hcju/isabelleps huggingface dataset:
cone_is_limit_if_pointwise_limit: fixes x :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and \<chi> :: "'j \<Rightarrow> ('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "YoD.cone x \<chi>"   and "\<forall>a. Cop.ide a \<longrightarrow> limit_cone J (\<cdot>\<^sub>S) (YoD.at a (local.map \<circ> D)) (\<chi>.\<Phi>.Ya.Cop_S.Map x a) (YoD.at a \<chi>)" shows "YoD.limit_cone x \<chi>"
right_neutral: fixes a :: "'a" shows "a \<odot> (1::'a) = a"
nat_pos2: fixes k :: "nat"   and r :: "nat" assumes "k < r" shows "0 < r"
assms_2: fixes x :: "'a" assumes "x \<in> subst_domain s" shows "fv (s x) = {}"
product_topology_basis': fixes I :: "'d set"   and U :: "'d \<Rightarrow> 'e set"   and x :: "'d \<Rightarrow> 'f" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> open (U i)" shows "open {f. \<forall>i\<in>I. f (x i) \<in> U i}"
Act_eqvt: fixes p :: "perm"   and f :: "'d"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "p \<bullet> FL_Formula.Act f \<alpha> x = FL_Formula.Act (p \<bullet> f) (p \<bullet> \<alpha>) (p \<bullet> x)"
add_PR_CONST_to_def: fixes x :: "'a"   and y :: "'a" assumes "x \<equiv> y" shows "PR_CONST x \<equiv> y"
piv: shows "pivot_fun (jordan_block n (0::'a) ^\<^sub>m k) (\<lambda>i. min (k + i) n) n"
eventually_inf_principal: fixes P :: "'e \<Rightarrow> bool"   and F :: "'e filter"   and s :: "'e set" shows "eventually P (inf F (principal s)) = (\<forall>\<^sub>F x in F. x \<in> s \<longrightarrow> P x)"
lambda_dvd_nu: fixes i :: "nat" assumes "seq (i + i) = seq i" shows "lambda dvd i"
square_free_monic_poly: fixes p :: "'c poly"   and x :: "'c" assumes "monic p" shows "(poly (yun_gcd.square_free_monic_poly gcd p) x = (0::'c)) = (poly p x = (0::'c))"
fBall_funion: fixes A :: "'a fset"   and B :: "'a fset"   and P :: "'a \<Rightarrow> bool" shows "fBall (A |\<union>| B) P = (fBall A P \<and> fBall B P)"
wi2l: fixes r :: "'a wordinterval" shows "(\<Union>(i, j)\<in>set (wi2l r). {i..j}) = wordinterval_to_set r"
assms_5: shows "Prj\<^sub>0 \<cdot> h = Prj\<^sub>0 \<cdot> h'"
DisjointPaths_empty: shows "DisjointPaths G v0 v1 {}"
sd_chain_lottery: fixes n :: "nat"   and q :: "'alt pmf" assumes "sd_chain n = Some q" shows "q \<in> lotteries_on alts"
var_prm_var_def: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and a :: "NominalSequents.var" shows "pi \<bullet> a \<equiv> var_prm_var pi a"
Rats_number_of: fixes w :: "num" shows "numeral w \<in> \<rat>"
measurable_count_space_eq_countable: fixes A :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "countable A" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
\<delta>_states_finite: fixes \<delta> :: "('a, 'b) ta_rule set" assumes "finite \<delta>" shows "finite (\<delta>_states \<delta>)"
thm_noncont_propos_4: fixes v :: "i" shows "[NonContingent (p\<^sub>0\<^sup>-) in v]"
prefix_elem_to_carriers: fixes xs :: "('a \<times> 'b) event list"   and i :: "nat"   and x :: "('a \<times> 'b) event" assumes "xs prefix of i"   and "x \<in> set xs" shows "x \<in> set (history i)"
Compl_Diff_eq: fixes A :: "'d set"   and B :: "'d set" shows "- (A - B) = - A \<union> B"
interior_atLeastLessThan: fixes a :: "real"   and b :: "real" shows "interior {a..<b} = {a<..<b}"
Says_ticket_analz: fixes S :: "agent"   and A :: "agent"   and K :: "nat"   and SesKey :: "msg"   and B :: "msg"   and TimeStamp :: "msg"   and Ticket :: "msg"   and evs :: "event list" assumes "Says S A \<lbrace>Crypt K \<lbrace>SesKey, B, TimeStamp\<rbrace>, Ticket\<rbrace> \<in> set evs" shows "Ticket \<in> analz (knows Spy evs)"
unreachable: shows "\<lparr>[$Unreachable]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
snd_swap: fixes x :: "'c \<times> 'd" shows "snd (prod.swap x) = fst x"
foundation10': fixes \<tau> :: "'b state \<times> 'b state"   and A :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> A and B) = (\<tau> \<Turnstile> A \<and> \<tau> \<Turnstile> B)"
map_graph_mem: fixes x :: "'c"   and y :: "'d"   and f :: "'c \<Rightarrow> 'd option" shows "((x, y) \<in> map_graph f) = (f x = Some y)"
assms_2: assumes "ps" shows "snd (rep_nat_term u) = snd (rep_nat_term v)"
tpOf_subst: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and arOf :: "'f \<Rightarrow> 'e list"   and resOf :: "'f \<Rightarrow> 'e"   and \<pi> :: "Preliminaries.var \<Rightarrow> 'f trm"   and T :: "'f trm" assumes "Signature TYPE('e) wtFsym wtPsym"   and "Signature.wtSB wtFsym arOf resOf \<pi>"   and "Signature.wt wtFsym arOf resOf T" shows "Signature.tpOf resOf (subst \<pi> T) = Signature.tpOf resOf T"
abs_eq_iff: fixes x :: "'b ltl"   and y :: "'b ltl" shows "(Abs x = Abs y) = x \<equiv>\<^sub>P y"
one_less_nat_eq: fixes z :: "int" shows "(Suc 0 < nat z) = (1 < z)"
com_sifum_secure_def: fixes cmd :: "('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)" shows "com_sifum_secure cmd \<equiv> case cmd of (c, mds\<^sub>s) \<Rightarrow> c \<sim>\<^bsub>mds\<^sub>s\<^esub> c"
order_char_2: fixes y :: "'a"   and x :: "'a" shows "(y \<le> x) = (y \<sqinter> - x \<le> x \<sqinter> - x)"
reads_consistent_flush_all_until_volatile_write_aux: fixes xs :: "'b memref list"   and pending_write :: "bool"   and \<O>' :: "nat set"   and m :: "nat \<Rightarrow> nat"   and ts :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c memref list \<times> 'd \<times> 'e \<times> 'f) list" assumes "outstanding_refs is_volatile_Read\<^sub>s\<^sub>b xs = {}"   and "reads_consistent pending_write \<O>' m xs"   and "\<forall>i<length ts. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts ! i in outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) xs \<inter> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) = {}" shows "reads_consistent pending_write \<O>' (flush_all_until_volatile_write ts m) xs"
a2: shows "set_offending_flows \<lparr>nodes = V, edges = insert e E\<rparr> nP \<noteq> {}"
N: shows "Max {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> j \<and> 0 < offs_num n xs index key mi ma i} \<in> {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> k \<and> 0 < offs_num n xs index key mi ma i}"
exit_room: fixes s :: "state"   and g :: "guest"   and r :: "room" assumes "s \<in> reach"   and "g \<in> state.isin s r" shows "s\<lparr>isin := (state.isin s)(r := state.isin s r - {g})\<rparr> \<in> reach"
sint_eq_uint: fixes x :: "'a word" assumes "\<not> msb x" shows "sint x = uint x"
simps_1: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "nnf (And \<phi>\<^sub>1 \<phi>\<^sub>2) = And (nnf \<phi>\<^sub>1) (nnf \<phi>\<^sub>2)"
get_dom_component_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c" shows "ptr \<in> set c"
r1_def: shows "r1 = {((pointsTo, pointsTo l'), pointsTo, Link l') |l'. \<forall>l''. pointsTo l' \<noteq> Link l''}"
analz_insert_analz: fixes c :: "msg"   and Z :: "msg"   and H :: "msg set" assumes "c \<notin> parts {Z}"   and "\<forall>K. Key K \<notin> parts {Z}"   and "c \<in> analz (insert Z H)" shows "c \<in> analz H"
uniform_limit_tendsto_bcontfun: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow>\<^sub>C 'e"   and l :: "'d \<Rightarrow>\<^sub>C 'e"   and F :: "'c filter" assumes "uniform_limit UNIV (\<lambda>x. apply_bcontfun (f x)) (apply_bcontfun l) F" shows "(f \<longlongrightarrow> l) F"
machine_star_implies_cbv_star: fixes e :: "lam"   and e' :: "lam" assumes "<e,[]> \<mapsto>* <e',[]>" shows " e \<longrightarrow>cbv* e'"
subst_upd_uvar_def: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and x :: "'c \<Longrightarrow> 'b"   and v :: "('c, 'a) uexpr" shows "\<sigma>(x \<mapsto>\<^sub>s v) = (\<lambda>b. put\<^bsub>x\<^esub> (\<sigma> b) (\<lbrakk>v\<rbrakk>\<^sub>e b))"
rev_fimage_eqI: fixes x :: "'b"   and A :: "'b fset"   and b :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "x |\<in>| A"   and "b = f x" shows "b |\<in>| f |`| A"
lateEarlyTau: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'" shows "P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'"
that: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "linear f"   and "linear g"   and "f ` S = T"   and "g ` T = S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) = norm x"   and "\<And>x. x \<in> T \<Longrightarrow> norm (g x) = norm x"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "\<And>x. x \<in> T \<Longrightarrow> f (g x) = x" shows "thesis"
drop_eq_Nil: fixes n :: "nat"   and xs :: "'d list" shows "(drop n xs = []) = (length xs \<le> n)"
limit_induct: fixes p :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and X :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "p \<in> limit f X"   and "\<And>p. p \<in> X \<Longrightarrow> P p"   and "\<And>p Y. \<lbrakk>\<And>q. q \<in> Y \<Longrightarrow> P q; p \<in> f Y\<rbrakk> \<Longrightarrow> P p" shows "P p"
is_aligned_shiftr_shiftl: fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n" shows "w >> n << n = w"
card_injective_functions_range_permutation: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // range_permutation A B) = iverson (card A \<le> card B)"
Rep_game_cases: fixes y :: "ZF" assumes "y \<in> game" obtains x :: "game" where "y = Rep_game x"
base: fixes s2 :: "nat"   and f :: "nat \<Rightarrow> nat"   and s1 :: "nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and r1 :: "nat \<Rightarrow>f value option"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and r2 :: "nat \<Rightarrow>f value option" assumes "s2 = f s1" shows "trace_simulation f e1 s1 r1 e2 s2 r2 []"
monotone_spmf: fixes x :: "'c" shows "monotone (ord_spmf (=)) (\<le>) (\<lambda>p. spmf p x)"
prism_2: fixes q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain q S c" shows "singular_chain (Suc q) U (prism q c)"
entails_iff_unsatisfiable: fixes CC :: "'b literal multiset set"   and EE :: "'b literal multiset set" shows "(CC \<TTurnstile>e EE) = (\<forall>E\<in>EE. \<not> satisfiable (CC \<union> {{#- L#} |L. L \<in># E}))"
Sup_lim: fixes b :: "nat \<Rightarrow> 'a"   and s :: "'a set"   and a :: "'a" assumes "\<And>n. b n \<in> s"   and "b \<longlonglongrightarrow> a" shows "a \<le> Sup s"
scons_eq_siterate: fixes a :: "'b"   and w :: "'b stream"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "(a ## w = siterate f s) = (a = s \<and> w = siterate f (f s))"
tDouble: assumes "invar l \<and> invar r \<and> lv = lvl r \<and> Suc (lvl l) = lvl r \<and> sngl r" shows "thesis"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "S.ide a"   and "S.seq f a" shows "f \<cdot>\<^sub>S a = f"
arr_viewed_from_def: fixes accC :: "qtname"   and C :: "qtname" shows "arr_viewed_from accC C = {accC,Base,True}StatRef (ClassT C)..arr"
proper_it_def: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it' :: "('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" shows "proper_it it it' \<equiv> \<exists>l. it = foldli l \<and> it' = foldli l"
inv\<^sub>2E_5: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" shows "S \<subseteq> {u \<in> U. w u \<le> real c / 2} \<and> L \<subseteq> {u \<in> U. real c / 2 < w u}"
has_vector_derivative_real_field: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and a :: "real"   and s :: "real set" assumes "(f has_field_derivative f') (at (of_real a))" shows "((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
0: shows "Cring_Poly.compose R (Cring_Poly.compose R f (monom (UP R) d 1)) (monom (UP R) (inv d) 1) x = inv d [^] x \<otimes> Cring_Poly.compose R f (monom (UP R) d 1) x"
cring_simprules_1: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
bernpoly_altdef: fixes n :: "nat" shows "bernpoly n = (\<lambda>x. \<Sum>k\<le>n. of_nat (n choose k) * of_real (bernoulli (n - k)) * x ^ k)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_in_fv_trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s: fixes x :: "'c"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F" shows "x \<in> fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F)"
containsUnacceptableEdgeSnd'_def: fixes N :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and "is" :: "nat list" shows "containsUnacceptableEdgeSnd' N is \<equiv> \<exists>k<|is| - 2. let i0 = is ! k; i1 = is ! (k + 1); i2 = is ! (k + 2) in N i1 i2 \<and> i0 < i1 \<and> i1 < i2"
not_essentialD: fixes G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set"   and x :: "'k"   and p :: "'k list"   and y :: "'k" assumes "\<not> essential G B S x"   and "path G x p y"   and "y \<in> B" shows "x \<noteq> y \<and> (\<exists>z\<in>set p. z \<noteq> x \<and> z \<in> S)"
eventually_nhds: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "eventually P (nhds a) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
pdouble_def: fixes a :: "int"   and p :: "int \<times> int \<times> int" shows "pdouble a p = (let (x, y, z) = p in if z = \<zero> then p else let l = \<guillemotleft>2\<guillemotright> \<otimes> y \<otimes> z; m = \<guillemotleft>3\<guillemotright> \<otimes> x [^] 2 \<oplus> a \<otimes> z [^] 2 in (l \<otimes> (m [^] 2 \<ominus> \<guillemotleft>4\<guillemotright> \<otimes> x \<otimes> y \<otimes> l), m \<otimes> (\<guillemotleft>6\<guillemotright> \<otimes> x \<otimes> y \<otimes> l \<ominus> m [^] 2) \<ominus> \<guillemotleft>2\<guillemotright> \<otimes> y [^] 2 \<otimes> l [^] 2, l [^] 3))"
wf_J_prog_eq_wf_J_prog': shows "wf_J_prog = wf_J_prog'"
vrat_of_rat_in_vratI: fixes a :: "rat" shows "a\<^sub>\<rat> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
EWHILEIT_rule: fixes R :: "('c \<times> 'c) set"   and I :: "'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> ('d + 'c) nres"   and E :: "'d \<Rightarrow> bool"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "wf R"   and "I s\<^sub>0"   and "\<And>s. \<lbrakk>I s; b s; (s, s\<^sub>0) \<in> R\<^sup>*\<rbrakk> \<Longrightarrow> f s \<le> ESPEC E (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s; (s, s\<^sub>0) \<in> R\<^sup>*\<rbrakk> \<Longrightarrow> \<Phi> s" shows "EWHILEIT I b f s\<^sub>0 \<le> ESPEC E \<Phi>"
check_prime_factors_subset_correct: fixes n :: "nat"   and ps :: "nat list" assumes "check_prime_factors_subset n ps"   and "list_all prime ps" shows "prime_factors n \<subseteq> set ps"
deg_pm_single: fixes x :: "'b"   and k :: "'a" shows "deg_pm (Poly_Mapping.single x k) = k"
TS_finite: fixes V :: "'e list set"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and m :: "nat"   and n :: "nat" assumes "finite V"   and "finite (inputs M2)" shows "finite (TS M2 M1 \<Omega> V m n)"
Pi_cong: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>w. w \<in> A \<Longrightarrow> f w = g w" shows "(f \<in> Pi A B) = (g \<in> Pi A B)"
borel_eq_halfspace_le: shows "borel = sigma UNIV ((\<lambda>(a, i). {x. x \<bullet> i \<le> a}) ` (UNIV \<times> Basis))"
healthy_intros_10: fixes f :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "nearly_healthy (wlp f)"   and "nearly_healthy (wlp g)" shows "nearly_healthy (wlp (f \<Sqinter> g))"
Times_mset_Times: fixes A :: "'b multiset"   and B :: "'c multiset" shows "set_mset (A \<times># B) = set_mset A \<times> set_mset B"
elt_set_times_def: fixes a :: "'b"   and B :: "'b set" shows "a *o B = {c. \<exists>b\<in>B. c = a * b}"
minus_unique: fixes y :: "'a list"   and x :: "'a list"   and y' :: "'a list" assumes "y \<oplus>\<^bsub>poly_ring R\<^esub> x = \<zero>\<^bsub>poly_ring R\<^esub>"   and "x \<oplus>\<^bsub>poly_ring R\<^esub> y' = \<zero>\<^bsub>poly_ring R\<^esub>"   and "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)"   and "y' \<in> carrier (poly_ring R)" shows "y = y'"
simps: fixes n :: "nat"   and A_pos :: "nat set list"   and A_neg :: "nat set list"   and L :: "'b option list set list" shows "mmulti_join n A_pos A_neg L = (if length A_pos + length A_neg \<noteq> length L then {} else let L_pos = take (length A_pos) L; L_neg = drop (length A_pos) L in case dominate_True A_pos L_pos of None \<Rightarrow> case dominate_False A_pos L_pos A_neg L_neg of None \<Rightarrow> mmulti_join' A_pos A_neg L | Some (((A_zs, A_x, A_xs), A_ws, A_y, A_ys), (zs, xa, xs), ws, y, ys) \<Rightarrow> mmulti_join n (A_zs @ A_x # A_xs) (A_ws @ A_ys) ((zs @ bin_join n A_x xa False A_y y # xs) @ ws @ ys) | Some ((A_zs, A_x, A_xs, A_y, A_ys), zs, xa, xs, y, ys) \<Rightarrow> mmulti_join n (A_zs @ (A_x \<union> A_y) # A_xs @ A_ys) A_neg ((zs @ bin_join n A_x xa True A_y y # xs @ ys) @ L_neg))"
Alist_inverse: fixes y :: "('b \<times> 'c) list" assumes "y \<in> {xs. (distinct \<circ> map fst) xs}" shows "alist.impl_of (Alist y) = y"
decr_grading_p_keys_subset: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (decr_grading_p d n p) \<subseteq> decr_grading_term d n ` keys p"
set_drop_if_index: fixes xs :: "'b list"   and x :: "'b"   and i :: "nat" assumes "distinct xs"   and "index xs x < i" shows "x \<notin> set (drop i xs)"
is_UP_monomE_2: fixes f :: "nat \<Rightarrow> 'a" assumes "is_UP_monom f" shows "f = monom P (f (deg R f)) (deg R f)"
homr_def: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "homr G H = {h \<in> hom G H. h \<in> extensional (carrier G)}"
underS_I: fixes i :: "'c"   and j :: "'c"   and R :: "('c \<times> 'c) set" assumes "i \<noteq> j"   and "(i, j) \<in> R" shows "i \<in> underS R j"
edges_sym': fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "(u, v) \<in> edges g" shows "(v, u) \<in> edges g"
Liminf_le_Limsup: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" assumes "F \<noteq> bot" shows "Liminf F f \<le> Limsup F f"
len: fixes f :: "'a poly" assumes "f \<noteq> 0" shows "length (coeffs f) = Suc (degree f)"
cp_valid: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" shows "(\<upsilon> X) \<tau> = (\<upsilon> (\<lambda>_. X \<tau>)) \<tau>"
test1: fixes g :: "nat list list" assumes "g \<in> set_tries gs" shows "\<exists>h\<in>set arch. iso_test g h"
elims_3: fixes x :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and xa :: "nat"   and xb :: "int vec list" assumes "\<not> LLL_impl_inv x xa xb" obtains f :: "int vec list \<times> int vec list" and mu :: "int iarray iarray" and ds :: "int iarray" and i :: "nat" and fs :: "int vec list" where "x = (f, mu, ds)" and "xa = i" and "xb = fs" and "\<not> (list_repr i f (map ((!) fs) [0..<m]) \<and> d_repr ds fs \<and> mu_repr mu fs)"
length_last_fusion_def: fixes g :: "('c, 'd) generator"   and s :: "'d" shows "length_last_fusion g s = length_last (list.unfoldr g s)"
piecewise_differentiable_on_subset: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a set" assumes "f piecewise_differentiable_on S"   and "T \<subseteq> S" shows "f piecewise_differentiable_on T"
lsc_liminf_at: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (f x0 \<le> Liminf (at x0) f)"
rep_mult_scalar: fixes c :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.pmdl.rep (MPoly_Type_Class.punit.monom_mult c 0 r) = local.punit.mult_scalar c (local.punit.pmdl.rep r)"
evnodd_finite: fixes A :: "(nat \<times> nat) set"   and b :: "nat" assumes "finite A" shows "finite (evnodd A b)"
convex_hull_set_sum: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "convex hull sum B A = (\<Sum>i\<in>A. convex hull B i)"
assms: shows "st \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) I"
AAcc1Throw1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and i :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<lfloor>i\<rceil>,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
assms_3: shows "(h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)"
hd_if: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and b :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e1 :: "expr"   and g1 :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and e2 :: "expr"   and g2 :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "({}, V \<union> V', \<Gamma>, \<lambda>_. 1) \<turnstile>\<^sub>d b \<Rightarrow> f"   and "(V, V', \<Gamma>, if_dens \<delta> f True) \<turnstile>\<^sub>d e1 \<Rightarrow> g1"   and "(V, V', \<Gamma>, if_dens \<delta> f False) \<turnstile>\<^sub>d e2 \<Rightarrow> g2" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d IF b THEN e1 ELSE e2 \<Rightarrow> (\<lambda>\<rho> x. g1 \<rho> x + g2 \<rho> x)"
index_transpose_mat_2: fixes A :: "'c mat" shows "dim_row A\<^sup>T = dim_col A"
drinks2_2_invalid: fixes a :: "String.literal \<times> value list"   and r :: "nat \<Rightarrow>f value option" assumes "fst a = STR ''coin'' \<longrightarrow> length (snd a) \<noteq> 1"   and "a \<noteq> (STR ''vend'', [])" shows "possible_steps drinks2 2 r (fst a) (snd a) = {||}"
lex_two: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s2 :: "('b \<times> 'b) set" assumes "ns O s \<subseteq> s"   and "SN s"   and "SN s2" shows "SN (lex_two s ns s2)"
sorted_merge_all: fixes xss :: "'b list list" assumes "\<forall>xs\<in>set xss. sorted xs" shows "sorted (merge_all xss)"
distinct_13: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "BOOL \<noteq> PRODUCT x51 x52"
sup_state_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "sup_state P \<equiv> Product.le (Listn.le (subtype P)) (Listn.le (sup_ty_opt P))"
prems_2: shows "P t (app w\<^sub>1 w\<^sub>2)"
sbintr_lt: fixes n :: "nat"   and w :: "int" shows "signed_take_bit n w < 2 ^ n"
subst_fm_IndP: fixes t :: "tm"   and i :: "name"   and x :: "tm" shows "(IndP t)(i::=x) = IndP (subst i x t)"
normalize_match_correct: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and a :: "action"   and m :: "'b match_expr"   and s :: "state" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) (normalize_match m)) s = approximating_bigstep_fun \<gamma> p [Rule m a] s"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(\<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>y1\<^bold>\<rangle>\<^sub>0) = (x1 = y1)"
proj_plus: fixes k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "proj_poly k (p + q) = proj_poly k p + proj_poly k q"