hcju/isabelleps huggingface dataset:
l11_52: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Cong A C A' C'"   and "Cong B C B' C'"   and "Tarski_neutral_dimensionless.Le Bet Cong B C A C" shows "Cong B A B' A' \<and> Tarski_neutral_dimensionless.CongA Bet Cong B A C B' A' C' \<and> Tarski_neutral_dimensionless.CongA Bet Cong B C A B' C' A'"
nat_mono: fixes x :: "int"   and y :: "int" assumes "x \<le> y" shows "nat x \<le> nat y"
nonpos_Reals_Real: fixes x :: "'a" assumes "x \<in> \<real>\<^sub>\<le>\<^sub>0" shows "x \<in> \<real>"
cancel_semigroup_add_ow_axioms: shows "cancel_semigroup_add_ow U (+\<^sub>o\<^sub>w)"
connex_subset: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "connex A less_eq"   and "B \<subseteq> A" shows "connex B less_eq"
scaleR_scaleC: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "real" assumes "class.scaleC scaleR scaleC" shows "scaleR r = scaleC (complex_of_real r)"
assms_1: shows "f ` {- 1..1} \<subseteq> cbox (- 1) 1"
det_zero_column_2: fixes j :: "'c"   and F :: "(('b, 'c) vec, 'c) vec" assumes "column j F = 0" shows "det F = (0::'b)"
usubst_upd_pr_var_id: fixes x :: "'a \<Longrightarrow> 'b" assumes "vwb_lens x" shows "[x \<mapsto>\<^sub>s &x] = id"
fps_of_poly_linear: fixes a :: "'b" shows "fps_of_poly [:a, 1::'b:] = fps_X + fps_const a"
nths_atLeastAtMost_0_take: fixes xs :: "'b list"   and m :: "nat" shows "nths xs {0..m} = take (Suc m) xs"
kernel_dim: shows "kernel_dim (gauss_jordan_single A) = K.dim"
M1: shows "(\<lambda>x. f x * indicat_real {g a..g b} x) \<in> borel_measurable borel"
in_cong: fixes L :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and L' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and Lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and h :: "'b \<Rightarrow> 'c" assumes "landau_symbol L L' Lr"   and "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> L F h) = (g \<in> L F h)"
intensional_rews_2: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'c \<Rightarrow> 'b"   and w :: "'c" shows "(f<x>) w \<equiv> f (x w)"
step_thread_Some_NoneD: fixes update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and t :: "'t"   and \<sigma>' :: "'a" assumes "step_thread update_state s t' = \<lfloor>(t, None, \<sigma>')\<rfloor>" shows "\<exists>x ln n. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> 0 < ln $ n \<and> \<not> waiting (wset s t) \<and> may_acquire_all (locks s) t ln \<and> \<sigma>' = update_state (K$ [], [], [], [], [], convert_RA ln)"
poly_Pred: fixes x :: "'a" shows "Pred x \<and> (Pred [x] \<or> \<not> Pred [x])"
simps_3: fixes c :: "poly"   and n :: "nat"   and p :: "poly"   and c' :: "int \<times> int" shows "CN c n p +\<^sub>p C c' = CN (c +\<^sub>p C c') n p"
dupe: shows "dupe_monic_dynamic m D H S T U = (A, B)"
generalise_output_subsumes_original: fixes t :: "transition"   and r :: "nat"   and v :: "value"   and c :: "nat \<Rightarrow>f value option"   and p :: "nat" assumes "Outputs t ! r = L v"   and "c $ p = Some v" shows "subsumes (generalise_output t p r) c t"
commute_bpp_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" shows "commute_bpp f A = (\<forall>x\<in>addition_set f A. \<forall>y\<in>addition_set f A. x \<^sub>f+ y = y \<^sub>f+ x)"
FullSpec_def: shows "FullSpec \<equiv> FullInit \<and> \<box>[FullNxt]_vars \<and> WF(Deq inp q1 mid)_vars \<and> WF(Deq mid q2 out)_vars"
multiset_inter_single: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b" shows "\<LM> a  \<RM> \<inter># \<LM> b  \<RM> = \<Empt>"
absorb_iff2: fixes a :: "'a word"   and b :: "'a word" shows "(a \<le>s b) = (signed.max a b = b)"
A3: shows "nat_to_ce_set n = nat_to_ce_set m"
simps_107: fixes x101 :: "nat"   and x102 :: "msg"   and x7 :: "rat" shows "Crypt x101 x102 \<noteq> Pri_AgrK x7"
abelian_monoidE_1: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
Zp_residue_a_inv_1: fixes a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp" shows "(\<ominus> a) k = \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> a k"
that: fixes a :: "'out" assumes "obsp \<sigma> i = Some a" shows "thesis"
cycle_free_dest: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "cycle_free m n"   and "i \<le> n"   and "j \<le> n"   and "set xs \<subseteq> {0..n}" shows "len m i j (rem_cycles i j xs) \<le> len m i j xs"
insert_Diff: fixes a :: "'k"   and A :: "'k set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
field_def: fixes C :: "cname" shows "field C \<equiv> class_rec C flds"
map_poly_comm_ring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_comm_ring_hom hom \<equiv> comm_ring_hom hom"
vp_split_snoc: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<surd>* n'"   and "as \<noteq> []" obtains a' :: "'edge" and as' :: "'edge list" where "as = as' @ [a']" and "n -as'\<rightarrow>\<^sub>\<surd>* sourcenode a'" and "valid_edge a'" and "n' = targetnode a'"
l1_l2_def_2: shows "l2 = L ! 1"
tendsto_null_sum: fixes I :: "'a set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>i. i \<in> I \<Longrightarrow> ((\<lambda>x. f x i) \<longlongrightarrow> (0::'c)) F" shows "((\<lambda>i. sum (f i) I) \<longlongrightarrow> (0::'c)) F"
dickson_grading_lcs: fixes d :: "'c \<Rightarrow> nat"   and s :: "'c"   and t :: "'c" assumes "dickson_grading d" shows "d (lcs s t) \<le> ord_class.max (d s) (d t)"
sel_4: fixes x21 :: "nat"   and x22 :: "'c" shows "iAnnot' (H x21 x22) = x21"
isometry_on_uniformly_continuous: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "uniformly_continuous_on X f"
lnth_lappend2: fixes xs :: "'a llist"   and k :: "nat"   and n :: "nat"   and ys :: "'a llist" assumes "llength xs = enat k"   and "k \<le> n" shows "lnth (lappend xs ys) n = lnth ys (n - k)"
card_Un_le: fixes A :: "'g set"   and B :: "'g set" shows "card (A \<union> B) \<le> card A + card B"
from_nat_eq_imp_eq: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and x :: "nat"   and xa :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.from_nat Abs x = mod_type.from_nat Abs xa"   and "x < CARD('b)"   and "xa < CARD('b)" shows "x = xa"
power_one_right: fixes a :: "'b list" shows "a \<^sup>@ 1 = a"
wf_exec_is_write_seenD: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "P \<turnstile> (E, ws) \<surd>" shows "is_write_seen P E ws"
quotient_of_denom_add_dvd: fixes x :: "rat"   and y :: "rat" shows "snd (quotient_of (x + y)) dvd snd (quotient_of x) * snd (quotient_of y)"
prl: fixes g :: "'f \<times> nat"   and f :: "'f \<times> nat" assumes "prl g" shows "snd (prc f g) = True"
suc_def: fixes A :: "'a set" shows "local.suc A \<equiv> local.minim (local.AboveS A)"
ideD_1: fixes a :: "'a" assumes "ide a" shows "arr a"
splitAt_simpF: fixes ram :: "'a"   and a :: "'a list"   and b :: "'a list" assumes "ram \<notin> set a" shows "snd (splitAt ram (a @ b)) = snd (splitAt ram b)"
all_consts_def: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset" shows "pre_constants.all_consts C_info heads = heads |\<union>| constructors.C C_info"
Rule_DecisionE: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and a :: "action"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Decision X" obtains chain :: "char list" where "matches \<gamma> m p" and "a = Call chain"   | "matches \<gamma> m p" and "X = FinalAllow \<Longrightarrow> a = Accept" and "X = FinalDeny \<Longrightarrow> a = Drop \<or> a = Reject"
asTerm_def: fixes qX :: "('e, 'f, 'varSort, 'var, 'g) qTerm" shows "asTerm qX \<equiv> proj alphaGood qX"
find_slot_is_l_find_slot: shows "l_find_slot find_slot"
anonymous_profile_def: shows "anonymous_profile (permute_profile (permutation_of_list xs) R) = image_mset (weak_ranking \<circ> permute_profile (permutation_of_list xs) R) (mset_set agents)"
full_0_elim: fixes t :: "'b tree23" assumes "full 0 t" obtains "t = \<langle>\<rangle>"
act_emp: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c" assumes "unital_quantale_module act" shows "act x (\<Squnion> {}) = \<bottom>"
algebraic_int_power: fixes x :: "'a"   and n :: "nat" assumes "algebraic_int x" shows "algebraic_int (x ^ n)"
circ_def: shows "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
cases: fixes a :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "a \<in> deadlocked s" obtains t :: "'t" and x :: "'x" where "a = t__" and "thr s t__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "t__ \<turnstile> \<langle>x__, shr s\<rangle> \<wrong>" and "wset s t__ = None" and "\<And>LT. t__ \<turnstile> \<langle>x__, shr s\<rangle> LT \<wrong> \<Longrightarrow> \<exists>lt\<in>LT. must_wait s t__ lt (deadlocked s \<union> final_threads s)"   | t :: "'t" and x :: "'x" and ln :: "'l \<Rightarrow>f nat" where "a = ta__" and "thr s ta__ = \<lfloor>(xa__, ln__)\<rfloor>" and "all_final_except s (deadlocked s)" and "waiting (wset s ta__)"   | t :: "'t" and x :: "'x" and l :: "'l" and t' :: "'t" and ln :: "'l \<Rightarrow>f nat" where "a = t" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "0 < ln $ l" and "has_lock (locks s $ l) t'" and "t' \<noteq> t" and "t' \<in> deadlocked s \<or> t' \<in> final_threads s"
length_concat: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xss :: "'b list list" assumes "class.monoid_add plus zero" shows "length (concat xss) = sum_list (map length xss)"
valid_imp_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(imp_PL p q \<in> Cn {}) = (q \<in> Cn {p})"
in_set_subseqs: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs \<in> set (subseqs ys)) = subseq xs ys"
map_rec: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map f xs = rec_list [] (\<lambda>x _. (#) (f x)) xs"
integrable_cong_AE: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integrable M f = integrable M g"
SoundWhile: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and n :: "nat" assumes "\<And>m. G \<Turnstile>\<^sub>m c : B"   and "\<And>s. \<not> evalB b s \<Longrightarrow> A s s"   and "\<And>s r t. \<lbrakk>evalB b s; B s r; A r t\<rbrakk> \<Longrightarrow> A s t" shows "G \<Turnstile>\<^sub>n While b c : (\<lambda>s t. A s t \<and> \<not> evalB b t)"
Final_def: shows "Final \<equiv> {(X, mrk). mrk = reach root}"
bpred_def: shows "bpred \<equiv> \<lambda>(b, f, i) (pc, s, stk) (pc', s', stk'). pc' = pc + size (bcomp (b, f, i)) + (if bval b s = f then i else 0) \<and> s' = s \<and> stk' = stk"
uminus_carrier_iff_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" shows "(- A \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
LstSeq_1: fixes y :: "hf" shows "LstSeq \<lbrace>\<langle>0, y\<rangle>\<rbrace> 0 y"
mod_mult_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times (modulo a c) (modulo b c)) c = modulo (times a b) c"
assms_2: shows "f \<subseteq>\<^sub>f g"
BinOpThrow2: fixes E :: "char list \<Rightarrow> ty option"   and v\<^sub>1 :: "val"   and bop :: "bop"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> Throw r,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
dd0: fixes v :: "'a" assumes "v \<in> S" shows "dd v = 0"
TerminationPartial_sound: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<Gamma>\<turnstile>c \<down> Normal s"   and "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
assms: shows "set (coeffs p) \<subseteq> \<real>"
mult_cancel_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times c a = times c b) = (c = zero \<or> a = b)"
Sigma_fm_Iff: fixes B :: "fm"   and A :: "fm" assumes "{} \<turnstile> B IFF A"   and "supp A \<subseteq> supp B"   and "Sigma_fm A" shows "Sigma_fm B"
abs_of_diff_eq: fixes x :: "nat"   and y :: "nat" shows "(abs_of_diff x y = 0) = (x = y)"
comp_FSwImorph: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'k, 'm, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'l, 'n, 'q) model_scheme"   and h' :: "'l \<Rightarrow> 'r"   and hA' :: "'n \<Rightarrow> 's"   and MOD'' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'r, 's, 't) model_scheme" assumes "FSwImorph h hA MOD MOD'"   and "FSwImorph h' hA' MOD' MOD''" shows "FSwImorph (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
abs_conv_abscissa_diff_le: shows "abs_conv_abscissa (f - g) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa g)"
systemIN_noOUT: fixes x :: "chanID"   and i :: "AbstrLevelsID" assumes "systemIN x i" shows "\<not> systemOUT x i"
remove_vertex_V': fixes x :: "'a" shows "V\<^bsub>G.remove_vertex x\<^esub> \<subseteq> V\<^bsub>remove_vertex v\<^esub>"
epsclo_def: fixes Q :: "hf set" shows "ST.epsclo Q \<equiv> nfa.states \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr> \<inter> (\<Union>q\<in>Q. {q'. (q, q') \<in> (eps \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>)\<^sup>*})"
G'_reaches_V: fixes x :: "'a"   and y :: "'a" assumes "G'.reaches x y"   and "V x" shows "V y"
\<alpha>_l: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs)" shows "\<forall>b\<in>set (\<alpha> p). numbound0 b \<and> isint b (a # bs)"
eq_size_0: fixes t :: "'b tree" shows "(size t = 0) = (t = \<langle>\<rangle>)"
mat_col_last_def: fixes A :: "'b mat"   and n :: "nat" shows "mat_col_last A n \<equiv> (mat_row_last A\<^sup>T n)\<^sup>T"
explode_inject: fixes x :: "String.literal"   and y :: "String.literal" shows "(literal.explode x = literal.explode y) = (x = y)"
deltaG_real_def: fixes uu :: "real itself" shows "deltaG uu = 0"
1: shows "local.iso (some_unit a) \<and> \<guillemotleft>some_unit a : a \<star> a \<Rightarrow> a\<guillemotright>"
hyps_1: fixes xs :: "nat list" assumes "n = length xs - 1"   and "0 < length xs" shows "tuple_decode (length xs - 1) (tuple_encode (length xs - 1) xs) = xs"
empty_tree_empty_V: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>T\<^esub> = {}" shows "V\<^bsub>G\<^esub> = {}"
eigenvector_pow: fixes A :: "'b mat"   and n :: "nat"   and v :: "'b vec"   and k :: "'b"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "eigenvector A v k" shows "A ^\<^sub>m i *\<^sub>v v = k ^ i \<cdot>\<^sub>v v"
eventually_def: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>F \<equiv> \<not> \<box>\<not> F"
Tree\<^sub>\<alpha>_rep_abs: fixes t :: "('e, 'f, 'g) Tree" shows "rep_Tree\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> t) =\<^sub>\<alpha> t"
ts_ord_antisym: fixes x :: "'a"   and y :: "'a" assumes "!x \<sqsubseteq> !y"   and "!y \<sqsubseteq> !x" shows "!x = !y"
Pow_Int_eq: fixes A :: "'a set"   and B :: "'a set" shows "Pow (A \<inter> B) = Pow A \<inter> Pow B"
image_eq: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on g A" shows "\<Prod>(g ` A) = prod g A"
HAInitStates_InitState_trancl2: fixes S :: "'d"   and HA :: "'g \<Rightarrow> ('d, 'e, 'f) hierauto"   and ST :: "'g"   and A :: "('d, 'e, 'f) seqauto"   and x :: "'d" assumes "S \<in> HAStates (HA ST)"   and "A \<in> the (CompFun (HA ST) S)"   and "(x, S) \<in> (Restr (ChiRel (HA ST)) (HAInitStates (HA ST)))\<^sup>+" shows "(x, InitState A) \<in> (Restr (ChiRel (HA ST)) (HAInitStates (HA ST)))\<^sup>+"
tauOutputTrans: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "\<tau>.P \<longmapsto> a[b] \<prec> P'" shows "False"
mojmir_axioms: shows "mojmir \<Sigma> \<delta> q\<^sub>0 w F"
bind_return: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and N :: "'c measure"   and x :: "'b" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "x \<in> space M" shows "return M x \<bind> f = f x"
simps_1: fixes F :: "'b formula"   and G :: "'b formula" shows "is_cnf (F \<^bold>\<and> G) = (is_cnf F \<and> is_cnf G)"
scale_sum_left: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set"   and x :: "'b" shows "sum f A *\<^sub>C x = (\<Sum>a\<in>A. f a *\<^sub>C x)"
qq: shows "b = lincomb (ls q) (a ` Is q) \<and> hs q \<in> Is q \<and> ls q (a (hs q)) < (0::'a) \<and> (\<forall>h'. h' \<in> Is q \<longrightarrow> h' < hs q \<longrightarrow> (0::'a) \<le> ls q (a h')) \<and> cs q \<in> carrier_vec n \<and> local.span (a ` (Is q - {hs q})) = {x \<in> carrier_vec n. cs q \<bullet> x = (0::'a)} \<and> cs q \<bullet> b < (0::'a) \<and> r < m \<and> cs q \<bullet> a r < (0::'a) \<and> (\<forall>k'<r. (0::'a) \<le> cs q \<bullet> a k') \<and> Is (Suc q) = insert r (Is q - {hs q})"
fg: shows "f = g \<or> g \<in> set (P f)"
diff: fixes w :: "'a" assumes "v \<noteq> w" shows "f w = g w"
dfs_invar_def: fixes \<Sigma>i :: "'d set"   and R :: "('d \<times> 'd) set" shows "dfs_invar \<Sigma>i R \<equiv> dfs_invar_add \<inter> {s. dfs_\<alpha> s \<in> sse_invar \<Sigma>i R}"
absolutely_integrable_absolutely_integrable_ubound: fixes f :: "'b \<Rightarrow> real"   and S :: "'b set"   and g :: "'b \<Rightarrow> real" assumes "f integrable_on S"   and "g absolutely_integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x" shows "f absolutely_integrable_on S"
compare_Maybe_simps_1: shows "compare\<cdot>Nothing\<cdot>Nothing = EQ"
subgroup_finite: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)"   and "finite (carrier (G\<lparr>carrier := H__\<rparr>))" shows "finite H"
inner_zero_right: fixes x :: "'b" shows "inner x (0::'b) = (0::'a)"
right_respects_isomorphic: fixes f :: "'a"   and g :: "'a" assumes "f \<cong> g" shows "right f = right g"
fail_converter_def: shows "fail_converter = Converter (\<lambda>_. Fail)"
word_plus_mono_right: fixes y :: "'b word"   and z :: "'b word"   and x :: "'b word" assumes "y \<le> z"   and "x \<le> x + z" shows "x + y \<le> x + z"
max_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
cs_not_nil_implies_postrecording_event: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "fst (cs (S t i) cid) \<noteq> []"   and "i \<le> length t"   and "channel cid = Some (p, q)" shows "\<exists>j<i. postrecording_event t j"
update_update_1: fixes k :: "'c"   and v :: "'d"   and w :: "'d"   and m :: "('c, 'd) mapping" shows "Mapping.update k v (Mapping.update k w m) = Mapping.update k v m"
simps_1: fixes M :: "object atom formula set"   and M' :: "object atom formula set" shows "ground_action_path M [] M' = (M = M')"
well_valued0_append: fixes t :: "'b tree"   and as :: "'b list"   and as' :: "'b list" assumes "well_shaped t"   and "well_valued0 as t" shows "well_valued0 (as @ as') t"
D_UU: shows "D \<bottom> = {d. front_tickFree d}"
image_cong: fixes M :: "'a set"   and N :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
coprime2: shows "coprime v (u\<^sup>2 - v\<^sup>2)"
assms: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t" assumes "P c0"   and "next_propagate' c0 c1 loc t" shows "P c1"
permute_list_def: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list" shows "permute_list f xs = map (\<lambda>i. xs ! f i) [0..<length xs]"
scope_cases: fixes v :: "'form \<times> nat list"   and i :: "('form, 'var) antecedent"   and v' :: "'form \<times> nat list" assumes "scope' v i v'" obtains c :: "'form" and is' :: "nat list" and "is" :: "nat list" and ia :: "nat" where "v = (c, is)" and "i = in_port_at (c, is) ia" and "v' = (c, is')" and "c \<in> set conclusions" and "is' \<in> (#) 0 ` it_paths (it' c)" and "prefix (is @ [ia]) is'"
at_calc_1: fixes a :: "'a"   and b :: "'a"   and pi :: "('a \<times> 'a) list"   and x :: "'a" assumes "at TYPE('a)" shows "((a, b) # pi) \<bullet> x = swap (a, b) (pi \<bullet> x)"