hcju/isabelleps huggingface dataset:
summatory_divisor_count_asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>n. real (divisor_count n)) x - (x * ln x + (2 * euler_mascheroni - 1) * x)) \<in> O(sqrt)"
assms: shows "\<forall>(k, \<gamma>)\<in>boundary twoC. valid_path \<gamma>"
no_leading_dropWhile: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "no_leading P (dropWhile P xs)"
measurable_restrict_mono: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and A :: "'a set"   and N :: "'b measure"   and B :: "'a set" assumes "f \<in> restrict_space M A \<rightarrow>\<^sub>M N"   and "B \<subseteq> A" shows "f \<in> restrict_space M B \<rightarrow>\<^sub>M N"
hensel_lifting_def: fixes p :: "int"   and n :: "nat"   and f :: "int poly"   and gs :: "int poly list" shows "hensel_lifting p n f gs = (let lc = lead_coeff f; ilc = inverse_mod lc (p ^ n); g = Polynomial.smult ilc f in hensel_lifting_monic p n g gs)"
fun_pair_subst: fixes d :: "('fun, 'b) Term.term \<times> ('fun, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('fun, 'a) Term.term" shows "pair d \<cdot> \<delta> = pair (d \<cdot>\<^sub>p \<delta>)"
integrable: fixes y :: "'a"   and a' :: "real"   and b :: "real" assumes "y \<in> A"   and "a \<le> a'"   and "a' \<le> b" shows "(\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
success_def: fixes f :: "'a Heap"   and h :: "heap" shows "success f h = (execute f h \<noteq> None)"
initState_subset_knows: fixes A :: "agent"   and evs :: "event list" shows "initState A \<subseteq> knows A evs"
quote_all_MonPon_PfP_ssubst: fixes \<alpha> :: "fm"   and \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> \<alpha> IMP \<beta>"   and "V \<subseteq> {}"   and "supp \<alpha> \<subseteq> atom ` {}"   and "supp \<beta> \<subseteq> atom ` {}" shows "quote_all 0 V \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V (make_F {} 0)) IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V (make_F {} 0))"
finlsts_predI: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "(r, a ## r) \<in> finlsts_pred"
openE: fixes S :: "'d set"   and x :: "'d" assumes "open S"   and "x \<in> S" obtains e :: "real" where "0 < e" and "ball x e \<subseteq> S"
prems: shows "eval (list_conj (map (fm.Atom \<circ> Eq) (p # A') @ map fm.Atom L @ F)) (xs @ x # \<Gamma>)"
compP\<^sub>1_pres_wf: fixes P :: "(char list list \<times> char list exp) prog" assumes "wf_J_prog P" shows "wf_J\<^sub>1_prog (compP\<^sub>1 P)"
arccos_arcsin_sqrt_pos: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "arccos x = arcsin (sqrt (1 - x\<^sup>2))"
Rep_inverse: fixes x :: "'a" shows "Abs (Rep x) = x"
is_zero_double_cases: fixes x :: "double" assumes "Code_Double.is_zero x" obtains "x = 0"   | "x = - 0"
unwrapExist: fixes j :: "nat"   and A :: "atom fm"   and L :: "real list" shows "eval (ExN (j + 1) A) L = eval (ExQ (ExN j A)) L"
dim_dual_space: fixes B :: "'c set"   and S :: "'c set" assumes "independent B"   and "span B = S"   and "finite B" shows "dim (dual_space S) = dim S"
set_pmf_of_list_eq: fixes xs :: "('b \<times> real) list" assumes "pmf_of_list_wf xs"   and "\<And>x. x \<in> snd ` set xs \<Longrightarrow> 0 < x" shows "set_pmf (pmf_of_list xs) = fst ` set xs"
cnf2bhv_mono_strict: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<and> t\<rangle> \<le> n"   and "n < n'" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n < \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n'"
sums_zero: shows "(\<lambda>n. 0::'a) sums (0::'a)"
h2: shows "maximal (set B) (\<lambda>T. T \<subseteq> set (cols (mat_of_cols n B)) \<and> lin_indpt T)"
circ_circ_mult_sub: fixes x :: "'a" shows "x\<^sup>\<star> * (1::'a)\<^sup>\<star> \<le> x\<^sup>\<star>\<^sup>\<star>"
distinct_67: fixes x31 :: "char list"   and x32 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "\<lparr>x31\<rparr>x32 \<noteq> Call x101 x102 x103 x104"
is_valid_problem_sas_plus_then_4: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>dom (\<Psi>\<^sub>I\<^sub>+). the ((\<Psi>\<^sub>I\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v"
SD_efficient_def': fixes p :: "'alt pmf" shows "SD_efficient R p = (\<not> (\<exists>q\<in>lotteries_on alts. (\<forall>i\<in>agents. SD (R i) p q) \<and> (\<exists>i\<in>agents. p \<prec>[SD (R i)] q)))"
well_founded_singleton: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "\<not> r x x" shows "well_founded {x} r"
scaleRat_scaleRat: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "rat"   and b :: "rat"   and x :: "'b" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat a (scaleRat b x) = scaleRat (a * b) x"
retraction_maps_compose: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'d \<Rightarrow> 'c"   and Z :: "'e topology"   and g :: "'d \<Rightarrow> 'e"   and g' :: "'e \<Rightarrow> 'd" assumes "retraction_maps X Y f f'"   and "retraction_maps Y Z g g'" shows "retraction_maps X Z (g \<circ> f) (f' \<circ> g')"
arity_apps: fixes s :: "('s, 'v) tm"   and ts :: "('s, 'v) tm list" shows "arity (apps s ts) = arity s - enat (length ts)"
getMinTree_min_prio: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and y :: "'a \<times> 'b" assumes "SkewBinomialHeapStruc.queue_invar bq"   and "y \<in># SkewBinomialHeapStruc.queue_to_multiset bq" shows "SkewBinomialHeapStruc.prio (SkewBinomialHeapStruc.getMinTree bq) \<le> snd y"
SPEC_nofail: fixes X :: "'c nres"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "X \<le> SPEC \<Phi>" shows "nofail X"
constant_functionE: fixes a :: "'a"   and b :: "'c" assumes "a \<in> carrier R"   and "b \<in> S" shows "constant_function S a b = a"
kernel_from_subgroup_generated: fixes S :: "'a set"   and G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c" assumes "subgroup S G" shows "kernel (subgroup_generated G S) H f = kernel G H f \<inter> S"
strip_eq_Assign: fixes c :: "'c acom"   and x :: "char list"   and e :: "aexp" shows "(strip c = x ::= e) = (\<exists>P. c = x ::= e {P})"
s_t_stop: fixes a :: "real"   and v :: "real"   and s0 :: "real" assumes "braking_movement a v" shows "movement.s a v s0 (movement.t_stop a v) = movement.p_max a v s0"
plus_iarray_component: fixes i :: "nat"   and A :: "'a iarray"   and B :: "'a iarray" assumes "i < IArray.length A"   and "i < IArray.length B" shows "(A + B) !! i = A !! i + B !! i"
subst_elimD'': fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and v :: "'c"   and w :: "'a" assumes "subst_elim \<sigma> v" shows "v \<notin> fv (\<sigma> w)"
Sup_inf: fixes B :: "'a set"   and a :: "'a" shows "\<Squnion>B \<sqinter> a = (SUP b\<in>B. b \<sqinter> a)"
l_get_owner_document_wf_axioms: shows "l_get_owner_document_wf heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes get_owner_document get_parent"
distinct_degree_factorization_i_def: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and f :: "'a list" shows "distinct_degree_factorization_i p ff_ops f = (let ze = arith_ops_record.zero ff_ops; on = arith_ops_record.one ff_ops in if degree_i f = 1 then [(1, f)] else dist_degree_factorize_main_i p ff_ops ze on (degree_i f) f [ze, on] 0 [])"
Pdrop_conforms: fixes n :: "nat" shows "path_conforms_with_strategy p (ldropn n P) \<sigma>"
ed_ou_eqs_2: fixes X :: "'a set" assumes "ed_ou X" shows "K \<circ> I \<circ> K = K \<circ> I"
identity_map: fixes U :: "'a set"   and x :: "'b" assumes "is_open U"   and "x \<in> id.im_sheaf U" shows "id.im_sheaf_morphisms U U x = x"
ENF_offending_members: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "('v \<times> 'v) set" assumes "\<not> sinvar G nP"   and "sinvar_all_edges_normal_form P"   and "f \<in> set_offending_flows G nP" shows "f \<subseteq> edges G \<and> (\<forall>(e1, e2)\<in>f. \<not> P (nP e1) (nP e2))"
spec_of_waiting: shows "\<forall>s a. waiting s (current s) a \<longrightarrow> kstep s a = s"
Equiv_Exec_sym2_ex: fixes m :: "'f"   and equiv_states :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and localState1 :: "'g \<Rightarrow> 'e"   and localState2 :: "'h \<Rightarrow> 'd"   and input_fun1 :: "'f \<Rightarrow> 'i message_af"   and input_fun2 :: "'f \<Rightarrow> 'j message_af"   and output_fun1 :: "'g \<Rightarrow> 'k message_af"   and output_fun2 :: "'h \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and trans_fun2 :: "'j message_af \<Rightarrow> 'h \<Rightarrow> 'h"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'g"   and c2 :: "'h" shows "\<exists>equiv_states_sym. Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2 = Equiv_Exec m equiv_states_sym localState2 localState1 input_fun2 input_fun1 output_fun2 output_fun1 trans_fun2 trans_fun1 k2 k1 c2 c1"
add_pairs_def: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" shows "add_pairs gs bs ps hs data = (let ps1 = apply_ncrit ncrit data gs bs hs (apply_icrit icrit data gs bs hs (np gs bs hs data)) in Let (apply_ocrit ocrit data hs ps1 ps) (comb (map snd (filter (\<lambda>x. \<not> fst x) ps1))))"
enum_not0: fixes v :: "'a" assumes "v \<in> verts G"   and "v \<noteq> s"   and "enum v \<noteq> \<infinity>" shows "enum v \<noteq> enat 0"
leq_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
seq_g'g: shows "A.seq (snd Fg') (snd Fg)"
that: fixes q :: "'a list" assumes "poly p2 = poly (p1 *** q)" shows "thesis"
rotate1_is_Nil_conv: fixes xs :: "'b list" shows "(rotate1 xs = []) = (xs = [])"
assms: shows "h \<turnstile> ok get_scdom_component ptr"
filter_chan_set_def: fixes a :: "'c"   and cs :: "'c set" shows "filter_chan_set a cs = (\<not> (\<exists>e\<in>cs. ev_eq a e))"
sylvester_mat_dim_1: fixes p :: "'c poly"   and q :: "'c poly" shows "dim_row (sylvester_mat p q) = degree p + degree q"
inf_right_idem: fixes x :: "'d upred"   and y :: "'d upred" shows "((x \<and> y) \<and> y) = (x \<and> y)"
gdpr_ref: shows "[\<N>\<^bsub>(Igdpr, sgdpr)\<^esub>] \<oplus>\<^sub>\<and>\<^bsup>(Igdpr, sgdpr)\<^esup> \<sqsubseteq> [\<N>\<^bsub>(Igdpr, GDPR')\<^esub>, \<N>\<^bsub>(GDPR', sgdpr)\<^esub>] \<oplus>\<^sub>\<and>\<^bsup>(Igdpr, sgdpr)\<^esup>"
bind_rpv_Done: fixes f :: "('c, 'd, 'b) rpv" shows "Done \<bind> f = f"
negligible: fixes S :: "'a set" shows "negligible S = (\<forall>T. (indicat_real S has_integral 0) T)"
atLeastLessThan_empty_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(subset_mset.atLeastLessThan a b = {}) = (\<not> a \<subset># b)"
zmultiset_eq_iff: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" shows "(M = N) = (\<forall>a. zcount M a = zcount N a)"
bounded: shows "\<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
semilat_errI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Semilat A r f" shows "semilat (err A, le r, lift2 (\<lambda>x y. OK (f x y)))"
strip_pfp_wn: fixes f :: "'a option acom \<Rightarrow> 'a option acom"   and c :: "com"   and c' :: "'a option acom" assumes "\<forall>c. strip (f c) = strip c"   and "pfp_wn f c = Some c'" shows "strip c' = c"
hlog_as_starfun: fixes a :: "real star"   and x :: "real star" assumes "0 < a"   and "a \<noteq> 1" shows "hlog a x = (*f* real_ln) x / (*f* real_ln) a"
fun_eq_onI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "fun_eq_on f g A"
simps: fixes \<alpha> :: "'b \<Rightarrow> 'b"   and \<beta> :: "'a \<Rightarrow> 'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr" shows "\<R>\<^sub>L \<alpha> \<beta> (\<sigma>, \<tau>, e) = (\<R>\<^sub>S \<alpha> \<beta> \<sigma>, \<R>\<^sub>S \<alpha> \<beta> \<tau>, \<R>\<^sub>E \<alpha> \<beta> e)"
induct: fixes P :: "(nat \<times> 'b) list \<Rightarrow> bool"   and a0 :: "(nat \<times> 'b) list" assumes "P []"   and "\<And>n a ps. P ((n, a) # ps) \<Longrightarrow> P ((Suc n, a) # ps)"   and "\<And>a ps. P ps \<Longrightarrow> P ((0, a) # ps)" shows "P a0"
ideD_1: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide a" shows "J_C.arr a"
IH: fixes i :: "nat" assumes "i < n" shows "\<bar>v $ i\<bar>\<^sup>2 \<le> \<parallel>v\<parallel>\<^sup>2"
convex_closed_segment: fixes a :: "'a"   and b :: "'a" shows "convex (closed_segment a b)"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (S.UP f) = arr f"
vec_vec_Tensor_right_empty: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_vec_Tensor f x [] = []"
C_2: fixes B :: "c \<Rightarrow> w \<Rightarrow> bool"   and A :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c. (\<^bold>O\<langle>A|B\<rangle>) c \<sqsubseteq> (\<lambda>w. \<I>(B \<^bold>\<and> A) c)"
prv_exi_inst_same: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and x :: "'a" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var" shows "prv (imp \<phi> (exi x \<phi>))"
supporters_nonagent1: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "election agents alts"   and "pref_profile_wf agents alts R"   and "x \<notin> alts" shows "supporters R x y = {}"
Inf_eq_Sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf A = Sup {b. \<forall>a\<in>A. less_eq b a}"
Red_fv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "fv e' \<subseteq> fv e"
16: shows "infs (gaccepting A) (gtrace (stake n (fromN 1 ||| r) @- sdrop n (fromN 1 ||| r)) (0, p))"
filterlim_at_infinity_imp_eventually_ne: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and c :: "'d" assumes "filterlim f at_infinity F" shows "\<forall>\<^sub>F z in F. f z \<noteq> c"
path_set_finite: shows "finite {p. \<exists>f'a l'a. ((f', l'), p, f'a, l'a) \<in> trcl pr_algo_lts}"
span_in_category_axioms: shows "span_in_category (\<cdot>) \<lparr>Leg0 = f0, Leg1 = f1\<rparr>"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "G (g \<cdot>\<^sub>A f) = G g \<cdot>\<^sub>B G f"
F_anal_valid: shows "\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i"
InvGlobPointstampsEq_def: fixes c :: "('a, 't, 'loc) Combined.configuration" shows "InvGlobPointstampsEq c = (\<forall>p loc t. zcount (c_pts (prop_config c p) loc) t = zcount (c_glob (exchange_config c) p) (loc, t))"
fps_shift_0: fixes f :: "'a fps" shows "fps_shift 0 f = f"
size_insert_d: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat"   and t' :: "'a tree"   and dl' :: "nat" assumes "insert_d x (t, dl) = (t', dl')" shows "size t \<le> size t'"
E_TAbs: fixes T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and t\<^sub>1\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type" shows "(\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2 \<longmapsto> t\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]"
round_down_correct: fixes f :: "real"   and e :: "int" shows "f - round_down e f \<in> {0..2 powr - real_of_int e}"
neg_eq_iff_add_eq_0: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (plus a b = zero)"
swap_id_eq: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "(a \<rightleftharpoons>\<^sub>F b) x = (if x = a then b else if x = b then a else x)"
dotprod_Cons: fixes x :: "nat"   and xs :: "nat list"   and y :: "nat"   and ys :: "nat list" shows "(x # xs) \<bullet> (y # ys) = x * y + xs \<bullet> ys"
"3_1": assumes "\<And>j1 j2. \<lbrakk>j1 < j2; j2 < length zs\<rbrakk> \<Longrightarrow> zs ! j1 < zs ! j2"   and "\<And>x. x \<in> set zs \<Longrightarrow> x < Suc i"   and "j1 < j2"   and "j2 < length (snd (mk_eqcl xs zs (Suc i) T))" shows "snd (mk_eqcl xs zs (Suc i) T) ! j1 < snd (mk_eqcl xs zs (Suc i) T) ! j2"
i_Exec_Stream_Acc_Output__State_Idle_eq: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState l localState trans_fun input c n)" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c = i_Exec_Comp_Stream_Acc_Output l output_fun trans_fun input c"
eta_cfun: fixes f :: "'d \<rightarrow> 'e" shows "(\<Lambda> x. f\<cdot>x) = f"
of_nat_eq_1_iff: fixes n :: "nat" shows "(of_nat n = (1::'a)) = (n = 1)"
sz: shows "\<forall>i\<ge>n. s i = (0::'a)"
le_alg_mult_imp_pdivides: fixes x :: "'a list"   and p :: "'a list list"   and n :: "nat" assumes "x \<in> carrier (poly_ring R)"   and "p \<in> carrier (poly_ring (poly_ring R))"   and "n \<le> UP.alg_mult p x" shows "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> x] [^]\<^bsub>poly_ring (poly_ring R)\<^esub> n pdivides\<^bsub>poly_ring R\<^esub> p"
case_2: fixes f1 :: "'h \<Rightarrow> 'g"   and f2 :: "'i \<Rightarrow> 'g"   and x2 :: "'i" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
self_codomain_iff_ide: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "partial_magma C" shows "(a \<in> partial_magma.codomains C a) = partial_magma.ide C a"
loweq_def: fixes \<sigma> :: "'var \<Rightarrow> 'a"   and \<sigma>' :: "'var \<Rightarrow> 'a" shows "(\<sigma> =\<^sub>L \<sigma>') = (\<sigma> \<restriction> (- local.hvars) = \<sigma>' \<restriction> (- local.hvars))"
1: shows "Suc (card X - 3) = card X - 2"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "P F"   and "P T"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (And v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
coset_mult_one: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set" assumes "Group.group G"   and "M \<subseteq> carrier G" shows "M #>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = M"
vec_vs_col: fixes A :: "'b mat" shows "vectorspace class_ring (module_vec TYPE('b) (dim_row A)\<lparr>carrier := LinearCombinations.module.span class_ring (module_vec TYPE('b) (dim_row A)) (set (cols A))\<rparr>)"
assms: shows "univalent f_a \<or> univalent f_b"
S2_def: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma> :: "bool"   and b\<^sub>\<sigma> :: "bool" assumes "ETP_base I domain range F F\<^sub>i\<^sub>n\<^sub>v" shows "ETP_base.S2 I range B F\<^sub>i\<^sub>n\<^sub>v \<sigma> b\<^sub>\<sigma> = I \<bind> (\<lambda>(\<alpha>, \<tau>). ETP.etp.S range \<alpha> \<bind> (\<lambda>x\<^sub>\<sigma>. ETP.etp.S range \<alpha> \<bind> (\<lambda>y\<^sub>\<sigma>'. let x\<^sub>\<sigma>' = F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>'; \<beta>\<^sub>\<sigma> = B \<alpha> x\<^sub>\<sigma> \<oplus> b\<^sub>\<sigma>; \<beta>\<^sub>\<sigma>' = B \<alpha> x\<^sub>\<sigma>' in return_spmf (\<sigma>, \<alpha>, \<beta>\<^sub>\<sigma>, \<beta>\<^sub>\<sigma>'))))"
max_var_value_def: shows "max_var_value = 2 ^ 31 - 1"
add_opp_double_opp: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "ell_field_class.add a p\<^sub>1 p\<^sub>2 = opp p\<^sub>1" shows "p\<^sub>2 = ell_field_class.add a (opp p\<^sub>1) (opp p\<^sub>1)"
map_poly_hom_smult: fixes c :: "rat"   and p :: "rat poly" shows "map_poly of_rat (Polynomial.smult c p) = Polynomial.smult (of_rat c) (map_poly of_rat p)"
g1_leadsto_a1: shows "\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2) \<and> SF(N1)_(x, y, sem, pc1, pc2) \<and> \<box>#True \<longrightarrow> (pc1 = #g \<leadsto> pc1 = #a)"
inject_2: fixes x2 :: "'b match_expr"   and y2 :: "'b match_expr" shows "(MatchNot x2 = MatchNot y2) = (x2 = y2)"
content_1_mult: fixes f :: "'b poly"   and g :: "'b poly" assumes "content f = (1::'b)"   and "content g = (1::'b)" shows "content (f * g) = (1::'b)"
left_unit: fixes a :: "'a" assumes "a \<in> R" shows "\<zero> + a = a"
homotopy_equivalent_space_contractibility: fixes X :: "'a topology"   and Y :: "'b topology" assumes "X homotopy_equivalent_space Y" shows "contractible_space X = contractible_space Y"
lt_setI: fixes f :: "'t \<Rightarrow>\<^sub>0 'd"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "f \<in> F"   and "f \<noteq> 0" shows "lt f \<in> lt_set F"
map_comp0: fixes g1 :: "'g \<Rightarrow> 'e"   and f1 :: "'c \<Rightarrow> 'g"   and g2 :: "'h \<Rightarrow> 'f"   and f2 :: "'d \<Rightarrow> 'h" shows "map_G0 (g1 \<circ> f1) (g2 \<circ> f2) = map_G0 g1 g2 \<circ> map_G0 f1 f2"
assms: shows "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u"