hcju/isabelleps huggingface dataset:
rComm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (K\<lparr>N\<rparr>) Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "distinct xvec"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
iactionI: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<times> 'b"   and s1 :: "'b"   and a1 :: "'c"   and ou1 :: "'a"   and s1' :: "'b"   and \<phi> :: "('b, 'c, 'a) trans \<Rightarrow> bool"   and f :: "('b, 'c, 'a) trans \<Rightarrow> 'd"   and vl1 :: "'d list"   and vl1' :: "'d list"   and \<gamma> :: "('b, 'c, 'a) trans \<Rightarrow> bool"   and \<Delta> :: "'e \<Rightarrow> 'f \<Rightarrow> 'b \<Rightarrow> 'd list \<Rightarrow> bool"   and s :: "'e"   and vl :: "'f" assumes "step s1 a1 = (ou1, s1')"   and "\<phi> (Trans s1 a1 ou1 s1')"   and "BD_Security.consume \<phi> f (Trans s1 a1 ou1 s1') vl1 vl1'"   and "\<not> \<gamma> (Trans s1 a1 ou1 s1')"   and "\<Delta> s vl s1' vl1'" shows "BD_Security.iaction step \<phi> f \<gamma> \<Delta> s vl s1 vl1"
e_type_const_unwrap: fixes e :: "e" assumes "is_const e" shows "\<exists>v. e = $C v"
assms_6: shows "aux = compl gs bs (ps -- sps) sps (snd data)"
tendsto_minus: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c"   and F :: "'d filter" assumes "class.topological_group_add minus plus zero uminus open"   and "filterlim f (topological_space.nhds open a) F" shows "LIM x F. uminus (f x) :> topological_space.nhds open (uminus a)"
intros_2: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and k' :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "cs\<^bsup>\<pi>'\<^esup> k' = cs\<^bsup>\<pi>\<^esup> k"   and "\<pi> = path \<sigma>"   and "\<pi>' = path \<sigma>'"   and "\<sigma>\<^bsup>k\<^esup> \<restriction> local.reads (\<pi> k) \<noteq> \<sigma>'\<^bsup>k'\<^esup> \<restriction> local.reads (\<pi> k)" shows "(\<sigma>', k') \<cc> (\<sigma>, k)"
monom_of_set_plus_monom_of_set: fixes A :: "nat set"   and B :: "nat set" assumes "A \<inter> B = {}"   and "finite A"   and "finite B" shows "monom_of_set A + monom_of_set B = monom_of_set (A \<union> B)"
assms_1: shows "VectorSpace (\<sharp>\<cdot>) W"
intros_3: fixes u :: "'b list"   and C :: "'b list set"   and b :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> no_head C b"   and "v \<in> \<langle>C\<rangle>" shows "u \<cdot> v \<in> no_head C b"
sumCompose: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and R :: "pi"   and T :: "pi"   and Rel' :: "(pi \<times> pi) set" assumes "P \<leadsto><Rel> Q"   and "R \<leadsto><Rel> T"   and "Rel \<subseteq> Rel'" shows "P \<oplus> R \<leadsto><Rel'> Q \<oplus> T"
simps_9: fixes f :: "'m \<Rightarrow> 'l"   and x21 :: "'m"   and x22 :: "'m list" shows "map f (x21 # x22) = f x21 # map f x22"
aff_dim_translation_eq_subtract: fixes a :: "'a"   and S :: "'a set" shows "aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S"
rep_eq: fixes x :: "'c word" shows "signed x = of_int (signed_take_bit (LENGTH('c) - Suc 0) (Word.rep x))"
arg_cong: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "x = y" shows "f x = f y"
encL: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> SourceTerm S"
ET_Delta_target: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<subseteq> ET ST"   and "t \<in> TS"   and "target t \<in> States A"   and "A \<in> SAs (HA ST)" shows "t \<in> Delta A"
finite_orbit: fixes f :: "'a perm"   and a :: "'a" shows "finite (orbit f a)"
Domain_def: shows "Domain \<equiv> \<lambda>r. {x. Domainp (\<lambda>x xa. (x, xa) \<in> r) x}"
prime_dvd_prod_iff: fixes A :: "'c set"   and p :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "finite A"   and "prime p" shows "(p dvd prod f A) = (\<exists>x\<in>A. p dvd f x)"
length_shift_mopup: fixes n :: "nat"   and ss :: "nat" shows "length (shift (mopup n) ss) = 4 * n + 12"
MatchOr_def: fixes m1 :: "'c match_expr"   and m2 :: "'c match_expr" shows "MatchOr m1 m2 = MatchNot (MatchAnd (MatchNot m1) (MatchNot m2))"
l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Shadow_DOM.remove"
hyps: assumes "fst ` set xs \<subseteq> fst ` upds" shows "A (ListUpdate l'' xs q) = A l''"
dom_comp: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq g f" shows "local.dom (local.comp g f) = local.dom f"
Lim_bounded: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b"   and M :: "nat"   and C :: "'b" assumes "f \<longlonglongrightarrow> l"   and "\<forall>n\<ge>M. f n \<le> C" shows "l \<le> C"
universal_property: fixes f :: "'a \<Rightarrow> 'b" shows "\<exists>!g. g \<circ> Abs_freeletter = f \<and> UGroupHom g"
intro: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow \<Delta> g"   and "value_flow \<Delta> g \<noteq> \<top>"   and "\<^bold>V\<^bsub>\<Delta>\<^esub> - {source \<Delta>, sink \<Delta>} \<noteq> {}" shows "flow_network_axioms \<Delta> g"
IH1: shows "simple_fw (rmshadow rs P) p = simple_fw rs p"
integrable_restrict_UNIV: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "S \<in> sets lebesgue" shows "integrable lebesgue (\<lambda>x. if x \<in> S then f x else (0::'c)) = integrable (lebesgue_on S) f"
simps_6: fixes f :: "'d \<Rightarrow> 'c"   and i :: "'d"   and p :: "('b, 'd) fm" shows "sub f (\<^bold>@ i p) = (\<^bold>@ f i sub f p)"
context_equiv_trans: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and P :: "'BExp set"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and P' :: "'BExp set"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option" assumes "\<Gamma> =:\<^sub>P \<Gamma>'"   and "\<Gamma>' =:\<^sub>P' \<Gamma>''"   and "P \<turnstile> P'" shows "\<Gamma> =:\<^sub>P \<Gamma>''"
lhd_ltakeWhile: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<not> lnull xs"   and "P (lhd xs)" shows "lhd (ltakeWhile P xs) = lhd xs"
simplex: fixes n :: "int"   and S :: "'b set" shows "(n simplex S) = (\<exists>C. finite C \<and> \<not> affine_dependent C \<and> int (card C) = n + 1 \<and> S = convex hull C)"
take_bit_nat_less_self_iff: fixes n :: "nat"   and m :: "nat" shows "(take_bit n m < m) = (2 ^ n \<le> m)"
order_OrdSum_def: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "a \<le> b \<equiv> a l\<rightarrow> b = 1"
scopes_not_refl: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent" assumes "v |\<in>| vertices" shows "v \<notin> scope (v, p)"
comp_mkarr: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Dom t = Cod u" shows "\<F>C.mkarr t \<cdot> \<F>C.mkarr u = \<F>C.mkarr (t \<^bold>\<cdot> u)"
collapse_1: fixes sum :: "'h + 'i" assumes "isl sum" shows "Inl (projl sum) = sum"
sees_method_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "'b prog"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M, b :  Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
bin_clr_le: fixes n :: "nat"   and w :: "int" shows "bin_sc n False w \<le> w"
even_minus: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.ring_parity plus minus zero times one divide modulo uminus" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (uminus a) = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"
transpose_mat_append_rows: fixes A :: "'a mat"   and a :: "nat"   and n :: "nat"   and B :: "'a mat"   and b :: "nat" assumes "A \<in> carrier_mat a n"   and "B \<in> carrier_mat b n" shows "(A @\<^sub>r B)\<^sup>T = A\<^sup>T @\<^sub>c B\<^sup>T"
uniform_lim_divide: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set"   and l :: "'b \<Rightarrow> 'c"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and m :: "'b \<Rightarrow> 'c"   and b :: "real" assumes "uniform_limit S f l F"   and "uniform_limit S g m F"   and "bounded (l ` S)"   and "\<And>x. x \<in> S \<Longrightarrow> b \<le> norm (m x)"   and "0 < b" shows "uniform_limit S (\<lambda>a b. f a b / g a b) (\<lambda>a. l a / m a) F"
oLimit_leI: fixes f :: "nat \<Rightarrow> ordinal"   and x :: "ordinal" assumes "\<forall>n. f n \<le> x" shows "oLimit f \<le> x"
permute_set_eq_image: fixes p :: "perm"   and X :: "'b set" shows "p \<bullet> X = permute p ` X"
intro: fixes invar :: "'b \<Rightarrow> bool"   and mset :: "'b \<Rightarrow> 'c multiset"   and merge :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>q1 q2. \<lbrakk>invar q1; invar q2\<rbrakk> \<Longrightarrow> mset (merge q1 q2) = mset q1 + mset q2"   and "\<And>q1 q2. \<lbrakk>invar q1; invar q2\<rbrakk> \<Longrightarrow> invar (merge q1 q2)" shows "Priority_Queue_Merge_axioms invar mset merge"
nor: shows "\<forall>r. \<not> (\<exists>(a', b', c')\<in>set c. ((a' \<noteq> 0 \<or> b' \<noteq> 0) \<and> a' * r\<^sup>2 + b' * r + c' = 0) \<and> (\<forall>(d, e, f)\<in>set a. d * r\<^sup>2 + e * r + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * r\<^sup>2 + e * r + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * r\<^sup>2 + e * r + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * r\<^sup>2 + e * r + f \<noteq> 0))"
yz1: shows "det2 a11 a12 a21 a22 * y1 + det2 a11 a13 a21 a23 * z1 = (0::'a)"
Collect_closed_imp_pred_borel: fixes P :: "'c \<Rightarrow> bool" assumes "closed {x. P x}" shows "Measurable.pred borel P"
conj_simps_4: fixes P :: "fm" shows "Ferrack.conj P T = P"
epred_Max: fixes A :: "enat set" assumes "finite A"   and "A \<noteq> {}" shows "epred (Max A) = Max (epred ` A)"
setops\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'e) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
sqsubset_is_subset: fixes A :: "'c set"   and B :: "'c set" shows "(A \<sqsubseteq> B) = (A \<subseteq> B)"
nsClose: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows " n \<times>\<^bsub>R\<^esub> x \<in> carrier R"
level_shift: fixes l :: "nat"   and lc :: "nat"   and x :: "real" assumes "l < lc" shows "x / 2 ^ (lc - Suc l) = x * 2 / 2 ^ (lc - l)"
assms_2: fixes i :: "nat" assumes "i < j" shows "P (\<sigma> |\<^sub>s i)"
sup_right_unit_idempotent: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - x = - x \<squnion> - (- y \<squnion> - - y)"
fls_X_inv_pow_nonzero: fixes n :: "nat" shows "fls_X_inv ^ n \<noteq> 0"
inconsistent_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "inconsistent A b = (\<not> consistent A b)"
ex_iff_abs: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>n. P (abs_nat n))"
Class_is_Left_Coset: fixes g :: "'a" assumes "g \<in> G" shows "Class g = g \<cdot>| K"
pr_rec_last: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec3" shows "PrimRecOp_last g h \<in> PrimRec2"
unitary11_gen_cis_inversion_rev: fixes k' :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and \<phi> :: "real" assumes "k' \<noteq> 0"   and "M = k' *\<^sub>s\<^sub>m (cis \<phi>, 0, 0, 1) *\<^sub>m\<^sub>m (0, 1, 1, 0)" shows "\<exists>k a b. k \<noteq> 0 \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)"
subring_inter: fixes I :: "'a set"   and J :: "'a set" assumes "subring I R"   and "subring J R" shows "subring (I \<inter> J) R"
well_ordering: shows "\<exists>r. Well_order r \<and> Field r = UNIV"
add_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c + a \<le> c + b"
remdups_list_of_dlist: fixes dxs :: "'d dlist" shows "remdups (list_of_dlist dxs) = list_of_dlist dxs"
tests: fixes ws :: "int poly multiset" assumes "ws \<subseteq># mset vs"   and "ws \<noteq> {#}"   and "size ws < d \<or> size ws = d \<and> ws \<notin> (mset \<circ> snd) ` set cands" shows "test_dvd u ws"
distinct_3: fixes x1 :: "('e, 'f) Term.term"   and x31 :: "poscheckvariant"   and x32 :: "('e, 'f) Term.term"   and x33 :: "('e, 'f) Term.term" shows "send\<langle>x1\<rangle> \<noteq> \<langle>x31: x32 \<doteq> x33\<rangle>"
wf_digraph_del_vert: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c" assumes "wf_digraph G" shows "wf_digraph (pre_digraph.del_vert G u)"
holomorphic_on_subset: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and t :: "complex set" assumes "f holomorphic_on s"   and "t \<subseteq> s" shows "f holomorphic_on t"
subgraph: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and E' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Subgraph E E'"   and "E' a b" shows "E a b"
assms_4: shows "\<not> a[[top]] \<le> H * e * top"
esssup_AE_mono: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "AE x in M. f x \<le> g x" shows "esssup M f \<le> esssup M g"
left_commute: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b \<sqinter> (a \<sqinter> c) = a \<sqinter> (b \<sqinter> c)"
rep_eq: fixes x :: "('a, 'b) pfun"   and xa :: "'a"   and xb :: "'b" shows "Rep_pfun (x(xa \<mapsto> xb)\<^sub>p) = Rep_pfun x(xa \<mapsto> xb)"
star_involutive: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x\<^sup>\<odot> = x\<^sup>\<odot>\<^sup>\<odot>"
S3: shows "univ_ce = {z. c_snd z \<in> nat_to_ce_set (c_fst z)}"
FFE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form" assumes "G \<turnstile> \<bottom>" shows "G \<turnstile> a"
SUP_sup_distrib: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" shows "sup (Sup (f ` A)) (Sup (g ` A)) = (SUP a\<in>A. sup (f a) (g a))"
simps: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lfilter P xs = (case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then LCons x (lfilter P xs') else lfilter P xs')"
in_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b pmf"   and b :: "'c pmf" shows "rel_pmf R a b = (\<exists>z. z \<in> {x. set_pmf x \<subseteq> {(x, y). R x y}} \<and> map_pmf fst z = a \<and> map_pmf snd z = b)"
inline_sel: fixes callee :: "'i \<Rightarrow> 'l \<Rightarrow> ('m \<times> 'i, 'j, 'k) gpv"   and c :: "('h, 'l, 'm) gpv"   and s :: "'i" shows "the_gpv (inline callee c s) = map_spmf (\<lambda>result. case result of Inl xs \<Rightarrow> Pure xs | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. oracle input \<bind> (\<lambda>(x, s'). inline callee (rpv x) s'))) (inline1 callee c s)"
translation_apply: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "left.translation a b = a \<cdot> b"
mod_nonneg_pos_def: fixes x :: "integer"   and y :: "integer" assumes "0 \<le> x"   and "0 < y" shows "mod_nonneg_pos x y = x mod y"
sublens_refl: fixes X :: "'a \<Longrightarrow> 'b" shows "X \<subseteq>\<^sub>L X"
rep_eq: fixes x :: "('c, 'd) mapping_rbt" shows "keys x = set (RBT_Impl.keys (impl_of x))"
simps_12: fixes f :: "'c \<Rightarrow> 'b" shows "lmap f LNil = LNil"
mono2mono_eSuc: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> enat" assumes "monotone orda (\<le>) t" shows "monotone orda (\<le>) (\<lambda>x. eSuc (t x))"
HFinite_square_iff: fixes x :: "'a star" shows "(x * x \<in> HFinite) = (x \<in> HFinite)"
elims: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and xa :: "nat"   and y :: "nat option" assumes "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x xa = y" obtains xb :: "nat" and n :: "nat" where "x = xb" and "xa = n" and "y = (if xb \<le> n \<and> \<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w xb n) then Some (card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w xb n)) else None)"
nsqnr_def': fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "nsqn\<^sub>r r = (if \<pi>\<^sub>4 r = Aodv_Basic.inv then \<pi>\<^sub>2 r - 1 else \<pi>\<^sub>2 r)"
k1: fixes x :: "'a" shows "k (1, x) = g x"
lhd_LCons: fixes x21 :: "'a"   and x22 :: "'a llist" shows "lhd (LCons x21 x22) = x21"
comm_monoid_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "class.comm_monoid_add (sep_algebra.sep_conj plus sep_disj) (sep_algebra.sep_empty zero)"
vector_prop1: fixes v :: "'a" assumes "is_vector v" shows "- v\<^sup>T ; v = 0"
help_h: shows "reduction_signs_R p (qs1 @ qs2) (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))) = take_indices (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (find_nonzeros_from_input_vec (solve_for_lhs_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))))))"
drop_left_mult_eq: fixes r :: "('a \<times> 'a) set"   and N :: "'a multiset"   and M :: "'a multiset" assumes "trans r"   and "irrefl r"   and "(N + M, M) \<in> mul_eq r" shows "N = {#}"
power_diff_conv_inverse: fixes x :: "'b"   and m :: "nat"   and n :: "nat" assumes "x \<noteq> (0::'b)"   and "m \<le> n" shows "x ^ (n - m) = x ^ n * inverse x ^ m"
simps: fixes i :: "nat"   and j :: "nat" shows "f1.\<mu> i j = (if j < i then f1 ! i \<bullet> f1.gso j / \<parallel>f1.gso j\<parallel>\<^sup>2 else if i = j then 1::'a else (0::'a))"
multrow_carrier: fixes mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a"   and A :: "'a mat"   and n :: "nat"   and nc :: "nat" shows "(mat_multrow_gen mul k a A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
simps_12: fixes x3 :: "int"   and x2 :: "int" shows "Greater' x3 \<noteq> Intv' x2"
set_minus_mono_aux: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subseteq> B" shows "A - C \<subseteq> B - C"
rInput: fixes K :: "'a"   and Tvec :: "'a list" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "distinct xvec" shows "Prop (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec])"
valid_def: fixes P :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and Q :: "'b \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace> \<equiv> \<forall>s. P s \<longrightarrow> Q (c s)"
interval_lebesgue_integral_add_2: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "interval_lebesgue_integrable M a b f"   and "interval_lebesgue_integrable M a b g" shows "LINT x=a..b|M. (f x + g x) = interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g"
i_cut_Un_disj: fixes cut_op :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "cut_op = (\<down><) \<or> cut_op = (\<down>\<le>) \<or> cut_op = (\<down>>) \<or> cut_op = (\<down>\<ge>)" shows "cut_op (A \<union> B) t = cut_op A t \<union> cut_op B t"
trancl_into_rtrancl: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set" assumes "(a, b) \<in> r\<^sup>+" shows "(a, b) \<in> r\<^sup>*"
fact_split: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and k :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero"   and "k \<le> n" shows "semiring_char_0.fact one plus zero n = times (semiring_1.of_nat one plus zero (prod Suc {n - k..<n})) (semiring_char_0.fact one plus zero (n - k))"
M_wf_trms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"
unbounded_k_infinite: fixes k :: "nat"   and S :: "nat set" assumes "\<forall>m>k. \<exists>n>m. n \<in> S" shows "infinite S"