, b__) \<in> r"   | a :: "'c" and b :: "'c" and c :: "'c" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>+" and "(b, c) \<in> r"
simps_7: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "step ((Mn n f, xs, ls) # fs, rv) = (if ls = [] then ((f, 0 # xs, []) # (Mn n f, xs, [0]) # fs, None) else if rv \<down>= 0 then (fs, Some (hd ls)) else ((f, Suc (hd ls) # xs, []) # (Mn n f, xs, [Suc (hd ls)]) # fs, None))"
fine_Nil: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real" shows "fine \<delta> (a, a) []"
queue_invariant_def: fixes q :: "'a list \<times> 'a list" shows "queue_invariant q = (case q of ([], []) \<Rightarrow> True | ([], _) \<Rightarrow> True | (a # list, []) \<Rightarrow> False | (a # list, l # ls) \<Rightarrow> True)"
in_homE: fixes f :: "'c \<times> 'c \<times> 'c"   and a :: "'c \<times> 'c \<times> 'c"   and b :: "'c \<times> 'c \<times> 'c" assumes "C.VVV.in_hom f a b" obtains "C.VVV.arr f" and "C.VVV.dom f = a" and "C.VVV.cod f = b"
simps_2: fixes a :: "nat"   and b :: "nat"   and x :: "nat"   and xs :: "nat list" shows "arcs a b (x # xs) = (a, x) # arcs x b xs"
simps_3: fixes x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (ELam x e) \<rho> = {VFun f |f. \<forall>v1 v2. (v1, v2) \<in> fset f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)}"
AE_measure_pmf_iff: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a pmf" shows "almost_everywhere (measure_pmf M) P = (\<forall>y\<in>set_pmf M. P y)"
bres_galois_imp: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "less_eq_prod (times_prod' x y) z \<longrightarrow> less_eq_prod y (prod.bres x z)"
simps_1: fixes ao :: "'a option" shows "set_less_aux_Compl ao [] [] = proper_interval ao None"
l7_20: fixes A :: "'p"   and M :: "'p"   and B :: "'p" assumes "Col A M B"   and "Cong M A M B" shows "A = B \<or> M Midpoint A B"
kripke_def: shows "kripke = {(S, S0, T, L) |S S0 T L. Kripke S S0 T L}"
inject_2: fixes x21 :: "String.literal"   and x22 :: "'a OCL_Types.type option"   and x23 :: "'a expr"   and x24 :: "'a expr"   and y21 :: "String.literal"   and y22 :: "'a OCL_Types.type option"   and y23 :: "'a expr"   and y24 :: "'a expr" shows "(expr.Let x21 x22 x23 x24 = expr.Let y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
F_seq: fixes P :: "'a process"   and Q :: "'a process" shows "F (P `;` Q) = {(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"
assms_1: shows "(x, y) \<in> r\<^sup>*"
AA1: shows "Esigma 87633 = Esigma (3\<^sup>2 * 7 * 13 * 107)"
adopt_node_removes_child: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr"   and h2 :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h2"   and "h2 \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "node_ptr \<notin> set children"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "'b"   and a2 :: "('b \<times> color) tree" assumes "\<And>t1 a t2 b t3 c t4. P t1 a (R t2 b (R t3 c t4))"   and "\<And>t1 a t2 b t3 c. P t1 a (R (R t2 b t3) c \<langle>\<rangle>)"   and "\<And>t1 a t2 b t3 c v vc vb. P t1 a (R (R t2 b t3) c (B v vc vb))"   and "\<And>t1 a. P t1 a \<langle>\<rangle>"   and "\<And>t1 a v vc vb. P t1 a (B v vc vb)"   and "\<And>t1 a va. P t1 a \<langle>\<langle>\<rangle>, va, \<langle>\<rangle>\<rangle>"   and "\<And>t1 a vb ve vd va. P t1 a \<langle>B vb ve vd, va, \<langle>\<rangle>\<rangle>"   and "\<And>t1 a v vb. P t1 a (B v vb \<langle>\<rangle>)"   and "\<And>t1 a va vc vf ve. P t1 a \<langle>\<langle>\<rangle>, va, B vc vf ve\<rangle>"   and "\<And>t1 a vb vh vg va vc vf ve. P t1 a \<langle>B vb vh vg, va, B vc vf ve\<rangle>"   and "\<And>t1 a v vb vc vf ve. P t1 a (B v vb (B vc vf ve))" shows "P a0 a1 a2"
accepts_rexp2nae: fixes r :: "'b rexp"   and w :: "'b list" shows "NAe.accepts (rexp2nae r) w = (w \<in> lang r)"
simps: fixes a :: "letter list list" shows "good a = ((\<exists>w ws. a = w # ws \<and> L w ws) \<or> (\<exists>ws w. a = w # ws \<and> good ws))"
univ_commute': fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (r `` {x}) = f x"
thin0: fixes A :: "fm"   and H :: "fm set" assumes "{} \<turnstile> A" shows "H \<turnstile> A"
admissible\<^sub>1_def: fixes w :: "face \<Rightarrow> nat"   and g :: "graph" shows "admissible\<^sub>1 w g \<equiv> \<forall>f\<in>\<F> g. \<d> |vertices f| \<le> w f"
not_identical\<^sub>E_def: shows "(\<^bold>\<noteq>\<^sub>E) \<equiv> exe2 (\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P)\<^sup>-)"
convex_hull_caratheodory: fixes p :: "'b set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('b) + 1 \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
spanLN: shows "local.span (set Ls \<union> set Ns) = carrier_vec n"
next_plane_def: fixes p :: "nat"   and g :: "graph" shows "next_plane\<^bsub>p\<^esub> g \<equiv> let fs = nonFinals g in if fs = [] then [] else let f = minimalFace fs; v = minimalVertex g f in concat (map (\<lambda>i. generatePolygon i v f g) [3..<Suc (maxGon p)])"
degree_bound_def: fixes vs :: "'b poly list" shows "degree_bound vs = max_factor_degree (map degree vs)"
Sup_to_Inf: shows "Sup = uminus \<circ> Inf \<circ> \<P> uminus"
cases: fixes x :: "(real \<times> real) \<times> (real \<times> real) \<times> real \<times> real" obtains xp :: "real" and yp :: "real" and xq :: "real" and yq :: "real" and xr :: "real" and yr :: "real" where "x = ((xp, yp), (xq, yq), xr, yr)"
dom_res_addition_not_in: fixes m\<^sub>2 :: "('d, 'c) fmap"   and k :: "'d"   and v :: "'e"   and m\<^sub>1 :: "('d, 'e) fmap" assumes "m\<^sub>2 $$ k = None" shows "fmdom' (m\<^sub>1(k $$:= v)) \<lhd> m\<^sub>2 = fmdom' m\<^sub>1 \<lhd> m\<^sub>2"
simps_1: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t' :: "'f"   and x :: "'i"   and m :: "'j" shows "redT_updT' ts (NewThread t' x m) = ts(t' \<mapsto> (undefined, no_wait_locks))"
open_real_def: fixes U :: "real set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
distinct_9: fixes x2 :: "real mpoly"   and x4 :: "real mpoly" shows "Eq x2 \<noteq> Neq x4"
resolutionD3: fixes S :: "'a set multiset" assumes "S \<in># \<P>" shows "resolution_class (set_mset S)"
cindexE_def: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" shows "cindexE a b f = (\<Sum>x | jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < b. jumpF f (at_right x)) - (\<Sum>x | jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> b. jumpF f (at_left x))"
D_dest'': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "x = linordered_ab_monoid_add.D plus less_eq m i j k" shows "x \<in> {linordered_ab_monoid_add.len plus m i j xs |xs. set xs \<subseteq> {0..k}}"
prefix_of_psliceI: fixes \<pi> :: "'b prefix"   and \<sigma> :: "'b trace"   and S :: "'a list set" assumes "prefix_of \<pi> \<sigma>" shows "prefix_of (pslice S \<pi>) (slice S \<sigma>)"
gbm_delete_node_def: fixes v :: "'V"   and g :: "'m1" shows "gbm_delete_node v g \<equiv> Let (m1.delete v g) (m1_mvif (\<lambda>_ m2. Some (m2.delete v m2)))"
TW: shows "{0..1} \<times> T' \<subseteq> W"
fls_lr_inverse_X_intpow'_2: fixes i :: "int"   and y :: "'a" shows "fls_right_inverse (fls_X_intpow i) y = fls_const y * fls_X_intpow (- i)"
Complexs_of_int: fixes z :: "int" shows "of_int z \<in> \<complex>"
rep_eq: fixes x :: "('d, 'e) mapping"   and xa :: "('d, 'e) mapping" shows "mapping.rep (mapping_add x xa) = mapping.rep x ++ mapping.rep xa"
fresh_imp_notin_env: fixes name :: "'a"   and e :: "'a list" assumes "atom name \<sharp> e" shows "name \<notin> set e"
da_elim_cases_4: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (Expr e)\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"