hcju/isabelleps huggingface dataset:
state_rank_initial: fixes q :: "'b" shows "state_rank q 0 = initial q"
factors_of_int_poly_2: fixes p :: "int poly"   and qs :: "int poly list"   and x :: "'a" assumes "factors_of_int_poly p = qs"   and "p \<noteq> 0" shows "(ipoly p x = (0::'a)) = (\<exists>q\<in>set qs. ipoly q x = (0::'a))"
Power_dfa_def: shows "Power_dfa = \<lparr>dfa.states = {HF (epsclo q) |q. q \<in> Pow (nfa.states M)}, init = HF (epsclo (nfa.init M)), final = {HF (epsclo Q) |Q. Q \<subseteq> nfa.states M \<and> Q \<inter> nfa.final M \<noteq> {}}, nxt = \<lambda>Q x. HF (\<Union>q\<in>epsclo (hfset Q). epsclo (nfa.nxt M q x))\<rparr>"
trms\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_eq_rcv: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"   and "\<S>' = update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<A>' = \<A> @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)]" shows "\<Union> (trms\<^sub>s\<^sub>t ` \<S>) \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>) = \<Union> (trms\<^sub>s\<^sub>t ` \<S>') \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>')"
upper_remainder: fixes B :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "B \<subseteq> A"   and "\<not> B \<turnstile> \<phi>" shows "\<exists>B'. B \<subseteq> B' \<and> B' \<in> A .\<bottom>. \<phi>"
permutes_inv_o_2: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "inv' p \<circ> p = id"
le_iff_diff_le_0: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a b = less_eq (minus a b) zero"
in_qle_def: fixes x :: "'a"   and M :: "'a set" shows "x \<in>\<^sub>\<preceq> M \<equiv> \<exists>y\<in>M. x \<preceq> y"
gen_igba_impl_rel_eext_def: fixes Rm :: "('k \<times> 'n) set"   and Rl :: "('j \<times> ('l \<Rightarrow> 'm \<Rightarrow> bool)) set" shows "\<langle>Rm, Rl\<rangle>gen_igba_impl_rel_eext = {(\<lparr>igbai_L = Li, \<dots> = mi\<rparr>, \<lparr>igba_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
bind_get: fixes f :: "('v \<Rightarrow> int option) \<Rightarrow> 'm"   and g :: "int \<Rightarrow> 'm" shows "bind (get f) g = get (\<lambda>s. bind (f s) g)"
split_Edge1_eq_0I: fixes y :: "'v vertex"   and x :: "'v"   and f :: "'v \<times> 'v \<Rightarrow> ennreal"   and z :: "'v" assumes "\<And>z. y \<noteq> Vertex x" shows "split f (Edge z x, y) = 0"
in_homI: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
supr_inField: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "local.Above B \<noteq> {}" shows "supr B \<in> Field r"
eventually_iff_iff: fixes P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in \<U>. P x = Q x) = (eventually P \<U> = eventually Q \<U>)"
less_flat_complete_lattice_def: fixes x :: "'a flat_complete_lattice"   and y :: "'a flat_complete_lattice" shows "(x < y) = (case x of flat_complete_lattice.Value v1 \<Rightarrow> y = flat_complete_lattice.Top | flat_complete_lattice.Bot \<Rightarrow> y \<noteq> flat_complete_lattice.Bot | flat_complete_lattice.Top \<Rightarrow> False)"
nth_mono_on: fixes xs :: "'b list"   and A :: "'b set" assumes "sorted xs"   and "distinct xs"   and "set xs = A" shows "mono_on ((!) xs) {..<card A}"
FAssRed2: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 v :: "val"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>F{D} := e,s\<rangle> \<rightarrow> \<langle>Val v\<bullet>F{D} := e',s'\<rangle>"
s_wf_trm: shows "wf\<^sub>t\<^sub>r\<^sub>m s"
P'_def: shows "P' = exi yy (cnj (PPf (Var yy) (Var xx)) (RR (Var yy) (Var xx)))"
se_as_ce: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c' :: "('c, 'd) conf" assumes "se c l c'" shows "states c' = {\<sigma>'. \<exists>\<sigma>\<in>states c. ce \<sigma> l \<sigma>'}"
Re_Arccos: fixes z :: "complex" shows "Re (Arccos z) = Im (Ln (z + \<i> * csqrt (1 - z\<^sup>2)))"
a_repr_independence': fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and y :: "'e"   and x :: "'e" assumes "abelian_subgroup H G"   and "y \<in> H +>\<^bsub>G\<^esub> x"   and "x \<in> carrier G" shows "H +>\<^bsub>G\<^esub> x = H +>\<^bsub>G\<^esub> y"
implies_galois_swap: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> y \<leadsto> z) = (y \<le> x \<leadsto> z)"
\<gamma>_def: shows "\<gamma> \<equiv> (\<beta>, in_doubt_deny)"
ih: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "F = set_mset M \<union> set_mset N"   and "count (image_mset f M) (f a) + count N a < count (image_mset f N) (f a) + count M a" shows "\<exists>b. f b = f a \<and> count M b < count N b"
simps: fixes a1 :: "typ list"   and a2 :: "Core.term"   and a3 :: "typ" shows "a1 \<turnstile>\<^sub>\<tau> a2 : a3 = ((\<exists>uu_ uv_ T. a1 = uu_ \<and> a2 = Ct uv_ T \<and> a3 = T) \<or> (\<exists>i Ts. a1 = Ts \<and> a2 = Bv i \<and> a3 = Ts ! i \<and> i < length Ts) \<or> (\<exists>uw_ ux_ T. a1 = uw_ \<and> a2 = Fv ux_ T \<and> a3 = T) \<or> (\<exists>T Ts t T'. a1 = Ts \<and> a2 = Abs T t \<and> a3 = T \<rightarrow> T' \<and> T # Ts \<turnstile>\<^sub>\<tau> t : T') \<or> (\<exists>Ts u U t T. a1 = Ts \<and> a2 = t $ u \<and> a3 = T \<and> Ts \<turnstile>\<^sub>\<tau> u : U \<and> Ts \<turnstile>\<^sub>\<tau> t : U \<rightarrow> T))"
measurable_def: fixes F :: "'b set set"   and G :: "'c set set" shows "measurable F G = {f. \<forall>g\<in>G. f -` g \<in> F}"
pbval_bval_vars: fixes b :: "bexp"   and ps :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" assumes "vars b \<subseteq> dom ps" shows "pbval b ps = bval b (emb ps s)"
simps_2: fixes cx :: "int"   and cy :: "int"   and more :: "'b" shows "cy \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = cy"
distinct_1: fixes x2 :: "'b" shows "extC.Bot \<noteq> extC.Val x2"
suc_cong1: fixes t :: "trm"   and t' :: "trm" shows "{} \<turnstile> t EQ t' IMP suc t EQ suc t'"
tree_interval_inv_def: fixes t :: "interval_tree" shows "tree_interval_inv t = (\<forall>p\<in>tree_set t. is_interval (idx_interval.int p))"
ring_preimage: shows "ring (f \<^sup>\<inverse> A B) addA multA zeroA oneA"
admissible_le_mono: fixes f :: "'a \<Rightarrow> 'a" assumes "monotone (\<le>) (\<le>) f" shows "ccpo.admissible Sup (\<le>) (\<lambda>x. x \<le> f x)"
trancl_list_impl: fixes r :: "('b \<times> 'b) list"   and as :: "'b list" shows "set (trancl_list_impl r as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set r)\<^sup>+}"
assms_3: fixes x :: "'a" assumes "x \<in> space M" shows "\<bar>f x\<bar> \<le> g x"
iso_iff_mono_and_retraction: fixes f :: "'d" shows "Dop.iso f = (Dop.mono f \<and> Dop.retraction f)"
StableBox: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> stable P \<longrightarrow> \<box>(Init P \<longrightarrow> \<box>P)"
eq_Eq_conv: fixes x :: "'b"   and y :: "'b" shows "(c1 x y = Eq) = (x = y)"
initial_in_F_token_run: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and y :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "q\<^sub>0 \<in> F" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x y \<in> F"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>ip = ip r, sn = sn r, rt = rt r, rreqs = rreqs r, store = store r, msg = msg r, data = data r, dests = dests r, pre = pre r, rreqid = rreqid r, dip = dip r, oip = oip r, hops = hops r, dsn = dsn r, dsk = dsk r, osn = osn r, sip = sip r, handled = handled r, \<dots> = state.more r\<rparr>"
Gromov_product_geodesic_segment: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and t :: "real" assumes "geodesic_segment_between G x y"   and "t \<in> {0..dist x y}" shows "Gromov_product_at x y (geodesic_segment_param G x t) = t"
assms_2: fixes k :: "'a"   and l :: "'a" assumes "k \<in> A"   and "l \<in> A"   and "k \<le> l" shows "f k \<le> f l"
magQ_def: fixes x :: "'d['e, 'f]" shows "\<lbrakk>x\<rbrakk>\<^sub>Q = mag (fromQ x)"
distinct_map': fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex"   and f :: "Vertex" assumes "r \<in> complex_rotations"   and "distinct [a, b, c, d]"   and "\<forall>e\<in>{a, b, c}. r f \<noteq> e" shows "r f = d"
pairs_iff_elts: fixes x :: "V"   and y :: "V"   and z :: "V" shows "((x, y) \<in> pairs z) = (\<langle>x, y\<rangle> \<in>\<^sub>\<circ> z)"
less: fixes w :: "'a list" assumes "\<^bold>|w\<^bold>| < \<^bold>|wa__\<^bold>|" shows "w \<in> G \<longrightarrow> w \<in> \<langle>\<BB> G\<rangle>"
residual_eq_iff_perm_renaming: fixes \<alpha>1 :: "'b"   and P1 :: "'c"   and \<alpha>2 :: "'b"   and P2 :: "'c" shows "(\<langle>\<alpha>1,P1\<rangle> = \<langle>\<alpha>2,P2\<rangle>) = (\<exists>p. supp (\<alpha>1, P1) - bn \<alpha>1 = supp (\<alpha>2, P2) - bn \<alpha>2 \<and> (supp (\<alpha>1, P1) - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> (\<alpha>1, P1) = (\<alpha>2, P2) \<and> p \<bullet> bn \<alpha>1 = bn \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1)"
latestActEx: fixes n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'. latestAct_cond nid t n n'" shows "\<exists>n'. latestAct_cond nid t n n' \<and> (\<forall>n''. latestAct_cond nid t n n'' \<longrightarrow> n'' \<le> n')"
wt_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and T :: "'fsym trm" assumes "wtE \<xi>"   and "wt T" shows "D (int \<xi> T)"
att_orD1: fixes x1 :: "'a attree"   and x2 :: "'a attree list"   and x :: "'a set \<times> 'a set" assumes "\<turnstile>x1 # x2 \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>" shows "\<turnstile>x1"
nn_integral_pmf_eq_1: fixes p :: "'b pmf" shows "\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space UNIV = 1"
rep_eq: fixes x :: "mini_alg_unique" shows "mau_ge_0 x = ma_ge_0 (Rep_mini_alg_unique x)"
mult_left_cancel: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c \<noteq> (0::'b)" shows "(c * a = c * b) = (a = b)"
le_utrans_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "le_utrans t u = (\<forall>P. unitary P \<longrightarrow> t P \<tturnstile> u P)"
BSI_stable: fixes X :: "X5 set" shows "BSI.stable_on UNIV X = (X = {X5.Xd1, X5.Xd2'} \<or> X = {X5.Xd1', X5.Xd2})"
Z_def: shows "Z \<equiv> Matrix.mat 2 2 (\<lambda>(i, j). if i \<noteq> j then 0 else if i = 0 then 1 else - 1)"
max2_def: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "wo_rel r" shows "max2 r a b \<equiv> if (a, b) \<in> r then b else a"
distr_distr: fixes g :: "'d \<Rightarrow> 'e"   and N :: "'d measure"   and L :: "'e measure"   and f :: "'f \<Rightarrow> 'd"   and M :: "'f measure" assumes "g \<in> N \<rightarrow>\<^sub>M L"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "distr (distr M N f) L g = distr M L (g \<circ> f)"
p31: shows "\<exists>D\<in>MajoritySet. \<forall>d\<in>D. mbal (disk s' d p) = bal (dblock s' p)"
ivl_disj_un_26: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m..u} = {l..u}"
CR_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "CR_on r A = (\<forall>a\<in>A. \<forall>b c. (a, b) \<in> r\<^sup>* \<and> (a, c) \<in> r\<^sup>* \<longrightarrow> (b, c) \<in> r\<^sup>\<down>)"
elementary_bounded: fixes p :: "'a set set"   and S :: "'a set" assumes "p division_of S" shows "bounded S"
authKeys_not_insert: fixes ev :: "event"   and evs :: "event list" assumes "\<forall>A Ta akey Peer. ev \<noteq> Says Kas A (Crypt (shrK A) \<lbrace>akey, Agent Peer, Ta, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, akey, Ta\<rbrace>\<rbrace>)" shows "authKeys (ev # evs) = authKeys evs"
good_card_imp_finite: fixes A :: "'b set" assumes "of_nat (card A) \<noteq> (0::'a)" shows "finite A"
sort_oalist_raw_id: fixes xs :: "('a \<times> 'b) list \<times> 'a key_order" assumes "ko.oalist_inv xs" shows "sort_oalist_ko xs = xs"
Rangep_Grp: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "Rangep (BNF_Def.Grp A f) = (\<lambda>x. x \<in> f ` A)"
reduce_root_nonzero: fixes a :: "'a"   and p :: "'a poly" assumes "a \<noteq> (0::'a)"   and "p \<noteq> 0" shows "reduce_root a p \<noteq> 0"
prems_9: shows "set uus \<inter> snd ` set (tx # txs) = {}"
idx_of_code_def: fixes node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat" shows "idx_of_code node_eq_impl node_hash_impl \<equiv> \<lambda>(x, xa, xb, xc) xd. let y' = case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xd xb of Some i \<Rightarrow> if 0 \<le> i then nat i else undefined in Let (find_max_nat (as_length xa) (\<lambda>j. as_get xa j \<le> y')) RETURN"
sdrop_stl: fixes n :: "nat"   and s :: "'b stream" shows "sdrop n (stl s) = stl (sdrop n s)"
lipschitz: fixes t :: "real" assumes "t \<in> T" shows "B'-lipschitz_on X' (f t)"
counit_in_vhom: shows "\<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> src g\<guillemotright>"
mem_Collect_eq: fixes a :: "'g"   and P :: "'g \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
expand: fixes octo :: "octo"   and octo' :: "octo" assumes "Ree octo = Ree octo' \<and> Im1 octo = Im1 octo' \<and> Im2 octo = Im2 octo' \<and> Im3 octo = Im3 octo' \<and> Im4 octo = Im4 octo' \<and> Im5 octo = Im5 octo' \<and> Im6 octo = Im6 octo' \<and> Im7 octo = Im7 octo'" shows "octo = octo'"
rep_eq: fixes x :: "('c, 'd) chart" shows "apply_chart x = (case Rep_chart x of (d, d', f, f') \<Rightarrow> f)"
fdisjnt_Union_left: fixes S :: "'b fset fset"   and T :: "'b fset" shows "fdisjnt (ffUnion S) T = fBall S (\<lambda>S. fdisjnt S T)"
closure_greaterThanLessThan: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "closure {a<..<b} = {a..b}"
ch2ch_cont: fixes f :: "'a \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'a" assumes "cont f"   and "chain Y" shows "chain (\<lambda>i. f (Y i))"
lvars_distinct: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" shows "distinct (map lhs (Tableau (preprocess' cs start)))"
split_paired_All: fixes P :: "'d \<times> 'e \<Rightarrow> bool" shows "(\<forall>x. P x) = (\<forall>a b. P (a, b))"
pd_antisym: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a"   and y :: "'a" assumes "IFC prob"   and "IFC_def.is_pd prob x y"   and "IFC_def.is_pd prob y x" shows "x = y"
append_prefixD: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "prefix (xs @ ys) zs" shows "prefix xs zs"
HcompNml_Obj_Nml: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Obj t"   and "E.Nml u"   and "E.Src t = E.Trg u" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = u"
a_neigh_neq_a: shows "a_neigh \<noteq> a"
a_factorgroup_is_comm_group: shows "comm_group (R A_Mod I)"
DX_2: shows "(5, Dec 2 2 1) \<in> X"
card_eq_sum: fixes A :: "'b set" shows "card A = (\<Sum>x\<in>A. 1)"
wp2_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>2 c Q = (\<lambda>s. if \<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> Q t < \<infinity> then enat (\<down>\<^sub>t (c, s)) + Q (\<down>\<^sub>s (c, s)) else \<infinity>)"
1: shows "{..<length cl} = theFT cl \<union> theNFT cl"
filterlim_at_top_ge: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and c :: "'d" shows "filterlim f at_top F = (\<forall>Z\<ge>c. \<forall>\<^sub>F x in F. Z \<le> f x)"
Rep_subject_inverse: fixes x :: "subject" shows "Abs_subject (Rep_subject x) = x"
simps_21: fixes t :: "'b local.term"   and u :: "'b local.term"   and va :: "'b local.term"   and vb :: "'b local.term" shows "(t \<^bold>\<otimes> u) \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> va \<^bold>\<otimes> vb = t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> va \<^bold>\<otimes> vb"
converse_Un: fixes r :: "('c \<times> 'b) set"   and s :: "('c \<times> 'b) set" shows "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
intro: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>t. ord (0::'b) t"   and "\<And>s t u. ord s t \<Longrightarrow> ord (s + u) (t + u)" shows "ordered_powerprod_axioms ord"
impl_elimC: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "(P \<longrightarrow> Q) x" obtains "\<not> P x"   | "Q x"
alpha_Tree_FL_valid_Tree: fixes t1 :: "('a, 'pred, 'act, 'effect) Tree"   and t2 :: "('a, 'pred, 'act, 'effect) Tree"   and P :: "'state" assumes "t1 =\<^sub>\<alpha> t2" shows "FL_valid_Tree P t1 = FL_valid_Tree P t2"
dropWhile_eq_drop: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "dropWhile P xs = drop (length (takeWhile P xs)) xs"
nprv_exuI: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "nprv F (subst \<phi> t x)"   and "nprv (insert \<phi> F) (eql (Var x) t)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "x \<notin> FvarsT t"   and "x \<notin> \<Union> (Fvars ` F)" shows "nprv F (exu x \<phi>)"
ideI: fixes a :: "'a" assumes "right (src f) a"   and "ide a" shows "Right.ide a"
eq: shows "(\<lambda>x. ((f \<circ> X) x, X x)) ` space M = (\<lambda>x. (f x, x)) ` X ` space M"
cod_assoc': fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "D.cod (D.assoc' a b c) = (a \<otimes>\<^sub>D b) \<otimes>\<^sub>D c"
proj2_line_abs_mult: fixes k :: "real"   and v :: "(real, 3) vec" assumes "k \<noteq> 0" shows "proj2_line_abs (k *\<^sub>R v) = proj2_line_abs v"
fpxs_const_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(fpxs_const x = fpxs_const y) = (x = y)"
CatCod: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" assumes "PreFunctor F" shows "Category (CatCod F)"
Afix_eq: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysis.Afix Aexp \<Gamma>\<cdot>ae = (\<mu> x. ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>x \<squnion> ae)"
has_witness_two: fixes n :: "nat"   and S :: "'c set" assumes "has n S"   and "2 \<le> n" shows "\<exists>x y. hasw [x, y] S"
rep_eq: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "'e bset" shows "set_bset (map_bset x xa) = x ` set_bset xa"
val_Zp_p_nat_unit: fixes n :: "nat" assumes "int n mod p \<noteq> 0" shows "val_Zp ([n] \<cdot> \<one>) = 0"
orientable_4_implies_1: fixes x :: "'a" assumes "orientable_4 (x \<sqinter> - (1::'a))" shows "orientable_1 x"
Call_lemma: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and C' :: "char list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T \<and> P \<turnstile> C' \<preceq>\<^sup>* D' \<and> is_type P T' \<and> (\<forall>T\<in>set Ts'. is_type P T) \<and> wf_md P D' (M, Ts', T', m')"
irreducibleI': fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<not> a dvd (1::'a)"   and "\<And>b. b dvd a \<Longrightarrow> a dvd b \<or> b dvd (1::'a)" shows "irreducible a"
intros: fixes R :: "('g \<times> 'g) set" assumes "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R" shows "asym R"
strict_trans: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "a \<subset># b"   and "b \<subset># c" shows "a \<subset># c"
abs_convergent_prod_def: fixes f :: "nat \<Rightarrow> 'b" shows "abs_convergent_prod f = convergent_prod (\<lambda>i. 1 + norm (f i - (1::'b)))"
is_const_def: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_const unconst t = (unconst t \<noteq> None)"
bA: shows "b \<in> local.underS (local.suc A)"
hyps_2: fixes s :: "'a list"   and v :: "'a val" assumes "s \<in> fst (simp r2) \<rightarrow> v" shows "s \<in> r2 \<rightarrow> snd (simp r2) v"
simps_14: fixes b :: "nat"   and \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "fvi b (Since \<phi> I \<psi>) = fvi b \<phi> \<union> fvi b \<psi>"
dens'_3: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (state_measure (V \<union> V') \<Gamma> \<Otimes>\<^sub>M stock_measure t)"
strict_iff_order: fixes b :: "'e"   and a :: "'e" shows "(b < a) = (b \<le> a \<and> a \<noteq> b)"
helper_optimize_shallow: fixes r :: "'b wordinterval" shows "wordinterval_empty_shallow (wordinterval_optimize_empty r) = wordinterval_empty (wordinterval_optimize_empty r)"
jb: assumes "poly (char_poly A) 1 = 0"   and "d + 1 < order 1 (char_poly A)" shows "dim_gen_eigenspace A 1 (d + 1) = order 1 (char_poly A)"
test_add_distr: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "test p"   and "test q"   and "test r" shows "p + q \<cdot> r = (p + q) \<cdot> (p + r)"
intros_1: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "typed_model arity public Ana \<Gamma>"   and "t \<in> M" shows "t \<in> typed_model.SMP arity Ana \<Gamma> M"
sublist_lengths: fixes xs :: "'a list"   and ys :: "'a list"   and i :: "nat" assumes "sublist_at xs ys i" shows "i + length xs \<le> length ys"
Conj_E1: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A AND B" shows "H \<turnstile> A"
enat_ord_code_4: fixes m :: "nat" shows "(enat m < \<infinity>) = True"
fls_prpart_plus: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_prpart (f + g) = fls_prpart f + fls_prpart g"
prime_elem_primitive_part_fract: fixes p :: "'b fract poly" assumes "irreducible p" shows "prime_elem (primitive_part_fract p)"