hcju/isabelleps huggingface dataset:
syc_cong_ax_holds: fixes e :: "(name, nat) finfun" shows "eval_fmla e syc_cong_ax"
transp_less: shows "transp (\<prec>\<^sub>t)"
vec_to_iarray_row': fixes i :: "'c"   and A :: "(('a, 'b) vec, 'c) vec" shows "vec_to_iarray (row i A) = matrix_to_iarray A !! mod_type_class.to_nat i"
subst_upd_uvar_def: fixes \<sigma> :: "'e \<Rightarrow> 'f"   and x :: "'g \<Longrightarrow> 'f"   and v :: "('g, 'e) uexpr" shows "\<sigma>(x \<mapsto>\<^sub>s v) = (\<lambda>b. put\<^bsub>x\<^esub> (\<sigma> b) (\<lbrakk>v\<rbrakk>\<^sub>e b))"
time_closest_pair_conv_closest_pair_recurrence: fixes ps :: "(int \<times> int) list" assumes "distinct ps" shows "real (time (closest_pair_tm ps)) \<le> closest_pair_time (length ps)"
verit_comp_simplify1_2: fixes a :: "'i" shows "a \<le> a"
commute: fixes a :: "'a"   and b :: "'a" shows "a OR b = b OR a"
assms_1: shows "a \<in> r\<^sup>* `` A"
BinOp_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and bop :: "bop"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e \<guillemotleft>bop\<guillemotright> e') h s s'"
univalent_composes: fixes R :: "('e \<times> 'f) set"   and S :: "('f \<times> 'g) set" assumes "univalent R"   and "univalent S" shows "univalent (R O S)"
simps_2: fixes m :: "nat \<Rightarrow> nat"   and r :: "'b memref"   and rs :: "'b memref list" shows "volatile_reads_consistent m (r # rs) = (case r of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> volatile_reads_consistent (m(a := v)) rs | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> (volatile \<longrightarrow> v = m a) \<and> volatile_reads_consistent m rs | _ \<Rightarrow> volatile_reads_consistent m rs)"
simps_1: fixes G :: "'b graph" shows "generate_valid_topology [] G = G"
ivls_of_set_FORWEAK: fixes X :: "'c set"   and z :: "'c set nres" assumes "SPEC ((\<subseteq>) X) \<le> z" shows "ivls_of_sets X \<le> z"
sc_r_distr: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and a :: "'i"   and m :: "'h"   and n :: "'h" assumes "R module M"   and "a \<in> carrier R"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> (m \<plusminus>\<^bsub>M\<^esub> n) = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> n"
seq_app_second: fixes s :: "'b"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(s ## \<sigma>) 1 = \<sigma> 0"
round_robin_\<alpha>_def: fixes queue_\<alpha> :: "'a \<Rightarrow> 'b list" shows "round_robin_base.round_robin_\<alpha> queue_\<alpha> = apfst queue_\<alpha>"
cases: fixes x :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'num, 'bool) instr" obtains n :: "'num" where "xa__ = IPushUbx1 n"   | b :: "'bool" where "xa__ = IPushUbx2 b"   | uu_ :: "Unboxed.type" and x :: "'d" where "xa__ = ILoadUbx uu_ xb__"   | uv_ :: "Unboxed.type" and x :: "'d" where "xa__ = IStoreUbx uv_ x"   | opubx :: "'opubx" where "xa__ = IOpUbx opubx"   | v :: "'dyn" where "xa__ = IPush v__"   | "xa__ = IPop"   | v :: "'d" where "xa__ = ILoad va__"   | v :: "'d" where "xa__ = IStore vb__"   | v :: "'f" where "xa__ = IOp vc__"   | v :: "'opinl" where "xa__ = IOpInl vd__"   | v :: "nat" where "xa__ = ICJump ve__"   | v :: "'e" where "xa__ = ICall v"
vectorize_minus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "vectorize_poly (p - q) = vectorize_poly p - vectorize_poly q"
dom_nodes: fixes fP :: "('c \<Rightarrow> nat option) \<times> 'c set"   and A :: "('d, 'c) nba" assumes "fP \<in> nodes (complement A)" shows "dom (fst fP) \<subseteq> nodes A"
YFtor_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme" shows "YFtor C \<equiv> MakeFtor (YFtor' C)"
lNactLe: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "dynamic_component.lNAct active c t n \<le> n"
list_induct2: fixes xs :: "'f list"   and ys :: "'g list"   and P :: "'f list \<Rightarrow> 'g list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
ode_na_def: fixes ode_ops :: "ode_ops" shows "ode_operations.ode_na ode_ops = (\<lambda>a. ode_operations.ode ode_ops (snd a))"
assms: shows "R - 1\<^sub>\<pi> \<noteq> {}"
homotopic_with_subset_left: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and Z :: "'a set" assumes "homotopic_with_canon P X Y f g"   and "Z \<subseteq> X" shows "homotopic_with_canon P Z Y f g"
i4: fixes x :: "('a, 'rows) vec" assumes "inner_rows x x = (0::'a)" shows "x = 0"
inter_sorted_correct: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "Sorted_Less.sorted (rev l1)"   and "Sorted_Less.sorted (rev l2)" shows "Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \<and> set (inter_sorted_rev l1 l2) = set l1 \<inter> set l2"
4: shows "(x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * (z * Rf y (x\<^sup>\<star>))\<^sup>\<star> \<le> z\<^sup>\<star> * (x\<^sup>\<star> \<squnion> y * x\<^sup>\<star>) * z\<^sup>\<star>"
pt_open_on_def: fixes Q :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and A :: "('b \<Rightarrow> 'c) set" shows "pt_open_on Q A = (\<forall>f\<in>A. Q f = (\<exists>n. \<forall>g\<in>A. (\<forall>i<n. g i = f i) \<longrightarrow> Q g))"
hyps: assumes "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1))"   and "\<And>vv s val. \<lbrakk>v = In2 vv; normal s1\<rbrakk> \<Longrightarrow> dom (locals (snd s)) \<subseteq> dom (locals (snd (snd vv val s)))" shows "P"
has_ell2_norm_L2_set: fixes x :: "'c \<Rightarrow> complex" shows "has_ell2_norm x = bdd_above (L2_set (cmod \<circ> x) ` Collect finite)"
norm_triangle_ineq3: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a - norm b\<bar> \<le> norm (minus a b)"
coF_dtor_o_unfold: fixes s :: "'e \<Rightarrow> ('f, 'e) coF0" shows "dtor_coF \<circ> dtor_unfold_coF s = map_pre_coF id (dtor_unfold_coF s) \<circ> s"
project_coord_on_plane: fixes x :: "'a"   and n :: "'a"   and c :: "real" assumes "x \<in> plane_of (Sctn n c)" shows "project_coord x n c = x"
length_stake: fixes n :: "nat"   and s :: "'a stream" shows "length (stake n s) = n"
Ch_lad: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "ContractsWithLAD Xd Xh Pd Ch" shows "\<forall>h. lad (Ch h)"
le_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "le r = list_all2 (\<lambda>x. lesub x r)"
eq_below_trans: fixes a :: "'c"   and b :: "'c"   and below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c" assumes "a = b"   and "below b c" shows "below a c"
T4: fixes w :: "i"   and \<psi> :: "\<o>" shows "[\<^bold>\<not>\<psi> in w] = (\<not> [\<psi> in w])"
rhs_def: shows "rhs \<equiv> \<Prod>i<n. 2 ^ n - 2 ^ i"
lran_empty_1: fixes a :: "int \<Rightarrow> 'b"   and l :: "int" shows "lran a l l = []"
True_step_conc: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and a :: "'b option" shows "((True # p, q) \<in> step (RegExp2NAe.conc L R) a) = ((\<exists>r. q = True # r \<and> (p, r) \<in> step L a) \<or> fin L p \<and> a = None \<and> q = False # start R)"
EX_lem0r: fixes x :: "'a"   and g :: "'a set"   and f :: "'a set" assumes "x \<in> EX' g" shows "x \<in> EX' (f \<union> g)"
B: fixes x :: "'a"   and y :: "'a" assumes "x \<in> N <#> H"   and "y \<in> N <#> H" shows "x \<otimes> y \<in> N <#> H"
disc_eq_case_2: fixes sum :: "'a + 'b" shows "(\<not> isl sum) = (case sum of Inl uu_ \<Rightarrow> False | Inr uu_ \<Rightarrow> True)"
arr_def: fixes f :: "'c" shows "C.arr f \<equiv> C.domains f \<noteq> {} \<or> C.codomains f \<noteq> {}"
e: shows "abc_step_l (sa, lma) (abc_fetch sa (A @ B)) = abc_step_l (sa, lma) (abc_fetch sa A)"
add_less_cancel_left: fixes plus :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and minus :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and c :: "'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "class.ordered_ab_semigroup_add_imp_le (plus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (minus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((plus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (c::'b::type) (a::'b::type)) (plus c (b::'b::type)) = less a b"
finite_measure_finite_Union: fixes S :: "'a set"   and A :: "'a \<Rightarrow> ('s \<times> 's option) stream set"   and s :: "'s \<times> 's option" assumes "finite S"   and "A ` S \<subseteq> KN.T.events s"   and "disjoint_family_on A S" shows "KN.T.prob s (\<Union> (A ` S)) = (\<Sum>i\<in>S. KN.T.prob s (A i))"
continuous_on_closure: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on (closure S) f = (\<forall>x e. x \<in> closure S \<and> 0 < e \<longrightarrow> (\<exists>d>0. \<forall>y. y \<in> S \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e))"
cl: fixes s :: "'b"   and t :: "'b"   and a :: "'a"   and u :: "'b"   and b :: "'a" assumes "(s, t) \<in> L a"   and "(s, u) \<in> E b" shows "(t, u) \<in> conversion' L L E (under r a) O E b O conversion' L L E (under r a) O ((L a)\<inverse>)\<^sup>= O conversion' L L E (under r a \<inter> under r b) \<or> (t, u) \<in> conversion' L L E (under r a \<union> under r b) O ((L a)\<inverse>)\<^sup>= O conversion' L L E (under r b)"
simps_5: fixes f1 :: "'l"   and f2 :: "'m \<Rightarrow> 'l"   and x2 :: "'m" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
smcf_eqI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<GG> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<FF> : \<CC> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<DD>"   and "\<GG>\<lparr>ObjMap\<rparr> = \<FF>\<lparr>ObjMap\<rparr>"   and "\<GG>\<lparr>ArrMap\<rparr> = \<FF>\<lparr>ArrMap\<rparr>"   and "\<AA> = \<CC>"   and "\<BB> = \<DD>" shows "\<GG> = \<FF>"
S0'_props_3: shows "S0 = \<alpha> ` S0'"
simps: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and uu :: "Core.term set" shows "sig (\<Sigma>, uu) = \<Sigma>"
simps_2: fixes a :: "String.literal"   and Ts :: "typ list"   and b :: "String.literal"   and Us :: "typ list"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option" shows "raw_match' (Ty a Ts) (Ty b Us) subs = (if a = b \<and> length Ts = length Us then fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip Ts Us) (Some subs) else None)"
butlast_append: fixes xs :: "'c list"   and ys :: "'c list" shows "butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
disjoint: shows "\<not> O y v \<and> \<not> O w x"
has_typ1_fun_typ: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and R :: "typ"   and U :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> t $ u : R"   and "Ts \<turnstile>\<^sub>\<tau> t : U \<rightarrow> R" shows "Ts \<turnstile>\<^sub>\<tau> u : U"
truncate_down_eq_truncate_up: fixes p :: "nat"   and x :: "real" shows "truncate_down p x = - truncate_up p (- x)"
finite_UN_D: fixes S :: "'c set set" assumes "finite (\<Union> S)" shows "\<forall>A\<in>S. finite A"
max_lp_in_sat_primal: fixes x :: "'a vec"   and c :: "'a vec"   and A :: "'a mat"   and b :: "'a vec" assumes "x \<in> max_lp A b c" shows "x \<in> sat_primal A b"
S1: shows "c_assoc_have_key (pr_gr (loc_upb (c_fst (c_snd n)) x)) (c_pair (c_fst (c_snd n)) x) = 0"
minusinf_inf: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list" assumes "islin p" shows "\<exists>z. \<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_apply (x - xa) = (\<lambda>xb. cblinfun_apply x xb - cblinfun_apply xa xb)"
order_trans: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b"   and z :: "'a \<times> 'b" assumes "less_eq_prod x y"   and "less_eq_prod y z" shows "less_eq_prod x z"
smult: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. complex_of_real c * f n $$ (i, j)) \<longlonglongrightarrow> matrix_seq.lowner_lub (\<lambda>n. complex_of_real c \<cdot>\<^sub>m f n) $$ (i, j)"
linear_span_def: fixes R :: "('i, 'j) Ring_scheme"   and M :: "('h, 'i, 'k) Module_scheme"   and A :: "'i set"   and H :: "'h set" shows "linear_span R M A H = (if H = {} then {\<zero>\<^bsub>M\<^esub>} else {x. \<exists>n. \<exists>f\<in>{j. j \<le> n} \<rightarrow> H. \<exists>s\<in>{j. j \<le> n} \<rightarrow> A. x = l_comb R M n s f})"
decr_grading_term_def: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat"   and v :: "'c" shows "Quasi_PM_Power_Products.punit.decr_grading_term d n v = fst (decr_grading d n (MPoly_Type_Class.punit.pp_of_term v), MPoly_Type_Class.punit.component_of_term v)"
ri: shows "Ri (Sup_llist Ns - Rf (Sup_llist Ns)) \<subseteq> Ri (Liminf_llist Ns)"
elims: fixes key_gen :: "('b \<times> 'c, 'd, 'e) gpv"   and encrypt :: "'b \<Rightarrow> 'f \<Rightarrow> ('g, 'd, 'e) gpv"   and valid_plains :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "('h \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'h) spmf) \<times> 'h"   and xa :: "('b \<Rightarrow> (('f \<times> 'f) \<times> 'i, 'd, 'e) gpv) \<times> ('g \<Rightarrow> 'i \<Rightarrow> (bool, 'd, 'e) gpv)"   and y :: "real" assumes "advantage key_gen encrypt valid_plains x xa = y" obtains "oracle" :: "'h \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'h) spmf" and \<sigma> :: "'h" and \<A> :: "('b \<Rightarrow> (('f \<times> 'f) \<times> 'i, 'd, 'e) gpv) \<times> ('g \<Rightarrow> 'i \<Rightarrow> (bool, 'd, 'e) gpv)" where "x = (oracle, \<sigma>)" and "xa = \<A>" and "y = \<bar>spmf (run_gpv oracle (game key_gen encrypt valid_plains \<A>) \<sigma>) True - 1 / 2\<bar>"
sem: shows "sterm_sem I \<theta>1 = sterm_sem J \<theta>2"
cases: fixes x :: "('a, 'b) ta_rule" obtains q :: "'a" and l :: "'b" and qs :: "'a list" where "x = q \<rightarrow> l qs"
abs_add_one_gt_zero: 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 x :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (plus one (abs x))"
aborting_dom_independent: fixes s :: "rstate_t"   and d :: "thread_id_t"   and a :: "int_point_t"   and a' :: "int_point_t" assumes "rcurrent s \<noteq> d" shows "raborting (rstep s a) d a' = raborting s d a'"
of_nat_inverse: fixes r :: "nat"   and a :: "'b word" assumes "word_of_nat r = a"   and "r < 2 ^ LENGTH('b)" shows "unat a = r"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. fst (f, w) \<star> snd (f, w)) local.map (\<lambda>f. \<rr> f \<cdot> (local.dom f \<star> \<phi>))"
iso_iff_section_and_epi: fixes f :: "'a" shows "local.iso f = (section f \<and> epi f)"
big_unclocked_notimeout_3: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r3 :: "(v, v) result" assumes "evaluate False env s e (s', r3)" shows "r3 \<noteq> Rerr (Rabort Rtimeout_error)"
simps_18: fixes n :: "float"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and slpl :: "nat" shows "slp_of_fa_rev (Num n) M slp slpl = slp_of_fa_rev_cnst (Num n) (Num n) M slp slpl"
step_pres_bounded: shows "\<forall>(q, \<tau>)\<in>set (step (hd w) (ss ! hd w)). q < length ss \<and> \<tau> \<in> A"
assms_1: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
B: shows "offs_set_prev ns (x # xs) index key mi ma i \<noteq> {}"
assms: shows "finite (reach (set \<Sigma>) \<delta> q\<^sub>0)"
a: shows "R[x\<turnstile>n>y] = AndL1 v.N u"
fs'_def: shows "fs' = fs[k := fs ! (k - 1), k - 1 := fs ! k]"
welltyped_commute: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and empty_heap :: "'c"   and hconf :: "'c \<Rightarrow> bool"   and t :: "'b"   and \<sigma> :: "'a option \<times> 'c \<times> 'a frame list"   and spurious_wakeups :: "bool"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and \<sigma>' :: "'a option \<times> 'c \<times> 'a frame list" assumes "JVM_heap_conf_base' addr2thread_id thread_id2addr allocate typeof_addr heap_write P"   and "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "JVM_heap_conf_base.correct_state thread_id2addr empty_heap allocate typeof_addr hconf P \<Phi> t \<sigma>" shows "JVM_heap_base.exec_1_d addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t (Normal \<sigma>) ta (Normal \<sigma>') = JVM_heap_base.exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t \<sigma> ta \<sigma>'"
WT_gpv_OutD: fixes \<Gamma> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
cr_prod1_simps: fixes c' :: "'f"   and A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h"   and c :: "'f" shows "cr_prod1 c' A a (b, c) = (A a b \<and> c' = c)"
assms_4: shows "g -` {g z} \<in> sets (F n)"
fds_nth_fds: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "fds_nth (fds f) n = (if n = 0 then 0::'a else f n)"
assms_3: fixes i1 :: "'b"   and i2 :: "'b" assumes "i1 \<in> I"   and "i2 \<in> I"   and "p i1 \<noteq> 0"   and "p i2 \<noteq> 0"   and "lt (p i1) = lt (p i2)" shows "i1 = i2"
cohomologous_approx_cond_exp_aux: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 < real_cond_exp M Invariants f x" shows "\<exists>u g. u \<in> borel_measurable M \<and> integrable M g \<and> (AE x in M. 0 \<le> g x \<and> g x \<le> max 0 (f x)) \<and> (\<forall>x. f x = g x + u x - u (T x))"
AE_cong: fixes M :: "'a measure"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x = Q x" shows "almost_everywhere M P = almost_everywhere M Q"
hS_sub: shows "h ` S \<subseteq> topspace (Euclidean_space n)"
visible_methods_exist: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option"   and M :: "char list"   and m :: "ty list \<times> ty \<times> 'b"   and D :: "char list" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>m\<rfloor>"
find_adds_SomeD_red: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "find_adds fs (lt p) = Some f" shows "red (set fs) p (tail p - monom_mult (lc p / lc f) (lp p - lp f) (tail f))"
rep_eq: shows "comp_fun_idem_apply insert_clause' = insert_clause"
fst3_snd3_trd3: fixes abc :: "'a \<times> 'b \<times> 'c" shows "abc = (fst3 abc, snd3 abc, trd3 abc)"
continuous_linepath_at: fixes x :: "real"   and a :: "'b"   and b :: "'b" shows "isCont (linepath a b) x"
check_poly_gt_def: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "('c monom \<times> 'b) list"   and q :: "('c monom \<times> 'b) list" shows "check_poly_gt gt p q \<equiv> let (a1, p1) = poly_split 1 p; (b1, q1) = poly_split 1 q in gt a1 b1 \<and> check_poly_ge p1 q1"
eval_wp_AC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" shows "wp (a \<Squnion> b) P = (\<lambda>s. max (wp a P s) (wp b P s))"
induct: fixes x1 :: "'var \<Rightarrow> 'val"   and x2 :: "nat"   and x3 :: "'var \<Rightarrow> 'val"   and x4 :: "nat"   and P :: "('var \<Rightarrow> 'val) \<Rightarrow> nat \<Rightarrow> ('var \<Rightarrow> 'val) \<Rightarrow> nat \<Rightarrow> bool" assumes "((x1, x2), x3, x4) \<in> cop"   and "\<And>\<sigma> n \<sigma>' n'. \<lbrakk>((\<sigma>, n), \<sigma>', n') \<in> cp; path \<sigma> n \<in> dom local.att\<rbrakk> \<Longrightarrow> P \<sigma> n \<sigma>' n'"   and "\<And>\<sigma> k \<sigma>' k' n. \<lbrakk>((\<sigma>, k), \<sigma>', k') \<in> cp; n cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> k; path \<sigma> (Suc k) \<noteq> path \<sigma>' (Suc k'); path \<sigma> n \<in> dom local.att\<rbrakk> \<Longrightarrow> P \<sigma> n \<sigma>' k'" shows "P x1 x2 x3 x4"
frag_extend_def: fixes b :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and x :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_extend b x \<equiv> \<Sum>i\<in>keys x. frag_cmul (lookup x i) (b i)"
assms_1: shows "fst cf \<approx>01 fst df"
3: shows "\<Union> {if u \<in> FvarsT \<chi> then FvarsT t else {} |t u. (t, u) \<in> set (zip (map fst txs) us)} = \<Union> {if x \<in> FvarsT r then FvarsT t else {} |t x. (t, x) \<in> set txs}"