hcju/isabelleps huggingface dataset:
oalist_tc_eqI: fixes xs :: "('a, 'b) oalist_tc"   and ys :: "('a, 'b) oalist_tc" assumes "list_of_oalist_tc xs = list_of_oalist_tc ys" shows "xs = ys"
cases: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "('c, 'e) resource"   and a2 :: "('d, 'f) resource" assumes "rel_resource A B a1 a2" obtains res1 :: "('c, 'e) resource" and res2 :: "('d, 'f) resource" where "a1 = res1" and "a2 = res2" and "rel_fun A (rel_spmf (rel_prod B (rel_resource A B))) (run_resource res1) (run_resource res2)"
cryptcorrect: fixes M :: "nat"   and e :: "nat"   and n :: "nat" shows "rsa_crypt M e n = M ^ e mod n"
member_is_static_simp: fixes m :: "'a member_scheme" shows "is_static m = static m"
syncSameHdTl: fixes a :: "'a event list"   and t :: "'a event list"   and A :: "'a event set"   and u :: "'a event list" assumes "a setinterleaves ((t, u), A) \<and> hd t \<in> A \<and> hd u \<in> A" shows "hd t = hd u \<and> hd a = hd t \<and> tl a setinterleaves ((tl t, tl u), A)"
at_fin_set_supp: fixes X :: "'a set" assumes "at TYPE('a)"   and "finite X" shows "supp X = X"
derived_quot_is_comm_group: shows "comm_group (G Mod derived G (carrier G))"
Class_self: fixes a :: "'a" assumes "a \<in> G" shows "a \<in> contained.Class a"
prim_pyth_triple_def: fixes x :: "nat"   and y :: "nat"   and z :: "nat" shows "prim_pyth_triple x y z = (0 < x \<and> 0 < y \<and> coprime x y \<and> x\<^sup>2 + y\<^sup>2 = z\<^sup>2)"
set_dom_\<Phi>: fixes f :: "'s" assumes "S.arr f" shows "S'.set (S'.dom (\<Phi> f)) = \<phi> ` S.set (S.dom f)"
fold_invariant_remove: fixes A :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "finite A"   and "I A s"   and "\<And>x s A'. \<lbrakk>x \<in> A'; I A' s; A' \<subseteq> A\<rbrakk> \<Longrightarrow> I (A' - {x}) (f x s)" shows "I {} (Finite_Set.fold f s A)"
empty: shows "Gcd\<^sub>f\<^sub>i\<^sub>n {} = (0::'a)"
rep_eq: fixes x :: "real_alg_3" shows "rep_real_alg_3 (inverse_3 x) = inverse_2 (rep_real_alg_3 x)"
Body: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and D :: "qtname"   and Q :: "'b assn"   and c :: "stmt"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} .Init D. {Q}"   and "G,A\<turnstile>{Q} .c. {\<lambda>s.. abupd (absorb Ret) .; R\<leftarrow>In1 (the (locals s Result))}" shows "G,A\<turnstile>{Normal P} Body D c-\<succ> {R}"
sup_idem: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "sup x x = x"
trm_rep_preserves_eq_semantics: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and t1 :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and t2 :: "'b trm"   and s :: "'b trm" assumes "fo_interpretation I"   and "I (t1 \<lhd> \<sigma>) (t2 \<lhd> \<sigma>)"   and "validate_ground_eq I (subst_equation (Eq t1 s) \<sigma>)" shows "validate_ground_eq I (subst_equation (Eq t2 s) \<sigma>)"
mult_less_cancel_right2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times a c) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
map_empty: fixes empty :: "'c"   and update :: "'d \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and delete :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and lookup :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool" assumes "Map empty update delete lookup invar" shows "lookup empty = Map.empty"
hom_mult: fixes x :: "'a"   and y :: "'a" shows "[:x * y:] = [:x:] * [:y:]"
WHILEI_refine: fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres" assumes "I' x' \<Longrightarrow> (x, x') \<in> R"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I' x'\<rbrakk> \<Longrightarrow> I x"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I x; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "WHILE\<^bsup>I\<^esup> b f x \<le> \<Down> R (WHILE\<^bsup>I'\<^esup> b' f' x')"
l_new_document_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<equiv> l_new_document type_wf \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_new_document_get_shadow_root_axioms get_shadow_root_locs"
Agreement: fixes i :: "('p, 'v, 's) configuration"   and c :: "('p, 'v, 's) configuration" shows "agreementInit i c"
tern_rel_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a,b\<triangleright>c \<equiv> a ## b \<and> a + b = c"
fmupd_lookup: fixes a :: "'e"   and b :: "'d"   and m :: "('e, 'd) fmap"   and a' :: "'e" shows "fmlookup (fmupd a b m) a' = (if a = a' then Some b else fmlookup m a')"
transitive_closurep_trans'_8: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>+\<^sup>+ a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
comp_ext: fixes z :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" shows "(z \<cdot> w1 \<bowtie> z \<cdot> w2) = (w1 \<bowtie> w2)"
prems: shows "\<Gamma>\<turnstile> \<langle>sequence Seq ((x # xs) @ ys),Normal s\<rangle> =n\<Rightarrow> t"
fls_base_factor_def: fixes f :: "'b fls" shows "fls_base_factor f = fls_shift (fls_subdegree f) f"
norm_mult_ineq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.real_normed_algebra plus zero minus uminus scaleR times dist norm sgn uniformity open" shows "norm (times x y) \<le> norm x * norm y"
gx0_3: fixes x :: "nat" assumes "gx0 \<le> x" shows "0 \<le> g' (real x)"
well_formed_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.well_formed problem \<equiv> length (ast_problem.astI problem) = ast_problem.numVars problem \<and> (\<forall>x<ast_problem.numVars problem. ast_problem.astI problem ! x < ast_problem.numVals problem x) \<and> ast_problem.wf_partial_state problem (ast_problem.astG problem) \<and> distinct (map fst (ast_problem.ast\<delta> problem)) \<and> (\<forall>\<pi>\<in>set (ast_problem.ast\<delta> problem). ast_problem.wf_operator problem \<pi>)"
nth_index: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "xs ! index xs x = x"
S: fixes V :: "('a \<times> real) set" assumes "openin (top_of_set U) V"   and "T retract_of V"   and "U \<noteq> {}" shows "locally path_connected S"
HMM3_axioms: shows "HMM3 \<O> \<S> state_list \<O>\<^sub>s \<K>"
coeff_prod_linear_factors: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and e :: "'a \<Rightarrow> nat"   and i :: "nat" assumes "finite A" shows "poly.coeff (\<Prod>x\<in>A. [:- f x, 1::'b:] ^ e x) i = (\<Sum>X | X \<in> Pow (SIGMA x:A. {..<e x}) \<and> i = sum e A - card X. (- (1::'b)) ^ card X * (\<Prod>x\<in>X. f (fst x)))"
components_nonoverlap: fixes c :: "'b set"   and s :: "'b set"   and c' :: "'b set" assumes "c \<in> components s"   and "c' \<in> components s" shows "(c \<inter> c' = {}) = (c \<noteq> c')"
quicksort_correct_case6: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and xs0 :: "'d list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'d list"   and xs' :: "'d list"   and p :: "nat"   and xs'' :: "'d list" assumes "quicksort_pre R h xs0 lo hi xs"   and "partition_spec R h xs lo hi xs' p"   and "\<not> p - Suc 0 \<le> lo"   and "\<not> hi \<le> Suc p"   and "quicksort_post R h lo (p - Suc 0) xs' xs''" shows "quicksort_pre R h xs0 (Suc p) hi xs''"
norm_quat_def: fixes z :: "quat" shows "norm z = sqrt ((quat.Re z)\<^sup>2 + (Im1 z)\<^sup>2 + (Im2 z)\<^sup>2 + (Im3 z)\<^sup>2)"
move_follow_read: fixes c :: "'b"   and r :: "(bool \<times> 'b) rexp"   and m :: "bool" shows "move c r m = follow m (read c r)"
Int_interval_cart: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and d :: "(real, 'a) vec" shows "cbox a b \<inter> cbox c d = {\<chi>i. max (a $ i) (c $ i)..\<chi>i. min (b $ i) (d $ i)}"
simps_1: fixes h :: "'b heap" shows "merge h heap.Empty = h"
interaction_bounded_by'_exec_gpv_bad_count: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and "consider" :: "'b \<Rightarrow> bool"   and gpv :: "('d, 'b, 'c) gpv"   and n :: "nat"   and s :: "'a"   and bad :: "'a \<Rightarrow> bool"   and count :: "'a \<Rightarrow> nat"   and k :: "real" assumes "callee_invariant_on callee I \<I>"   and "interaction_bounded_by' consider gpv n"   and "I s"   and "\<not> bad s"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; \<not> consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<And>s' x. \<lbrakk>I s'; \<not> bad s'; count s' < n + count s; consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> spmf (map_spmf (bad \<circ> snd) (callee s' x)) True \<le> k"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; \<not> bad s; bad s'; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> consider x"   and "0 \<le> k"   and "\<I> \<turnstile>g gpv \<surd>" shows "spmf (map_spmf (bad \<circ> snd) (exec_gpv callee gpv s)) True \<le> k * real n"
gtt: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "2 \<le> card bl"
subset_empty: fixes A :: "ZF" shows "HOLZF.subset Empty A"
12: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
as_left_supdistl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "\<alpha> x \<cdot> (y + z) \<le> \<alpha> x \<cdot> y + \<alpha> x \<cdot> z"
psubstT_atrm: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm" shows "psubstT r txs \<in> atrm"
runit\<^sub>F\<^sub>M\<^sub>C_def: fixes a :: "'c local.term set" shows "\<r>[a] = mkarr \<^bold>\<r>\<^bold>[rep a\<^bold>]"
lemVertexOnCone: fixes cone :: "'a Cone" shows "onCone (vertex cone) cone"
to_bl_inverse: fixes cbl :: "(('b \<Rightarrow> bool list) \<times> (bool list \<Rightarrow> 'b)) option"   and to_bl :: "'b \<Rightarrow> bool list"   and from_bl :: "bool list \<Rightarrow> 'b"   and a :: "'b" assumes "class.cbl cbl"   and "ID cbl = Some (to_bl, from_bl)" shows "from_bl (to_bl a) = a"
Object_is_acc_class: fixes G :: "prog"   and S :: "pname" assumes "wf_prog G" shows "is_acc_class G S Object"
characteristic_iso_1: fixes f :: "'a" assumes "Right.ide f" shows "Right.in_hom \<a>[f, src f, src f] ((f \<star> src f) \<star> src f) (f \<star> src f \<star> src f)"
l_set_disconnected_nodes_get_disconnected_nodes_axioms_def: fixes 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 set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_disconnected_nodes_get_disconnected_nodes_axioms get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs \<equiv> (\<forall>h document_ptr disc_nodes h'. h \<turnstile> set_disconnected_nodes document_ptr disc_nodes \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes) \<and> (\<forall>ptr ptr' w h h' r. ptr \<noteq> ptr' \<longrightarrow> w \<in> set_disconnected_nodes_locs ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h')"
char_eq_0: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c" assumes "character G \<chi>"   and "a \<notin> carrier G" shows "\<chi> a = 0"
set_product: fixes xs :: "'e list"   and ys :: "'f list" shows "set (List.product xs ys) = set xs \<times> set ys"
supp_Pair: fixes x :: "'a"   and y :: "'b" shows "supp (x, y) = supp x \<union> supp y"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (trg f)"
abs_norm1: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "\<bar>norm1 v\<bar> = norm1 v"
remdups_sort_3: fixes xs :: "'a list" shows "distinct (remdups_sort xs)"
lifting: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and C\<^sub>1' :: "fterm literal set"   and C\<^sub>2' :: "fterm literal set"   and L\<^sub>1' :: "fterm literal set"   and L\<^sub>2' :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "unification"   and "finite C\<^sub>1 \<and> finite C\<^sub>2"   and "vars\<^sub>l\<^sub>s C\<^sub>1 \<inter> vars\<^sub>l\<^sub>s C\<^sub>2 = {}"   and "instance_of\<^sub>l\<^sub>s C\<^sub>1' C\<^sub>1 \<and> instance_of\<^sub>l\<^sub>s C\<^sub>2' C\<^sub>2"   and "applicable C\<^sub>1' C\<^sub>2' L\<^sub>1' L\<^sub>2' \<sigma>" shows "\<exists>L\<^sub>1 L\<^sub>2 \<tau>. applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<tau> \<and> instance_of\<^sub>l\<^sub>s (resolution C\<^sub>1' C\<^sub>2' L\<^sub>1' L\<^sub>2' \<sigma>) (resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<tau>)"
set_tag_name_get_tag_name_is_l_set_tag_name_get_tag_name: shows "l_set_tag_name_get_tag_name ShadowRootClass.type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"
mat_mul_finsum_alt: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B = mat\<^sub>r nr nc (\<lambda>i. finsum_vec TYPE('b) nc (\<lambda>k. A $$ (i, k) \<cdot>\<^sub>v Matrix.row B k) {0..<n})"
pair_measure_countable: fixes S1 :: "'a set"   and S2 :: "'b set" assumes "countable S1"   and "countable S2" shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
sorted_list_of_set_sort_remdups: fixes xs :: "'d Word.word list" shows "signed.sorted_list_of_set (set xs) = signed.sort (remdups xs)"
finite_measure_Union: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets N"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure N (A \<union> B) = Sigma_Algebra.measure N A + Sigma_Algebra.measure N B"
in_KeysE: fixes s :: "'d"   and F :: "('d \<Rightarrow>\<^sub>0 'e) set" assumes "s \<in> Keys F" obtains f :: "'d \<Rightarrow>\<^sub>0 'e" where "s \<in> keys f" and "f \<in> F"
assms_1: shows "s \<in> reachable (pnet np n) TT"
ennreal_lessI: fixes q :: "real"   and r :: "real" assumes "0 < q"   and "r < q" shows "ennreal r < ennreal q"
assms_2: shows "simplex (mat_x_leq_vec A b) = Inr X"
dyadics_levels: shows "dyadics = (\<Union>K. \<Union>k<K. \<Union>m. {of_nat m / (2::'b) ^ k})"
Rep_ref_cases: fixes y :: "nat" assumes "y \<in> ref" obtains x :: "ref" where "y = Rep_ref x"
x: shows "x \<in> set (y # xs) \<and> P x"
differentiable_minus: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "f differentiable F" shows "(\<lambda>x. - f x) differentiable F"
maximum_def: fixes x :: "'c"   and y :: "'c" shows "maximum x y = (if x \<sqsubseteq> y then y else x)"
surj_pair: fixes p :: "'e \<times> 'f" shows "\<exists>x y. p = (x, y)"
lookupEvalHeap_other: fixes v :: "'a"   and \<Gamma> :: "('a \<times> 'b) list"   and f :: "'b \<Rightarrow> 'c" assumes "v \<notin> domA \<Gamma>" shows "evalHeap \<Gamma> f v = \<bottom>"
index_mat_multrow_4: fixes mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "nat"   and a :: "'c"   and A :: "'c mat" shows "dim_row (mat_multrow_gen mul k a A) = dim_row A"
disjnt_Times2_iff: fixes A :: "'c set"   and C :: "'d set"   and B :: "'c set" shows "disjnt (A \<times> C) (B \<times> C) = (C = {} \<or> disjnt A B)"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Can (t \<^bold>\<cdot> u) = (Can t \<and> Can u \<and> Dom t = Cod u)"
i3: fixes r :: "'a"   and x :: "('a, 'n) vec"   and y :: "('a, 'n) vec" shows "inner (r *s x) y = r * inner x y"
or_nonnegative_int_iff: fixes k :: "int"   and l :: "int" shows "(0 \<le> k OR l) = (0 \<le> k \<and> 0 \<le> l)"
leq_methods_subtypes: fixes D :: "char list"   and C :: "char list"   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 Ds :: "char list list"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "char list list \<times> expr"   and M :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> D \<preceq>\<^sup>* C"   and "(Ds, Ts', T', m') \<in> MinimalMethodDefs P D M"   and "wf_prog wf_md P" shows "\<forall>Ts T m Cs Cs'. P \<turnstile> Path D to C via Cs'  \<and> P,D \<turnstile> Ds \<sqsubseteq> Cs' @\<^sub>p Cs \<and> Cs \<noteq> [] \<and> P \<turnstile> C has M = (Ts, T, m) via Cs \<longrightarrow> Ts = Ts' \<and> P \<turnstile> T' \<le> T"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>F f : F a \<rightarrow>\<^sub>A F b\<guillemotright>"
safe_future: fixes s :: "event list"   and r :: "room"   and s' :: "event list" assumes "safe\<^sub>0 s r"   and "no_Check_in s' r" shows "safe\<^sub>0 (s' @ s) r"
scale_cancel_right: fixes a :: "'c"   and x :: "('c, 'd) vec"   and b :: "'c" shows "(a *s x = b *s x) = (a = b \<or> x = 0)"
coll_scale: fixes r :: "real \<times> real"   and q :: "real \<times> real" assumes "coll 0 r q"   and "r \<noteq> 0" shows "\<exists>x. q = x *\<^sub>R r"
exhaust: fixes y :: "'c arctic_delta" obtains "y = MinInfty_delta"   | x2 :: "'c" where "y = Num_arc_delta x2"
agg_sum_not_bot: fixes f :: "'d \<times> 'd \<Rightarrow> 'c"   and i :: "'d"   and j :: "'d" assumes "f (i, j) \<noteq> bot" shows "aggregation.sum_0 (\<lambda>k. aggregation.sum_0 (\<lambda>l. f (k, l)) {l. True}) {k. True} \<noteq> bot"
main_real: fixes b :: "nat"   and a :: "nat" assumes "b < a" shows "real (valid_countings a b) = (real a - real b) / (real a + real b) * real (all_countings a b)"
to_tree_order_same_root: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r root_ptr"   and "h \<turnstile> Shadow_DOM.to_tree_order root_ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "h \<turnstile> Shadow_DOM.get_root_node ptr' \<rightarrow>\<^sub>r root_ptr"
closure_Un_frontier: fixes S :: "'a set" shows "closure S = S \<union> frontier S"
enabled_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "Enabled F \<equiv> %s. \<exists>t. F (first s ## t)"
simps_1: shows "inorder \<langle>\<rangle> = []"
induct: fixes P :: "Ferrack.num \<Rightarrow> bool"   and a0 :: "Ferrack.num" assumes "\<And>j. P (C j)"   and "\<And>n c a. (\<And>x. P a) \<Longrightarrow> P (CN n c a)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)" shows "P a0"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Right.in_hom f a b = (Right.arr a \<and> Right.arr b \<and> Right.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
preserves_comp: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "seq g f" shows "trg (g \<bullet> f) = trg g \<bullet> trg f"
delete_idem: fixes k :: "'d"   and al :: "('d \<times> 'e) list" shows "delete k (delete k al) = delete k al"
less_imp_not_eq2: fixes y :: "'a"   and x :: "'a" assumes "y \<in> U"   and "x <\<^sub>o\<^sub>w y" shows "(y = x) = False"
list_insert_mset: fixes x :: "'c"   and xs :: "'c list" shows "mset (list_insert x xs) = {#x#} + mset xs"
z: assumes "\<forall>s. case s of (ps, n) \<Rightarrow> (Q \<and>* (\<lambda>s. True)) (ps, n div k)" shows "\<forall>s. case s of (ps, n) \<Rightarrow> (Q \<and>* (\<lambda>s. True)) (ps, n)"
compet_mono: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and c :: "real"   and S0 :: "'state set"   and c' :: "real" assumes "compet A c S0"   and "c \<le> c'" shows "compet A c' S0"
distinct_95: fixes x4 :: "val"   and x131 :: "expr"   and x132 :: "expr"   and x133 :: "expr" shows "Val x4 \<noteq> if (x131) x132 else x133"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "map_option f (Some x2) = Some (f x2)"
simps_4: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form"   and q :: "('c, 'b) form" shows "psubst f (And p q) = And (psubst f p) (psubst f q)"
assms_2: shows "(Neg x, Pos x) \<in> (imp_graph cnf)\<^sup>*"
fmdrop_fmrestrict_fset: fixes b :: "'c"   and S :: "'c fset"   and m :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmdrop b (fmrestrict_fset S m) = fmrestrict_fset (S |-| {|b|}) m"