hcju/isabelleps huggingface dataset:
plus_wb_lens: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e" assumes "wb_lens x"   and "wb_lens y"   and "x \<bowtie> y" shows "wb_lens (x +\<^sub>L y)"
stieltjes_gamma_aux5: fixes k :: "nat" shows "(\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * (deriv ^^ k) g 1"
length_map2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b list"   and b :: "'c list" shows "length (map2 f a b) = min (length a) (length b)"
lim: fixes x :: "'a" assumes "x \<in> space M" shows "(\<lambda>i. f i x) \<longlonglongrightarrow> g x"
elims_2: fixes x :: "'b trm"   and xa :: "indices list"   and xb :: "'b trm" assumes "subterm x xa xb" obtains T :: "'b trm" and S :: "'b trm" where "x = T" and "xa = []" and "xb = Sa__" and "T = Sa__"   | xc :: "'b trm" and y :: "'b trm" and "next" :: "indices list" and S :: "'b trm" where "x = xc__ \<cdot> y__" and "xa = indices.Left # next__" and "xb = Sb__" and "subterm xc__ next__ Sb__"   | xc :: "'b trm" and y :: "'b trm" and "next" :: "indices list" and S :: "'b trm" where "x = xc \<cdot> y" and "xa = indices.Right # next" and "xb = S" and "subterm y next S"
subset_cls_val_process': fixes A :: "'c set"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and n :: "nat"   and w :: "'d" assumes "finite A"   and "support_set p \<subseteq> A" shows "cls_val_process Mkt p (Suc n) w = (\<Sum>x\<in>A. prices Mkt x (Suc n) w * p x (Suc n) w)"
AG_lem2: fixes x :: "'b"   and s :: "'b set" assumes "x \<in> AG s" shows "x \<in> s \<inter> AX (AG s)"
sel: fixes x2 :: "'m" shows "the (Some x2) = x2"
hyps_3: assumes "ws'' t = \<lfloor>PostWS w\<rfloor>"   and "ws' t = \<lfloor>PostWS w\<rfloor>"   and "\<And>w. Suspend w \<notin> set was" shows "Notified \<notin> set was \<and> WokenUp \<notin> set was"
push_bit_of_1: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n one = power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n"
from_list_ga_aux_rule: fixes is_set :: "'a set \<Rightarrow> 'b \<Rightarrow> assn"   and ins :: "'a \<Rightarrow> 'b \<Rightarrow> 'b Heap"   and s :: "'a set"   and p :: "'b"   and l :: "'a list" assumes "imp_set_ins is_set ins" shows "<is_set s p> from_list_ga_aux ins l p <is_set (set l \<union> s)>\<^sub>t"
aboveS_def: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
return_sets_cong: fixes M :: "'d measure"   and N :: "'d measure" assumes "sets M = sets N" shows "return M = return N"
openin_Times_eq: fixes S :: "'b set"   and T :: "'c set"   and S' :: "'b set"   and T' :: "'c set" shows "openin (top_of_set (S \<times> T)) (S' \<times> T') = (S' = {} \<or> T' = {} \<or> openin (top_of_set S) S' \<and> openin (top_of_set T) T')"
span_minimal: fixes Sa :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a"   and T :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set" assumes "Sa \<subseteq> cotangent_space p"   and "T \<subseteq> cotangent_space p"   and "Sa \<subseteq> T"   and "subspace T" shows "span Sa \<subseteq> T"
le_imp_less_or_eq: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less"   and "less_eq x y" shows "less x y \<or> x = y"
two_div_sqr_of_cmd_sqrt_two: shows "2 * (cmod (1 / complex_of_real (sqrt 2)))\<^sup>2 = 1"
IH: shows "fps_conv_radius f \<le> fps_conv_radius (f ^ n)"
Cnotzero_UNIV: shows "(|UNIV|, czero) \<notin> ordIso \<and> Card_order |UNIV|"
conv_one_strongly_connected: fixes x :: "'a" shows "one_strongly_connected x = one_strongly_connected (x\<^sup>T)"
assms_1: shows "lookup p v \<noteq> (0::'b)"
Agent_Aenc: fixes X :: "agent"   and X' :: "msg"   and Y' :: "msg" shows "Messages.Agent X \<noteq> Aenc X' Y'"
R_G_K_cfg_valid_cfgD_1: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and cfg' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' = cont cfg (state cfg')"
string_rel_def: shows "string_rel \<equiv> \<langle>char_rel\<rangle>list_rel"
simps: fixes a :: "fmla" shows "special_axiomsp a = (\<exists>A i x. a = A(i::=x) IMP exi i A)"
token_run_merge: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat"   and m :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n"   and "y \<le> n"   and "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = semi_mojmir_def.token_run \<delta> q\<^sub>0 w y n" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (n + m) = semi_mojmir_def.token_run \<delta> q\<^sub>0 w y (n + m)"
order_finite_set_exists_foundation: fixes M :: "'a set"   and t :: "'a" assumes "finite M"   and "t \<in> M" shows "\<exists>s\<in>M. s \<le> t \<and> (\<forall>u\<in>M. \<not> u < s)"
intro: fixes carrier :: "'a set" assumes "finite carrier" shows "finite_total_preorder_on_axioms carrier"
bv_prepend_add: fixes x :: "nat"   and b :: "'a"   and y :: "nat" shows "bv_prepend x b [] @ bv_prepend y b [] = bv_prepend (x + y) b []"
absolutely_integrable_on_null: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "content (cbox a b) = 0" shows "f absolutely_integrable_on cbox a b"
inverts_mat_symm: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "inverts_mat A B" shows "inverts_mat B A"
show_l_zero: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier R"   and "a \<in> carrier R" shows "(a \<oplus> x = x) = (a = \<zero>)"
cutoff_branch_internal': fixes T :: "tree"   and red :: "bool list \<Rightarrow> bool" assumes "anybranch T red" shows "anyinternal (cutoff red [] T) (\<lambda>p. \<not> red p) \<and> anybranch (cutoff red [] T) red"
singular_chain_cmul: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and a :: "int" assumes "singular_chain p X c" shows "singular_chain p X (frag_cmul a c)"
evalF_FNot: fixes M :: "model"   and phi :: "vbl \<Rightarrow> object"   and A :: "formula" shows "evalF M phi (FNot A) = (\<not> evalF M phi A)"
1: shows "hbt \<langle>l, (a, n), r\<rangle>"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>B) src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C F"
antisym_r: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> x" shows "x = y"
prems_1: shows "\<forall>x\<in>insert x F. i < IArray.length (f x)"
IH: assumes "distinct (map fst xs @ map fst ys)" shows "distinct (map fst xs @ map fst (butlast ys))"
AE_Invariant_func_is_AE_constant: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "ergodic_qmpt M T"   and "f \<in> borel_measurable M"   and "AE x in M. f (T x) = f x" shows "\<exists>y. AE x in M. f x = y"
image_mset_remove1_mset_if: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset"   and a :: "'d" shows "f `# remove1_mset a M = (if a \<in># M then remove1_mset (f a) (f `# M) else f `# M)"
list_rel_id: fixes R :: "('a \<times> 'a) set" assumes "R = Id" shows "\<langle>R\<rangle>list_rel = Id"
R1_comp: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is R1 healthy"   and "Q is R1 healthy" shows "P ;; Q is R1 healthy"
delete1_ssubseqs: fixes as :: "'b list"   and bs :: "'b list"   and a :: "'b" shows "as @ bs \<in> ssubseqs (as @ [a] @ bs)"
left_impl_ded: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
subset_box_imp_3: fixes a :: "'b"   and c :: "'b"   and d :: "'b"   and b :: "'b" assumes "\<forall>i\<in>Basis. a \<bullet> i \<le> c \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i" shows "box c d \<subseteq> cbox a b"
card_order_cprod: fixes r1 :: "('p \<times> 'p) set"   and r2 :: "('q \<times> 'q) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 *c r2)"
intro: fixes Q :: "'a set"   and Bot :: "'b set"   and entails_q :: "'a \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool" assumes "Q \<noteq> {}"   and "\<forall>q\<in>Q. consequence_relation Bot (entails_q q)" shows "consequence_relation_family Bot Q entails_q"
game\<^sub>2_game\<^sub>3: fixes x :: "nat"   and y :: "nat" shows "game\<^sub>2 x y = game\<^sub>3 (\<lambda>b m\<^sub>1 m\<^sub>2. map_spmf (([\<oplus>]) (if b then m\<^sub>1 else m\<^sub>2)) (spmf_of_set (nlists UNIV len_plain))) x y"
prime_imp_solovay_strassen_witness: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "odd p"   and "a \<in> {2..<p}" shows "solovay_strassen_witness p a"
eq_Nil_appendI: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs = ys" shows "xs = [] @ ys"
fmdomI: fixes m :: "('b, 'a) fmap"   and x :: "'b"   and y :: "'a" assumes "m $$ x = Some y" shows "x |\<in>| fmdom m"
cases: fixes a :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and H :: "chan set" assumes "a \<in> extr bad IK H" obtains M :: "msg" where "a = Ma__" and "Ma__ \<in> IK"   | c :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "a = M" and "Chan c A B M \<in> H" and "c = insec \<or> c = auth \<or> A \<in> bad \<or> B \<in> bad"
nat_prod_le: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "m \<noteq> 0"   and "m * n \<le> k" shows "n \<le> k"
pd_node2: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and y :: "'a"   and x :: "'a" assumes "IFC prob"   and "IFC_def.is_pd prob y x" shows "x \<in> IFC_def.nodes prob"
all_lemma: fixes rf :: "recf"   and xs :: "nat list"   and rt :: "recf" assumes "primerec rf (Suc (length xs))"   and "primerec rt (length xs)" shows "rec_exec (rec_all rt rf) xs = (if \<forall>x\<le>rec_exec rt xs. 0 < rec_exec rf (xs @ [x]) then 1 else 0)"
df_cos: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real"   and g' :: "'b" assumes "(f has_vector_derivative f') (at x)"   and "(cos has_field_derivative g') (at (f x))" shows "((\<lambda>x. cos (f x)) has_vector_derivative f' * g') (at x)"
diff_self: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.group_add minus plus zero uminus" shows "minus a a = zero"
disc_2: fixes x2 :: "'h" shows "\<not> isl (Inr x2)"
intersection_calculus_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a inference set"   and Red_F_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set" shows "intersection_calculus Bot Inf Q entails_q Red_I_q Red_F_q \<equiv> consequence_relation_family Bot Q entails_q \<and> intersection_calculus_axioms Bot Inf Q entails_q Red_I_q Red_F_q"
l2_inv1_trans: shows "{l2_inv1} TS.trans l2 {> l2_inv1}"
bn_tau_fresh: fixes P :: "'a" shows "bn \<tau> \<sharp>* P"
f: shows "f = fpxs_compose_power (fpxs_of_fls f') (1 / rat_of_nat d)"
HEndPhase0_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4c s p" shows "HInv4c s' p"
sets_eq: fixes p :: "real"   and n :: "nat" assumes "edge_space p" shows "sets (edge_space.P n p) = Pow (Pow (edge_space.S_edges n))"
bij_fv_UV_2: shows "bij_betw fv GV HV"
InvariantQCharacterizationAfterApplyLearn: fixes state :: "State" assumes "getConflictFlag state"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)" shows "let state' = applyLearn state in InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
line_integral_exists_smooth_one_base: fixes \<gamma> :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a"   and b :: "'a" assumes "\<gamma> C1_differentiable_on {0..1}"   and "continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> b)" shows "line_integral_exists F {b} \<gamma>"
self_list_diff: fixes xs :: "'a list" shows "xs - xs = replicate (length xs) (0::'a)"
monom_mult_uminus_right: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult c t (- p) = - monom_mult c t p"
PsiInv_gamma1: shows "\<turnstile> gamma1 \<and> $PsiInv \<longrightarrow> PsiInv$"
hyps_3: shows "l cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> k"
finite_irreflp_transp_imp_wfp: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite {(x, y). p x y}"   and "irreflp p"   and "transp p" shows "wfP p"
funpow_simps_right_1: fixes f :: "'d \<Rightarrow> 'd" shows "f ^^ 0 = id"
spectral_radius_poly_bound: fixes A :: "complex mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n n"   and "Spectral_Radius_Theory.spectral_radius A \<le> 1"   and "\<And>ev k. \<lbrakk>poly (char_poly A) ev = 0; cmod ev = 1\<rbrakk> \<Longrightarrow> order ev (char_poly A) \<le> d" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ (d - 1))"
Nil2: fixes xs :: "'c list"   and zs :: "'b list" assumes "as = xs"   and "bs = []"   and "cs = zs" shows "P"
op_norm_eq_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "(\<parallel>A\<parallel>\<^sub>o\<^sub>p = 0) = (A = 0)"
uw\<theta>w'\<theta>'\<beta>: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<iota> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and w' :: "'c"   and \<theta>' :: "'c"   and \<beta> :: "'c" assumes "uw\<theta>w'\<theta>'\<beta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<beta>" shows "partial_magma.ide V u \<and> partial_magma.ide V w \<and> partial_magma.ide V w' \<and> partial_magma.in_hom V \<theta> (H f w) u \<and> partial_magma.in_hom V \<theta>' (H f w') u \<and> partial_magma.in_hom V \<beta> (H g w) (H g w') \<and> V (H r \<theta>) (V (\<a> r f w) (H \<rho> w)) = V (H r \<theta>') (V (\<a> r f w') (V (H \<rho> w') \<beta>))"
right_collapsed_case: assumes "s\<^sub>2' = \<epsilon>" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
gt_diff: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t >\<^sub>h\<^sub>d head s"   and "chkvar t s"   and "chksubs (>\<^sub>t) t s" shows "t >\<^sub>t s"
Ker_equality: shows "Ker = {a |a. a \<in> G \<and> \<eta> a = \<one>'}"
eq: fixes x :: "real" shows "(2 + 2 * real n) * x / 2 = (1 + real n) * x"
independent_bound_general: fixes S :: "'c set" assumes "vs2.independent S" shows "finite S \<and> card S \<le> vs2.dim S"
update_convs_2: fixes return_status' :: "bool \<Rightarrow> bool"   and break_status :: "bool"   and return_status :: "bool"   and more :: "'b" shows "return_status_update return_status' \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = \<lparr>break_status = break_status, return_status = return_status' return_status, \<dots> = more\<rparr>"
sorted_finfun_to_list: fixes f :: "('a, 'b) finfun" shows "sorted (finfun_to_list f)"
atLeastLessThan_empty_iff: fixes a :: "'b word"   and b :: "'b word" shows "(signed.atLeastLessThan a b = {}) = (\<not> a <s b)"
distinct_1: fixes x21 :: "char list"   and x22 :: "expr" shows "Skip \<noteq> x21:=x22"
listsp_simps_2: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "listsp A (x # xs) = (A x \<and> listsp A xs)"
find_remove_subset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and res :: "'a option"   and xs' :: "'a list" assumes "find_remove P xs = (res, xs')" shows "set xs' \<subseteq> set xs"
op_elem_def: fixes oper :: "('f, 'e) operation" shows "op_elem oper \<equiv> case oper of Create i e \<Rightarrow> e | Delete is e \<Rightarrow> e | Append i e \<Rightarrow> e | Expunge e mo i \<Rightarrow> e | Store e mo i \<Rightarrow> e"
fterm_to_iterm_cancel: fixes t :: "(char list, char list) Term.term" shows "fterm_to_iterm (iterm_to_fterm t) = t"
chosen_adjoint_equivalence: fixes a :: "'a" assumes "C.obj a" shows "adjoint_equivalence_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a) (map\<^sub>0 a) (unit a) (counit a)"
Diag_Diagonalize_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diag C (monoidal_language.Diagonalize C t)"
prob_T_split: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s"   and n :: "nat" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "T.prob s {\<omega> \<in> space (T s). P \<omega>} = T.expectation s (\<lambda>\<omega>. T.prob ((s ## \<omega>) !! n) {\<omega>' \<in> space (T ((s ## \<omega>) !! n)). P (stake n \<omega> @- \<omega>')})"
takeWhile_eq_Nil_iff: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "(takeWhile P xs = []) = (xs = [] \<or> \<not> P (hd xs))"
q1: fixes f :: "'a" assumes "f \<in> M"   and "h - k \<in> orthogonal_complement M \<and> k \<in> M" shows "dist h k \<le> dist h f"
convergentD: fixes X :: "nat \<Rightarrow> 'c" assumes "convergent X" shows "\<exists>L. X \<longlonglongrightarrow> L"
simps: fixes d :: "nat"   and amount :: "nat"   and f :: "atom fm" shows "liftFm d amount f = map_fm_binders (\<lambda>a x. liftAtom (d + x) amount a) f 0"
m1: fixes m :: "int" assumes "poly_mod_type TYPE('a) m" shows "1 < m"
ys_2: shows "degree_aforms_err XS \<le> n"
simps_14: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case B of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f2"
coeff_mult_sum: fixes p :: "'b poly"   and m :: "nat"   and q :: "'b poly"   and n :: "nat" assumes "degree p \<le> m"   and "degree q \<le> n" shows "coeff (p * q) (m + n) = coeff p m * coeff q n"
const_gbij: shows "gbij\<^bsub>(\<natural>{\<one>}),(\<natural>{\<one>})\<^esub> (1\<^bsub>(\<natural>{\<one>}),(\<natural>{\<one>})\<^esub>)"
in_rel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a set['c]"   and b :: "'b set['c]" shows "rel_bset R a b = (\<exists>z. z \<in> {x. set_bset x \<subseteq> {(x, y). R x y}} \<and> map_bset fst z = a \<and> map_bset snd z = b)"
empty_interior_lowdim: fixes S :: "'a set" assumes "dim S < DIM('a)" shows "interior S = {}"
A_B_step: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and a' :: "'d" assumes "Simulation A B sim"   and "A a b"   and "sim a a'" shows "\<exists>b'. B a' b' \<and> sim b b'"
assms: shows "\<exists>p. eval cid t t' n (\<gamma> p)"
subst_eqI: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "\<And>t. t \<cdot> \<sigma> = t \<cdot> \<theta>" shows "\<sigma> = \<theta>"
set_subset_Cons: fixes xs :: "'g list"   and x :: "'g" shows "set xs \<subseteq> set (x # xs)"
A': fixes S :: "nat set" shows "A ` S \<subseteq> lambda_system \<Omega> M f"
nsubst_mkv: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and ODE :: "('sf + 'b, 'sz) ODE"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<nu>' :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "OadmitFO \<sigma> ODE U"   and "osafe ODE"   and "\<And>i. dsafe (\<sigma> i)" shows "mk_v I (OsubstFO ODE \<sigma>) \<nu> (fst \<nu>') = mk_v (adjointFO I \<sigma> \<nu>') ODE \<nu> (fst \<nu>')"
is_lub_lub: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<forall>n. f n << x" shows "lub f << x"
augment_edge_cf': fixes u :: "nat"   and v :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and \<Delta> :: "'a" assumes "(u, v) \<in> cfE_of f" shows "cf_of (augment_edge f (u, v) \<Delta>) = (cf_of f)((u, v) := cf_of f (u, v) - \<Delta>, (v, u) := cf_of f (v, u) + \<Delta>)"
preserves_arr: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "D.arr (E.map f)"
keep_coeff: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and x :: "char list multiset \<times> int" assumes "normalize_poly_p p q" shows "normalize_poly_p (add_mset x p) (add_mset x q)"
mult_of_simps_1: fixes R :: "('c, 'd) ring_scheme" shows "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
C1: fixes \<Psi> :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "(\<Psi>, S, T) \<in> Rel"   and "guarded S"   and "guarded T" shows "(\<Psi>, U \<parallel> !S, U \<parallel> !T) \<in> Rel'"
degree_oneE: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and K :: "'c set" assumes "ring R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "degree p = 1" obtains a :: "'c" and b :: "'c" where "a \<in> K" and "a \<noteq> \<zero>\<^bsub>R\<^esub>" and "b \<in> K" and "p = [a, b]"
eigenvector_def: fixes A :: "'b mat"   and v :: "'b Matrix.vec"   and k :: "'b" shows "eigenvector A v k = (v \<in> carrier_vec (dim_row A) \<and> v \<noteq> 0\<^sub>v (dim_row A) \<and> A *\<^sub>v v = k \<cdot>\<^sub>v v)"
take_bit_int_less_eq: fixes n :: "nat"   and k :: "int" assumes "2 ^ n \<le> k"   and "0 < n" shows "take_bit n k \<le> k - 2 ^ n"
assms_3: fixes i :: "nat"   and j :: "nat" assumes "i \<in> set xs" shows "(s j = i) = (j = i)"
get_env_def: fixes get :: "('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> ('g, 'h) envT" shows "get_env get f = EnvT (\<lambda>r. get (\<lambda>s. run_env (f s) r))"
translation_map: fixes a :: "'a" assumes "a \<in> M" shows "(a)\<^sub>L \<in> M \<rightarrow>\<^sub>E M"
comparator_prod: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and comp\<^sub>'\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> order" assumes "comparator comp\<^sub>'\<^sub>a"   and "comparator comp\<^sub>'\<^sub>b" shows "comparator (comparator_prod comp\<^sub>'\<^sub>a comp\<^sub>'\<^sub>b)"
vectorize_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "vectorize_poly (p + q) = vectorize_poly p + vectorize_poly q"
trancl_minus_refl_idem: fixes ts :: "('a \<times> 'a) set" shows "{(a, b). (a, b) \<in> {(a, b). (a, b) \<in> ts\<^sup>+ \<and> a \<noteq> b}\<^sup>+ \<and> a \<noteq> b} = {(a, b). (a, b) \<in> ts\<^sup>+ \<and> a \<noteq> b}"
minimal_unsat_core_tabl_atoms_def: fixes I :: "'b set"   and t :: "(nat \<times> linear_poly) list"   and as :: "('b \<times> 'c atom) set" shows "minimal_unsat_core_tabl_atoms I t as = (I \<subseteq> fst ` as \<and> (\<nexists>v. v \<Turnstile>\<^sub>t t \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as) \<and> (distinct_indices_atoms as \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. v \<Turnstile>\<^sub>t t \<and> (J, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as)))"