hcju/isabelleps huggingface dataset:
x_axis_intersection_equation: fixes H :: "circline"   and A :: "real"   and B :: "complex"   and C :: "complex"   and D :: "real"   and z :: "real" assumes "H = mk_circline (cor A) B C (cor D)"   and "(cor A, B, C, cor D) \<in> hermitean_nonzero" shows "(of_complex (cor z) \<in> circline_set x_axis \<inter> circline_set H) = (A * z\<^sup>2 + 2 * Re B * z + D = 0 \<and> is_real (cor z))"
modify_roundup_ge1: fixes emBits :: "nat" assumes "8 \<le> roundup emBits 8 * 8 - 168" shows "176 \<le> roundup emBits 8 * 8"
lookup_base_Cons: fixes d :: "nat"   and ds :: "nat list"   and v :: "'b list"   and i :: "nat"   and "is" :: "nat list" shows "lookup_base (d # ds) v (i # is) = lookup_base ds (fixed_length_sublist v (prod_list ds) i) is"
guaranteeStripPair_def: fixes f :: "'d"   and g :: "'e set" shows "guaranteeStripPair f g = (f, g)"
0: shows "r \<in> carrier (residue_ring (p ^ n))"
Abs_List_inject: fixes x :: "('b, unit) Old_Datatype.node set"   and y :: "('b, unit) Old_Datatype.node set" assumes "x \<in> List"   and "y \<in> List" shows "(Abs_List x = Abs_List y) = (x = y)"
defined_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<delta> X \<equiv> \<lambda>\<tau>. if X \<tau> = \<bottom> \<tau> \<or> X \<tau> = null \<tau> then false \<tau> else true \<tau>"
oassignT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and u :: "'e \<Rightarrow> 'e"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = u (\<sigma> i)" shows "((\<sigma>, {l}\<lbrakk>u\<rbrakk> p), \<tau>, \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
unconst_rel: fixes t :: "'a"   and u :: "'b" assumes "P t u" shows "unconst t = unconst u"
min_elems2: fixes s :: "'c event list"   and P :: "'c process"   and c :: "'c event"   and S :: "'c process"   and Q :: "'c process" assumes "s \<notin> D P"   and "s @ [c] \<in> D P"   and "P \<sqsubseteq> S"   and "Q \<sqsubseteq> S" shows "(s @ [c], {}) \<in> F Q"
gcsort_out_def: shows "gcsort_out \<equiv> snd \<circ> snd"
deg_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e" shows "deg R S X p = (if p = \<zero>\<^bsub>R\<^esub> then - \<infinity> else an (deg_n R S X p))"
simps_3: fixes x :: "three" shows "f2 x Two = Two"
rank_le_i: shows "rank_iarray (matrix_to_iarray A) \<le> i"
\<dd>_Star: fixes r :: "'a language" shows "\<dd> (Star r) = (\<lambda>a. Times (\<dd> r a) (Star r))"
step_CFG_SSA_Transformed_notriv: shows "i.inst g (i.uses' g) (i.phis' g)"
bind_return'': fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "M \<bind> return N = M"
us: shows "us \<equiv> getFrN (map snd txs) (map fst txs) [eql t1 t2] (length txs)"
ord_le_eq_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
IF: fixes \<sigma> :: "'a" assumes "I {} \<sigma>" shows "P \<sigma>"
borel_measurable_times: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
r_constn: fixes xs :: "nat list"   and n :: "nat"   and c :: "nat" assumes "length xs = Suc n" shows "eval (r_constn n c) xs \<down>= c"
const_onI: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "\<And>y. y \<in> S \<Longrightarrow> f y = x" shows "const_on f S x"
add_increasing2: fixes c :: "'b multiset"   and b :: "'b multiset"   and a :: "'b multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
Col_def: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec" shows "real_euclid.Col a b c \<equiv> B\<^sub>\<real> a b c \<or> B\<^sub>\<real> b c a \<or> B\<^sub>\<real> c a b"
bind_left_unit': fixes c :: "'c"   and P :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "unit\<^sub>S\<^sub>E c ;- P = P"
assms: shows "sorted_sublist_map R h xs 0 (length xs - 1)"
three\<^sub>1_def: shows "\<^bold>1 = fst threes"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<lparr>x\<rparr> \<in>\<^sub>\<circ> A"
mult_zero_ow_def: fixes U :: "'b set"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "mult_zero_ow U times zero \<equiv> times_ow U times \<and> zero_ow U zero \<and> mult_zero_ow_axioms U times zero"
s_type_unfold_1: fixes \<S> :: "s_context"   and rs :: "t list option"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list" assumes "\<S>\<bullet>rs \<tturnstile>_ i vs;es : ts" shows "i < length (s_inst \<S>)"
is_unit_Lcm_fin_iff: 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 A :: "'a set" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (semiring_gcd.Lcm_fin one zero lcm A) one = (semiring_gcd.Lcm_fin one zero lcm A = one)"
assms_4: shows "\<forall>U\<in>top.neighborhoods \<pp>. \<forall>s\<in>\<O> U. (\<phi> \<circ> st.canonical_fun U) s = local.key_map U s"
inj_vimage_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "f -` \<P> f A = A"
baIANow: fixes \<phi> :: "'KS \<Rightarrow> bool"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'KS" assumes "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "sb.eval c t t' n [\<phi>]\<^sub>b"
diff_strict_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and d :: "'c"   and c :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b"   and "less d c" shows "less (minus a c) (minus b d)"
DiaP: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and p :: "('c, 'b) fm"   and i :: "'b" assumes "Hintikka A H"   and "\<nexists>a. p = Nom a"   and "(\<^bold>\<diamond> p) at i in' H" shows "\<exists>j. (\<^bold>\<diamond> Nom j) at i in' H \<and> (\<^bold>@ j p) at i in' H"
iSince_conj_not: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "finite I" shows "(P t1 \<and> \<not> Q t1. t1 \<S> t2 I. Q t2) = (P t1. t1 \<S> t2 I. Q t2)"
subset_implies_mult: fixes A :: "'c multiset"   and B :: "'c multiset"   and r :: "('c \<times> 'c) set" assumes "A \<subset># B" shows "(A, B) \<in> mult r"
msb_neg: fixes w :: "32 Word.word" assumes "\<not> msb (- w)"   and "msb w" shows "uint w \<in> {2 ^ (LENGTH(32) - 1) + 1..2 ^ LENGTH(32) - 1}"
lcm_mult_distrib': fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "normalize c * lcm a b = lcm (c * a) (c * b)"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "R.isomorphic a a' = (\<exists>f. R.in_hom f a a' \<and> R.iso f)"
sum_code: fixes g :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "sum g (set xs) = sum_list (map g (remdups xs))"
encode_problem_serializable_sound_i: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat"   and ops :: "'b strips_operator list" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t"   and "k < t"   and "ops \<in> set (subseqs ((\<Phi>\<inverse> \<Pi> \<A> t) ! k))" shows "are_all_operators_non_interfering ops"
simps: fixes d :: "nat"   and amount :: "nat"   and f :: "atom fm" shows "lowerFm d amount f = map_fm_binders (\<lambda>a x. lowerAtom (d + x) amount a) f 0"
set_cover_bound: fixes w :: "nat \<Rightarrow> real"   and m :: "nat"   and S :: "nat \<Rightarrow> 'b set"   and C :: "nat set"   and C' :: "nat set" assumes "Set_Cover w m S"   and "Set_Cover.inv w m S C {}"   and "Set_Cover.sc m S C' (Set_Cover.U m S)" shows "sum w C \<le> harm (Set_Cover.d_star m S) * sum w C'"
lcp_len: fixes u :: "'b list"   and v :: "'b list" assumes "\<^bold>|u\<^bold>| \<le> \<^bold>|u \<and>\<^sub>p v\<^bold>|" shows "u \<le>p v"
coprime_poly_x_minus_y_poly_lift: fixes p :: "'a poly"   and q :: "'a poly" assumes "0 < degree p"   and "0 < degree q"   and "primitive p" shows "comm_monoid_mult_class.coprime (poly_x_minus_y p) (poly_lift q)"
i_Exec_Stream_Acc_LocalState__State_Idle_eq: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState l localState trans_fun input c n)" shows "i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c = i_Exec_Comp_Stream_Acc_LocalState l localState trans_fun input c"
induct: fixes x :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "fsigma x"   and "\<And>F. (\<And>n. closed (F n)) \<Longrightarrow> P (\<Union> (range F))" shows "P x"
rep_eq: fixes x :: "'b \<Longrightarrow> 'a" shows "\<lbrakk>x\<rbrakk>\<^sub>e = get\<^bsub>x\<^esub>"
CJNB_def: fixes f :: "fm \<Rightarrow> fm"   and p :: "fm" shows "CJNB f p \<equiv> let cjs = conjuncts p; (yes, no) = partition bound0 cjs in Parametric_Ferrante_Rackoff.conj (decr0 (list_conj yes)) (f (list_conj no))"
finite_dom_rbt_lookup: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less" shows "finite (dom (ord.rbt_lookup less t))"
wordinterval_to_set_ipcidr_tuple_to_wordinterval_uncurry: fixes ipcidr :: "'a word \<times> nat" shows "wordinterval_to_set (ipcidr_tuple_to_wordinterval ipcidr) = uncurry ipset_from_cidr ipcidr"
ufvs_V: shows "set ufvs \<subseteq> V"
dep_two: fixes T :: "'a set"   and A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "dep3_event T A B x" shows "dep_path T {A, B} x"
awalk_verts_in_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "u \<in> pverts G"   and "set p \<subseteq> parcs G"   and "v \<in> set (pawalk_verts u p)" shows "v \<in> pverts G"
that: fixes s :: "('a \<times> 'a) set" assumes "linear_order_on A s"   and "order_consistent r s" shows "thesis"
RDirProd_list_is_ring: fixes Rs :: "('a, 'b) ring_scheme list" assumes "\<And>i. i < length Rs \<Longrightarrow> ring (Rs ! i)" shows "ring (RDirProd_list Rs)"
tensor_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a"   and d :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>f \<otimes>\<^sub>1 g : a \<otimes>\<^sub>1 c \<rightarrow> b \<otimes>\<^sub>1 d\<guillemotright>"
ord_rec_0: fixes T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf" shows "ord_rec T G H 0 = T"
add_1_Suc_conv: fixes m :: "nat" shows "m + 1 = Suc m"
intF: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "list_all2 intT (arOf f) al" shows "intT (resOf f) (intF f al)"
direct_decompD_2: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss" shows "inj_on sum_list (listset ss)"
no_distrib_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "no_distrib a b c = (a \<sqinter> b \<squnion> c \<sqinter> a < a \<sqinter> (b \<squnion> c))"
congruent_triangle_sas_19: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "angle b1 a1 c1 = angle c2 a2 b2"
convergent_prod_iff_convergent: fixes f :: "nat \<Rightarrow> 'e" assumes "\<And>i. f i \<noteq> (0::'e)" shows "convergent_prod f = (convergent (\<lambda>n. prod f {..n}) \<and> lim (\<lambda>n. prod f {..n}) \<noteq> (0::'e))"
simps_1: fixes P :: "'c" shows "pre \<lbrace> P \<rbrace> = P"
transitive: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<simeq>\<^sup>s Q"   and "Q \<simeq>\<^sup>s R" shows "P \<simeq>\<^sup>s R"
span_list_as_span: fixes vs :: "'a vec list" assumes "set vs \<subseteq> carrier_vec n" shows "span_list vs = local.span (set vs)"
assms_2: shows "e = Cvtop t1 cvtop t sx"
simps_1: fixes f :: "'fsym" shows "TE_arOf (Oldf f) = arOf f"
not_basis_inD_aux: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "\<not> indep_system.basis_in indep \<E> X" shows "\<not> indep_system.basis \<E> (indep_system.indep_in indep \<E>) X"
assms: shows "(g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
ennreal_sum_less_top: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I < \<top>) = (\<forall>i\<in>I. f i < \<top>)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_monoid_add_hom hom"   and "monoid_mult_hom hom" shows "semiring_hom hom"
assms: shows "p \<in> {p. p partitions n \<and> (\<forall>i. p i \<le> 1)}"
compare_EQ_dest: fixes x :: "'a"   and y :: "'a" assumes "compare\<cdot>x\<cdot>y = EQ" shows "x = y"
fact_ge_zero: fixes n :: "nat" shows "(0::'a) \<le> fact n"
c2_1: shows "A2 \<in> carrier_mat n1 nc1"
generate_empty: shows "generate (G\<lparr>carrier := H\<rparr>) {} = {\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>}"
integer_hull_def: fixes P :: "'a vec set" shows "integer_hull P = convex_hull (P \<inter> \<int>\<^sub>v)"
rel_simps_28: fixes n :: "num" shows "\<not> numeral n \<le> (0::'c)"
induct: fixes x1 :: "'var \<Rightarrow> 'val"   and x2 :: "nat"   and x3 :: "'var \<Rightarrow> 'val"   and x4 :: "nat"   and P :: "('var \<Rightarrow> 'val) \<Rightarrow> nat \<Rightarrow> ('var \<Rightarrow> 'val) \<Rightarrow> nat \<Rightarrow> bool" assumes "((x1, x2), x3, x4) \<in> cp"   and "\<And>\<sigma> \<sigma>' n n' h. \<lbrakk>\<sigma> =\<^sub>L \<sigma>'; cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'; h \<in> local.reads (path \<sigma> n); (\<sigma>\<^bsup>n\<^esup>) h \<noteq> (\<sigma>'\<^bsup>n'\<^esup>) h; \<forall>k<n. h \<notin> local.writes (path \<sigma> k); \<forall>k'<n'. h \<notin> local.writes (path \<sigma>' k')\<rbrakk> \<Longrightarrow> P \<sigma> n \<sigma>' n'"   and "\<And>\<sigma> k \<sigma>' k' n v n'. \<lbrakk>((\<sigma>, k), \<sigma>', k') \<in> cp; P \<sigma> k \<sigma>' k'; n dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> k; n' dd\<^bsup>path \<sigma>',v\<^esup>\<rightarrow> k'; cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'; (\<sigma>\<^bsup>n\<^esup>) v \<noteq> (\<sigma>'\<^bsup>n'\<^esup>) v\<rbrakk> \<Longrightarrow> P \<sigma> n \<sigma>' n'"   and "\<And>\<sigma> k \<sigma>' k' n v l n'. \<lbrakk>((\<sigma>, k), \<sigma>', k') \<in> cp; P \<sigma> k \<sigma>' k'; n dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> l; l cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> k; cs\<^bsup>path \<sigma>\<^esup> n = cs\<^bsup>path \<sigma>'\<^esup> n'; path \<sigma> (Suc k) \<noteq> path \<sigma>' (Suc k'); (\<sigma>\<^bsup>n\<^esup>) v \<noteq> (\<sigma>'\<^bsup>n'\<^esup>) v; \<forall>j'\<in>{LEAST i'. k' < i' \<and> (\<exists>i. cs\<^bsup>path \<sigma>\<^esup> i = cs\<^bsup>path \<sigma>'\<^esup> i')..<n'}. v \<notin> local.writes (path \<sigma>' j')\<rbrakk> \<Longrightarrow> P \<sigma> n \<sigma>' n'"   and "\<And>\<sigma> k \<sigma>' k'. \<lbrakk>((\<sigma>, k), \<sigma>', k') \<in> cp; P \<sigma> k \<sigma>' k'\<rbrakk> \<Longrightarrow> P \<sigma>' k' \<sigma> k" shows "P x1 x2 x3 x4"
ss_rewrite1: fixes h :: "'a \<times> 'b"   and a :: "'a"   and e :: "nat"   and z :: "nat"   and e' :: "nat"   and z' :: "nat" assumes "fst h \<in> carrier \<G>"   and "a \<in> carrier \<G>"   and "e < order \<G>"   and "a \<otimes> fst h [^] e = \<^bold>g [^] z"   and "e' < e"   and "a \<otimes> fst h [^] e' = \<^bold>g [^] z'" shows "fst h = \<^bold>g [^] ((int z - int z') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>))"
False: shows "\<not> (a \<in> S \<or> b \<in> S)"
dom_iter_properties: fixes ss :: "nat list list"   and w :: "nat list" shows "iter f step start (unstables r step start) = (ss, w) \<longrightarrow> ss ! 0 = [] \<and> (\<forall>p<n. p \<noteq> 0 \<longrightarrow> ss ! p \<noteq> rev [0..<n])"
is_tangent_def: fixes p :: "'b point"   and q :: "'b point" shows "is_tangent p q = (p \<noteq> Infinity \<and> p = q \<and> p \<noteq> opp q)"
individually_rational_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Pds'.individually_rational_on ds X = (Pds'.CD_on ds X = X \<and> Pdds'.CH X = X)"
limsup_shift: fixes u :: "nat \<Rightarrow> 'b" shows "limsup (\<lambda>n. u (n + 1)) = limsup u"
learn_U0: shows "learn_bc \<phi> U\<^sub>0 s'"
anonymous_neutral_indifference: fixes A :: "'c set"   and Is :: "'d set"   and scf :: "('d \<Rightarrow> ('c \<times> 'c) set) \<Rightarrow> ('c \<times> 'c) set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "May A Is scf"   and "profile A Is P"   and "hasw [x, y] A"   and "card {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y} = card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x}" shows "x \<^bsub>(scf P)\<^esub>\<approx> y"
assoc_naturality: fixes f0 :: "'a"   and a0 :: "'a"   and b0 :: "'a"   and f1 :: "'a"   and a1 :: "'a"   and b1 :: "'a"   and f2 :: "'a"   and a2 :: "'a"   and b2 :: "'a" assumes "\<guillemotleft>f0 : a0 \<rightarrow> b0\<guillemotright>"   and "\<guillemotleft>f1 : a1 \<rightarrow> b1\<guillemotright>"   and "\<guillemotleft>f2 : a2 \<rightarrow> b2\<guillemotright>" shows "assoc b0 b1 b2 \<cdot> local.prod (local.prod f0 f1) f2 = local.prod f0 (local.prod f1 f2) \<cdot> assoc a0 a1 a2"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_1: fixes t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>)"
assms_7: shows "s\<^sub>2' r'' = Some (\<sigma>', \<tau>', VE v)"
inverse: fixes x :: "nat"   and q :: "nat" assumes "gcd x q = 1" shows "[int x * Number_Theory_Aux.inverse x q = 1] (mod int q)"
SigmaI: fixes a :: "'b"   and A :: "'b set"   and b :: "'d"   and B :: "'b \<Rightarrow> 'd set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"
lexordp_def: shows "ord_class.lexordp \<equiv> lfp (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> x < y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> x < y \<and> \<not> y < x \<and> p xs ys))"
chainD: fixes ord :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'g set"   and x :: "'g"   and y :: "'g" assumes "Complete_Partial_Order.chain ord S"   and "x \<in> S"   and "y \<in> S" shows "ord x y \<or> ord y x"
intros_1: fixes n :: "nat" shows "i_set_indp [n\<dots>]"
cosh_conv_cos: fixes z :: "complex" shows "cosh z = cos (\<i> * z)"
pair_bidirected_digraph_def: fixes G :: "'b pair_pre_digraph" shows "pair_bidirected_digraph G \<equiv> pair_sym_digraph G \<and> pair_loopfree_digraph G"
outputPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<sim> Q" shows "a{b}.P \<sim> a{b}.Q"
eigenvalue_def: fixes A :: "'a mat"   and k :: "'a" shows "eigenvalue A k = (\<exists>v. eigenvector A v k)"
empty_not_add_mset: fixes a :: "'d"   and A :: "'d multiset" shows "{#} \<noteq> add_mset a A"
assoc_ide_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "Cop.src f = Cop.trg g"   and "Cop.src g = Cop.trg h" shows "Cop.\<a> f g h = \<a>\<^sup>-\<^sup>1[h, g, f]"
simps_1: fixes p :: "fm"   and q :: "fm" shows "minusinf (And p q) = MIR.conj (minusinf p) (minusinf q)"
scene_indep_sym: fixes X :: "'a scene"   and Y :: "'a scene" assumes "X \<bowtie>\<^sub>S Y" shows "Y \<bowtie>\<^sub>S X"
space_extend_measure: fixes G :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and \<Omega> :: "'c set"   and \<mu> :: "'d \<Rightarrow> ennreal" assumes "G ` I \<subseteq> Pow \<Omega>" shows "space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
vec_zero_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "A *\<^sub>m\<^sub>v vec_zero = vec_zero"
Cons: fixes xs :: "expr list"   and ys :: "val list"   and zs :: "expr list"   and y :: "val" assumes "map_val2 xs ys zs" shows "map_val2 (Val y # xs) (y # ys) zs"
not_in_kmin: shows "i \<in> PR - kmin f PR"
B: shows "(evsFR5, S, A, U) \<in> protocol"
fold_insert_idem: fixes A :: "'b fset set set"   and z :: "'b fset set"   and x :: "'b fset set" assumes "finite A" shows "Finite_Set.fold (\<otimes>\<^sub>m) z (insert x A) = x \<otimes>\<^sub>m Finite_Set.fold (\<otimes>\<^sub>m) z A"
iSp_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n i-[]\<rightarrow>\<^sub>d* n"
2: shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t]) = wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t \<A>"
gen_sa_impl_rel_sv: fixes Rm :: "('a \<times> 'b) set"   and Rf :: "('c \<times> ('d \<Rightarrow> 'e)) set" assumes "single_valued Rm"   and "single_valued Rf" shows "single_valued (\<langle>Rm, Rf\<rangle>gen_sa_impl_rel_eext)"
is_nth_power_upto_unit_0_left: fixes x :: "'a" shows "is_nth_power_upto_unit 0 x = is_unit x"
Nil_is_rev_conv: fixes xs :: "'b list" shows "([] = rev xs) = (xs = [])"
sel_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (TimesLR r s) = (\<oo> r \<and> \<oo> s)"
S1_and'_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool"   and \<sigma> :: "bool" shows "S1_and' A B \<sigma> = (let (a1, a2) = A; (b1, b2) = B; s0 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False); s1 = \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True); s2 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False); s3 = \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True) in S1_14_OT (s0, s1, s2, s3) () \<bind> (\<lambda>V. return_spmf (((a1, a2), \<sigma>, V), \<sigma>, \<sigma> \<oplus> (a1 \<oplus> b1 \<and> a2 \<oplus> b2))))"
conforms_allocL_aux: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and vn :: "lname" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "G,s\<turnstile>v\<Colon>\<preceq>T" shows "G,s\<turnstile>locals s(vn \<mapsto> v)[\<sim>\<Colon>\<preceq>]L(vn \<mapsto> T)"
derivable_well_ordered: assumes "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"   and "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x" shows "well_ordered_set (Collect derivable) (\<sqsubseteq>)"
ordinal_le_oLog: fixes b :: "ordinal"   and y :: "ordinal"   and x :: "ordinal" assumes "oSuc 0 < b"   and "b ** y \<le> x" shows "y \<le> oLog b x"
powers_distr: fixes x :: "'a"   and y :: "'a" shows "qstar x \<cdot> y = (\<Squnion>i. x ^ i \<cdot> y)"
t'_def: shows "t' = (\<lambda>y. (y - x0) /\<^sub>R (e / r) + apply_chart c x)"
rel_pred_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set" shows "rel_pred R A B = (R ===> (=)) (\<lambda>x. x \<in> A) (\<lambda>y. y \<in> B)"