hcju/isabelleps huggingface dataset:
walk_take: fixes xs :: "'a list"   and n :: "nat" assumes "walk xs" shows "walk (take n xs)"
closed: fixes g :: "'a"   and h :: "'a" assumes "g \<in> G"   and "h \<in> G" shows "binop g h \<in> G"
hcomp_conc_unit2: fixes P :: "conc_interface" shows "P \<otimes>\<^sub>c \<epsilon>\<^sub>c \<simeq> P"
transp_combine_per: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "transp R" shows "transp (combine_per P R)"
fp_cop_F_group_strategy_proof: fixes ds :: "'b set" shows "Y.group_strategy_proof ds (\<lambda>Pd Ch ds. Contracts.CH Ch (Contracts.fp_cop_F Pd Ch ds))"
agree_imp_submap: fixes f1 :: "('a, 'b) fmap"   and f2 :: "('a, 'b) fmap" assumes "f1 \<subseteq>\<^sub>f f2" shows "agree f1 f2"
pr_simps_6: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod \<pp>\<^sub>1[a, b] = a"
disj_def: fixes p :: "fm"   and q :: "fm" shows "MIR.disj p q \<equiv> if p = T \<or> q = T then T else if p = F then q else if q = F then p else if p = q then p else Or p q"
triv_neg_inf_h: assumes "sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} = []" shows "sga = sign_vec qs (real_of_int (- crb (prod_list (cast_rat_list qs))))"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "cycle (x # xs) = (x \<notin> set xs \<and> cycle xs)"
float_neg_add: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "is_finite b"   and "is_finite (a - b)" shows "valof a + valof (- b) = valof a - valof b"
integrate_vars_def: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "integrate_vars \<Gamma> = foldr (integrate_var \<Gamma>)"
has_derivative_def: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (netlimit F) - f' (y - netlimit F)) /\<^sub>R norm (y - netlimit F)) \<longlongrightarrow> (0::'d)) F)"
1: shows "[\<^bold>\<forall>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. Null (x\<^sup>P)) \<^bold>\<rightarrow> Null (y\<^sup>P) in v]"
l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs"
DERIV_chain_s: fixes s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and x :: "'a" assumes "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x)"   and "(f has_field_derivative f') (at x)"   and "f x \<in> s" shows "((\<lambda>x. g (f x)) has_field_derivative f' * g' (f x)) (at x)"
sup_nil: fixes x :: "'a" shows "x \<squnion> \<top> = \<top>"
max_flow_min_cut: shows "\<exists>f S. flow \<Delta> f \<and> MFMC_Network.cut \<Delta> S \<and> orthogonal \<Delta> f S"
nth_tl: fixes xs :: "'b list"   and n :: "nat" assumes "xs \<noteq> []" shows "tl xs ! n = xs ! Suc n"
sum_mono_gt: fixes K :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and i :: "'b" assumes "\<And>i. i \<in> K \<Longrightarrow> g i \<le> f i"   and "i \<in> K"   and "f i \<succ> g i"   and "finite K" shows "sum f K \<succ> sum g K"
is_methd_def: fixes G :: "prog" shows "is_methd G = (\<lambda>C sig. is_class G C \<and> methd G C sig \<noteq> None)"
inj_onTr0: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and x :: "'c"   and y :: "'c" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A"   and "y \<in> A"   and "inj_on f A"   and "f x = f y" shows "x = y"
RR_OK: fixes x :: "'S"   and y :: "'S" assumes "x \<in> S"   and "y \<in> S"   and "RR x y" shows "RR' (\<alpha> x) (\<alpha> y)"
two_pred_simps_3: shows "two_pred\<cdot>none = none"
succ_pred: fixes x :: "hf" assumes "Ord x"   and "x \<noteq> 0" shows "succ (pred x) = x"
subprob_space_measure_pmf: fixes x :: "'a pmf" shows "subprob_space (measure_pmf x)"
simps_1: fixes x :: "'e"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "Var x \<cdot> \<sigma> = \<sigma> x"
DecomposeR: fixes M :: "('fun, 'var) Term.term set"   and Q :: "('fun, 'var) Term.term \<Rightarrow> bool"   and t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and t\<^sub>i :: "('fun, 'var) Term.term" assumes "\<langle>M;Q\<rangle> \<turnstile>\<^sub>r t"   and "Ana t = (K, T)"   and "\<And>k. k \<in> set K \<Longrightarrow> \<langle>M;Q\<rangle> \<turnstile>\<^sub>r k"   and "t\<^sub>i \<in> set T" shows "\<langle>M;Q\<rangle> \<turnstile>\<^sub>r t\<^sub>i"
nat_lessSucSuc0_conv: fixes n :: "nat" shows "(n < Suc (Suc 0)) = (n = 0 \<or> n = Suc 0)"
inverse_arrowsE: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.inverse_arrows f g" obtains "VoV.ide (VoV.comp g f)" and "VoV.ide (VoV.comp f g)"
ide_char: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VVV.ide a = (VVV.arr a \<and> VxVxV.ide a)"
mono: fixes c :: "'c"   and a :: "'c"   and d :: "'c"   and b :: "'c" assumes "c \<le> a"   and "d \<le> b" shows "c + d \<le> a + b"
simps_2: fixes bs :: "'a list"   and accs :: "'a list" shows "merge_tailrec [] bs accs = rev accs @ bs"
zero_coeff_zero: fixes p :: "linear_poly" shows "(p = 0) = (\<forall>v. coeff p v = 0)"
remove_induct: fixes P :: "'d set \<Rightarrow> bool"   and B :: "'d set" assumes "P {}"   and "infinite B \<Longrightarrow> P B"   and "\<And>A. \<lbrakk>finite A; A \<noteq> {}; A \<subseteq> B; \<And>x. x \<in> A \<Longrightarrow> P (A - {x})\<rbrakk> \<Longrightarrow> P A" shows "P B"
starfunNat_inverse_real_of_nat_Infinitesimal: fixes N :: "nat star" assumes "N \<in> HNatInfinite" shows "(*f* (\<lambda>x. inverse (real x))) N \<in> Infinitesimal"
getFrN_Fvars: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and \<phi> :: "'fmla"   and n :: "nat" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s" shows "set (getFrN xs ts \<phi>s n) \<inter> Fvars \<phi> = {}"
frechet_derivative_sqrt: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "f differentiable at x"   and "f x \<noteq> 0" shows "frechet_derivative (\<lambda>x. sqrt (f x)) (at x) = (\<lambda>v. (if 0 < f x then 1 else - 1) / (2 * sqrt (f x)) * frechet_derivative f (at x) v)"
intros: fixes cfg :: "'s cfg"   and s :: "'s" assumes "state cfg = s"   and "action cfg \<in> K s"   and "\<And>t. t \<in> set_pmf (action cfg) \<Longrightarrow> cfg_onp t (cont cfg t)" shows "cfg_onp s cfg"
gen_step_def: fixes s :: "('a, 'd) simple_state_scheme" shows "a.c.gen_step s \<equiv> if gds_is_empty_stack a.c.gds s then SPEC (\<lambda>v0. v0 \<in> a.a.V0 \<and> \<not> gds_is_discovered a.c.gds v0 s) \<bind> (\<lambda>v0. gds_new_root a.c.gds v0 s) else gds_get_pending a.c.gds s \<bind> (\<lambda>(u, Vs, s). case Vs of None \<Rightarrow> gds_finish a.c.gds u s | Some v \<Rightarrow> if gds_is_discovered a.c.gds v s then if gds_is_finished a.c.gds v s then gds_cross_edge a.c.gds u v s else gds_back_edge a.c.gds u v s else gds_discover a.c.gds u v s)"
r_zero: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<oplus> \<zero> = x"
eucl_le: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
simps_2: fixes x2 :: "'c" shows "None \<noteq> Some x2"
assms_1: shows "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"
nn_expl_cond_exp_integrable: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "\<forall>w\<in>space M. 0 \<le> X w"   and "integrable M X"   and "disc_fct Y"   and "point_measurable M (space N) Y" shows "integrable M (expl_cond_expect M Y X)"
mean_divisor_count_asymptotics: shows "(\<lambda>m. M m - (ln (real m) + 2 * euler_mascheroni - 1)) \<in> O(\<lambda>x. 1 / sqrt (real x))"
before_dist_not_r2: fixes vs :: "'a list"   and r1 :: "'a"   and r2 :: "'a" assumes "distinct vs"   and "before vs r1 r2" shows "r2 \<notin> set (fst (splitAt r1 vs))"
assms: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
is_cparser_length: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list"   and s :: "char list"   and x :: "'a"   and r :: "char list" assumes "is_cparser p"   and "p s = Inr (x, r)" shows "length r < length s"
assms_2: shows "wf_formula n (FOr \<phi> \<psi>)"
uint64_shiftr_def: fixes x :: "uint64"   and n :: "integer" shows "uint64_shiftr x n = (if n < 0 \<or> 64 \<le> n then undefined drop_bit x n else drop_bit (nat_of_integer n) x)"
relcompEpair: fixes a :: "'d"   and c :: "'e"   and r :: "('d \<times> 'f) set"   and s :: "('f \<times> 'e) set" assumes "(a, c) \<in> r O s" obtains b :: "'f" where "(a, b) \<in> r" and "(b, c) \<in> s"
Cauchy_altdef2: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity s = (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (s n) (s N) < e)"
assms_1: shows "nprv F (dsj \<phi> \<psi>)"
seqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.VV.arr f"   and "B.VV.arr g"   and "B.VV.dom g = B.VV.cod f" shows "B.VV.seq g f"
step: fixes u :: "'a list"   and v :: "'a list" assumes "u \<in> A"   and "v \<in> star A"   and "P v" shows "P (u @ v)"
lconf_upd2: fixes 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and V :: "char list" assumes "P,h \<turnstile> l (:\<le>) E"   and "P,h \<turnstile> v :\<le> T" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>) E(V \<mapsto> T)"
simps_2: fixes x :: "'a"   and l :: "'a list" shows "free_tv (x # l) = free_tv x \<union> free_tv l"
match_final_backward: fixes s1 :: "('fenv_std, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state" assumes "s1 \<sim> s2"   and "Sinca.final s2" shows "Sstd.final s1"
G2_def: shows "G2 = (\<lambda>x. ln x * sum_upto (\<lambda>xa. F2 (x / real xa)) x)"
d: shows "\<exists>m\<in>monomials p. lookup m x = n + 1"
nat_pow_closed: fixes x :: "nat \<times> nat"   and n :: "nat" assumes "x \<in> carrier S" shows "x [^]\<^bsub>S\<^esub> n \<in> carrier S"
cosh_dist: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "cosh_dist u v = poincare_distance_formula' (to_complex u) (to_complex v)"
language_def: shows "language \<equiv> {smap int (p ## trace w p) |p w. p = init \<and> run w p}"
whilefalse: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and c :: "('exp, 'id) MWLfCom" assumes "BMap (E b m) = False" shows "SRabbr (while b do c od) m [] m"
repcNodes_in_Nodes: shows "repc ` Nodes (Suc nb) levellista \<subseteq> Nodes (Suc nb) levellista"
\<R>_to_\<R>\<^sub>3: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
hyps_3: shows "mds\<^sub>1 \<le> mds\<^sub>2''"
P2: fixes j :: "nat" assumes "0 < j" shows "foldl1 f1 (replicate j Zero) = Zero"
derivative_is_holomorphic: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and f' :: "complex \<Rightarrow> complex" assumes "open S"   and "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)" shows "f' holomorphic_on S"
ptr_kinds_M_pure: fixes h :: "(_) heap" shows "pure element_ptr_kinds_M h"
poincare_between_x_axis_u0v: fixes u' :: "complex"   and v' :: "complex" assumes "is_real u'"   and "u' \<noteq> 0"   and "v' \<noteq> 0" shows "poincare_between (of_complex u') 0\<^sub>h (of_complex v') = (is_real v' \<and> Re u' * Re v' < 0)"
measurable_space: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and A :: "'e measure"   and x :: "'d" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "x \<in> space M" shows "f x \<in> space A"
not_lfinite_llength: fixes xs :: "'b llist" assumes "linfinite xs" shows "llength xs = \<infinity>"
latestAct_prop_1: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'<n. active nid (t n')" shows "active nid (t (dynamic_component.latestAct active nid t n))"
ExS: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<exists>x. \<phi> x in v] = (\<exists>y. [\<phi> y in v])"
failure_refine_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<sqsubseteq>\<^sub>F Q \<equiv> F Q \<subseteq> F P"
cond_know_b: fixes b :: "'b \<times> 'b \<Rightarrow> bool"   and P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(b \<and> (P \<triangleleft> b \<triangleright>  Q)) = (b \<and> P)"
fun_add_zeroR: fixes a :: "'a \<Rightarrow> 'a" assumes "a \<in> carrier (function_ring (carrier R) R)" shows "a \<oplus>\<^bsub>function_ring (carrier R) R\<^esub> \<zero>\<^bsub>function_ring (carrier R) R\<^esub> = a"
rsquarefree_smult_iff: fixes s :: "'a"   and p :: "'a poly" assumes "s \<noteq> (0::'a)" shows "rsquarefree (smult s p) = rsquarefree p"
polyfun_tensor_deep_model: fixes rs :: "nat list"   and y :: "nat"   and "is" :: "nat list"   and shared_weights :: "bool" assumes "deep_model_correct_params_y rs y"   and "is \<lhd> input_sizes (deep_model_l rs)" shows "polyfun {..<deep_model_correct_params.weight_space_dim shared_weights rs} (\<lambda>f. Tensor.lookup (tensors_from_net (insert_weights shared_weights (deep_model_l rs) f) $ y) is)"
simps_1: fixes x :: "'b"   and x' :: "'b"   and xs :: "'b list" shows "l_sorted (x # x' # xs) = (x \<le> x' \<and> l_sorted (x' # xs))"
rep_eq: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a multiset" shows "count (filter_mset x xa) = (\<lambda>xb. if x xb then count xa xb else 0)"
has_parametrized_subprob_densityD_3: fixes M :: "'a measure"   and N :: "'a \<Rightarrow> 'b measure"   and R :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "has_parametrized_subprob_density M N R f" shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)"
case_eq_if: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and option :: "'f option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
term_p_subid_var: fixes x :: "('a \<times> 'a set) set" shows "(x \<cdot> {} = x) = (x \<subseteq> 1\<^sub>\<pi>)"
simps_26: fixes x1 :: "'b"   and x101 :: "'b local.term"   and x102 :: "'b local.term"   and x103 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[x101, x102, x103\<^bold>]"
reflp_onI: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>a. a \<in> A \<Longrightarrow> P a a" shows "reflp_on P A"
rev_conv_rev_it': fixes xs :: "'a list"   and acc :: "'a list" shows "rev xs @ acc = rev_it' acc xs"
DynCastRedsNull: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>"
S_relator_freeword: fixes s :: "'w" assumes "s \<in> S" shows "pair_relator_freeword s s = s [+] s"
sets_Pair1: fixes A :: "('c \<times> 'd) set"   and M1 :: "'c measure"   and M2 :: "'d measure"   and x :: "'c" assumes "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" shows "Pair x -` A \<in> sets M2"
meet_compl2_bot: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> \<^bold>-\<^bsub>L\<^esub> x = \<bottom>\<^bsub>L\<^esub>"
Sup_upto_llist_infinity: fixes Xs :: "'b set llist" shows "Sup_upto_llist Xs \<infinity> = Sup_llist Xs"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "float interval" shows "bounds_of_interval (power_float_interval x xa xb) = (case bounds_of_interval xb of (l, u) \<Rightarrow> float_power_bnds x xa l u)"
strict_trans1: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "b \<preceq>\<^sub>t a"   and "c \<prec>\<^sub>t b" shows "c \<prec>\<^sub>t a"
init_lvars_def: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and mode :: "inv_mode"   and a' :: "val"   and pvs :: "val list" shows "init_lvars G C sig mode a' pvs = (\<lambda>(x, s). let m = mthd (the (methd G C sig)); l = \<lambda>k. case k of EName (VNam v) \<Rightarrow> [pars m [\<mapsto>] pvs] v | EName Res \<Rightarrow> None | This \<Rightarrow> if mode = Static then None else Some a' in set_lvars l (if mode = Static then x else np a' x, s))"
ORRIGHT1_elim: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "<a>:M \<in> ORRIGHT1 (B OR C) (\<parallel><B>\<parallel>)" obtains a' :: "coname" and M' :: "trm" where "M = OrR1 <a'>.M' a" and "fic (OrR1 <a'>.M' a) a" and "<a'>:M' \<in> \<parallel><B>\<parallel>"
permute: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "p permutes S" shows "prod g S = prod (g \<circ> p) S"
list_pmf: fixes xs :: "bool list"   and n :: "nat" assumes "length xs = n" shows "pmf (bv n) xs = (1 / 2) ^ n"
funion_finsert_right: fixes A :: "'l fset"   and a :: "'l"   and B :: "'l fset" shows "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
tagged_division_of_finite: fixes s :: "('b \<times> 'b set) set"   and i :: "'b set" assumes "s tagged_division_of i" shows "finite s"
le_vec_mono_left: fixes B :: "((real, 'n) Finite_Cartesian_Product.vec, 'n) Finite_Cartesian_Product.vec"   and x :: "(real, 'n) Finite_Cartesian_Product.vec"   and y :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "\<And>i j. 0 \<le> B $h i $h j"   and "le_vec x y" shows "le_vec (B *v x) (B *v y)"
assms: fixes as :: "'a list"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "dec as xs y ys \<equiv> y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys"
H: fixes y :: "'a" assumes "t !! j = Some y" shows "Q"
msort_monoms_impl: shows "(msort_monoms_impl, merge_monoms_poly) \<in> monom_rel \<rightarrow> monom_rel"
B: shows "g \<le>\<^sub>n SPEC \<Phi>"
prems: shows "[:a, - (1::'a):] * p \<noteq> 0"
eq_iff_1: fixes trm1 :: "trm"   and trm2 :: "trm"   and trm1a :: "trm"   and trm2a :: "trm" shows "(trm1 EQ trm2 = trm1a EQ trm2a) = (trm1 = trm1a \<and> trm2 = trm2a)"
subset_Collect_iff: fixes B :: "'b set"   and A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "B \<subseteq> A" shows "(B \<subseteq> {x \<in> A. P x}) = (\<forall>x\<in>B. P x)"
Suc: shows "sqrt (real_of_float x) < real_of_float (sqrt_iteration prec n x)"
zero_eq_add_iff_both_eq_0: fixes x :: "'b multiset"   and y :: "'b multiset" shows "({#} = x + y) = (x = {#} \<and> y = {#})"
gim: shows "g ` sphere 0 1 \<subseteq> S"
power_Suc2: fixes a :: "'b \<Rightarrow> 'b"   and n :: "nat" shows "fpower a (Suc n) = fpower a n \<circ> a"
left_diff_distrib': fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and b :: "'k"   and c :: "'k"   and a :: "'k" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times (minus b c) a = minus (times b a) (times c a)"
Exists_imp_exists: fixes p :: "hf"   and q :: "hf"   and e :: "(name, hf) finfun" assumes "Exists p q" shows "\<exists>A B i. p = \<lbrakk>\<guillemotleft>A IMP B\<guillemotright>\<rbrakk>e \<and> q = \<lbrakk>\<guillemotleft>SyntaxN.Ex i A IMP B\<guillemotright>\<rbrakk>e \<and> atom i \<sharp> B"
distinct_37: fixes x4 :: "('a, 'b, 'c) expr"   and x5 :: "('a, 'b, 'c) expr" shows "Ref x4 \<noteq> Read x5"
integrable_eq: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on s"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "g integrable_on s"
dictator_def: fixes i :: "indi" shows "dictator i \<equiv> \<forall>a b. i dictates a < b"
rHom_def: fixes A :: "('c, 'e) Ring_scheme"   and R :: "('d, 'f) Ring_scheme" shows "rHom A R = {f \<in> aHom A R. (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. f (x \<cdot>\<^sub>r\<^bsub>A\<^esub> y) = f x \<cdot>\<^sub>r\<^bsub>R\<^esub> f y) \<and> f 1\<^sub>r\<^bsub>A\<^esub> = 1\<^sub>r\<^bsub>R\<^esub>}"
Un_carrier_def: fixes I :: "'e set"   and A :: "'e \<Rightarrow> ('d, 'f) aGroup_scheme" shows "Un_carrier I A = \<Union> {X. \<exists>i\<in>I. X = carrier (A i)}"
dependent_def: fixes P :: "('b \<times> 'c) set" shows "p.dependent P = (\<exists>a\<in>P. a \<in> p.span (P - {a}))"
composition: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "compose A g f \<in> A \<rightarrow> C"
dependent_live_vars_same_pathsI: fixes V :: "'var"   and n' :: "'node"   and as :: "'edge list" assumes "V \<in> Use n'"   and "\<forall>as' a as''. as = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n'"   and "as \<noteq> [] \<longrightarrow> n' = last (targetnodes as)" shows "(V, as, as) \<in> dependent_live_vars n'"
f_closed: fixes B :: "'c set"   and D :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "LCD B D f"   and "x \<in> B"   and "y \<in> D" shows "f x y \<in> D"
unique_resr: fixes z :: "'a"   and x :: "'a"   and z' :: "'a"   and y :: "'a" assumes "D z x"   and "D z' x"   and "z \<oplus> x = y"   and "z' \<oplus> x = y" shows "z = z'"
limsup_ereal_mult_right: fixes c :: "real"   and f :: "nat \<Rightarrow> ereal" assumes "0 \<le> c" shows "limsup (\<lambda>n. f n * ereal c) = limsup f * ereal c"
Var_def: fixes x :: "nat"   and t :: "ty" shows "Var x t = Abs_var (Atom (sort_of_ty t) x)"
fix_eq: fixes F :: "'a \<rightarrow> 'a" shows "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
distinctness: fixes xs :: "'a list" assumes "op_history xs" shows "distinct xs"
idr: shows "(\<Otimes>a\<in>insert a' A \<times> B. case a of (a, b) \<Rightarrow> g a b) = finprod G (g a') B \<otimes> (\<Otimes>a\<in>A \<times> B. case a of (a, b) \<Rightarrow> g a b)"