hcju/isabelleps huggingface dataset:
mismatchId: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "[a\<noteq>b]P \<sim>\<^sub>e P"
assms: shows "f \<in>\<^sub>\<circ> vproduct (set {c}) A"
split_by_degree_correct_3: fixes l :: "float poly"   and r :: "float poly"   and ord :: "nat"   and p :: "float poly" assumes "(l, r) = split_by_degree ord p" shows "num_params l \<le> num_params p"
lossless_init: shows "\<forall>h w. lossless_spmf (init h w)"
rel_compp_Grp: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "fmrel R = (BNF_Def.Grp {x. fmran' x \<subseteq> {(x, y). R x y}} (fmmap fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. fmran' x \<subseteq> {(x, y). R x y}} (fmmap snd)"
isoI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.inverse_arrows f g" shows "VoV.iso f"
rep_endofun_def: shows "rep_endofun \<equiv> quot_type.rep Rep_endofun"
twoCisTypeI_9: shows "(\<lambda>x. twoC (x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
bdd_node_valid_def: fixes bdd :: "bdd"   and n :: "nat" shows "bdd_node_valid bdd n \<equiv> n \<in> Domain (Rmi bdd)"
iOp_mHom_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'f" shows "iOp_mHom R M N f = (\<lambda>x\<in>carrier M. -\<^sub>a\<^bsub>N\<^esub> f x)"
a: shows " t \<longrightarrow>\<^sub>1 s"
lookup_in: fixes x :: "name"   and e :: "name list"   and n :: "nat" assumes "x \<in> set e" shows "\<exists>k. lookup e n x = DBInd k \<and> n \<le> k \<and> k < n + length e"
Down_id_eq: fixes x :: "'c nres" shows "\<Down> Id x = x"
Un_Diff_cancel2: fixes B :: "'b set"   and A :: "'b set" shows "B - A \<union> A = B \<union> A"
enum_word_def: shows "enum_class.enum = map word_of_nat [0..<2 ^ LENGTH('b)]"
STCal_steps_2: fixes T :: "'b"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus"   and P' :: "('a, 'b) Proc" shows "TargetTerm T \<longmapsto>STCal Source Target* P' = (\<exists>T'. P' = TargetTerm T' \<and> T \<longmapsto>Target* T')"
gcd_neg2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd a (uminus b) = gcd a b"
Rep_status_tuple: fixes ST :: "('d, 'e, 'f) status" shows "Rep_status ST = (HA ST, Conf ST, Events ST, Value ST)"
is_nonterminal: shows "is_nonterminal (\<gamma> ! j)"
eta_cases_2: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s $ t \<rightarrow>\<^sub>\<eta> u" obtains t :: "Core.term" where "u = tb__ $ ta__" and "s \<rightarrow>\<^sub>\<eta> tb__"   | t :: "Core.term" where "u = s $ t" and "ta__ \<rightarrow>\<^sub>\<eta> t"
open_UN: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'a set" assumes "\<forall>x\<in>A. open (B x)" shows "open (\<Union> (B ` A))"
finite_enumerate_Suc'': fixes S :: "'a set"   and n :: "nat" assumes "finite S"   and "Suc n < card S" shows "enum S (Suc n) = (LEAST s. s \<in> S \<and> enum S n < s)"
esl_def: fixes P :: "'b prog" shows "SemiType.esl P = (types P, widen P, SemiType.sup P)"
vars_s't: shows "vars_mset (App s t) \<subseteq># vars_mset (App s' t)"
simps_2: fixes v :: "int" shows "contours_in_d (DI v) = {}"
map_gpv_bind_gpv: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and gpv :: "('f, 'e, 'c) gpv"   and h :: "'f \<Rightarrow> ('d, 'e, 'c) gpv" shows "map_gpv f g (gpv \<bind> h) = map_gpv id g gpv \<bind> (\<lambda>x. map_gpv f g (h x))"
assms_2: shows "(fst a \<subseteq>\<^sub>f s) = (fst a' \<subseteq>\<^sub>f s')"
\<sigma>_def: shows "\<sigma> = HMA_Connect.spectral_radius (HMA_Connect.map_matrix complex_of_real B)"
sch: fixes s :: "'s" assumes "s \<in> S" shows "sc s \<in> K s"
cases: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a1 :: "'b set"   and a2 :: "'c" assumes "fold_graph f z a1 a2" obtains "a1 = {}" and "a2 = z"   | x :: "'b" and A :: "'b set" and y :: "'c" where "a1 = insert x A" and "a2 = f x y" and "x \<notin> A" and "fold_graph f z A y"
oFix_plus: fixes a :: "ordinal" shows "oFix ((+) a) 0 = a * \<omega>"
map_comp: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and p :: "'f spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
\<I>_wf_trm: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>)"
eucl_of_list_inner_eq: fixes xs :: "real list"   and ys :: "real list" assumes "length xs = DIM('b)"   and "length ys = DIM('b)" shows "eucl_of_list xs \<bullet> eucl_of_list ys = inner_lv_rel xs ys"
leI: fixes y :: "'d"   and x :: "'d" assumes "\<not> y < x" shows "x \<le> y"
atLeastatMost_subset_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subseteq> ord.atLeastAtMost less_eq c d) = (\<not> less_eq a b \<or> less_eq c a \<and> less_eq b d)"
l_neg: fixes x :: "'c" assumes "x \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> x \<oplus>\<^bsub>S\<^esub> x = \<zero>\<^bsub>S\<^esub>"
lsorted_LCons_LCons: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c llist" shows "ord.lsorted less_eq (LCons x (LCons y xs)) = (less_eq x y \<and> ord.lsorted less_eq (LCons y xs))"
sub2: fixes B :: "'b"   and k :: "'b"   and b :: "'b"   and a :: "'b" assumes "B \<equiv> k + b" shows "a - B \<equiv> - k + (a - b)"
push_bit_of_Suc_0: fixes n :: "nat" shows "push_bit n (Suc 0) = 2 ^ n"
eigenvector_def: fixes A :: "'c mat"   and v :: "'c vec"   and k :: "'c" shows "eigenvector A v k = (v \<in> carrier_vec (dim_row A) \<and> v \<noteq> 0\<^sub>v (dim_row A) \<and> A *\<^sub>v v = k \<cdot>\<^sub>v v)"
simps_11: fixes b :: "nat"   and y :: "nat"   and \<omega> :: "agg_type \<times> event_data"   and b' :: "nat"   and f :: "trm"   and \<phi> :: "formula" shows "fvi b (Agg y \<omega> b' f \<phi>) = fvi (b + b') \<phi> \<union> fvi_trm (b + b') f \<union> (if b \<le> y then {y - b} else {})"
assms_2: fixes i :: "nat" assumes "i < n" shows "p A i m"
int_or_extra_simps_2: fixes w :: "int" shows "w OR - 1 = - 1"
path_component_trans: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "path_component S x y"   and "path_component S y z" shows "path_component S x z"
h_def: shows "h = (\<lambda>(x, y). if x \<in> X \<and> edge \<Gamma> x y then h' x y else 0)"
zip_set_def: fixes A :: "('a \<Rightarrow> 'b) set"   and B :: "('a \<Rightarrow> 'c) set" shows "zip_set A B = {u. fst \<circ> u \<in> A \<and> snd \<circ> u \<in> B}"
punit_min_term: shows "min_term_punit = (0::'a)"
norm_eq_sqrt_cinner: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "norm x = sqrt (cmod (cinner x x))"
ordinal_plus_left_cancel_le: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" shows "(w + x \<le> w + y) = (x \<le> y)"
C: shows "Key (Auth_ShaKey n) \<notin> spied s"
distinct_23: fixes x31 :: "assn"   and x32 :: "exp"   and x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp" shows "LetA x31 x32 \<noteq> (x61 ? x62 : x63)"
smooth_on_imp_differentiable_on: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k \<noteq> 0" shows "f differentiable_on S"
all_shared_acquired_in: fixes a :: "nat"   and A :: "nat set"   and sb :: "'b memref list" assumes "a \<in> A"   and "a \<notin> all_shared sb" shows "a \<in> acquired True sb A"
inner_image_cbox: assumes "\<i> \<in> Basis"   and "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" shows "(\<lambda>x. x \<bullet> \<i>) ` cbox a b = {a \<bullet> \<i>..b \<bullet> \<i>}"
m_lcomm: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "x * (y * z) = y * (x * z)"
bigomega_real_nat_transfer: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "f \<in> \<Omega>(g)" shows "(\<lambda>x. f (real x)) \<in> \<Omega>(\<lambda>x. g (real x))"
minors_def: fixes A :: "'b mat"   and k :: "nat" shows "minors A k = {Determinant.det (submatrix A I J) |I J. I \<subseteq> {0..<dim_row A} \<and> J \<subseteq> {0..<dim_col A} \<and> card I = k \<and> card J = k}"
inres_def: fixes S :: "'c nres"   and x :: "'c" shows "inres S x \<equiv> RETURN x \<le> S"
Nset_le: fixes x :: "'c"   and n :: "'c" assumes "x \<in> {i. i \<le> n}" shows "x \<le> n"
rep_eq: fixes x :: "'a"   and xa :: "'a['c, 'b]" shows "fromQ (x *\<^sub>Q xa) = \<lparr>mag = x \<cdot> mag (fromQ xa), dim = QD('c), unit_sys = unit\<rparr>"
canonical_retraction: fixes A :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A" shows "ChamberComplexRetraction X (canonical_retraction A C)"
ope: shows "openin (top_of_set (S \<union> T)) V"
charts_subset_atlas: fixes k :: "enat" shows "charts_eucl \<subseteq> manifold_eucl.atlas k"
inject_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) tllist"   and y21 :: "'a"   and y22 :: "('a, 'b) tllist" shows "(TCons x21 x22 = TCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
Coset_def: fixes t :: "('b, unit) RBT.rbt" shows "Coset t = - Set t"
Pi_split_domain: fixes x :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and J :: "'d set"   and X :: "'d \<Rightarrow> 'e set" shows "(x \<in> Pi (I \<union> J) X) = (x \<in> Pi I X \<and> x \<in> Pi J X)"
llist_all2_lappendI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist"   and xs' :: "'a llist"   and ys' :: "'b llist" assumes "llist_all2 P xs ys"   and "\<lbrakk>lfinite xs; lfinite ys\<rbrakk> \<Longrightarrow> llist_all2 P xs' ys'" shows "llist_all2 P (lappend xs xs') (lappend ys ys')"
collect_pair_finite'': fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite {(x, y). P x \<and> Q y}" shows "finite {(x, y). P x \<and> Q y \<and> R x y}"
assms: shows "x \<in> fv (t \<cdot> rm_vars X \<theta>) - X"
intros: fixes F1 :: "'fenv_std"   and F2 :: "'fenv_inca"   and H :: "'a"   and st :: "'b list" assumes "rel_fundefs (Fstd_get F1) (Finca_get F2)" shows "State F1 H st \<sim> State F2 H st"
bind_commute: fixes m :: "'m"   and m' :: "'m"   and f :: "int \<Rightarrow> int \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m' (f x)) = bind m' (\<lambda>y. bind m (\<lambda>x. f x y))"
Mprefix_refines_Mndet: fixes A :: "'d set"   and P :: "'d \<Rightarrow> 'd process" shows "mndet A P \<le> Mprefix A P"
top_sort_eff: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list"   and i :: "nat"   and j :: "nat" assumes "irreflp p\<^sup>+\<^sup>+"   and "sorted_wrt (to_ord p) x"   and "i < length x"   and "j < length x"   and "p\<^sup>+\<^sup>+ (x ! i) (x ! j)" shows "i < j"
univ_poly_is_cring: fixes K :: "'a set" assumes "subring K R" shows "cring (K [X])"
sorted_list_of_set_sort_remdups: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq (set xs) = linorder.sort_key less_eq (\<lambda>x. x) (remdups xs)"
tls1: shows "tls1 = tmap fst id stlsss1"
state_mem_mod_1011_low_equal_sub1: fixes s2 :: "'b sparc_state"   and s1 :: "'b sparc_state"   and va :: "word32"   and addr :: "word36"   and val :: "word8"   and pa :: "word36"   and b :: "word8" assumes "(\<forall>va. virt_to_phys va (mmu s2) (mem s1) = virt_to_phys va (mmu s2) (mem s2)) \<and> (\<forall>pa. (\<exists>va b. virt_to_phys va (mmu s2) (mem s2) = Some (pa, b) \<and> mmu_readable (get_acc_flag b) 10) \<longrightarrow> mem_equal s1 s2 pa) \<and> mmu s1 = mmu s2 \<and> virt_to_phys va (mmu s2) ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) = Some (pa, b) \<and> mmu_readable (get_acc_flag b) 10" shows "mem_equal s1 s2 pa"
strict_implies_order: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c" assumes "b \<prec>\<^sub>p a" shows "b \<preceq>\<^sub>p a"
U_function_ring_def: fixes R :: "('c, 'd) ring_scheme" shows "U_function_ring R \<equiv> ring R"
toQ_inverse: fixes y :: "('a, sdim, 'b) Measurement_System" assumes "y \<in> {x. Quantity.dim x = QD('c)}" shows "fromQ (toQ y) = y"
prefixes_Suc_length: fixes j :: "nat"   and t :: "nat" shows "e_length (local.prefixes j (Suc t)) = Suc (e_length (local.prefixes j t) + the (eval r_find_k [local.prefixes j t]))"
sum_val_process: fixes pf1 :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf2 :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market" assumes "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. val_process Mkt (qty_sum pf1 pf2) n w = val_process Mkt pf1 n w + val_process Mkt pf2 n w"
f1: fixes a :: "'a"   and aa :: "'a"   and ab :: "'a" shows "ad (ad (ad a \<cdot> ad aa) \<cdot> ad (ad a \<cdot> ad ab)) = ad a \<cdot> ad (ad (ad aa) \<cdot> ad (ad ab))"
leftderives1_implies_LeftDerives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftderives1 \<NN> \<TT> \<RR> u v" shows "\<exists>i r. CFG.LeftDerives1 \<NN> \<TT> \<RR> u i r v"
can_syncI: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x' :: "'x1"   and m' :: "'m1"   and LT :: "('l + 't + 't) set" assumes "r1.r_syntax t x m ta x' m'"   and "LT = collect_waits ta" shows "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>1"
null_setsD2: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> null_sets M" shows "A \<in> sets M"
simple_extesion_mem_imp_algebraic: fixes K :: "'a set"   and x :: "'a"   and y :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x"   and "y \<in> simple_extension K x" shows "(algebraic over K) y"
path_from_toE: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto> w" shows "path xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
interrupt_actions_ok_all_nthI: fixes ias :: "'b interrupt_action list"   and "is" :: "'b set" assumes "\<And>n. n < length ias \<Longrightarrow> interrupt_action_ok (redT_updIs is (take n ias)) (ias ! n)" shows "interrupt_actions_ok is ias"
upd_conv_take_nth_drop: fixes i :: "nat"   and xs :: "'g list"   and a :: "'g" assumes "i < length xs" shows "xs[i := a] = take i xs @ a # drop (Suc i) xs"
assms_1: shows "x \<in>\<^sub>\<circ> X"
next_p_frontier_change: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('a, 'c) configuration"   and c1 :: "('a, 'c) configuration"   and loc :: "'a"   and t :: "'c"   and loc' :: "'a" assumes "dataflow_topology summary results_in"   and "dataflow_topology.next_propagate' summary results_in c0 c1 loc t"   and "summary loc loc' \<noteq> {}\<^sub>A" shows "c_work c1 loc' = c_work c0 loc' + dataflow_topology.after_summary results_in (zmset_of (mset_set (set_antichain (frontier (c_imp c1 loc)))) - zmset_of (mset_set (set_antichain (frontier (c_imp c0 loc))))) (summary loc loc')"
intros_15: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> =n\<Rightarrow> Stuck"
inverse_inverse: fixes a :: "'b perm" shows "inverse (inverse a) = a"
abs_eq: fixes x :: "('d, 'e) RBT.rbt" shows "RBT_Mapping.Mapping x = mapping.Mapping (RBT.lookup x)"
contlub_cfun_fun: fixes F :: "nat \<Rightarrow> 'e \<rightarrow> 'f"   and x :: "'e" assumes "chain F" shows "(\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
rep_eq: fixes x :: "'c \<Rightarrow> 'c" shows "lgenerator (iterates_prod x) = iterates_raw x"
distinct_29: fixes x51 :: "'b formula"   and x52 :: "'b formula"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "x51 \<^bold>\<or> x52 \<noteq> x61 \<^bold>\<rightarrow> x62"
Basis_not_uminus: fixes i :: "'a" assumes "i \<in> Basis" shows "- i \<notin> Basis"
pdevs_apply_filter_pdevs: fixes I :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b pdevs"   and i :: "nat" shows "pdevs_apply (filter_pdevs I x) i = (if I i (pdevs_apply x i) then pdevs_apply x i else (0::'b))"
iwlsFSb_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "wlsFSb MOD" shows "iwlsFSb (fromMOD MOD)"
Cons_3: shows "vars (c' # F') \<subseteq> vars valuation"
fvi_subset: shows "fv \<phi> \<subseteq> fv \<psi>"
ring_simprules_14: fixes x :: "'a"   and y :: "'a" shows "x \<ominus> y = x \<oplus> \<ominus> y"
select_convs_2: fixes pset :: "'b set"   and order :: "('b \<times> 'b) set"   and more :: "'c" shows "order \<lparr>pset = pset, order = order, \<dots> = more\<rparr> = order"
j1: shows "j < card {j. j < dim_col (A * B) \<and> j \<in> J}"
mult_zero_right: fixes P :: "'a mpoly" shows "P * Const (0::'a) = Const (0::'a)"
mau_sqrt_main: fixes r :: "mini_alg_unique" assumes "mau_coeff r = 0" shows "sqrt (real_of_u r) = real_of_u (mau_sqrt r)"
as: shows "(c' # fctxt c' @ cs, s) \<rightarrow>\<^bsub>\<alpha>\<^esub> s'"
normalise_language_equivalent: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n id \<phi> = w \<Turnstile>\<^sub>n \<phi>"
surj_on: shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"
Sup_upper: fixes x :: "'e"   and A :: "'e set" assumes "x \<in> A" shows "x \<le> Sup A"
add_mset_eq_single: fixes b :: "'c"   and M :: "'c multiset"   and a :: "'c" shows "(add_mset b M = {#a#}) = (b = a \<and> M = {#})"
sort_eqv_def: fixes cs :: "('a \<times> 'a) set"   and s1 :: "'a set"   and s2 :: "'a set" shows "sort_eqv cs s1 s2 = (sort_leq cs s1 s2 \<and> sort_leq cs s2 s1)"
halt_lemma2''': fixes P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and Q :: "'a \<Rightarrow> bool"   and LE :: "('a \<times> 'a) set"   and na :: "nat" assumes "\<forall>n. \<not> P (f n) \<and> Q (f n) \<longrightarrow> Q (f (Suc n)) \<and> (f (Suc n), f n) \<in> LE"   and "Q (f 0)"   and "\<forall>i<na. \<not> P (f i)" shows "Q (f na)"
offs_enum_pred: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and n :: "nat"   and k :: "nat" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "offs_pred (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma"
times_eqpoll_cong: fixes A :: "'c set"   and C :: "'d set"   and B :: "'e set"   and D :: "'f set" assumes "A \<approx> C"   and "B \<approx> D" shows "A \<times> B \<approx> C \<times> D"
assms_4: shows "((\<lambda>_. 0) expands_to C) bs"
intros_2: fixes ys :: "'a list"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>y\<in>set ys. le x y"   and "linorder_list0.sortedP le ys" shows "linorder_list0.sortedP le (x # ys)"
restrict_map_insert: fixes f :: "'c \<Rightarrow> 'd option"   and a :: "'c"   and A :: "'c set" shows "f |` insert a A = (f |` A)(a := f a)"
finite_field_factorization_def: fixes f :: "'a mod_ring poly" shows "local.finite_field_factorization f = (if degree f = 0 then (lead_coeff f, []) else let a = lead_coeff f; u = Polynomial.smult (inverse a) f; gs = if use_distinct_degree_factorization then distinct_degree_factorization u else [(1, u)]; (irr, hs) = List.partition (\<lambda>(i, f). degree f = i) gs in (a, map snd irr @ concat (map (\<lambda>(i, g). berlekamp_monic_factorization i g) hs)))"
healthy_wp_SetPC: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>a s. a \<in> supp (p s) \<Longrightarrow> healthy (wp (f a))"   and "\<And>s. sound (p s)"   and "\<And>s. sum (p s) (supp (p s)) \<le> 1" shows "healthy (wp (SetPC f p))"
5: shows "\<forall>x\<in>set xs. h \<turnstile> P x \<rightarrow>\<^sub>r True \<longrightarrow> h' \<turnstile> P x \<rightarrow>\<^sub>r True"
Entry_no_call_source: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "sourcenode a = (_Entry_)" shows "False"
commute: fixes a :: "'b"   and b :: "'b" shows "min a b = min b a"
der_dif: shows "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
is_RB_inI2: fixes d :: "'a \<Rightarrow> nat"   and u :: "'t"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "is_syz_sig d u" shows "is_RB_in d rword G u"
owner_ii_unique: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner m, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s"   and "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s" shows "m = n"
witness_weights_def: shows "witness_weights = extract_weights shared_weights (witness_l rs)"
uniform_limit_le_const: fixes I :: "'a filter"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and X :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "I \<noteq> bot"   and "uniform_limit X f g I"   and "\<forall>\<^sub>F i in I. \<forall>x\<in>X. f i x \<le> h x"   and "x \<in> X" shows "g x \<le> h x"
ground_subst_apply: fixes t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "ground t" shows "subst_atm_abbrev t \<sigma> = t"
lepoll_iff: fixes A :: "'a set"   and B :: "'b set" shows "(A \<lesssim> B) = (\<exists>g. A \<subseteq> g ` B)"
rightInvertible: fixes R' :: "('a sequent list \<times> 'a sequent) set"   and R :: "('a sequent list \<times> 'a sequent) set"   and \<Gamma> :: "'a form multiset"   and F :: "'a"   and Fs :: "'a form list"   and n :: "nat"   and \<Gamma>' :: "'a form multiset"   and E :: "'a form" assumes "R' \<subseteq> upRules \<and> R = Ax \<union> R'"   and "( \<Gamma> \<Rightarrow>* Compound F Fs, n) \<in> derivable R*"   and "\<forall>r'\<in>R. rightPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* E) \<in> set (fst r')"   and "E \<noteq> Em" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* E, m) \<in> derivable R*"
transI: fixes r :: "('c \<times> 'c) set" assumes "\<And>x y z. \<lbrakk>x \<^bsub>r\<^esub>\<preceq> y; y \<^bsub>r\<^esub>\<preceq> z\<rbrakk> \<Longrightarrow> x \<^bsub>r\<^esub>\<preceq> z" shows "trans r"
hyps: assumes "\<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0" shows "\<exists>y'>q. \<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"