hcju/isabelleps huggingface dataset:
RedFAss: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and v :: "val"   and l :: "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" assumes "h a = \<lfloor>(C, fs)\<rfloor>" shows "P \<turnstile> \<langle>addr a\<bullet>F{D} := Val v,(h, l)\<rangle> \<rightarrow> \<langle>unit,(h(a \<mapsto> (C, fs((F, D) \<mapsto> v))), l)\<rangle>"
rev_compower: fixes n :: "nat"   and A :: "'a list set" shows "rev ` (A ^^ n) = rev ` A ^^ n"
y_before_x': shows "\<not> x < cs ! i in s_TS init h \<sigma> (length (as @ [x] @ bs @ [x]) + i + 1)"
neg_le: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "neg\<cdot>(Ord.le compare\<cdot>x\<cdot>y) = Ord.lt compare\<cdot>y\<cdot>x"
map_def: fixes f :: "'a" shows "B.map f = (if B.arr f then f else B.null)"
card_Diff1_less: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) < card A"
INF_eq_bot_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(Inf (f ` A) = bot) = (\<forall>x>bot. \<exists>i\<in>A. f i < x)"
norm_minus_cancel: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (uminus x) = norm x"
inverse_add_Infinitesimal_approx2: fixes x :: "'a star"   and h :: "'a star" assumes "x \<in> HFinite - Infinitesimal"   and "h \<in> Infinitesimal" shows "inverse (h + x) \<approx> inverse x"
ennreal_max_0: fixes x :: "real" shows "ennreal (max 0 x) = ennreal x"
zipWithS_fix_ind: fixes x :: "'b"   and f :: "'b \<rightarrow> 'b"   and y :: "'c"   and g :: "'c \<rightarrow> 'c"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "x = fix\<cdot>f"   and "y = fix\<cdot>g"   and "adm (\<lambda>x. P (fst x) (snd x))"   and "adm (\<lambda>x. Q (fst x) (snd x))"   and "\<And>b. P \<bottom> b"   and "\<And>a b. \<lbrakk>P a b; Q a b\<rbrakk> \<Longrightarrow> P (f\<cdot>a) b"   and "\<And>a. Q a \<bottom>"   and "\<And>a b. \<lbrakk>P a b; Q a b\<rbrakk> \<Longrightarrow> Q a (g\<cdot>b)" shows "P x y \<and> Q x y"
tailrec_transform_aux3: fixes a :: "'d \<Rightarrow> 'd nres"   and s :: "'d"   and b :: "'d \<Rightarrow> 'c nres" shows "REC (\<lambda>D s. sup (a s \<bind> D) (RETURN s)) s \<bind> b \<le> REC (\<lambda>D s. sup (a s \<bind> D) (b s)) s"
square2_disj: fixes n :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat"   and y :: "nat" shows "(square2 n i j \<inter> square2 n x y = {}) = ((2 ^ n + i \<le> x \<or> 2 ^ n + x \<le> i) \<or> 2 ^ n + j \<le> y \<or> 2 ^ n + y \<le> j)"
sup_subdistl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> y \<squnion> x \<cdot> z \<le> x \<cdot> (y \<squnion> z)"
disc_2: fixes x21 :: "'c"   and x22 :: "'c llist" shows "\<not> lnull (LCons x21 x22)"
assms_7: shows "\<guillemotleft>Map F : EVAL (Dom F) \<rightarrow>\<^sub>B EVAL (Cod F)\<guillemotright>"
rel_option_map2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and f :: "'e \<Rightarrow> 'd"   and y :: "'e option" shows "rel_option R x (map_option f y) = rel_option (\<lambda>x y. R x (f y)) x y"
disc2_noProt: shows "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated is_Src_Ports False m"
rev_cases: fixes xs :: "'c list" obtains "xs = []"   | ys :: "'c list" and y :: "'c" where "xs = ys @ [y]"
huets_us_simps_2: fixes y :: "nat list"   and i :: "nat" shows "\<^bold>u y (Suc i) = inc y 0 (\<^bold>u y i)"
is_Guard_basic_simps_5: fixes b :: "'d set"   and c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" shows "is_Guard (Cond b c1 c2) = False"
rep_eq: shows "Rep_stone_phi_pair bot = stone_phi.pairs_bot"
extend_fun: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and B1 :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "B \<subseteq> B1" shows "f \<in> A \<rightarrow> B1"
ord_of_nat_le_omega: fixes n :: "nat" shows "n\<^sub>\<nat> \<subseteq>\<^sub>\<circ> \<omega>"
ce_rel_to_set_def: shows "ce_rel_to_set = (\<lambda>R. {c_pair x y |x y. (x, y) \<in> R})"
truncate_down_uminus_eq: fixes prec :: "nat"   and x :: "real" shows "truncate_down prec (- x) = - truncate_up prec x"
uminus_uminus_mat: fixes A :: "'a Matrix.mat" shows "- (- A) = A"
F_def: shows "F = (\<lambda>x. (A\<^sup>\<dagger>) x + (B\<^sup>\<dagger>) x)"
call_indirect_Some: fixes s :: "s"   and i :: "nat"   and c :: "i32"   and cl :: "cl"   and j :: "nat"   and tf :: "tf"   and vs :: "v list" assumes "stab s i (nat_of_int c) = Some cl"   and "stypes s i j = tf"   and "cl_type cl = tf" shows "\<lparr>s;vs;[$C ConstInt32 c, $Call_indirect j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Callcl cl]\<rparr>"
carrier_update_2: fixes S :: "'a set" shows "\<zero>\<^bsub>R\<lparr>carrier := S\<rparr>\<^esub> = \<zero>"
exec_step_ind_Getstatic_Init: fixes D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and D :: "char list"   and F :: "char list"   and C :: "char list"   and sh :: "sheap"   and ics :: "init_call_status"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "(D', b, t) = field P D F"   and "P \<turnstile> C has F,Static:t in D"   and "\<forall>sfs. sh D' \<noteq> \<lfloor>(sfs, Done)\<rfloor>"   and "\<forall>Cs. ics \<noteq> Called Cs" shows "exec_step_ind (StepI (Getstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Calling D' []) # frs, sh)"
Ints_vec_vec_set: fixes v :: "'b vec" shows "(v \<in> \<int>\<^sub>v) = (set\<^sub>v v \<subseteq> \<int>)"
assms: shows "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps"
tms_cong: fixes H :: "fmla set"   and t :: "trm"   and t' :: "trm"   and u :: "trm"   and u' :: "trm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> tms t u EQ tms t' u'"
subset_segment_1: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a--b} \<subseteq> {c--d}) = (a \<in> {c--d} \<and> b \<in> {c--d})"
satisfies_atom'_stronger: fixes v :: "nat \<Rightarrow> 'b"   and a :: "'b atom" assumes "v \<Turnstile>\<^sub>a\<^sub>e a" shows "v \<Turnstile>\<^sub>a a"
oid_is_typerepr: shows "\<tau> \<Turnstile> X->forAll\<^sub>S\<^sub>e\<^sub>t(a|not (StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x a))"
valid_iff_valid_S_transform: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and x :: "('d, 'b, 'c) formula" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid satisfies transition P x = nominal_ts.valid nominal_ts.S_satisfies (nominal_ts.S_transition satisfies transition) P (S_transform x)"
chfindom_monofun2cont: fixes f :: "'d \<Rightarrow> 'e" assumes "monofun f" shows "cont f"
assms_1: shows "compl_fam (\<lambda>g. generate G {g}) gs"
moebius_inverse': fixes a :: "'a"   and d :: "'a"   and b :: "'a"   and c :: "'a"   and z :: "'a" assumes "a * d \<noteq> b * c"   and "c * z - a \<noteq> (0::'a)" shows "moebius a b c d (moebius d (- b) (- c) a z) = z"
split_list: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys \<cdot> x # zs"
rep_eq: fixes x :: "'m fset"   and xa :: "'m fset" shows "(x |\<subseteq>| xa) = (fset x \<subseteq> fset xa)"
promela_run_in_language_iff: fixes \<phi> :: "expr ltlc"   and APs :: "expr iarray"   and \<phi>\<^sub>i :: "nat ltlc"   and \<xi> :: "nat \<Rightarrow> gState" assumes "ltl_convert \<phi> = (APs, \<phi>\<^sub>i)" shows "(promela_props \<circ> \<xi> \<in> language_ltlc \<phi>) = (promela_props_ltl APs \<circ> \<xi> \<in> language_ltlc \<phi>\<^sub>i)"
relcomp_unfold: fixes r :: "('g \<times> 'i) set"   and s :: "('i \<times> 'h) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
1: shows "make_vert_equiv n = blockmat (make_vert_block n)"
mult_im_sheaf_def: shows "mult_im_sheaf \<equiv> \<lambda>V. mult_str\<^sub>X (f \<^sup>\<inverse> X V)"
multiplicity_le_imp_dvd': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "x \<noteq> zero"   and "\<And>p. p \<in># factorial_semiring.prime_factorization zero times one normalize x \<Longrightarrow> factorial_semiring.multiplicity times one p x \<le> factorial_semiring.multiplicity times one p y" shows "dvd.dvd times x y"
trans: fixes b :: "'e list"   and a :: "'e list"   and c :: "'e list" assumes "prefix b a"   and "prefix c b" shows "prefix c a"
ideD_1: fixes a :: "'a" assumes "S.ide a" shows "S.arr a"
simps_2: fixes \<alpha> :: "'b \<Rightarrow> 'a"   and a :: "'a" shows "eval_tpoly \<alpha> (PNum a) = a"
\<gamma>_eq_sym: fixes a :: "real list"   and b :: "real list" shows "\<gamma>_eq (a, b) = \<gamma>_eq (b, a)"
simps_3: fixes X :: "'d list"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "simple\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t = (\<exists>\<I>. ineq_model \<I> X F)"
simps_4: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "nnf (And \<phi>\<^sub>1 \<phi>\<^sub>2) = And (nnf \<phi>\<^sub>1) (nnf \<phi>\<^sub>2)"
get_proc_get_return_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and a' :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "a' \<in> get_return_edges a" shows "get_proc (sourcenode a) = get_proc (targetnode a')"
initial_stateD: fixes sys :: "('e, 'f, 'g, 'h, 'i, 'j) pre_system_scheme"   and s :: "'g \<Rightarrow> ('e, 'f, 'h, 'i) com list \<times> 'f option \<times> 'i" assumes "initial_state sys s" shows "AT \<lparr>GST = s, HST = []\<rparr> = atC \<circ> PGMs sys \<and> INIT sys \<lparr>GST = s, HST = []\<rparr>\<down> \<and> (\<forall>p l. \<not> taken p l \<lparr>GST = s, HST = []\<rparr>)"
guard_def: fixes P :: "heap \<Rightarrow> bool"   and f :: "heap \<Rightarrow> 'a \<times> heap" shows "Heap_Monad.guard P f = Heap.Heap (\<lambda>h. if P h then Some (f h) else None)"
comp_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x * z \<le> y * z"
is_field: fixes p :: "nat" assumes "residues_prime p" shows "field (residue_ring (int p))"
has_integral_localized_vector_derivative: fixes f :: "'a \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and i :: "'a" shows "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} = ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
invertible_inverse_closed: fixes u :: "'a" assumes "f.source.additive.invertible u"   and "u \<in> A" shows "f.source.additive.inverse u \<in> A"
Cauchy_Schwarz_ineq2: fixes x :: "'b"   and y :: "'b" shows "cmod \<langle>x, y\<rangle> \<le> norm x * norm y"
dim_subspace_orthogonal_to_vectors: fixes A :: "'a set"   and B :: "'a set" assumes "subspace A"   and "subspace B"   and "A \<subseteq> B" shows "dim {y \<in> B. \<forall>x\<in>A. orthogonal x y} + dim A = dim B"
Graph5: fixes T :: "nat"   and E :: "(nat \<times> nat) list"   and M :: "node list"   and R :: "nat" assumes "T \<in> Reach E"   and "Roots \<subseteq> Blacks M"   and "\<forall>i<R. \<not> BtoW (E ! i, M)"   and "T < length M"   and "R < length E"   and "M ! fst (E ! R) = Black"   and "M ! snd (E ! R) = Black"   and "M ! T \<noteq> Black" shows "\<exists>r>R. r < length E \<and> BtoW (E[R := (fst (E ! R), T)] ! r, M)"
coeff_poly_of_vec: fixes v :: "'a vec"   and i :: "nat" shows "coeff (poly_of_vec v) i = (if i < dim_vec v then v $ (dim_vec v - Suc i) else (0::'a))"
commute: fixes a :: "'m"   and b :: "'m" shows "a * b = b * a"
pNode_4: shows "bdt (Node lt p rt) var = Some bdt1"
supr_under: fixes a :: "'a" assumes "a \<in> Field r" shows "a = r.supr (r.under a)"
idem: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "semilattice f" shows "f a a = a"
continuous_add: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x + g x)"
r_minus: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
semilattice_order_axioms: shows "semilattice_order (\<cdot>) (\<le>) (<)"
strict_not_outside_2: fixes x :: "'b"   and y :: "'b" assumes "x \<prec>[Pareto R] y" shows "y \<in> alts"
All_mapping_def: fixes m :: "('b, 'c) mapping"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "All_mapping m P = (\<forall>x. case Mapping.lookup m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
wf_since_aux_def: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and n :: "nat"   and R :: "'b list set"   and pos :: "bool"   and \<phi> :: "'b MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'b MFOTL.formula"   and aux :: "(nat \<times> 'b option list set) list"   and ne :: "nat" shows "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux ne = (sorted_wrt (\<lambda>x y. fst y < fst x) aux \<and> (\<forall>t X. (t, X) \<in> set aux \<longrightarrow> ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) (ne - 1) (Sincep pos \<phi> (point (\<tau> \<sigma> (ne - 1) - t)) \<psi>)) X) \<and> (\<forall>t. ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<longrightarrow> (\<exists>X. (t, X) \<in> set aux)))"
sop_closed1: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and z :: "'d"   and r :: "'a"   and x :: "'d" assumes "fgmodule_condition R f i s A z"   and "r \<in> carrier R"   and "x \<in> addition_set f (aug_pm_set z i (s_set R s A))" shows "r \<^sub>s\<cdot> x \<in> addition_set f (aug_pm_set z i (s_set R s A))"
1: shows "(w \<sqinter> - (top * e * w\<^sup>T\<^sup>\<star>)) * e\<^sup>T * top = bot"
that: assumes "r = fst (Ref.alloc x h)"   and "h' = snd (Ref.alloc x h)" shows "thesis"
lm086: fixes N :: "'d set"   and G :: "'e list"   and bids :: "'d \<times> 'e set \<Rightarrow> 'f" assumes "N \<noteq> {}"   and "finite N"   and "distinct G"   and "set G \<noteq> {}" shows "(\<lambda>x. winningAllocationRel N (set G) ((\<in>) x) bids) -` {True} \<inter> set (allAllocationsAlg N G) \<noteq> {}"
that: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "P (f i) (f j)" shows "thesis"
rep_eq: fixes x :: "'a monom"   and xa :: "'a monom" shows "Rep_monom (x * xa) = monom_mult_list (Rep_monom x) (Rep_monom xa)"
lF_simps_1: fixes f :: "'a" assumes "ide f" shows "S.arr (UP.lF f)"
single_is_union: fixes a :: "'d"   and M :: "'d multiset"   and N :: "'d multiset" shows "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
rAct: fixes \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)"   and "\<alpha> \<noteq> \<tau>" shows "\<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>', P', Q') \<in> Rel)"
step_elim_cases_7: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], css, Normal s) \<rightarrow> u" obtains nrms :: "('e, 'd, 'f) com list" and abrs :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" where "css = (nrms, abrs) # cssa" and "u = (nrms, cssa, Normal s)"
two_chain_integral_def: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and F :: "real \<times> real \<Rightarrow> real" shows "two_chain_integral twoChain F \<equiv> \<Sum>C\<in>twoChain. integral (cubeImage C) F"
iprev_image2: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and I :: "nat set" assumes "strict_mono f" shows "iprev (f n) (f ` I) = f (iprev n I)"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'d" assumes "D.ide x" shows "Adj.\<epsilon> x = \<psi> x (G x)"
scaleR: fixes f :: "'b \<Rightarrow> 'c"   and r :: "real"   and b :: "'b" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
prod_emb_UN: fixes M :: "'d set"   and L :: "'d \<Rightarrow> 'e measure"   and K :: "'d set"   and F :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) set"   and I :: "'f set" shows "prod_emb M L K (\<Union> (F ` I)) = (\<Union>i\<in>I. prod_emb M L K (F i))"
sum_emeasure: fixes F :: "'b \<Rightarrow> 'a set"   and I :: "'b set"   and M :: "'a measure" assumes "F ` I \<subseteq> sets M"   and "disjoint_family_on F I"   and "finite I" shows "(\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union> (F ` I))"
FixSyn_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature" shows "FixSyn TYPE('a) Delta \<equiv> (var_infinite undefined \<and> var_regular undefined \<and> varSortAsSort_inj Delta) \<and> arityOf_lt_var undefined Delta \<and> barityOf_lt_var undefined Delta \<and> sort_lt_var undefined undefined"
take_bit_nat_less_exp: fixes n :: "nat"   and m :: "nat" shows "take_bit n m < 2 ^ n"
inv_neg_one: shows "inv\<^bsub>K\<^esub> (\<ominus>\<^bsub>K\<^esub> \<one>\<^bsub>K\<^esub>) = \<ominus>\<^bsub>K\<^esub> \<one>\<^bsub>K\<^esub>"
s_def: shows "s \<equiv> (a + b + c) / 2"
is_zeroth_power: fixes x :: "'a" shows "is_nth_power 0 x = (x = (1::'a))"
restrict_compose_right: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and S :: "'a set" shows "restrict (g \<circ> restrict f S) S = restrict (g \<circ> f) S"
attract: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiattractive A less_eq"   and "less_eq x y"   and "less_eq y x"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
sum_1: assumes "vars p1 \<subseteq> {..<n}" shows "(\<lambda>f. insertion f p1) \<in> borel_measurable (lborel_f n)"
map_sq_matrix_smult: fixes c :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "('d, 'c) sq_matrix" shows "c *\<^sub>S map_sq_matrix f A = map_sq_matrix (\<lambda>x. c * f x) A"
abort_def: fixes uu :: "String.literal"   and f :: "unit \<Rightarrow> 'a" shows "Code.abort uu f = f ()"
IA: shows "Iagree I J {Inl x |x. x \<in> SIGT (Times t1 t2)}"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "cblinfun_apply (x o\<^sub>C\<^sub>L xa) = cblinfun_apply x \<circ> cblinfun_apply xa"
scale_minus_left: fixes a :: "real"   and x :: "'a" shows "- a *\<^sub>R x = - (a *\<^sub>R x)"
inf_flat_complete_lattice_def: fixes x :: "'a flat_complete_lattice"   and y :: "'a flat_complete_lattice" shows "inf x y = (case x of flat_complete_lattice.Value v1 \<Rightarrow> case y of flat_complete_lattice.Value v2 \<Rightarrow> if v1 = v2 then x else flat_complete_lattice.Bot | flat_complete_lattice.Bot \<Rightarrow> flat_complete_lattice.Bot | flat_complete_lattice.Top \<Rightarrow> x | flat_complete_lattice.Bot \<Rightarrow> flat_complete_lattice.Bot | flat_complete_lattice.Top \<Rightarrow> y)"
cc_restr_noop: fixes G :: "CoCalls"   and S :: "Vars.var set" assumes "ccField G \<subseteq> S" shows "G G|` S = G"
right_diff_distrib: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "x \<times> (y - z) = x \<times> y - x \<times> z"
Par2F: fixes Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>l \<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l \<alpha> \<prec> P \<parallel> Q'"
mset_right_cancel_elem: fixes a :: "'c"   and A :: "'c multiset"   and b :: "'c" assumes "a \<in># A + {#b#}"   and "a \<noteq> b" shows "a \<in># A"
rows_transpose: fixes A :: "'a mat" shows "rows A\<^sup>T = cols A"
sin_pi_minus: fixes x :: "real" shows "sin (pi - x) = sin x"
quorum_for_def: fixes Quorum :: "process set set"   and Q :: "process set"   and v :: "val"   and v_f :: "process \<Rightarrow> val option" assumes "quorum_process Quorum" shows "quorum_process.quorum_for Quorum Q v v_f = (Q \<in> Quorum \<and> (\<forall>p\<in>Q. v_f p = Some v))"
measurable_dens_ctxt_measure_eq: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val" shows "(\<rightarrow>\<^sub>M) (dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho>) = (\<rightarrow>\<^sub>M) (state_measure (V \<union> V') \<Gamma>)"
IrrI: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V"   and "\<And>U. RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V \<Longrightarrow> U = 0 \<or> U = V" shows "IrrFinGroupRepresentation G smult V"
IDirProd_def: fixes G :: "('c, 'd) monoid_scheme"   and Y :: "'c set"   and Z :: "'c set" shows "IDirProd G Y Z = generate G (Y \<union> Z)"
canonical_subs_def: fixes n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" shows "canonical_subs n I m = (\<forall>i j k. i \<le> n \<and> k \<le> n \<and> j \<in> I \<longrightarrow> m i k \<le> m i j + m j k)"