hcju/isabelleps huggingface dataset:
IH_3: shows "((l, \<langle>c: t \<doteq> s\<rangle>) \<in> set A) = ((l, \<langle>c: t \<doteq> s\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
src_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>C (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "a \<sqsubseteq> linorder.Max (\<sqsubseteq>) A"
bounded_clinear_finite_dim: fixes f :: "'a \<Rightarrow> 'b" assumes "clinear f" shows "bounded_clinear f"
simps_3: fixes x2 :: "'c" shows "Num_arc_delta x2 \<noteq> MinInfty_delta"
top_on_step': fixes C :: "ivl st option acom" assumes "top_on\<^sub>c C (- vars C)" shows "top_on\<^sub>c (step_ivl \<top> C) (- vars C)"
rep_iso_class: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C f" shows "category.isomorphic C (iso_class_rep (category.iso_class C f)) f"
defect_ge_1: fixes N :: "'c quasinorm" shows "1 \<le> defect N"
cases: fixes s :: "'b + 'c"   and a :: "'c" assumes "setrp s a" obtains x :: "'c" where "a = x" and "s = Inr x"
oghoare_if_valid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P\<^sub>1 :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P\<^sub>2 :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and r :: "'e set"   and b :: "'e set" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q, A"   and "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q, A"   and "r \<inter> b \<subseteq> P\<^sub>1"   and "r \<inter> - b \<subseteq> P\<^sub>2" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> r Cond b c\<^sub>1 c\<^sub>2 Q, A"
lookup_raw_tl_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a" assumes "oalist_inv xs" shows "lookup_raw (tl_raw xs) k = (if \<forall>k'\<in>fst ` set (fst xs). le (snd xs) k k' then 0::'c else lookup_raw xs k)"
simps_1: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t' :: "'f"   and x :: "'g"   and m :: "'i" shows "redT_updT ts (NewThread t' x m) = ts(t' \<mapsto> (x, no_wait_locks))"
head: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "prod g {m..n} = g m * prod g {m<..n}"
assms_3: fixes y :: "'a" assumes "y \<in> Y" shows "set_borel_measurable borel X' (f y)"
simple_append: fixes S :: "('c, 'd) strand_step list"   and S' :: "('c, 'd) strand_step list" assumes "simple S"   and "simple S'" shows "simple (S @ S')"
correctDeCompositionVARSYSTEM_def: shows "correctDeCompositionVARSYSTEM \<equiv> \<forall>S. correctDeCompositionVAR S"
gen_igbg_impl_rel_eext_def: fixes Rm :: "('f \<times> 'h) set"   and Racc :: "('e \<times> ('g \<Rightarrow> nat set)) set" shows "\<langle>Rm, Racc\<rangle>gen_igbg_impl_rel_eext = {(\<lparr>igbgi_num_acc = num_acci, igbgi_acc = acci, \<dots> = mi\<rparr>, \<lparr>igbg_num_acc = num_acc, igbg_acc = acc, \<dots> = m\<rparr>) |num_acci acci mi num_acc acc m. (num_acci, num_acc) \<in> nat_rel \<and> (acci, acc) \<in> Racc \<and> (mi, m) \<in> Rm}"
irreducible\<^sub>d_mD_2: fixes f :: "int poly"   and g :: "int poly"   and h :: "int poly" assumes "irreducible\<^sub>d_m f"   and "degree_m g < degree_m f"   and "degree_m h < degree_m f" shows "Mp f \<noteq> Mp (g * h)"
nat_term_order_eqI: fixes dg :: "bool"   and ps :: "bool"   and to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order" assumes "\<And>u v. \<lbrakk>dg \<Longrightarrow> deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v)); ps \<Longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v)\<rbrakk> \<Longrightarrow> nat_term_compare to1 u v = nat_term_compare to2 u v" shows "nat_term_order_eq to1 to2 dg ps"
tree_type: fixes t :: "'a bro"   and h :: "nat" assumes "t \<in> Bp h" shows "tree t \<in> B h \<union> B (Suc h)"
M1_def: shows "M1 = M.summset {#(if ia = j then lc j + lc i * c else lc ia) \<cdot>\<^sub>v us ! ia. ia \<in># mset_set ({0..<length us} - {i, j})#}"
a1: shows "s' = snd (fst (jmpl_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
Wpl_def: fixes a :: "int \<Rightarrow> nat"   and b :: "int \<Rightarrow> nat"   and i :: "int"   and t :: "int tree" shows "Wpl.Wpl a b i t = (\<Sum>(k, c)\<in>nodes t. c * b k) + (\<Sum>(k, c)\<in>leaves i t. c * a k)"
0: shows "Zp_converges_to (f \<circ> t) (res_lim (f \<circ> t))"
lookup_mult_scalar: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.mult_scalar pair_of_term term_of_pair p q) v = lookup (p * term_powerprod.proj_poly term_of_pair (term_powerprod.component_of_term pair_of_term v) q) (term_powerprod.pp_of_term pair_of_term v)"
aND_sortQ: fixes p :: "('b set set, 'c) Combinators list"   and l :: "'b set set set list" assumes "singleCombinators p"   and "allNetsDistinct p" shows "allNetsDistinct (qsort p l)"
eln_ennexp: fixes x :: "ereal" shows "eln (ennexp x) = x"
interval_real_split_2: fixes a :: "real"   and b :: "real"   and c :: "real" shows "{a..b} \<inter> {x. c \<le> x} = {max a c..b}"
consequence: shows "P \<subseteq> {s. \<exists>Z. s \<in> I Z \<and> (\<forall>t. t \<in> I Z \<inter> - b \<longrightarrow> t \<in> Q)}"
Osum_minus_Id2: fixes r' :: "('b \<times> 'b) set"   and r :: "('b \<times> 'b) set" assumes "r' \<subseteq> Id" shows "(r \<union>o r') - Id \<subseteq> r - Id \<union> Field r \<times> Field r'"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and x :: "Preliminaries.var" shows "int \<xi> (Var x) = \<xi> x"
prob_T: fixes P :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream \<Rightarrow> bool"   and s :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "T.prob s {\<omega> \<in> space (T s). P \<omega>} = measure_pmf.expectation (local.trans s) (\<lambda>t. T.prob t {\<omega> \<in> space (T t). P (t ## \<omega>)})"
T_merge_log: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" assumes "ltree l"   and "ltree r" shows "real (T_merge l r) \<le> log 2 (real (size1 l)) + log 2 (real (size1 r)) + 1"
privateKey_neq_publicKey: fixes b :: "keymode"   and A :: "agent"   and c :: "keymode"   and A' :: "agent" shows "privateKey b A \<noteq> publicKey c A'"
SMP_wt_subst_subset: fixes t :: "('fun, 'var) Term.term"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> SMP (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>)"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>)" shows "t \<in> SMP M"
assms_3: shows "(d, t) \<rightarrow>c (d', t')"
xy_1: shows "c - a = x *\<^sub>R (b - a)"
cont: fixes q :: "nat set"   and i :: "nat"   and j :: "nat" assumes "q \<in> P"   and "q \<noteq> I0"   and "i \<in> q"   and "j \<in> F q" shows "local.cont (fst cf1) (snd cf1) i \<approx>01 local.cont (fst cf2) (snd cf2) j"
executionally_equivalent_trace_equivalent: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" assumes "\<forall>x. executionally_equivalent e1 0 <> e2 0 <> x" shows "trace_equivalent e1 e2"
simp3: shows "\<forall>y. - b * y + a * (- c / a) = - b * y - c"
advantage_def: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "discrete_log.advantage \<A> \<equiv> spmf (discrete_log.dis_log \<A>) True"
hd_in_awalk_verts_1: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" shows "u \<in> set (G.awalk_verts u p)"
pmc_extensionality: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "K \<div>\<^bsub>\<gamma>\<^esub> \<phi> = K \<div>\<^bsub>\<gamma>\<^esub> \<psi>"
set_integrable_Gamma_hurwitz_aux2_real: fixes r :: "real"   and a :: "real"   and s :: "real" assumes "0 < r"   and "0 < a" shows "set_integrable lborel {r..} (\<lambda>x. x powr s * exp (- a * x) / (1 - exp (- x)))"
sub_eq_inf: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z -\<^sub>h w = \<infinity>\<^sub>h" shows "z = \<infinity>\<^sub>h \<or> w = \<infinity>\<^sub>h"
that: fixes et :: "real"   and ex :: "real"   and B :: "real"   and L :: "real" assumes "0 < et"   and "0 < ex"   and "cball t0 et \<subseteq> T"   and "cball x0 ex \<subseteq> X"   and "unique_on_cylinder t0 (cball t0 et) x0 ex f B L" shows "thesis"
is_orthogonal_cspan: fixes S :: "'b set"   and a :: "'b"   and x :: "'b" assumes "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s"   and "x \<in> cspan S" shows "is_orthogonal a x"
filt_equiv_filt: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure"   and G :: "'d \<Rightarrow> 'c measure" assumes "filt_equiv F M N"   and "Filtration.filtration M G" shows "Filtration.filtration N G"
ideal_generated_0: shows "ideal_generated {0::'b} = {0::'b}"
val_size_mem_r: fixes a :: "val"   and b :: "val"   and t :: "(val \<times> val) list" assumes "(a, b) \<in> set t" shows "val_size b < fun_size t"
ccProd_belowI: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S'\<rbrakk> \<Longrightarrow> x--y\<in>G" shows "S G\<times> S' \<sqsubseteq> G"
member_filter: fixes x :: "'d"   and P :: "'d \<Rightarrow> bool"   and A :: "'d set" shows "(x \<in> Set.filter P A) = (x \<in> A \<and> P x)"
delta: fixes a :: "'c"   and g :: "'c \<Rightarrow> 'a" shows "(\<Sum>b. if b = a then g b else (0::'a)) = g a"
notStuck_DynComD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
min_maxsimpchainI_betw: fixes x :: "'a set"   and y :: "'a set"   and xs :: "'a set list" assumes "x \<noteq> y"   and "maxsimpchain (x # xs @ [y])"   and "\<And>ys. maxsimpchain (x # ys @ [y]) \<Longrightarrow> order.greater_eq (length ys) (length xs)" shows "min_maxsimpchain (x # xs @ [y])"
prime_elem_multiplicity_eq_zero_iff: 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 p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero" shows "(factorial_semiring.multiplicity times one p x = 0) = (\<not> dvd.dvd times p x)"
cond_conj_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "((P \<triangleleft> b \<triangleright>  Q) \<and> R) = (P \<and> R \<triangleleft> b \<triangleright>  Q \<and> R)"
get_disconnected_nodes_new_character_data: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) document_ptr" assumes "l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
comp_def: shows "replete_setcat.comp \<equiv> concrete_category.COMP {A. |A| <o cardSuc (cmax |UNIV| natLeq)} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)"
pi_eq: fixes h1 :: "real \<Rightarrow> complex"   and h2 :: "real \<Rightarrow> complex" shows "valid_path h1 \<and> valid_path h2 \<and> (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and> pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow> path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and> (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
run_ask_state: fixes ask :: "('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> ('b, 'a) stateT"   and s :: "'b" shows "run_state (ask_state ask f) s = ask (\<lambda>r. run_state (f r) s)"
Joints_product_aforms: fixes a :: "(real \<times> real pdevs) list"   and b :: "(real \<times> real pdevs) list" shows "Joints (product_aforms a b) = product_listset (Joints a) (Joints b)"
wf_fdecl_def: fixes P :: "'a prog" shows "wf_fdecl P \<equiv> \<lambda>(F, b, T). is_type P T"
SPN_is_weak_arrow_of_homs: shows "weak_arrow_of_homs (\<cdot>) src trg (\<bullet>) Span.src Span.trg SPN"
of_list_def: fixes xs :: "'b list" shows "Array.of_list xs = Heap_Monad.heap (Array.alloc xs)"
goodAbs_imp_qGoodAbs_pick: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodAbs A" shows "qGoodAbs (pick A)"
cyclesI: fixes w :: "'action list"   and p :: "'state" assumes "path w p"   and "target w p = p" shows "w \<in> cycles p"
rep_eq: fixes x :: "fin_string" shows "ilength x = length (Rep_fin_string x)"
q'_2: fixes K :: "'n set" assumes "K \<in> q" shows "K \<subseteq> cbox a b"
step: fixes y :: "V" assumes "y \<in> elts ya__" shows "rank (x + y) = rank x + rank y"
simps_7: fixes p :: "pvar"   and \<phi> :: "'aprop dfmla"   and env :: "pvar \<Rightarrow> nat" shows "sem (Exi p \<phi>) env = exi (\<lambda>\<pi> \<pi>l. sem \<phi> (env(p := length \<pi>l)) (\<pi>l @ [\<pi>]))"
dds_of_def: shows "dds_of \<equiv> \<lambda>uu uua. corec_dds (\<lambda>(f, s). map_prod id (\<lambda>uu. Inr (f, uu)) \<circ> f s) (uu, uua)"
eSwap_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and X :: "'c"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "\<forall>s. \<not> igWls MOD s X" shows "eSwap MOD zs z1 z2 (OK X) = ERR"
remove_run: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" assumes "finite (nodes A)"   and "w \<notin> language A"   and "V \<subseteq> gunodes A w"   and "clean A w V \<noteq> {}" obtains v :: "nat \<times> 'a" and r :: "(nat \<times> 'a) stream" where "grun A w V r v" and "sset (gtrace r v) \<subseteq> clean A w V" and "sset (gtrace r v) \<subseteq> - prune A w (clean A w V)"
reach_foldl_def: fixes \<Sigma> :: "'a set"   and \<delta> :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and q\<^sub>0 :: "'b" assumes "\<Sigma> \<noteq> {}" shows "reach \<Sigma> \<delta> q\<^sub>0 = {foldl \<delta> q\<^sub>0 w |w. set w \<subseteq> \<Sigma>}"
stake_shift: fixes i :: "nat"   and w :: "'d list"   and s :: "'d stream" shows "stake i (w @- s) = take i w @ stake (i - length w) s"
ideals_downsets: fixes X :: "'a set" assumes "idealp X" shows "X \<in> downsets"
pigeonhole: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "card (f ` A) < card A" shows "\<not> inj_on f A"
hinsert_iff: fixes z :: "hf"   and y :: "hf"   and x :: "hf" shows "(z = x \<triangleleft> y) = (\<forall>u. (u \<^bold>\<in> z) = (u \<^bold>\<in> x \<or> u = y))"
dgrad_set_leE: fixes d :: "'a \<Rightarrow> nat"   and S :: "'a set"   and T :: "'a set"   and s :: "'a" assumes "dgrad_set_le d S T"   and "s \<in> S" obtains t :: "'a" where "t \<in> T" and "d s \<le> d t"
semiring_simprules_11: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "semiring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
F1set2_Lev2: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xd :: "'e"   and xc :: "'d"   and xa :: "'d" assumes "xf \<in> Lev2 s1 s2 n xd"   and "rv2 s1 s2 xf xd = Inl xc"   and "xa \<in> F1set2 (s1 xc)" shows "xf @ [Inl (tobd_F12 s1 xc xa)] \<in> Lev2 s1 s2 (Suc n) xd"
qGood_qTerm_induct: fixes phi :: "('b, 'c, 'varSort, 'var, 'd) qTerm \<Rightarrow> bool"   and phiAbs :: "('b, 'c, 'varSort, 'var, 'd) qAbs \<Rightarrow> bool"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and A :: "('b, 'c, 'varSort, 'var, 'd) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>|{i. inp i \<noteq> None}| <o |UNIV|; |{i. binp i \<noteq> None}| <o |UNIV|; liftAll (\<lambda>X. qGood X \<and> phi X) inp; liftAll (\<lambda>A. qGoodAbs A \<and> phiAbs A) binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>qGood X; \<And>Y. qGood Y \<and> qSkel Y = qSkel X \<Longrightarrow> phi Y; \<And>Y. (X, Y) \<in> qSwapped \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "(qGood X \<longrightarrow> phi X) \<and> (qGoodAbs A \<longrightarrow> phiAbs A)"
weakenSimWeakSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi>'' R S. (\<Psi>, R, S) \<in> Rel \<Longrightarrow> \<Psi> \<rhd> R \<lessapprox>\<^sub>w<Rel> S"   and "\<And>\<Psi>'' R S. (\<Psi>, R, S) \<in> Rel \<Longrightarrow> \<Psi> \<rhd> R \<leadsto>\<^sub>w<Rel'> S"   and "\<And>\<Psi>'' R S \<Psi>'''. (\<Psi>, R, S) \<in> Rel' \<Longrightarrow> (\<Psi> \<otimes> \<Psi>''', R, S) \<in> Rel'"   and "\<And>\<Psi>'' R S. (\<Psi>, R, S) \<in> Rel \<Longrightarrow> (\<Psi>, S, R) \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel'> Q"
closedin_singleton: fixes U :: "'b set"   and a :: "'b" shows "closedin (top_of_set U) {a} = (a \<in> U)"
mem_val_alt_10_mem_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 (pa AND 68719476732) s1 = mem_val_alt 10 (pa AND 68719476732) s2 \<and> mem_val_alt 10 ((pa AND 68719476732) + 1) s1 = mem_val_alt 10 ((pa AND 68719476732) + 1) s2 \<and> mem_val_alt 10 ((pa AND 68719476732) + 2) s1 = mem_val_alt 10 ((pa AND 68719476732) + 2) s2 \<and> mem_val_alt 10 ((pa AND 68719476732) + 3) s1 = mem_val_alt 10 ((pa AND 68719476732) + 3) s2"
run_rel_def: fixes R :: "('h \<times> 'i) set" shows "run_rel R \<equiv> {(ra, rb). \<forall>i. (ra i, rb i) \<in> R}"
sumResLeft: fixes Rel :: "(pi \<times> pi) set"   and x :: "name"   and P :: "pi"   and Q :: "pi" assumes "Id \<subseteq> Rel"   and "eqvt Rel" shows "<\<nu>x>P \<oplus> <\<nu>x>Q \<leadsto>[Rel] <\<nu>x>(P \<oplus> Q)"
adds_termE: fixes u :: "'t"   and v :: "'t" assumes "u adds\<^sub>t v" obtains t :: "'a" where "v = t \<oplus> u"
infs_def: fixes x :: "'a"   and y :: "'a" shows "Sup_lattice.infs Inf (\<lambda>x y. y \<le> x) x y = Sup_lattice.Infs Inf (\<lambda>x y. y \<le> x) {x, y}"
MakeCatDom: fixes f :: "'d"   and C :: "('e, 'd, 'f) Category_scheme" assumes "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> f = dom\<^bsub>MakeCat C\<^esub> f"
IH: fixes n :: "nat" assumes "distinct xs"   and "n < length xs" shows "length (insert_spec xs (oid, Some (xs ! n))) = Suc (length xs)"
power2_norm_eq_inner: 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 inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(norm x)\<^sup>2 = inner x x"
F1in_mono23': fixes A2 :: "'d set"   and A2a :: "'d set"   and A3 :: "'e set"   and A3a :: "'e set"   and c :: "('f, 'd, 'e) F1"   and A1 :: "'f set" assumes "A2 \<subseteq> A2a"   and "A3 \<subseteq> A3a"   and "c \<in> F1in A1 A2 A3" shows "c \<in> F1in A1 A2a A3a"
alw_invar: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "\<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> nxt \<phi> xs) xs" shows "alw \<phi> xs"
simps: fixes AB :: "('b \<times> bal) tree"   and c :: "'b"   and bc :: "bal"   and C' :: "('b \<times> bal) tree alt" shows "baldR AB c bc C' = (case C' of Same C \<Rightarrow> Same \<langle>AB, (c, bc), C\<rangle> | Diff C \<Rightarrow> case bc of Lh \<Rightarrow> case AB of \<langle>A, (a, Lh), B\<rangle> \<Rightarrow> Diff \<langle>A, (a, Bal), \<langle>B, (c, Bal), C\<rangle>\<rangle> | \<langle>A, (a, Bal), B\<rangle> \<Rightarrow> Same \<langle>A, (a, Rh), \<langle>B, (c, Lh), C\<rangle>\<rangle> | \<langle>A, (a, Rh), B\<rangle> \<Rightarrow> Diff (rot2 A a B c C) | Bal \<Rightarrow> Same \<langle>AB, (c, Lh), C\<rangle> | Rh \<Rightarrow> Diff \<langle>AB, (c, Bal), C\<rangle>)"
intro: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a"   and B :: "'a" assumes "Semilat A r f"   and "lbv_axioms A r T B" shows "lbv A r f T B"
kernel_is_closed_csubspace: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "closed_csubspace (f -` {0::'b})"
encode_problem_with_operator_interference_exclusion_valid: fixes sv :: "sat_plan_variable" assumes "sv \<in> atoms (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop abs_prob  h)" shows "valid_state_var sv"
io_12: shows "frontier outside = path_image (complex_of \<circ> c)"
llength_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "llength (lappend xs ys) = llength xs + llength ys"
devExt_bc_geq: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a"   and n' :: "nat"   and n\<^sub>s :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<not> Option.is_none (Blockchain.devBC active cmp bc mining t n nid n')"   and "n\<^sub>s \<le> n'" shows "Blockchain.devExt active cmp bc mining t n nid n\<^sub>s (n' - n\<^sub>s) = bc (cmp (the (Blockchain.devBC active cmp bc mining t n nid n')) (t n'))"
indRelRSTPO_refl: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "refl SRel"   and "refl TRel" shows "refl (indRelRSTPO SRel TRel)"
tail_alt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "tail p = except p {lt p}"
While: fixes r :: "(state \<times> state) set"   and C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c :: "com" assumes "wf r"   and "\<forall>s'. C \<turnstile>\<^sub>t {\<lambda>z s. P z s \<and> b s \<and> s' = s} c {\<lambda>z s. P z s \<and> (s, s') \<in> r}" shows "C \<turnstile>\<^sub>t {P} WHILE b DO c {\<lambda>z s. P z s \<and> \<not> b s}"
obj_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>C a = a"
fin_mono: fixes A :: "'a fset"   and B :: "'a fset"   and x :: "'a" assumes "A |\<subseteq>| B" shows "x |\<in>| A \<longrightarrow> x |\<in>| B"