hcju/isabelleps huggingface dataset:
two_pers': fixes w :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "w \<le>p u \<cdot> w"   and "w \<le>p v \<cdot> w"   and "\<^bold>|u\<^bold>| + \<^bold>|v\<^bold>| \<le> \<^bold>|w\<^bold>|" shows "u \<cdot> v = v \<cdot> u"
pdg_digraph_dg_prod: shows "digraph \<alpha> (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>2\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)"
o_def: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'f" shows "f \<circ> g = (\<lambda>x. f (g x))"
leD: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
pairwise_pairD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream" assumes "PTA_Reachability.pairwise P xs" shows "P (shd xs) (shd (stl xs))"
genby_diff_closed: fixes w :: "'a"   and S :: "'a set"   and w' :: "'a" assumes "w \<in> \<langle>S\<rangle>"   and "w' \<in> \<langle>S\<rangle>" shows "w - w' \<in> \<langle>S\<rangle>"
inner_sum_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. inner (f x) y)"
case_prodE': fixes c :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'a \<times> 'b"   and z :: "'c" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'a" and y :: "'b" where "p = (x, y)" and "c x y z"
induct: fixes P :: "pratt_tree \<Rightarrow> bool"   and a0 :: "pratt_tree" assumes "\<And>n a ts. (\<And>x. x \<in> set ts \<Longrightarrow> P x) \<Longrightarrow> P (Pratt_Node (n, a, ts))" shows "P a0"
P\<^sub>2_def: shows "P\<^sub>2 \<equiv> ts_process {[], [None], [Some a], [Some a, None]}"
lv_aforms_rel_def: shows "lv_aforms_rel = br eucl_of_list_aform (\<lambda>xs. length xs = DIM('a))"
bind_alt1: fixes m :: "'m"   and m' :: "'m"   and f :: "'a \<Rightarrow> 'm" shows "bind (alt m m') f = alt (bind m f) (bind m' f)"
Lq_num: shows "let LLq = \<lambda>t1 t2. psubst Lq [(t1, zz), (t2, yy)] in \<forall>\<phi>\<in>fmla. \<forall>q\<in>num. Fvars \<phi> = {zz} \<and> (\<forall>p\<in>num. bprv (subst \<phi> p zz)) \<longrightarrow> prv (all zz (imp (LLq (Var zz) q) \<phi>))"
intro: fixes f :: "'b mod_ring poly" assumes "irreducible\<^sub>d f" shows "poly_mod_type_irr_axioms f"
maxne0_impl_le: fixes x :: "'a list"   and a :: "nat list" shows "maxne0_impl x a \<le> Max (set a)"
associated_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" shows "a \<sim>\<^bsub>G\<^esub> b = (a divides\<^bsub>G\<^esub> b \<and> b divides\<^bsub>G\<^esub> a)"
c_tl_def: shows "c_tl = (\<lambda>u. list_to_nat (tl (nat_to_list u)))"
quotients_def: shows "comp\<HH>.quotients = map (\<lambda>i. G\<lparr>carrier := \<HH> ! (i + 1)\<rparr> Mod \<HH> ! i) [0..<length \<HH> - 1]"
check_add_alt_def: fixes p :: "nat"   and A :: "(nat, int mpoly) fmap"   and q :: "nat"   and i :: "nat"   and r :: "int mpoly"   and \<V> :: "nat set" shows "SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> q \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars r \<subseteq> \<V>) \<bind> (\<lambda>b. if \<not> b then RETURN False else ASSERT (p \<in># dom_m A) \<bind> (\<lambda>_. let p = the (fmlookup A p) in ASSERT (q \<in># dom_m A) \<bind> (\<lambda>_. let q = the (fmlookup A q) in add_poly_spec p q \<bind> (\<lambda>pq. weak_equality pq r \<bind> RETURN)))) \<le> check_add A \<V> p q i r"
execute_instr_sub1_privilege: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "snd (execute_instr_sub1 i s) = False \<and> s' = snd (fst (execute_instr_sub1 i s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
min_le_iff_disj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.linorder less_eq less" shows "less_eq (ord.min less_eq x y) z = (less_eq x z \<or> less_eq y z)"
weakBisimScopeExt: fixes x :: "name"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>"   and "x \<sharp> P" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<approx> P \<parallel> \<lparr>\<nu>x\<rparr>Q"
Phi_def: shows "\<Phi> \<equiv> \<lambda>w x y z. w ; y\<^sup>\<smile> \<cdot> x ; z\<^sup>\<smile>"
assms: shows "ves = map (Basic \<circ> EConst) vs"
card_ord_of_nat: fixes m :: "nat" shows "card (elts (ord_of_nat m)) = m"
zero_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "a *\<^sub>V (0::'b) = (0::'a)"
distinct_5: fixes x21 :: "rat"   and x22 :: "rat"   and x31 :: "int poly"   and x32 :: "nat" shows "Sqrt_Info x21 x22 \<noteq> Real_Alg_Info x31 x32"
image_Collect: fixes f :: "'h \<Rightarrow> 'g"   and P :: "'h \<Rightarrow> bool" shows "f ` {x. P x} = {f x |x. P x}"
get_scdom_component_owner_document_same: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" obtains owner_document :: "(_) document_ptr" where "h \<turnstile> Shadow_DOM.get_owner_document ptr' \<rightarrow>\<^sub>r owner_document" and "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r owner_document \<in> set sc"
poly_lift_hom_def: fixes R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and \<phi> :: "'e \<Rightarrow> 'f" shows "poly_lift_hom R S \<phi> = eval R (UP S) (to_polynomial S \<circ> \<phi>) (X_poly S)"
h: fixes x :: "'a" shows "h x = LCons x (lmap f (h x))"
nth_map_zip: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "i < length xs"   and "i < length ys" shows "map f (zip xs ys) ! i = f (xs ! i, ys ! i)"
downset_prop: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "ord.downset less_eq x = {y. less_eq y x}"
d: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> d) = (\<exists>k l u v. meets k l \<and> meets l p \<and> meets p u \<and> meets u v \<and> meets k q \<and> meets q v)"
accs_unique: fixes t :: "'L tree"   and q :: "'Q"   and q' :: "'Q" assumes "accs \<delta> t q"   and "accs \<delta> t q'" shows "q = q'"
cr_spmf_prob_optionTI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "('c, 'c spmf) optionT"   and q :: "'d spmf" assumes "rel_spmf A (run_option p) q" shows "cr_spmf_prob_optionT A p q"
basis_finite: fixes A :: "'a set" assumes "basis A" shows "finite A"
assms: shows "frees t |\<subseteq>| fmdom env\<^sub>2"
Union_image_empty: fixes A :: "'d set"   and f :: "'e \<Rightarrow> 'd set" shows "A \<union> \<Union> (f ` {}) = A"
cc_restr_mono1: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" assumes "S \<subseteq> S'" shows "G G|` S \<sqsubseteq> G G|` S'"
theFTOne_Int_theNFTBoth: fixes cl :: "('g, 'h, 'i) cmd list"   and dl :: "('j, 'k, 'l) cmd list" shows "theFTOne cl dl \<inter> theNFTBoth cl dl = {}"
c_5: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_Delete stateful_strand_step" shows "delete\<langle>the_elem_term stateful_strand_step,the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
ex_prop_def: fixes X :: "'b program set" shows "ex_prop X \<equiv> \<forall>F G. F ok G \<longrightarrow> F \<in> X \<or> G \<in> X \<longrightarrow> F \<squnion> G \<in> X"
abst_inj: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "(abst isom x = abst isom y) = (x = y)"
matchC_LC_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis" shows "PL_Indis.matchC_LC aval tval cval indis theta c d \<equiv> \<forall>s t. (s, t) \<in> indis \<longrightarrow> (\<exists>I0 P F. PL_Indis.mC_ZOC aval tval cval indis theta c d s t I0 P F)"
nl2: fixes ll :: "real"   and lr :: "real"   and r :: "real" assumes "1 \<le> ll"   and "1 \<le> lr"   and "1 \<le> r" shows "log \<alpha> (ll + lr) + \<beta> * log \<alpha> (lr + r) \<le> \<beta> * log \<alpha> (ll + lr) + log \<alpha> (ll + lr + r)"
P1: shows "Bet C' A' X' \<and> A' \<noteq> X'"
simps_2: fixes nat' :: "nat" shows "0 \<noteq> Suc nat'"
termMOD_igSubstIPresIGWls: shows "igSubstIPresIGWls termMOD"
incrtm0_tmbound: fixes n :: "nat"   and t :: "tm" assumes "tmbound n t" shows "tmbound (Suc n) (incrtm0 t)"
T2_axiom_implies_T0_axiom: fixes T :: "'b set set" assumes "\<forall>x\<in>carrier\<^bsub>T\<^esub>. \<forall>y\<in>carrier\<^bsub>T\<^esub>. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds\<^bsub>T\<^esub> x. \<exists>v\<in>nhds\<^bsub>T\<^esub> y. u \<inter> v = {})" shows "T0_axioms T"
closed_in_weaken': fixes T :: "ty"   and \<Gamma> :: "binding list"   and \<Delta> :: "binding list" assumes "T closed_in \<Gamma>" shows "T closed_in (\<Delta> @ \<Gamma>)"
that: fixes p :: "(name \<times> name) list" assumes "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "yvec = p \<bullet> xvec"   and "N = p \<bullet> M"   and "Q = p \<bullet> P" shows "thesis"
lcos_mult_one: fixes M :: "(Vertex \<Rightarrow> Vertex) set" assumes "M \<subseteq> carrier tetrahedral_group" shows "\<one>\<^bsub>tetrahedral_group\<^esub> <#\<^bsub>tetrahedral_group\<^esub> M = M"
parts_insert_spies: fixes X :: "msg"   and evs :: "event list" shows "parts (insert X (knows Spy evs)) = parts {X} \<union> parts (knows Spy evs)"
erfc_continuous_on: fixes A :: "'a set" shows "continuous_on A erfc"
IH: shows "image_mset fst (M - {#(e, f e)#}) = image_mset fst M - {#e#}"
inv_comp_left: fixes f :: "'d \<Rightarrow> 'd" assumes "bijection f" shows "inv f \<circ> f = id"
elts_VPow: fixes x :: "V" shows "elts (VPow x) = ZFC_in_HOL.set ` Pow (elts x)"
per_add_exp: fixes u :: "'b list"   and r :: "'b list"   and m :: "nat" assumes "u \<le>p r\<^sup>\<omega>"   and "m \<noteq> 0" shows "u \<le>p r \<^sup>@ m\<^sup>\<omega>"
sin_zero_iff_int: fixes x :: "real" shows "(sin x = 0) = (\<exists>i. even i \<and> x = real_of_int i * (pi / 2))"
floor_of_int: fixes z :: "int" shows "\<lfloor>of_int z\<rfloor> = z"
part_complement_reversal: fixes y :: "i"   and x :: "i" assumes "y \<noteq> u"   and "P x y" shows "P (\<midarrow> y) (\<midarrow> x)"
vsv_vintersection: fixes r :: "V"   and s :: "V" assumes "vsv r" shows "vsv (r \<inter>\<^sub>\<circ> s)"
residue_lmul: fixes s :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex"   and c :: "complex" assumes "open s"   and "z \<in> s"   and "f holomorphic_on s - {z}" shows "residue (\<lambda>z. c * f z) z = c * residue f z"
induct: fixes x1 :: "'c option list"   and x2 :: "'c option set"   and P :: "'c option process"   and Q :: "'c option process"   and Pa :: "'c option list \<Rightarrow> 'c option set \<Rightarrow> bool" assumes "(x1, x2) \<in> seq_comp_failures P Q"   and "\<And>xs X. \<lbrakk>xs \<notin> sentences P; (xs, X) \<in> failures P; None \<notin> set xs\<rbrakk> \<Longrightarrow> Pa xs X"   and "\<And>xs X Y. \<lbrakk>xs \<in> sentences P; (xs, X) \<in> failures P; ([], Y) \<in> failures Q\<rbrakk> \<Longrightarrow> Pa xs (insert None X \<inter> Y)"   and "\<And>xs ys Y. \<lbrakk>xs \<in> sentences P; (ys, Y) \<in> failures Q; ys \<noteq> []\<rbrakk> \<Longrightarrow> Pa (xs @ ys) Y"   and "\<And>xs X Y. \<lbrakk>(xs, X) \<in> seq_comp_failures P Q; Pa xs X; (xs, Y) \<in> seq_comp_failures P Q; Pa xs Y\<rbrakk> \<Longrightarrow> Pa xs (X \<union> Y)" shows "Pa x1 x2"
mem_interior: fixes x :: "'c"   and S :: "'c set" shows "(x \<in> interior S) = (\<exists>e>0. ball x e \<subseteq> S)"
fill_none: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and ns :: "nat list"   and ub :: "nat"   and i :: "nat" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "ns \<noteq> []"   and "offs_pred ns ub xs index key mi ma"   and "offs_none ns ub xs index key mi ma i" shows "fill xs ns index key ub mi ma ! i = None"
c_def: fixes n :: "nat" shows "c n = (g_sm ^^ n) 0"
powr_mono2: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "0 \<le> a"   and "0 \<le> x"   and "x \<le> y" shows "x powr a \<le> y powr a"
simps_2: fixes x :: "'b equation" shows "get_sign (Neg x) = neg"
sizes_def: shows "sizes = map fst (filter (\<lambda>(n, e). e = ev) n_as)"
cols_transpose: fixes A :: "'a Matrix.mat" shows "cols A\<^sup>t = rows A"
permutations_of_multiset_singleton: fixes x :: "'a" shows "permutations_of_multiset {#x#} = {[x]}"
SeqSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and R :: "'e set"   and c2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and c1 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c2 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c1 R,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Seq c1 c2 Q,A"
positions_in_row: fixes s :: "('b \<times> bool list) stream"   and i :: "nat" shows "positions_in_row s i = {p. snd (s !! p) ! i}"
of_int_closed: fixes R :: "('b, 'c) ring_scheme"   and i :: "int" assumes "ring R" shows "\<guillemotleft>i\<guillemotright>\<^bsub>R\<^esub> \<in> carrier R"
vec_to_lpoly_from_lpoly_coeff_dual1: fixes p :: "linear_poly"   and i :: "nat" shows "Abstract_Linear_Poly.coeff (vec_to_lpoly (lpoly_to_vec p)) i = Abstract_Linear_Poly.coeff p i"
exhaust: fixes y :: "'b formula" obtains x1 :: "'b" where "y = Atom x1"   | "y = \<bottom>"   | x3 :: "'b formula" where "y = \<^bold>\<not> x3"   | x41 :: "'b formula" and x42 :: "'b formula" where "y = x41 \<^bold>\<and> x42"   | x51 :: "'b formula" and x52 :: "'b formula" where "y = x51 \<^bold>\<or> x52"   | x61 :: "'b formula" and x62 :: "'b formula" where "y = x61 \<^bold>\<rightarrow> x62"
wf_prog_wwf_prog: fixes P :: "'a J_prog" assumes "wf_J_prog P" shows "wwf_J_prog P"
G_simp: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and f :: "'d" assumes "left_adjoint_functor D C F"   and "partial_magma.arr C f" shows "left_adjoint_functor.G D C F f = terminal_arrow_from_functor.the_coext D C F (left_adjoint_functor.Go D C F (partial_magma.cod C f)) (left_adjoint_functor.\<epsilon>o D C F (partial_magma.cod C f)) (left_adjoint_functor.Go D C F (partial_magma.dom C f)) (C f (left_adjoint_functor.\<epsilon>o D C F (partial_magma.dom C f)))"
not_outside_3: fixes i :: "'agent"   and x :: "'alt"   and y :: "'alt" assumes "R i x y" shows "y \<in> alts"
is_ord_mkt_bal_r: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" assumes "is_ord (MKT n l r h)" shows "is_ord (mkt_bal_r n l r)"
eq_refl: fixes x :: "'a"   and y :: "'a"   and xa :: "'o" assumes "x = y" shows "le xa x y"
assms_11: shows "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<beta>)"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and sum :: "'c + 'd" shows "h (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case sum of Inl x \<Rightarrow> h (f1 x) | Inr x \<Rightarrow> h (f2 x))"
i_expand_gr0: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k" shows "f \<odot> k = (\<lambda>n. if n mod k = 0 then f (n div k) else NoMsg)"
order_neq_le_trans: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
wf_dual_lderiv_dual: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and embed :: "'a \<Rightarrow> 'a list"   and wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp_dual"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a" assumes "Pi_Derivatives.embed \<Sigma> project embed"   and "alphabet.wf_dual wf_atom n r" shows "alphabet.wf_dual wf_atom n (embed.lderiv_dual lookup embed w r)"
preserves_dom: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "local.dom (trg f) = trg (local.dom f)"
A_def: shows "A = lang (\<Uplus>{r. Trn X r \<in> rhs})"
auxhyp_def: fixes e :: "nat" shows "auxhyp e \<equiv> eval r_auxhyp [e]"
hyps_7: shows "x\<^sub>4 = l\<^sup>2 - x\<^sub>1 - x\<^sub>2"
hd_map: fixes xs :: "'g list"   and f :: "'g \<Rightarrow> 'h" assumes "xs \<noteq> []" shows "hd (map f xs) = f (hd xs)"
nat_pow_0: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.monoid G" shows "x [^]\<^bsub>G\<^esub> 0 = \<one>\<^bsub>G\<^esub>"
point_indices_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and t :: "int" assumes "incidence_system point_set block_collection" shows "incidence_system.point_indices point_set block_collection t \<equiv> {int (block_collection index ps) |ps. int (card ps) = t \<and> ps \<subseteq> point_set}"
proves_eq_abstract_rule_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"   and "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0))"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> (f $ Bv 0)) (Abs \<tau> (g $ Bv 0))"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'b" assumes "\<phi>.A.ide a"   and "\<phi>.A.ide a'"   and "\<guillemotleft>g : F a' \<rightarrow>\<^sub>B F a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow>\<^sub>A a\<guillemotright> \<and> F f = g"
blues_seen_others_1: fixes w :: "'person \<Rightarrow> 'color"   and p' :: "'person"   and p :: "'person" assumes "w p' = blue"   and "p \<noteq> p'"   and "w p = blue" shows "card (local.blues_seen w p) = card (local.blues_seen w p')"
simps_2: fixes a :: "('a, 'b) atm" shows "M.wtL (Neg a) = M.wtA a"
Spec_def: fixes ic :: "state \<Rightarrow> 'b"   and oc :: "state \<Rightarrow> 'b" shows "Spec ic oc \<equiv> \<exists>\<exists> q. ISpec ic q oc"
extends_append: fixes S :: "'b list"   and L :: "'b list" shows "S \<lesssim> L @ S"
has_sort_sort_leq_osig: fixes sub :: "(String.literal \<times> String.literal) set"   and tcs :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option"   and T :: "typ"   and S :: "String.literal set"   and S' :: "String.literal set" assumes "wf_osig (sub, tcs)"   and "has_sort (sub, tcs) T S"   and "sort_leq sub S S'" shows "has_sort (sub, tcs) T S'"
select_convs_3: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d" shows "tj_stack \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = tj_stack"
update_convs_1: fixes bar1' :: "'a \<Rightarrow> 'a"   and bar1 :: "'a"   and bar2 :: "'b"   and bar3 :: "'c"   and bar21 :: "'b \<times> 'a"   and bar32 :: "'c \<times> 'b"   and bar31 :: "'c \<times> 'a"   and more :: "'d" shows "bar1_update bar1' \<lparr>bar1 = bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr> = \<lparr>bar1 = bar1' bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr>"
mult_comp_trading_strat: fixes pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy pf1"   and "borel_predict_stoch_proc F qty" shows "trading_strategy (qty_mult_comp pf1 qty)"
R3': fixes x :: "'e"   and y :: "'f" assumes "T3 x y" shows "R3 (Rep3 y) x"
left_treeI: fixes G :: "('c, 'd) Graph_scheme"   and u :: "'c"   and s :: "'c"   and t :: "'c" assumes "Tree G"   and "u \<in> V\<^bsub>G\<^esub>"   and "s \<in> set (u \<leadsto>\<^bsub>G\<^esub> t)" shows "u \<in> Tree.left_tree G s t"
xor_one_right: fixes x :: "'a" shows "x \<oplus> \<one> = \<sim> x"
reflpI: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
circ_decompose_5: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> = (y\<^sup>\<circ> * x\<^sup>\<circ>)\<^sup>\<circ>"
green_typeI_chain_axioms: shows "green_typeI_chain i j F two_chain s"
simps_4: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "Language.normalize (Seq c\<^sub>1 c\<^sub>2) = sequence Seq (flatten (Language.normalize c\<^sub>1) @ flatten (Language.normalize c\<^sub>2))"
f1: shows "col M N P \<or> (incid A (line B' C') \<or> incid C' (line A B)) \<and> (incid C (line A' B') \<or> incid B' (line A C)) \<and> (incid B (line A' C') \<or> incid A' (line B C))"
fresh_3: fixes a :: "name"   and x1 :: "ty" shows "a \<sharp> T x1 = a \<sharp> x1"
rcset_inject: fixes x :: "'c cset"   and y :: "'c cset" shows "(rcset x = rcset y) = (x = y)"
collapse_2: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "stateful_strand_step.is_Receive stateful_strand_step" shows "receive\<langle>the_msg stateful_strand_step\<rangle> = stateful_strand_step"
phi_Entry_empty: fixes g :: "'g"   and v :: "'val" assumes "defNode g v = Entry g" shows "phi g v = None"
mult_1_right: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.monoid_mult one times" shows "times a one = a"