hcju/isabelleps huggingface dataset:
injD: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "inj f"   and "f x = f y" shows "x = y"
awalk_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "awalk u [] v = (u = v \<and> u \<in> verts G)"
list_of_oalist_tc_cases: fixes y :: "('a \<times> 'b) list" assumes "y \<in> Collect tc.oalist_inv_raw" obtains x :: "('a, 'b) oalist_tc" where "y = list_of_oalist_tc x"
ballH_1: shows "cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S \<subseteq> cball s \<epsilon>\<^sub>1 \<inter> S"
s_r_as_upd: shows "s = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [Rfork e]))"
fun_pair_eq_subst: fixes d :: "('fun, 'c) Term.term \<times> ('fun, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('fun, 'b) Term.term"   and d' :: "('fun, 'd) Term.term \<times> ('fun, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('fun, 'b) Term.term" shows "(pair d \<cdot> \<delta> = pair d' \<cdot> \<theta>) = (d \<cdot>\<^sub>p \<delta> = d' \<cdot>\<^sub>p \<theta>)"
Skip: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>Q Skip Q,A"
zeta_partial_sum_gt_1_asymp_equiv: fixes s :: "real" assumes "1 < s" shows "sum_upto (\<lambda>n. real n powr - s) \<sim>[at_top] (\<lambda>x. Re (zeta (complex_of_real s)))"
dghm_inc_components_1: fixes \<BB> :: "V"   and \<CC> :: "V" shows "dghm_inc \<BB> \<CC>\<lparr>ObjMap\<rparr> = vid_on (\<BB>\<lparr>Obj\<rparr>)"
mat_hom_mult: fixes A :: "int mat"   and nr :: "nat"   and n :: "nat"   and B :: "int mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "of_int_hom.mat_hom (A * B) = of_int_hom.mat_hom A * of_int_hom.mat_hom B"
inputPermFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "set p \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* P"   and "Ys \<sharp>* P"   and "Xs \<sharp>* M"   and "Ys \<sharp>* M" shows "p \<bullet> \<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"
is_unit_prod_mset_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'a multiset" assumes "class.algebraic_semidom divide plus minus zero times one" shows "dvd.dvd times (comm_monoid_mult.prod_mset times one A) one = (\<forall>x\<in>#A. dvd.dvd times x one)"
simps_2: fixes uu :: "nat list"   and uv :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "step ((Z, uu, uv) # fs, rv) = (fs, Some 0)"
linorder_not_le: fixes x :: "'b"   and y :: "'b" shows "(\<not> x \<le> y) = (y < x)"
AEv: shows "AE x in M. subcocycle_lim v x = - subcocycle_lim u x"
T_App: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmark.type"   and T\<^sub>1\<^sub>2 :: "POPLmark.type"   and t\<^sub>2 :: "trm" assumes "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1\<^sub>1 \<rightarrow> T\<^sub>1\<^sub>2"   and "\<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1\<^sub>1" shows "\<Gamma> \<turnstile> t\<^sub>1 \<bullet> t\<^sub>2 : T\<^sub>1\<^sub>2"
case_eq_if: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "(case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
induct: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'c list"   and x3 :: "'b"   and P :: "'b \<Rightarrow> 'c list \<Rightarrow> 'b \<Rightarrow> bool" assumes "rtrancl3p r x1 x2 x3"   and "\<And>a. P a [] a"   and "\<And>a bs a' b a''. \<lbrakk>rtrancl3p r a bs a'; P a bs a'; r a' b a''\<rbrakk> \<Longrightarrow> P a (bs @ [b]) a''" shows "P x1 x2 x3"
leD: fixes y :: "'b list"   and x :: "'b list" assumes "y \<le>s x" shows "\<not> x <s y"
wf_subst_elim_append: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a"   and t :: "('b, 'a) Term.term"   and w :: "'a" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "subst_elim \<theta> v"   and "v \<notin> fv t" shows "subst_elim (\<theta>(w := t)) v"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "splice (x # xs) ys = x # splice ys xs"
adapt_sigma_sets: fixes n :: "'b"   and X :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "\<And>i. i \<le> n \<Longrightarrow> X i \<in> M \<rightarrow>\<^sub>M N" shows "sigma_algebra (space M) (sigma_sets (space M) (\<Union>i\<in>{m. m \<le> n}. {X i -` A \<inter> space M |A. A \<in> sets N}))"
fps_right_inverse': fixes x :: "'a"   and f :: "'a fps"   and y :: "'a" assumes "x * fps_nth f 0 = (1::'a)"   and "fps_nth f 0 * y = (1::'a)" shows "f * fps_left_inverse f x = 1"
fin: shows "finite ((\<lambda>c. LEQ_ns (lec_poly c) (lec_const c)) ` cs)"
shadow_root_ptr_kinds_eq_h3: shows "shadow_root_ptr_kinds h' = shadow_root_ptr_kinds h3"
multiset_of_partition_cards_respects_domain_and_range_permutation: fixes B :: "'c set"   and A :: "'d set" assumes "finite B" shows "(\<lambda>f. image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}))) respects domain_and_range_permutation A B"
bij_betw_inv_into_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A' :: "'d set"   and B :: "'c set"   and B' :: "'d set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "bij_betw (inv_into A f) B' B"
dep_not_circuit: fixes X :: "'a set" assumes "X \<subseteq> carrier"   and "\<not> indep X"   and "\<not> circuit X" shows "\<exists>x\<in>X. \<not> indep (X - {x})"
deadlocked'_def: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" shows "multithreaded_base.deadlocked' final r s \<equiv> \<forall>t. final_thread.not_final_thread final s t \<longrightarrow> t \<in> multithreaded_base.deadlocked final r s"
lprefix_ltlI: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lprefix xs ys" shows "lprefix (ltl xs) (ltl ys)"
prfOf_PPf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "prf \<in> proof"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prfOf prf \<phi>" shows "bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
SD_efficient_def': fixes p :: "'alt pmf" shows "SD_efficient R p = (\<not> (\<exists>q\<in>lotteries. (\<forall>i\<in>agents. SD (R i) p q) \<and> (\<exists>i\<in>agents. p \<prec>[SD (R i)] q)))"
valid_countings_set: fixes a :: "nat"   and b :: "nat" shows "valid_countings a b = card {V \<in> Pow {0..<a + b}. card V = a \<and> (\<forall>m\<in>{1..a + b}. m - card ({0..<m} \<inter> V) < card ({0..<m} \<inter> V))}"
abs_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (numeral.numeral one plus n) = numeral.numeral one plus n"
empty_def: shows "Trie2.empty = Trie2.trie.Trie empty_trie"
crel_vs_elim: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and dp :: "'a \<Rightarrow> 'c"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and v :: "'d"   and v\<^sub>T :: "('b, 'e) state"   and M :: "'b" assumes "dp_consistency lookup update P"   and "dp_consistency.crel_vs lookup P dp R v v\<^sub>T"   and "dp_consistency.cmem lookup dp M"   and "P M" obtains v' :: "'e" and M' :: "'b" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "dp_consistency.cmem lookup dp M'" and "P M'"
inject_4: fixes x4 :: "key"   and y4 :: "key" shows "(Key x4 = Key y4) = (x4 = y4)"
extremum_uniqueI: fixes a :: "'g" assumes "a \<le> \<bottom>" shows "a = \<bottom>"
assms_6: shows "rk {C, D, P} \<le> 2"
leg0_commutes: shows "r.cod.leg0 \<cdot> r.chine = r0"
linear_imp_scale: fixes D :: "complex \<Rightarrow> 'd" assumes "clinear D" obtains d :: "'d" where "D = (\<lambda>x. x *\<^sub>C d)"
of_rat_val_def: fixes v :: "nat \<Rightarrow> rat"   and x :: "nat" shows "of_rat_val v x = real_of_rat (v x)"
in_rel_Gr: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g" shows "in_rel (BNF_Def.Gr A f) = BNF_Def.Grp A f"
abs_eq_0: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(abs a = zero) = (a = zero)"
nonempty_subgraphs_finite: fixes G :: "nat set \<times> nat set set" assumes "finite_graph G" shows "finite (nonempty_subgraphs G)"
less_imp_neq: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subset># y" shows "x \<noteq> y"
Int_stableI_image: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<exists>k\<in>I. A i \<inter> A j = A k" shows "Int_stable (A ` I)"
Derivs_simps_3: fixes s1 :: "'a list"   and s2 :: "'a list"   and A :: "'a list set" shows "Derivs (s1 @ s2) A = Derivs s2 (Derivs s1 A)"
LOC: fixes s :: "'s"   and ce :: "'s multiset" assumes "c = {#s#} + ce"   and "((s, ce), e, s', ce') \<in> gtrs" shows "P"
Pfunc_def: fixes A :: "'c set"   and B :: "'d set" shows "Pfunc A B \<equiv> {f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and> (\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
left_mat_compose: fixes ws :: "wall" assumes "is_tangle_diagram ws"   and "codomain_wall ws = 0" shows "kauff_mat ws = kauff_mat (ws \<circ> basic [])"
abs_eq: shows "Uprod.not_equal = Abs_commute (\<noteq>)"
induced_modulusD: fixes d :: "nat"   and a :: "nat" assumes "induced_modulus d"   and "coprime a n"   and "[a = 1] (mod d)" shows "\<chi> a = 1"
cast_inversions_2: fixes x :: "('b, 'a, 'c) unboxed"   and n :: "'a" assumes "cast_Ubx1 x = Some n" shows "x = OpUbx1 n"
uadmit_prog_fml_adjoint'_1: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<alpha> :: "('sf, 'sc, 'sz) hp" assumes "ssafe \<sigma>"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP \<alpha>)))"   and "hpsafe \<alpha>" shows "prog_sem (local.adjoint I \<sigma> \<nu>) \<alpha> = prog_sem (local.adjoint I \<sigma> \<omega>) \<alpha>"
flip_at_base_simps_4: fixes a :: "'b"   and b :: "'b"   and x :: "'c" assumes "sort_of (atom a) \<noteq> sort_of (atom b)" shows "(a \<leftrightarrow> b) \<bullet> x = x"
elims: fixes x :: "('c, 'b) arr"   and y :: "'b" assumes "Map x = y" obtains uu_ :: "'c" and uv_ :: "'c" and F :: "'b" where "x = MkArr uu_ uv_ F" and "y = F"   | "x = Null" and "y = undefined"
c: fixes i :: "Quotient_Int.int" assumes "P i" shows "P (i - 1)"
unaffected_domains_single_dom: fixes X :: "'b set"   and D :: "'b \<Rightarrow> 'c"   and I :: "('c \<times> 'c) set"   and u :: "'c"   and xs :: "'b list" shows "{x \<in> X. D x \<in> unaffected_domains I D {u} xs} = ipurge_ref I D u xs X"
minus_vec_def: fixes v\<^sub>1 :: "'b Matrix.vec"   and v\<^sub>2 :: "'b Matrix.vec" shows "v\<^sub>1 - v\<^sub>2 \<equiv> Matrix.vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $v i - v\<^sub>2 $v i)"
fmsubset_drop_fset: fixes S :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmdrop_fset S m \<subseteq>\<^sub>f m"
measurable_restrict_subset: fixes J :: "'a set"   and L :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" assumes "J \<subseteq> L" shows "(\<lambda>f. restrict f J) \<in> Pi\<^sub>M L M \<rightarrow>\<^sub>M Pi\<^sub>M J M"
integral_swap_continuous: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "continuous_on (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y)" shows "integral (cbox a b) (\<lambda>x. integral (cbox c d) (f x)) = integral (cbox c d) (\<lambda>y. integral (cbox a b) (\<lambda>x. f x y))"
lq_emp: fixes u :: "'b list" shows "\<epsilon>\<inverse>\<^sup>>u = u"
delay_def: fixes i :: "nat"   and x :: "nat" shows "delay i x \<equiv> Some (if \<exists>j<e_length x. 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x]) then Suc (GREATEST j. j < e_length x \<and> 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x])) else 0)"
max_min_same_1: fixes x :: "'b"   and y :: "'b" shows "max x (min x y) = x"
ae: shows "\<forall>x\<in>space M. Pxy (X x, Y x) = Px (X x) * Py (Y x)"
image_iff: fixes z :: "'g"   and f :: "'h \<Rightarrow> 'g"   and A :: "'h set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
sub_id: fixes p :: "('b, 'c) fm" shows "sub id p = p"
opI: fixes a :: "'a" assumes "a \<in> S" shows "\<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. P x \<longrightarrow> P y)"
zI: shows "(SOME z. z \<notin> lesvars upds \<union> varacom ({A} WHILE b DO C)) \<notin> support I"
impWitnessPath_def: fixes c :: "('loc, 't, 'a) configuration_scheme"   and loc1 :: "'loc"   and loc2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" shows "impWitnessPath c loc1 loc2 xs t = (path loc1 loc2 xs \<and> distinct xs \<and> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc1) \<and> t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) xs)) \<and> (\<forall>k<length xs. \<exists>t. t \<in>\<^sub>A frontier (c_imp c (case xs ! k of (s, l, t) \<Rightarrow> t)) \<and> t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) (take (k + 1) xs))))))"
hyperbola_method_sqrt: fixes x :: "real"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "0 \<le> x" shows "sum_upto (dirichlet_prod f g) x = sum_upto (\<lambda>n. f n * sum_upto g (x / real n)) (sqrt x) + sum_upto (\<lambda>n. sum_upto f (x / real n) * g n) (sqrt x) - sum_upto f (sqrt x) * sum_upto g (sqrt x)"
unique_to_invertible: fixes R' :: "('a sequent list \<times> 'a sequent) set"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "R' \<subseteq> upRules \<and> R = Ax \<union> R'"   and "uniqueConclusion R'" shows "invertible_set R"
remove_max_multiset_size: fixes l :: "'b"   and m :: "'a"   and l' :: "'b" assumes "\<not> is_empty l"   and "inv l"   and "(m, l') = remove_max l" shows "size (multiset l') < size (multiset l)"
simps: fixes a1 :: "'d Heap"   and a2 :: "heap option"   and a3 :: "heap option"   and a4 :: "'d" shows "run a1 a2 a3 a4 = ((\<exists>\<sigma> c r. a1 = c \<and> a2 = \<sigma> \<and> a3 = \<sigma> \<and> a4 = r \<and> is_exn \<sigma>) \<or> (\<exists>\<sigma> c r. a1 = c \<and> a2 = \<sigma> \<and> a3 = None \<and> a4 = r \<and> \<not> is_exn \<sigma> \<and> execute c (the_state \<sigma>) = None) \<or> (\<exists>\<sigma> c r h'. a1 = c \<and> a2 = \<sigma> \<and> a3 = Some h' \<and> a4 = r \<and> \<not> is_exn \<sigma> \<and> execute c (the_state \<sigma>) = Some (r, h')))"
gf_U'_nonneg: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "0 < z"   and "z < 1" shows "0 \<le> gf_U' x y z"
path_connected_continuous_image: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on S f"   and "path_connected S" shows "path_connected (f ` S)"
sum_power2_ge_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (plus (power.power one times x 2) (power.power one times y 2))"
mult_closed: fixes V :: "'b set"   and x :: "'b"   and a :: "real" assumes "vectorspace V"   and "x \<in> V" shows "a \<cdot> x \<in> V"
roofed_def: fixes G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set" shows "roofed_gen G B S = {x. \<forall>p. \<forall>y\<in>B. path G x p y \<longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S}"
AnalBinds_delete_to_fun_upd: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "FBinds (delete x \<Gamma>)\<cdot>ae = (FBinds \<Gamma>\<cdot>ae)(x := \<bottom>)"
poly_add_coeff_aux: fixes p2 :: "'a list"   and p1 :: "'a list" assumes "length p2 \<le> length p1" shows "local.coeff (poly_add p1 p2) = (\<lambda>i. local.coeff p1 i \<oplus> local.coeff p2 i)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and color :: "color" shows "P (case color of Red \<Rightarrow> f1 | Black \<Rightarrow> f2) = ((color = Red \<longrightarrow> P f1) \<and> (color = Black \<longrightarrow> P f2))"
game_sem_loop_unwind: fixes I :: "interp"   and \<alpha> :: "game" shows "game_sem I \<alpha>** = (\<lambda>X. X \<union> game_sem I (\<alpha> ;; \<alpha>**) X)"
simps_1: fixes k :: "'c" shows "delete_aux k [] = []"
Loss_lengthOut2: fixes a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and x :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list" assumes "\<forall>t. length (a t) \<le> Suc 0"   and "Loss lose a i2 x i" shows "\<forall>t. length (x t) \<le> Suc 0"
rel_conversep: fixes R1 :: "'c \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_prod R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_prod R1 R2)\<inverse>\<inverse>"
mat0: fixes nr :: "nat"   and nc :: "nat"   and ze :: "'c" shows "mat nr nc (mat0I ze nr nc)"
Cons_6: shows "\<forall>e\<in>pred c2. finite (Bexp.vars e)"
simps_26: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b"   and x5 :: "char list" shows "(case Class x5 of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f5 x5"
vec_to_iarray_row: fixes i :: "'d"   and A :: "(('b, 'c) vec, 'd) vec" shows "vec_to_iarray (row i A) = row_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)"
ASSIGN_def: fixes x :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and e :: "'d \<Rightarrow> 'e" shows "ASSIGN x e \<equiv> action_of (R (true \<turnstile> \<lambda>(S, S'). tr S' = tr S \<and> \<not> wait S' \<and> alpha_rp.more S' = update x (\<lambda>_. e (alpha_rp.more S)) (alpha_rp.more S)))"
B_eval2: fixes g :: "dB"   and f :: "dB" shows "\<B> \<degree> g \<degree> f \<rightarrow>\<^sub>\<beta>\<^sup>* Abs (lift g 0 \<degree> (lift f 0 \<degree> Var 0))"
assms: shows "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2"
assms_7: fixes e' :: "pexpr"   and n :: "nat" assumes "e = PExpr2 (PPow e' n)" shows "P"
le_st_def: fixes F :: "'a Abs_State.st"   and G :: "'a Abs_State.st" shows "(F \<sqsubseteq> G) = (\<forall>x\<in>set (Abs_State.dom G). lookup F x \<sqsubseteq> fun G x)"
below_trans_cong: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'d" assumes "a \<sqsubseteq> f x"   and "x \<sqsubseteq> y"   and "cont f" shows "a \<sqsubseteq> f y"
power_eq_0_iff: fixes a :: "'a"   and n :: "nat" shows "(a ^ n = (0::'a)) = (a = (0::'a) \<and> 0 < n)"
add: fixes f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'b" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
implied_frontier_alt_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c, 'd) configuration_scheme"   and loc :: "'a" assumes "dataflow_topology summary results_in" shows "dataflow_topology.implied_frontier_alt summary results_in c loc = frontier (\<Sum>loc'\<in>UNIV. dataflow_topology.after_summary results_in (zmset_of (mset_set (set_antichain (frontier (c_pts c loc'))))) (graph.path_weight summary loc' loc))"
pcancel1_cancel1_confluent: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "pcancel1 xs ys"   and "cancel1 zs ys" shows "\<exists>us. cancel us xs \<and> pcancel1 us zs"
Card_order_empty: fixes r :: "('n \<times> 'n) set" assumes "Card_order r" shows "|{}| \<le>o r"
JFcol_natural: fixes u :: "'d \<Rightarrow> 'e"   and n :: "nat" shows "\<forall>b1 b2. u ` JF1col n b1 = JF1col n (JF1map u b1) \<and> u ` JF2col n b2 = JF2col n (JF2map u b2)"
ConsCons2: fixes e1 :: "'a \<times> 'a"   and e2 :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list" assumes "\<not> (e1 = (fst e, w) \<and> e2 = (w, snd e)) \<and> e1 = (snd e, w) \<and> e2 = (w, fst e)"   and "P e w es" shows "P e w (e1 # e2 # es)"
prems_2: shows "region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'} \<noteq> {}"
interchange_same_rows: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f" shows "interchange_rows A a a = A"
inv\<^sub>2_def: 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" shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L = (inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L) \<and> (L \<noteq> {} \<longrightarrow> (\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> {u \<in> U. real c / 2 < w u} \<noteq> {})) \<and> bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2)) \<and> 2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2) \<and> S \<subseteq> {u \<in> U. w u \<le> real c / 2} \<and> L \<subseteq> {u \<in> U. real c / 2 < w u})"
simps: fixes a1 :: "enat"   and a2 :: "enat" shows "((a1, a2) \<in> Le_enat) = ((\<exists>n. a1 = 0 \<and> a2 = n) \<or> (\<exists>m n k. a1 = eSuc m \<and> a2 = n + k \<and> (m, n) \<in> Le_enat \<and> k \<noteq> 0))"
unpadr_padr: fixes m :: "nat"   and v :: "'b vec" shows "vardim.unpadr m (v @\<^sub>v 0\<^sub>v m) = v"
codomain_nonnegative: fixes x :: "brick" shows "0 \<le> codomain x"
units_of_mult: fixes G :: "('a, 'b) monoid_scheme" shows "(\<otimes>\<^bsub>units_of G\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
wf_J\<^sub>1_mdecl_def: fixes P :: "nat exp prog"   and C :: "char list" shows "wf_J\<^sub>1_mdecl P C \<equiv> \<lambda>(M, b, Ts, T, body). \<not> sub_RI body \<and> (case b of Static \<Rightarrow> (\<exists>T'. P,Ts \<turnstile>\<^sub>1 body :: T' \<and> P \<turnstile> T' \<le> T) \<and> \<D> body \<lfloor>{..<length Ts}\<rfloor> \<and> \<B> body (length Ts) | NonStatic \<Rightarrow> (\<exists>T'. P,Class C # Ts \<turnstile>\<^sub>1 body :: T' \<and> P \<turnstile> T' \<le> T) \<and> \<D> body \<lfloor>{..length Ts}\<rfloor> \<and> \<B> body (length Ts + 1))"