hcju/isabelleps huggingface dataset:
lambda_ex_emb: fixes v :: "i"   and \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>" shows "[\<lparr>\<^bold>\<lambda>x. \<phi> (x\<^sup>P),x\<^sup>P\<rparr> \<^bold>\<equiv> (\<^bold>\<exists>y. (\<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) \<^bold>& \<phi> (y\<^sup>P)) in v]"
Gromov_extension_isometry_inv_2: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "bij (Gromov_extension f)"
subsequence_empty: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "(w [i \<rightarrow> j] = []) = (j \<le> i)"
NormalizeD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Language.normalize c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
i_Exec_Stream_Init_Pre_Post2: fixes c_n :: "'a"   and trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat"   and x_n :: "'b"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "c_n = i_Exec_Comp_Stream_Init trans_fun input c n"   and "x_n = input n" shows "(P (input n) (i_Exec_Comp_Stream_Init trans_fun input c n) \<longrightarrow> Q (i_Exec_Comp_Stream_Init trans_fun input c (Suc n))) = (P x_n c_n \<longrightarrow> Q (trans_fun x_n c_n))"
info_rel_br: fixes a :: "'e \<Rightarrow> 'h"   and I :: "'e \<Rightarrow> bool"   and b :: "'f \<Rightarrow> 'g"   and J :: "'f \<Rightarrow> bool" shows "\<langle>br a I, br b J\<rangle>info_rel = br (\<lambda>y. b (snd y)) (\<lambda>x. I (fst x) \<and> J (snd x))"
fset_of_list_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "fset_of_list (filter P xs) = ffilter P (fset_of_list xs)"
mkArr_restrict_eq: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B F)" shows "S'.mkArr A B (restrict F A) = S'.mkArr A B F"
restrict_relp_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "R \<upharpoonleft> P \<otimes> Q = (\<lambda>x y. R x y \<and> P x \<and> Q y)"
le_list_antisym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "order r"   and "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys"   and "ys [\<sqsubseteq>\<^bsub>r\<^esub>] xs" shows "xs = ys"
upto_esl_def: fixes m :: "nat" shows "upto_esl m = (\<lambda>(A, r, f). (\<Union> {list n A |n. n \<le> m}, Listn.le r, Listn.sup f))"
simps_1: fixes uu :: "typ"   and t :: "Core.term" shows "eta_reducible (Abs uu (t $ Bv 0)) = (\<not> is_dependent t \<or> eta_reducible t)"
index_row_2: fixes A :: "'b mat"   and i :: "nat" shows "dim_vec (row A i) = dim_col A"
hyps_1: fixes a :: "coname"   and M :: "trm" assumes "<a>:M \<in> \<parallel><A>\<parallel>" shows "SNa M"
rwof_in_bounded_WHILE: assumes "rwof m0 cond step x" shows "\<exists>n. RETURN x \<le> bounded_WHILE n cond step m0"
Abs_bit_inject: fixes x :: "bool"   and y :: "bool" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_bit x = Abs_bit y) = (x = y)"
Range_Collect_case_prod: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
owhile_def: fixes c :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'d \<Rightarrow> 'c \<Rightarrow> 'd option"   and a :: "'d" shows "owhile c b a \<equiv> \<lambda>s. option_while (\<lambda>a. c a s) (\<lambda>a. b a s) a"
distinct_mset_remove1_All: fixes C :: "'a multiset"   and L :: "'a" assumes "distinct_mset C" shows "remove1_mset L C = removeAll_mset L C"
pow_def: shows "pow = nat \<lceil>rat_of_int l2x / rat_of_nat p\<rceil>"
invar_reachable: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "system.is_invar s\<^sub>0 lstep I"   and "system.reachable s\<^sub>0 lstep s" shows "I s"
has_parametrized_subprob_densityD_1: fixes M :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and x :: "'c" assumes "has_parametrized_subprob_density M N R f"   and "x \<in> space M" shows "N x = density R (f x)"
WT_into_WT_code: fixes wf_md :: "'a J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'a expr) mdecl \<Rightarrow> bool"   and P :: "'a J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'a expr"   and T :: "ty" assumes "wf_prog wf_md P"   and "P,E \<turnstile> e :: T"   and "ran E \<subseteq> types P" shows "P,E \<turnstile> e ::' T"
Rep_Assertion_cases: fixes y :: "'c" assumes "y \<in> assertion" obtains x :: "'c Assertion" where "y = Rep_Assertion x"
a_l_coset_def: fixes G :: "('e, 'f) ring_scheme" shows "(<+\<^bsub>G\<^esub>) = (<#\<^bsub>add_monoid G\<^esub>)"
gWlsAbsIsInBar_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme" shows "gWlsAbsIsInBar MOD \<equiv> \<forall>us s A. gWlsAbs MOD (us, s) A \<longrightarrow> isInBar (us, s)"
cong: fixes isNode :: "'e \<Rightarrow> bool"   and memb :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and nodeAbs :: "'e \<Rightarrow> 'f" shows "DFS.rel isNode memb nodeAbs \<equiv> DFS.rel isNode memb nodeAbs"
sel_1: fixes x21 :: "'b"   and x22 :: "'b llist" shows "lhd (LCons x21 x22) = x21"
fmap_rel_fmupd_fmap_rel: fixes K :: "('a \<times> 'b) set"   and A :: "('a, 'c) fmap"   and B :: "('b, 'd) fmap"   and R :: "('c \<times> 'd) set"   and p :: "'a"   and p' :: "'b"   and q :: "'c"   and q' :: "'d" assumes "single_valued K"   and "single_valued (K\<inverse>)"   and "(A, B) \<in> \<langle>K, R\<rangle>fmap_rel"   and "(p, p') \<in> K"   and "(q, q') \<in> R" shows "(fmupd p q A, fmupd p' q' B) \<in> \<langle>K, R\<rangle>fmap_rel"
ord_eq_Suc_0_iff: fixes n :: "nat"   and x :: "nat" shows "(Pocklington.ord n x = Suc 0) = [x = 1] (mod n)"
empty_not_reachable_imp_not_unifiable: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, {#}) \<notin> unif\<^sup>*" shows "\<not> unifiable (set_mset E)"
induct: fixes P :: "'b upD \<Rightarrow> 'b \<Rightarrow> 'b tree23 \<Rightarrow> bool"   and a0 :: "'b upD"   and a1 :: "'b"   and a2 :: "'b tree23" assumes "\<And>t1 a t2. P (TD t1) a t2"   and "\<And>t1 a t2 b t3. P (UF t1) a \<langle>t2, b, t3\<rangle>"   and "\<And>t1 a t2 b t3 c t4. P (UF t1) a \<langle>t2, b, t3, c, t4\<rangle>"   and "\<And>v b. P (UF v) b \<langle>\<rangle>" shows "P a0 a1 a2"
h3ba: shows "length (filter (\<lambda>i. solve_for_lhs_R p qs subsets (M_mat_R signs subsets) $ i \<noteq> 0) [0..<length subsets]) \<le> length subsets"
rexp_of'_def: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.rexp_of' n \<phi> = rexp.Inter (formula.rexp_of_alt n \<phi>) (formula.ENC n (FOV \<phi>))"
simps: fixes u :: "'a Vector"   and v :: "'a Vector" shows "(u dot v) = tdir u * tdir v + xdir u * xdir v + ydir u * ydir v + zdir u * zdir v"
sign_changes_Cons_Cons_different: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x * y < (0::'a)" shows "sign_changes (x # y # xs) = 1 + sign_changes (y # xs)"
enat_0_less_mult_iff: fixes m :: "enat"   and n :: "enat" shows "(0 < m * n) = (0 < m \<and> 0 < n)"
fdia_d_simp: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "relfdia x (rel_ad (rel_ad y)) = relfdia x y"
int_power_div_base: fixes m :: "nat"   and k :: "int" assumes "0 < m"   and "0 < k" shows "k ^ m div k = k ^ (m - Suc 0)"
f2: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "a < b"   and "b < c"   and "c < d" shows "f2 {a, b, c, d} = f {(a, d), (b, c)}"
lunit_in_hom: fixes a :: "'c local.term set" assumes "ide a" shows "\<guillemotleft>\<l>[a] : \<I> \<otimes> a \<rightarrow> a\<guillemotright>"
return_reads: fixes x :: "'z"   and h :: "'x"   and h' :: "'x" shows "reads {} (return x) h h'"
scalar_prod_add_distrib: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec"   and v\<^sub>3 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (v\<^sub>2 + v\<^sub>3) = v\<^sub>1 \<bullet> v\<^sub>2 + v\<^sub>1 \<bullet> v\<^sub>3"
pw_EASSUME_3: fixes \<Phi> :: "bool"   and u :: "unit" shows "inres (EASSUME \<Phi>) (Inr u) = \<Phi>"
quasinorm_of_inverse: fixes y :: "real \<times> ('c \<Rightarrow> ennreal)" assumes "y \<in> {(x, N). 1 \<le> ennreal x \<and> (\<forall>x c. N (c *\<^sub>R x) = ennreal \<bar>c\<bar> * N x) \<and> (\<forall>xa y. N (xa + y) \<le> ennreal x * N xa + ennreal x * N y)}" shows "Rep_quasinorm (quasinorm_of y) = y"
simps_1: fixes Z :: "'a"   and M :: "nat" shows "shallow_model' Z M 0 = Conv (Z, M) (Input M)"
ev_fork_idx: fixes N :: "nat"   and e :: "dining_event"   and i :: "nat"   and s :: "nat" assumes "DiningPhilosophers N"   and "e \<in> DiningPhilosophers.fork_transitions N i s" shows "fork e = i"
dsr3: fixes x :: "'a" shows "d x + (1::'a) = (1::'a)"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "f x = g x"
maximum_modulus_principle: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and U :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "open U"   and "U \<subseteq> S"   and "\<xi> \<in> U"   and "\<And>z. z \<in> U \<Longrightarrow> cmod (f z) \<le> cmod (f \<xi>)" shows "f constant_on S"
2: shows "ord_class.filter (F \<inter> \<up>x) \<and> ord_class.filter (G \<inter> \<up>x)"
cSup_eq_cSup: fixes A :: "'b set"   and B :: "'b set" assumes "bdd_above A"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<le> b"   and "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b \<le> a" shows "\<Squnion> A = \<Squnion> B"
expandTauFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'f" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* C" obtains \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P'' :: "'b" where "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P''\<rangle>" and "\<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P''" and "A\<^sub>P' \<sharp>* C" and "A\<^sub>P' \<sharp>* P'" and "distinct A\<^sub>P'"
simps_1: fixes n :: "nat"   and l :: "real" shows "nxt Ins (n, l) = (n + 1, if real n + 1 \<le> f2 * l then l else e * l)"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list" shows "take n (take m xs) = take (min n m) xs"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelT Enc TRel"
simps_2: fixes m :: "'c monom"   and c :: "'d"   and m' :: "'c monom"   and d :: "'d"   and p :: "('c monom \<times> 'd) list" shows "monom_mult_poly (m, c) ((m', d) # p) = (if c * d = (0::'d) then monom_mult_poly (m, c) p else (m * m', c * d) # monom_mult_poly (m, c) p)"
UNIV_I: fixes x :: "'c" shows "x \<in> UNIV"
prems_1: shows "\<forall>x\<in>{1..Suc m}. \<forall>y\<in>{1..Suc m}. x \<noteq> y \<longrightarrow> A x \<inter> A y = {}"
frag_expansion: fixes a :: "'b \<Rightarrow>\<^sub>0 int" shows "a = frag_extend frag_of a"
assms: shows "u = 0\<^sub>v (dim_vec u)"
arrI: fixes f :: "'c local.term set" assumes "arr f \<and> (ide (local.dom f) \<and> Diag (DOM (local.dom f))) \<and> ide (cod f) \<and> Diag (DOM (cod f))" shows "\<F>\<^sub>SC.arr f"
sel_1: fixes x1 :: "'b list"   and x2 :: "nat" shows "body (Fundef x1 x2) = x1"
subset_subseqs: fixes X :: "'d set"   and xs :: "'d list" assumes "X \<subseteq> set xs" shows "X \<in> set ` set (subseqs xs)"
trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "Sublist.prefix a b"   and "Sublist.prefix b c" shows "Sublist.prefix a c"
surjective: fixes r :: "('b, 'c) Cone_scheme" shows "r = \<lparr>vertex = vertex r, slope = slope r, \<dots> = Cone.more r\<rparr>"
info_1: fixes p :: "int poly"   and n :: "nat" assumes "info_2 x = Inr (p, n)" shows "poly_cond p \<and> poly (real_of_int_poly p) (real_of_2 x) = 0 \<and> 2 \<le> degree p \<and> card {y. y \<le> real_of_2 x \<and> poly (real_of_int_poly p) y = 0} = n"
CallThrowParams: fixes es :: "'addr expr list"   and vs :: "'addr val list"   and a :: "'addr"   and es' :: "'addr expr list"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and v :: "'addr val"   and M :: "String.literal"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "es = map Val vs @ Throw a # es'" shows "extTA,P,t \<turnstile> \<langle>Val v\<bullet>M(es),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
intros_6: fixes P :: "complex mat"   and Q :: "complex mat"   and P' :: "complex mat"   and Q' :: "complex mat"   and S :: "com" assumes "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "is_quantum_predicate P'"   and "is_quantum_predicate Q'"   and "P \<le>\<^sub>L P'"   and "\<turnstile>\<^sub>p {P'} S {Q'}"   and "Q' \<le>\<^sub>L Q" shows "\<turnstile>\<^sub>p {P} S {Q}"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "\<chi> f = A_B.null"
WHATWHERE_Secure_Skip: fixes \<iota> :: "nat" shows "WHATWHERE_Secure [skip\<^bsub>\<iota>\<^esub>]"
surjective: fixes r :: "('c, 'd) list_graph_scheme" shows "r = \<lparr>nodesL = nodesL r, edgesL = edgesL r, \<dots> = list_graph.more r\<rparr>"
Polys_closed_monomial: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and c :: "'c" assumes "t \<in> .[X]" shows "monomial c t \<in> P[X]"
source_sink: shows "\<not> edge \<Delta> (source \<Delta>) (sink \<Delta>)"
IH: shows "real (catalan n) = I n / (2 * pi)"
interval_bounds'_2: fixes a :: "'a"   and b :: "'a" assumes "cbox a b \<noteq> {}" shows "interval_lowerbound (cbox a b) = a"
sum: fixes g :: "'e \<Rightarrow> 'b"   and S :: "'e set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
preserves_arr: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" shows "C.CC.arr (C.CC.map f)"
const_finfun: fixes a :: "'e" shows "(\<lambda>x. a) \<in> finfun"
reachable_inv: fixes v :: "'a"   and e :: "'a"   and t :: "'a"   and r :: "'a" assumes "v \<sqinter> top = v"   and "e \<le> v \<sqinter> - id v"   and "e \<sqinter> t = bot"   and "id v = id r \<sqinter> top" shows "id (v \<squnion> id e \<sqinter> top) = id r \<sqinter> top"
assms_1: shows "a \<in> carrier R - {\<zero>}"
len_bb: shows "Suc n < e_length (e_butlast (local.prefixes j (Suc (Suc n))))"
typ_of1_tsubst: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "typ_of1 Ts t = Some ty" shows "typ_of1 (map (\<lambda>T. tsubstT T \<rho>) Ts) (tsubst t \<rho>) = Some (tsubstT ty \<rho>)"
analz_Un_AgentSet: fixes H :: "msg set" shows "analz (AgentSet \<union> H) \<subseteq> AgentSet \<union> analz H"
simple_match_ip_conjunct: fixes ip1 :: "'a word \<times> nat"   and p_ip :: "'a word"   and ip2 :: "'a word \<times> nat" shows "(simple_match_ip ip1 p_ip \<and> simple_match_ip ip2 p_ip) = (case ipcidr_conjunct ip1 ip2 of None \<Rightarrow> False | Some ipx \<Rightarrow> simple_match_ip ipx p_ip)"
simps_11: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>n \<phi> \<and> (\<forall>j\<le>i. suffix j \<xi> \<Turnstile>\<^sub>n \<psi>))"
halfspace_simps_6: fixes sctns :: "'c sctn set" shows "sbelow_halfspaces sctns = \<Inter> (sbelow_halfspace ` sctns)"
unstream_appendS: fixes a :: "('b, 'c) Stream"   and b :: "('b, 'd) Stream" assumes "a \<noteq> \<bottom>"   and "b \<noteq> \<bottom>" shows "unstream\<cdot>(appendS\<cdot>a\<cdot>b) = appendL\<cdot>(unstream\<cdot>a)\<cdot>(unstream\<cdot>b)"
MN_imp_dfa: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set" assumes "MyhillNerode L R" obtains M :: "'b dfa" where "dfa M" and "dfa.language M = L" and "card (states M) = card (UNIV // R)"
OutOfMemoryC_def: shows "OutOfMemoryC \<equiv> (OutOfMemory, [], [], [])"
False: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3 {\<lambda>(ps, n). False} c { Q}"
bij_is_surj: fixes f :: "'c \<Rightarrow> 'd" assumes "bij f" shows "surj f"
returns_to_return_time_pos: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "closed P" shows "auto_ll_on_open.returns_to f X P x = (0 < auto_ll_on_open.return_time f X P x)"
2: fixes v :: "nat \<times> 'b" assumes "v \<in> V" shows "Bex (greachable A w V v) (gaccepting A)"
dg_prod_3_Cod_vsv: shows "vsv ((\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Cod\<rparr>)"
goedel_at: fixes f :: "nat \<Rightarrow> nat option"   and m :: "nat"   and k :: "nat" assumes "f \<in> \<R>" shows "\<exists>n\<ge>m. \<phi> n = (\<lambda>x. if x = k then Some n else f x)"
inf_K_S_next_behaviour: fixes a :: "'a"   and b :: "'a"   and s :: "'b"   and t :: "'b" assumes "a \<le>\<^sub>\<K> b \<and> s \<le>\<^sub>\<S> t" shows "s \<circ> a \<le>\<^sub>\<K> t \<circ> b"
can_step_def: fixes l :: "'l"   and s :: "'a" shows "A.can_step l s \<equiv> \<exists>s'. alstep l s s'"
not_outside_2: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and x :: "'b"   and y :: "'b" assumes "pref_profile_wf agents alts R"   and "R i x y" shows "x \<in> alts"
simps: fixes M :: "('c, 'd) insert_message set"   and x :: "('c, 'd) insert_message"   and y :: "('c, 'd) insert_message" shows "depends_on M x y = (x \<in> M \<and> y \<in> M \<and> I x \<in> deps (Insert y))"
Nil_in_shufflesI: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs = []"   and "ys = []" shows "[] \<in> shuffles xs ys"
f_def: shows "f \<equiv> \<lambda>x. if x \<le> b then f1 x else f2 x"
simps: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and e :: "('d, 'e, 'b) exp"   and x :: "'f"   and exs :: "(('d, 'e, 'b) exp \<times> 'g) list" shows "J1_heap_base.\<tau>Move1 typeof_addr P h ((e, x), exs) = (J1_heap_base.\<tau>move1 typeof_addr P h e \<or> final e)"
delta_min_mono: fixes as :: "(QDelta \<times> QDelta) list"   and bs :: "(QDelta \<times> QDelta) list" assumes "set as \<subseteq> set bs" shows "\<delta>_min bs \<le> \<delta>_min as"
star_inductr_var_eq: fixes y :: "'a"   and x :: "'a" assumes "y ; x = y" shows "y ; x\<^sup>\<star> \<le> y"
props_1: shows "{0..} \<subseteq> existence_ivl0 x"
Nil_lenlex_iff1: fixes ns :: "'b list"   and r :: "('b \<times> 'b) set" shows "(([], ns) \<in> lenlex r) = (ns \<noteq> [])"
subst_clss_def: fixes AA :: "'a literal multiset set"   and \<sigma> :: "'s" shows "AA \<cdot>cs \<sigma> = (\<lambda>A. A \<cdot> \<sigma>) ` AA"
FSwImorph_def: fixes h :: "'d \<Rightarrow> 'e"   and hA :: "'f \<Rightarrow> 'g"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'd, 'f, 'i) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'e, 'g, 'j) model_scheme" shows "FSwImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD'"
mult2_ns_def: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "mult2_ns ns s \<equiv> multpw ns O (mult s)\<^sup>="
iso_paired2: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'g"   and G' :: "('f, 'h) monoid_scheme"   and H' :: "('g, 'i) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "((\<lambda>(x, y). (f x, g y)) \<in> Group.iso (G \<times>\<times> H) (G' \<times>\<times> H')) = (f \<in> Group.iso G G' \<and> g \<in> Group.iso H H')"
assms_4: fixes d' :: "('test, 'atom) com"   and t' :: "'state" assumes "(d, t) \<rightarrow>*c (d', t')"   and "s' \<approx> t'"   and "c' \<approx>w d'" shows "thesis"
elims_3: fixes x :: "real \<times> real" assumes "\<not> invar x" obtains n :: "real" and l :: "real" where "x = (n, l)" and "\<not> (l0 \<le> l \<and> (l0 \<le> l / c \<longrightarrow> f1 * l \<le> n) \<and> n \<le> f2 * l)"
path_entry_last: fixes G :: "nat graph_rec"   and p :: "nat list"   and n :: "nat" assumes "cfg_doms G"   and "path_entry (g_E G) p n"   and "p \<noteq> []" shows "last p = 0"
in_hhom_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "horizontal_homs V src trg" shows "horizontal_homs.in_hhom V src trg \<mu> a b \<equiv> partial_magma.arr V \<mu> \<and> src \<mu> = a \<and> trg \<mu> = b"
pos: fixes bl :: "'a set" shows "0 \<le> (b1 |\<inter>| bl - \<Lambda>)\<^sup>2"
cone_not_czero: shows "(cone, czero) \<notin> ordIso"
Sups_dual_def: shows "\<partial> \<circ> Sup = Infs \<circ> (`) \<partial>"
vfun_le_inv: fixes t1 :: "(val \<times> val) fset"   and t2 :: "(val \<times> val) fset" assumes "VFun t1 \<sqsubseteq> VFun t2" obtains "fset t1 \<subseteq> fset t2"
foldr_aux_2_2: fixes ns :: "'node list"   and g :: "'g"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and n :: "'node" assumes "set ns \<subseteq> set (\<alpha>n g)"   and "distinct ns"   and "foldr (aux_2 g) ns (Mapping.empty, Mapping.empty) = (u', p')"   and "n \<notin> set ns" shows "Mapping.lookup u' n = None"
I_subst_pretty: fixes \<phi> :: "atom fm"   and r :: "real"   and cs :: "real list"   and xs :: "real list" assumes "qfree \<phi>" shows "R.I (subst \<phi> (r, cs)) xs = R.I \<phi> ((r + \<langle>cs,xs\<rangle>) # xs)"
assms_1: shows "Fn\<cdot>f \<triangleleft>\<triangleright> CFn\<cdot>g"
label_wrt_elt_image: fixes B :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and v :: "'a" assumes "label_wrt B f"   and "v \<in> \<Union> X" shows "f v \<in> B"
simps_1: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'c btree" assumes "split split" shows "bal_up\<^sub>i (T\<^sub>i t) = bal t"
emeasure_bind_const_prob_space: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'d set" assumes "prob_space M"   and "subprob_space N" shows "emeasure (M \<bind> (\<lambda>x. N)) X = emeasure N X"
update_simps_22: fixes T :: "(nat \<times> linear_poly) list"   and s :: "('b, 'a) state" shows "\<V> (\<T>_update T s) = \<V> s"
C_Rcd: fixes E :: "trm \<Rightarrow> (char list \<times> trm) list" assumes "E \<in> rctxt" shows "(\<lambda>t. Rcd (E t)) \<in> ctxt"
simps_16: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm" shows "Ifm bbs bs (E p) = (\<exists>x. Ifm bbs (x # bs) p)"
subterms_of_a_non_atomic_term: fixes t1 :: "'b trm"   and t2 :: "'b trm" shows "subterms_of (t1 \<cdot> t2) = subterms_of t1 \<union> subterms_of t2 \<union> {t1 \<cdot> t2}"
is_Vrecabs_def: fixes "value" :: "value" shows "is_Vrecabs value = (\<exists>x31 x32 x33. value = Vrecabs x31 x32 x33)"
length_prefixes: fixes xs :: "'b list" shows "\<^bold>|prefixes xs\<^bold>| = \<^bold>|xs\<^bold>| + 1"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> G"   and "y \<in> G" shows "kernel.Class (x \<cdot> y) = kernel.quotient_composition (kernel.Class x) (kernel.Class y)"
map_of_SomeD: fixes xs :: "('c \<times> 'b) list"   and k :: "'c"   and y :: "'b" assumes "table_of xs k = Some y" shows "(k, y) \<in> set xs"