hcju/isabelleps huggingface dataset:
Abs_traffic_inverse: fixes y :: "(cars \<Rightarrow> real) \<times> (cars \<Rightarrow> nat_int) \<times> (cars \<Rightarrow> nat_int) \<times> (cars \<Rightarrow> real \<Rightarrow> real) \<times> (cars \<Rightarrow> real) \<times> (cars \<Rightarrow> real)" assumes "y \<in> {ts. (\<forall>c. fst (snd ts) c \<sqinter> fst (snd (snd ts)) c = bot) \<and> (\<forall>c. 1 \<le> |fst (snd ts) c|) \<and> (\<forall>c. |fst (snd ts) c| \<le> 2) \<and> (\<forall>c. |fst (snd (snd ts)) c| \<le> 1) \<and> (\<forall>c. |fst (snd ts) c| + |fst (snd (snd ts)) c| \<le> 2) \<and> (\<forall>c. fst (snd (snd ts)) c \<noteq> bot \<longrightarrow> (\<exists>n. Rep_nat_int (fst (snd ts) c) \<union> Rep_nat_int (fst (snd (snd ts)) c) = {n, n + 1})) \<and> (\<forall>c. 0 < fst (snd (snd (snd (snd ts)))) c) \<and> (\<forall>c. 0 < snd (snd (snd (snd (snd ts)))) c)}" shows "Rep_traffic (Abs_traffic y) = y"
take_n_D: shows "take n D = E @ derivation_shift (take (n - m) F) 0 (Suc j)"
r_neg: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_group G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
inf_bot_right: fixes x :: "'b" shows "x \<sqinter> \<bottom> = \<bottom>"
fgmodule_is_module: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and s :: "'a \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z :: "'e" assumes "fgmodule_condition R f i s A z" shows "R module fgmodule R A z i f s"
lookup_sum: fixes pp :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'b"   and X :: "'d set"   and i :: "'c" shows "poly_mapping.lookup (sum pp X) i = (\<Sum>x\<in>X. poly_mapping.lookup (pp x) i)"
WT_restD_rfunc_usr: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and rest :: "('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme"   and y :: "'d"   and es :: "'f list"   and s' :: "'e"   and s :: "'e"   and x :: "'c" assumes "WT_rest \<I>_adv \<I>_usr I rest"   and "((y, es), s') \<in> set_spmf (rfunc_usr rest s x)"   and "x \<in> outs_\<I> \<I>_usr"   and "I s" shows "y \<in> responses_\<I> \<I>_usr x \<and> I s'"
alist_to_ns_def: fixes a :: "('b \<times> 'c) list" shows "alist_to_ns a = Bind a []"
lang_final: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and r :: "'b rexp"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat" assumes "project \<Sigma> project" shows "final r = ([] \<in> project.lang \<Sigma> project lookup n r)"
68018: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> y) \<sqinter> (x \<squnion> y \<sqinter> (z \<squnion> top)) = bot"
constructive_security2_axioms_def: fixes sim :: "nat \<Rightarrow> ('i, 'j, 'k, 'l) converter"   and \<I>_real :: "nat \<Rightarrow> ('i, 'j) \<I>"   and \<I>_ideal :: "nat \<Rightarrow> ('k, 'l) \<I>"   and w :: "nat \<Rightarrow> ('k \<Rightarrow> 'i) \<times> ('j \<Rightarrow> 'l)" shows "constructive_security2_axioms sim \<I>_real \<I>_ideal w \<equiv> \<exists>cnv. \<forall>\<eta>. wiring (\<I>_ideal \<eta>) (\<I>_real \<eta>) (cnv \<eta>) (w \<eta>) \<and> wiring (\<I>_ideal \<eta>) (\<I>_ideal \<eta>) (cnv \<eta> \<odot> sim \<eta>) (id, id)"
multiplicity_prime_cong_1_mod_4_aux: fixes p :: "gauss_int"   and c :: "nat" assumes "prime_elem p"   and "0 < ReZ p"   and "0 < ImZ p"   and "ImZ p \<noteq> ReZ p" shows "multiplicity p (of_nat c) = multiplicity (gauss_int_norm p) c"
maintained_holds: fixes G :: "('b, 'a) labeled_graph"   and e\<^sub>L :: "'b allegorical_term"   and e\<^sub>R :: "'b allegorical_term" assumes ":G:\<lbrakk>e\<^sub>L\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>R\<rbrakk>" shows "maintained (transl_rule (e\<^sub>L \<sqsubseteq> e\<^sub>R)) G"
Rep_coreflexive: fixes x :: "'b coreflexive" shows "Rep_coreflexive x \<in> \<down>(1::'b)"
cmult_in_iff: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>x. c * f x) \<in> \<Omega>[F](g)) = (f \<in> \<Omega>[F](g))"
length_eq_card_columns: fixes A :: "(('b, 'c) vec, 'd) vec" shows "IArray.length (matrix_to_iarray A !! 0) = CARD('c)"
S4: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "AxiomsSet6" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
lNact_active: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "singleton active" shows "dynamic_component.lNAct active (singleton.the_singleton active) t n = n"
vector_scaleR_matrix_ac: fixes x :: "(real, 'b) vec"   and k :: "real"   and A :: "((real, 'a) vec, 'b) vec" shows "x v* k *\<^sub>R A = k *\<^sub>R (x v* A)"
pair_wf_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_wf_digraph (subdivide G e w)"
det_selection_funD: fixes f :: "(nat \<times> 'a) list \<Rightarrow> nat"   and xs :: "(nat \<times> 'a) list" assumes "det_selection_fun f"   and "xs \<noteq> []" shows "f xs \<in> fst ` set xs"
rep_iso: fixes x :: "'a\<cdot>('b, 'c) stateT" shows "stateT_abs\<cdot>(stateT_rep\<cdot>x) = x"
928: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (- x \<sqinter> y) = - x \<sqinter> y"
pos_le_minus_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a \<le> - (b /\<^sub>R c)) = (c *\<^sub>R a \<le> - b)"
finite_norm_ACI_juncts_1: fixes B :: "('a, 'k) aformula set"   and f :: "('a, 'k) aformula \<Rightarrow> ('a, 'k) aformula" assumes "finite B" shows "finite {f \<phi> |\<phi>. nf_ACI \<phi> \<and> disjuncts \<phi> \<subseteq> B}"
poly_pos_between_less_less: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a < x \<and> x < b \<longrightarrow> 0 < poly p x) = (b \<le> a \<or> p \<noteq> 0 \<and> 0 < poly p ((a + b) / 2) \<and> count_roots_between p a b = (if poly p b = 0 then 1 else 0))"
c4: assumes "sqn rt ip = \<pi>\<^sub>2 r"   and "the (flag rt ip) = Aodv_Basic.inv" shows "P (rt(ip \<mapsto> r))"
O\<alpha>\<beta>: shows "Ord (\<alpha> \<up> \<beta>)"
sinks_aux_cons: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'c"   and xs :: "'c list" shows "sinks_aux I D U (x # xs) = (if \<exists>v\<in>U. (v, D x) \<in> I then sinks_aux I D (insert (D x) U) xs else sinks_aux I D U xs)"
OAlist_tc_lookup_inj: fixes xs :: "('b, 'c) oalist_tc"   and ys :: "('b, 'c) oalist_tc" assumes "OAlist_tc_lookup xs = OAlist_tc_lookup ys" shows "xs = ys"
CFG_intra_edge_no_Def_equal: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list"   and a :: "'g"   and V :: "'i"   and s :: "(('i \<Rightarrow> 'j option) \<times> 'k) list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "V \<notin> Def (sourcenode a)"   and "intra_kind (kind a)"   and "CFG.pred (kind a) s" shows "fst (hd (CFG.transfer procs (kind a) s)) V = fst (hd s) V"
R2_def: shows "R2 = {(W1, W2). \<exists>c1 c1' b b'. W1 = [while b do c1 od] \<and> W2 = [while b' do c1' od] \<and> [c1] \<approx>\<^bsub>d\<^esub> [c1'] \<and> b \<equiv>\<^bsub>d\<^esub> b'}"
cong_dvd_modulus: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and m :: "'b"   and n :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment"   and "unique_euclidean_semiring.cong modulo x y m"   and "dvd.dvd times n m" shows "unique_euclidean_semiring.cong modulo x y n"
True: fixes F :: "('l \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
Stirling_fps_nth: fixes m :: "nat"   and n :: "nat" shows "Stirling_fps m $ n = real (Stirling n m) / fact n"
ring_hom_ringI: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring R"   and "ring S"   and "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>" shows "ring_hom_ring R S h"
assertion_prop: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf (times x top) one = x"
domains_def: fixes f :: "('o, 'a) arr" shows "domains f \<equiv> {a. ide a \<and> COMP f a \<noteq> null}"
ssap: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {inp}" shows "ssap \<phi> \<in> F"
cod_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "partial_magma C" shows "partial_magma.cod C f = (if partial_magma.codomains C f \<noteq> {} then SOME b. b \<in> partial_magma.codomains C f else partial_magma.null C)"
outs'_gpv_Fail: shows "outs'_gpv Fail = {}"
lexordp_iff: fixes xs :: "'d word list"   and ys :: "'d word list" shows "signed.lexordp xs ys = ((\<exists>x vs. ys = xs @ x # vs) \<or> (\<exists>us a b vs ws. a <s b \<and> xs = us @ a # vs \<and> ys = us @ b # ws))"
isomorphic_implies_hpar_2: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong> f'" shows "ide f'"
pa_det_while_algo: fixes H1 :: "('a, 'b) hashedTa"   and H2 :: "('c, 'b) hashedTa" assumes "hashedTa H1"   and "hashedTa H2"   and "hta_has_idx_s H1"   and "hta_has_idx_sf H2" shows "det_while_algo (pa_det_algo H1 H2)"
bounded_linear_intro: fixes f :: "'b \<Rightarrow> 'c"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_linear f"
pullback_arr_cod_1: fixes f :: "'a" assumes "C.arr f" shows "C.inverse_arrows \<p>\<^sub>1[f, C.cod f] \<langle>C.dom f \<lbrakk>f, C.cod f\<rbrakk> f\<rangle>"
hb_broadcast_exists1: fixes m1 :: "'c"   and m2 :: "'c" assumes "trivial_network.hb m1 m2" shows "\<exists>i. Broadcast m1 \<in> set []"
self_le_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq one a"   and "0 < n" shows "less_eq a (power.power one times a n)"
monotone_w: fixes i :: "int"   and i' :: "int"   and j :: "int"   and j' :: "int" assumes "i \<le> i'"   and "i' < j"   and "j \<le> j'" shows "w i' j \<le> w i j'"
assms_4: fixes x :: "'c" assumes "x \<in> space P" shows "0 \<le> Py x"
Iagree_refl: fixes I :: "('d, 'e, 'f) interp"   and A :: "('d + 'e + 'f) set" shows "Iagree I I A"
pcompose_eq_0: fixes p :: "'a poly"   and q :: "'a poly" assumes "p \<circ>\<^sub>p q = 0"   and "0 < degree q" shows "p = 0"
insert_Key_singleton: fixes K :: "nat"   and H :: "msg set" shows "insert (Key K) H = Key ` {K} \<union> H"
tagged_division_ofI: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "finite s"   and "\<And>x K. (x, K) \<in> s \<Longrightarrow> x \<in> K"   and "\<And>x K. (x, K) \<in> s \<Longrightarrow> K \<subseteq> i"   and "\<And>x K. (x, K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"   and "\<And>x1 K1 x2 K2. \<lbrakk>(x1, K1) \<in> s; (x2, K2) \<in> s; (x1, K1) \<noteq> (x2, K2)\<rbrakk> \<Longrightarrow> interior K1 \<inter> interior K2 = {}"   and "\<Union> {K. \<exists>x. (x, K) \<in> s} = i" shows "s tagged_division_of i"
FGModuleHomSetD_FGModuleHom: fixes T :: "'a \<Rightarrow> 'b"   and G :: "'c set"   and fgsmult :: "('d, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set"   and fgsmult' :: "('d, 'c) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and W :: "'b set" assumes "T \<in> FGModuleHomSet G fgsmult V fgsmult' W" shows "FGModuleHom G fgsmult V fgsmult' T"
A34_ub: fixes t :: "'b tree"   and a :: "'b" assumes "bst t" shows "S34.A a t \<le> 3 / 2 * \<phi> t + 1"
ring_distribs_2: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "(a + b) \<cdot> c = a \<cdot> c + b \<cdot> c"
shuffle_fisher_yates_step: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "shuffle xs = pmf_of_set {..<length xs} \<bind> (\<lambda>i. let ys = swap xs 0 i in shuffle (tl ys) \<bind> (\<lambda>zs. return_pmf (hd ys # zs)))"
sup_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "sup x y = Sup (insert x (\<eta> y))"
islist_def: fixes h :: "'b \<Rightarrow> 'b ref"   and p :: "'b ref" shows "islist h p = (\<exists>as. List h p as)"
less_sfirst_iff: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and n :: "nat" assumes "((\<lambda>xs. \<not> P xs) until alw P) \<omega>" shows "(enat n < sfirst P \<omega>) = (\<not> P (sdrop n \<omega>))"
cases: fixes x :: "('c, 'd, 'a, 'b, 'e) qAbs" obtains xs :: "'a" and xa :: "'b" and X :: "('c, 'd, 'a, 'b, 'e) qTerm" where "x = qAbs xs xa X"
sum: fixes a :: "real"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "a *\<^sub>R sum g A = (\<Sum>x\<in>A. a *\<^sub>R g x)"
new_psubst_image'_1: fixes c :: "'a"   and t :: "'a FOL_Fitting.term"   and d :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "new_term c t"   and "d \<notin> f ` paramst t" shows "new_term d (psubstt (f(c := d)) t)"
par_qmsg_oreachable: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and \<zeta> :: "'e \<times> 'f list \<times> ('f list, 'f, unit, unit label) seqp"   and A :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and i :: "nat"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "(\<sigma>, \<zeta>) \<in> oreachable (A \<langle>\<langle>\<^bsub>i\<^esub> qmsg) (otherwith S {i} (orecvmsg R)) (other U {i})"   and "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). U (\<sigma> i) (\<sigma>' i))"   and "\<And>\<xi>. U \<xi> \<xi>"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'"   and "\<And>\<sigma> \<sigma>' m. \<lbrakk>\<forall>j. U (\<sigma> j) (\<sigma>' j); R \<sigma> m\<rbrakk> \<Longrightarrow> R \<sigma>' m" shows "(\<sigma>, fst \<zeta>) \<in> oreachable A (otherwith S {i} (orecvmsg R)) (other U {i}) \<and> snd \<zeta> \<in> reachable qmsg (recvmsg (R \<sigma>)) \<and> (\<forall>m\<in>set (fst (snd \<zeta>)). R \<sigma> m)"
MS_zero: fixes k :: "nat"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int"   and l :: "int"   and M :: "Var list \<times> (int \<times> Instr) list \<times> int" assumes "k = 0"   and "t = s"   and "ll = l" shows "(M, l, s, k, ll, t) \<in> MStep"
fail_lengths: shows "length (io_init @ [io_last]) = length (tr2 @ [fst FAIL]) \<and> length (tr2 @ [fst FAIL]) = length (tr1_init @ [snd FAIL])"
assms_9: shows "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"
emeasure_Un_Int: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A + emeasure M B = emeasure M (A \<union> B) + emeasure M (A \<inter> B)"
hyps: shows "poly_mult (poly_mult (local.monom a n) p) q = poly_mult (local.monom a n) (poly_mult p q)"
eventually_pos_imp_nz: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. (0::'b) < f x" shows "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)"
select_convs_3: fixes g_V :: "'e set"   and g_E :: "('e \<times> 'e) set"   and g_V0 :: "'e set"   and more :: "'f" shows "g_V0 \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_V0"
evaldjf_ex: fixes bbs :: "bool list"   and bs :: "int list"   and f :: "'a \<Rightarrow> fm"   and ps :: "'a list" shows "Ifm bbs bs (evaldjf f ps) = (\<exists>p\<in>set ps. Ifm bbs bs (f p))"
ltake_LNil: fixes i :: "nat" shows "LNil \<down> i = LNil"
T_V_Cons: fixes as :: "'s list"   and os :: "'t list"   and s :: "'s"   and o\<^sub>1 :: "'t"   and t :: "'s"   and o :: "'t" assumes "length as = length os" shows "emeasure (T (s, o\<^sub>1)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip (t # as) (o # os) @- \<omega>'} = ennreal (pmf (\<O> t) o * pmf (\<K> s) t) * emeasure (T (t, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'}"
augment_cf_refine_aux: fixes p :: "(nat \<times> nat) list"   and u :: "nat"   and v :: "nat" assumes "f.isAugmentingPath p" shows "cf_of (f.augment (f.augmentingFlow p)) (u, v) = (if (u, v) \<in> set p then f.cf (u, v) - f.resCap p else if (v, u) \<in> set p then f.cf (u, v) + f.resCap p else f.cf (u, v))"
1: shows "(cmod (nu / du - nv / dv))\<^sup>2 = cmod (d * cnj d * kk * kk / (du * cnj du * dv * cnj dv))"
surj_on_def: fixes f :: "'b \<Rightarrow> 'c"   and Y :: "'c set" shows "surj_on f Y = (\<forall>y\<in>Y. \<exists>x. y = f x)"
Act\<^sub>\<alpha>_eq_iff: fixes \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha>" shows "(Formula.Act\<^sub>\<alpha> \<alpha>1 t1 = Formula.Act\<^sub>\<alpha> \<alpha>2 t2) = (Formula.Tree.tAct \<alpha>1 (Formula.rep_Tree\<^sub>\<alpha> t1) =\<^sub>\<alpha> Formula.Tree.tAct \<alpha>2 (Formula.rep_Tree\<^sub>\<alpha> t2))"
assms_4: shows "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s"
pair_list_aux: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "rb_aux_inv (bs, ss, ps)"   and "p \<in> set ps" shows "sig_of_pair p = lt (poly_of_pair p) \<and> poly_of_pair p \<noteq> 0 \<and> poly_of_pair p \<in> dgrad_sig_set' (length fs) dgrad"
power_odd_eq: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (Suc (2 * n)) = times a (power.power one times (power.power one times a n) 2)"
subset_UNIV: fixes A :: "'c set" shows "A \<subseteq> UNIV"
weight_spmf_conv_pmf_None: fixes p :: "'a spmf" shows "weight_spmf p = 1 - pmf p None"
shconfD: fixes P :: "'a prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state" assumes "P,h \<turnstile>\<^sub>s sh \<surd>"   and "sh C = \<lfloor>(sfs, i)\<rfloor>" shows "P,h,C \<turnstile>\<^sub>s sfs \<surd>"
cases: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "('e, 'g, 'i) Vardi.var"   and a2 :: "('f, 'h, 'i) Vardi.var" assumes "rel_var R S a1 a2" obtains x :: "('e \<times> 'f, 'g \<times> 'h, 'i) Vardi.var" where "a1 = map_var fst fst x" and "a2 = map_var snd snd x" and "set1_var x \<subseteq> {(x, y). R x y}" and "set2_var x \<subseteq> {(x, y). S x y}"
distinct_9: fixes x1 :: "'b"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "Atom x1 \<noteq> x61 \<^bold>\<rightarrow> x62"
is_span: shows "C.span \<nu>.leg0 \<nu>.leg1"
wlp_close: fixes S :: "com"   and P :: "complex mat" assumes "ps_P.well_com S"   and "ps_P.is_quantum_predicate P" shows "ps_P.is_quantum_predicate (ps_P.wlp S P)"
simps: fixes st :: "nat"   and m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_inv st m rs (l, r) = (if st = Suc 0 then wadjust_start m rs (l, r) else if st = Suc (Suc 0) then wadjust_loop_start m rs (l, r) else if st = Suc (Suc (Suc 0)) then wadjust_loop_right_move m rs (l, r) else if st = 4 then wadjust_loop_check m rs (l, r) else if st = 5 then wadjust_loop_erase m rs (l, r) else if st = 6 then wadjust_loop_on_left_moving m rs (l, r) else if st = 7 then wadjust_loop_right_move2 m rs (l, r) else if st = 8 then wadjust_erase2 m rs (l, r) else if st = 9 then wadjust_on_left_moving m rs (l, r) else if st = 10 then wadjust_goon_left_moving m rs (l, r) else if st = 11 then wadjust_backto_standard_pos m rs (l, r) else if st = 0 then wadjust_stop m rs (l, r) else False)"
assms_1: shows "find_sub_True xs = Some (ys, w, ws, z, zs)"
circ_decompose_7: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * y\<^sup>\<circ> * (x \<squnion> y)\<^sup>\<circ>"
ab: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s"
secure_def: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd" shows "secure P I D \<equiv> \<forall>xs y ys Y zs Z. (y # ys, Y) \<in> futures P xs \<and> (zs, Z) \<in> futures P xs \<longrightarrow> (ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> futures P xs \<and> (y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> futures P xs"
assoc_simps_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.trg (S.\<a> f g h) = S.trg f"
lists_eq_set: fixes A :: "'a set" shows "lists A = {xs. order.greater_eq A (set xs)}"
arrE: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" obtains "C.arr (fst f) \<and> C.arr (snd f)"
complement_1: fixes x :: "'b"   and y :: "'b" shows "! (d (! (d x))) = ! (d (! (d x) \<squnion> ! (d y))) \<squnion> ! (d (! (d x) \<squnion> ! (d (! (d y)))))"
assms: shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}"
det_row_0: fixes k :: "'b"   and b :: "'b \<Rightarrow> ('a, 'b) vec" shows "det (\<chi>i. if i = k then 0 else b i) = (0::'a)"
subst_apply_eq_Var: fixes s :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and x :: "'b" assumes "s \<cdot> \<sigma> = Var x" obtains y :: "'c" where "s = Var y" and "\<sigma> y = Var x"