hcju/isabelleps huggingface dataset:
fconverse_fconverse: shows "((\<D>\<^sub>\<circ> f)\<inverse>\<^sub>\<bullet>)\<inverse>\<^sub>\<bullet> = \<D>\<^sub>\<circ> f"
assms_1: shows "G\<^sub>1 \<simeq> G\<^sub>2"
N_def: shows "(LEAST N. N \<in> Domain wl) = nextMax (Range f)"
trans: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
ran_def: fixes m :: "'i \<Rightarrow> 'h option" shows "ran m = {b. \<exists>a. m a = Some b}"
bind_is_OK_E3: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog" assumes "h \<turnstile> ok Heap_Error_Monad.bind f g"   and "pure f h" obtains x :: "'d" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> ok g x"
m_D: shows "P \<turnstile> D has F,Static:vT' in D"
hom_connected_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (H (partial_magma.cod V \<nu>) \<mu> \<noteq> partial_magma.null V)"
avl_insert: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t" shows "avl (AVL_Bal_Set.insert x t) \<and> height (AVL_Bal_Set.insert x t) = height t + (if incr t (AVL_Bal_Set.insert x t) then 1 else 0)"
fps_inverse_gp': shows "inverse (Abs_fps (\<lambda>n. 1::'a)) = 1 - fps_X"
Int_Collect: fixes x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
cSup_atLeastLessThan: fixes y :: "'b"   and x :: "'b" assumes "y < x" shows "Sup {y..<x} = x"
simps_1: fixes c :: "'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma> :: "'g" shows "foldli [] c f \<sigma> = \<sigma>"
brw_inner_invar_def: fixes rules :: "('d, 'e) ta_rule set"   and q :: "'d" shows "brw_inner_invar rules q \<equiv> \<lambda>(Q, W, rcm) it (Q', W', rcm'). br'_inner_invar rules q (brw_\<alpha> (Q, W, rcm)) it (brw_\<alpha> (Q', W', rcm')) \<and> Q' |` dom Q = Q \<and> (let dsqr = {r \<in> rules - it. the (rcm r) \<le> 1} in \<forall>q t. Q' q = Some t \<longrightarrow> Q q = Some t \<or> Q q = None \<and> (\<exists>r\<in>dsqr. q = lhs r \<and> t = construct_witness Q r))"
conj: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (MIR.conj p q) = Ifm bs (And p q)"
index_one_mat_2: fixes n :: "nat" shows "dim_row (1\<^sub>m n) = n"
digit_to_char_simps_5: shows "digit_to_char 4 = CHR ''4''"
coboundedI: fixes A :: "'t set"   and a :: "'t" assumes "finite A"   and "a \<in> A" shows "ord_term_lin.Min A \<preceq>\<^sub>t a"
simps_2: shows "negUni TrueFUni = FalseFUni"
lazycomm_var: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star> + y) = (y \<cdot> x\<^sup>\<star> \<le> x \<cdot> (x + y)\<^sup>\<star> + y)"
in_rangeCE: fixes y :: "'a"   and A :: "('b \<Rightarrow> 'a) set" assumes "y \<in> rangeC A" obtains f :: "'b \<Rightarrow> 'a" and x :: "'b" where "f \<in> A" and "y = f x"
filter_replicate: fixes P :: "'b \<Rightarrow> bool"   and n :: "nat"   and x :: "'b" shows "filter P (replicate n x) = (if P x then replicate n x else [])"
compact_translation_subtract: fixes s :: "'b set"   and a :: "'b" assumes "compact s" shows "compact ((\<lambda>x. x - a) ` s)"
add: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "additive f" shows "f (x + y) = f x + f y"
rotater_Suc: fixes n :: "nat"   and xs :: "'b list" shows "rotater (Suc n) xs = rotater1 (rotater n xs)"
a_def: shows "a \<equiv> \<lambda>k. LEAST x. \<phi> (the (\<phi> k (f k \<triangleright> x))) = f k"
IH: fixes f :: "'c \<Rightarrow> 'd" assumes "higher_differentiable_on S f n"   and "(0::'d) \<notin> f ` S"   and "open S" shows "higher_differentiable_on S (\<lambda>x. inverse (f x)) n"
nop_onto_vrange: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_onto A n f" shows "\<R>\<^sub>\<circ> f = A"
closure_of_subset_eq: fixes S :: "'c set"   and X :: "'c topology" shows "(S \<subseteq> topspace X \<and> X closure_of S \<subseteq> S) = closedin X S"
ta_prod_correct_aux1: fixes TA1 :: "('b, 'a, 'd) tree_automaton_rec_scheme"   and TA2 :: "('c, 'a, 'e) tree_automaton_rec_scheme" shows "ta_lang (ta_prod TA1 TA2) = ta_lang TA1 \<inter> ta_lang TA2"
subgroups_Inter: fixes A :: "'a set set" assumes "\<And>H. H \<in> A \<Longrightarrow> subgroup H (G\<lparr>carrier := S\<rparr>)"   and "A \<noteq> {}" shows "subgroup (\<Inter> A) (G\<lparr>carrier := S\<rparr>)"
simps_7: fixes x3 :: "nat \<times> nat" shows "SAT_PUSH' x3 \<noteq> NONSAT_PUSH'"
Bp_if_B: fixes t :: "'a bro"   and h :: "nat" assumes "t \<in> B h" shows "t \<in> Bp h"
subst_lift: fixes t :: "db"   and k :: "nat"   and s :: "db" shows "lift t k<s/k> = t"
prems_1: shows "wf\<^sub>s\<^sub>t V ((\<langle>assign: t' \<doteq> t''\<rangle>\<^sub>s\<^sub>t # S) @ S')"
mtf2_backwards_effect4': fixes xs :: "'c list"   and q :: "'c"   and n :: "nat"   and x :: "'c" assumes "index xs q < length xs"   and "q \<in> set xs"   and "distinct xs"   and "index (mtf2 n q xs) x < index xs q - n"   and "x \<in> set xs" shows "index xs x < index xs q - n"
pos_int_cases: fixes k :: "int" assumes "0 < k" obtains n :: "nat" where "k = int n" and "0 < n"
independent_card_le_dim: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set" assumes "B \<subseteq> V"   and "vec.independent B" shows "card B \<le> vec.dim V"
intros_4: fixes v :: "'a"   and p :: "Player"   and Ps :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<notin> VV p"   and "path_conforms_with_strategy p Ps \<sigma>" shows "path_conforms_with_strategy p (LCons v Ps) \<sigma>"
equalOn_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "equalOn A f g = (\<forall>x\<in>A. f x = g x)"
eucl_truncate_up_vec_def: fixes p :: "nat"   and x :: "('a, 'b) vec" shows "eucl_truncate_up p x = (\<chi>i. eucl_truncate_up p (x $ i))"
perms_comp: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "f \<in> perms"   and "g \<in> perms" shows "f \<circ> g \<in> perms"
assms: shows "e1 \<sqsubset>\<^sup>i e2"
simps_5: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "'a" shows "(case Any of Const x \<Rightarrow> f1 x | Any \<Rightarrow> f2) = f2"
get_via_put: fixes s :: "'b" shows "get s = (THE v. put s v = s)"
simps_2: fixes p :: "real mpoly"   and L :: "real list" shows "aEval (Less p) L = (insertion (nth_default 0 L) p < 0)"
CFG_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set" shows "CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry \<and> CFG_axioms \<alpha>n invar defs uses"
length_word_rsplit_exp_size': fixes w :: "'d word" shows "length (word_rsplit w) = (size w + LENGTH('c) - 1) div LENGTH('c)"
line_integral_on_path_eq_line_integral_on_equiv_chain'_1: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_path \<gamma> one_chain"   and "boundary_chain one_chain"   and "line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"   and "finite basis" shows "one_chain_line_integral F basis one_chain = line_integral F basis \<gamma>"
c1_on_open_axioms: shows "c1_on_open f f' X"
prems_2: fixes i :: "nat" assumes "i < length (s # ss)" shows "cone_decomp ((s # ss) ! i) ((ps # pss) ! i)"
le_fininf_empty: fixes w :: "'item stream" shows "[] \<preceq>\<^sub>F\<^sub>I w"
indep_sets_sigma: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set" assumes "indep_sets F I"   and "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
r_consistent_recfn: fixes r_psi :: "recf" assumes "recfn 2 r_psi" shows "recfn 2 (r_consistent r_psi)"
f_integrable: fixes i :: "nat" shows "set_integrable lborel {l i..u i} f"
vimage2p_mono': fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and f :: "'h \<Rightarrow> 'e"   and g :: "'i \<Rightarrow> 'f" assumes "R \<le> S" shows "vimage2p f g R \<le> vimage2p f g S"
subset_li_is_li: fixes A :: "'c set"   and B :: "'c set" assumes "lin_indpt A"   and "B \<subseteq> A" shows "lin_indpt B"
higher_deriv_ln_Gamma_complex: fixes x :: "complex"   and j :: "nat" assumes "x \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(deriv ^^ j) ln_Gamma x = (if j = 0 then ln_Gamma x else Polygamma (j - 1) x)"
total_def_var_1: fixes x :: "'a" shows "is_total x = (x ; 1 = 1)"
poly_exp: fixes p :: "'a list"   and n :: "nat"   and x :: "'a" shows "poly (p %^ n) x = poly p x ^ n"
induct: fixes P :: "parity \<Rightarrow> parity \<Rightarrow> bool"   and a0 :: "parity"   and a1 :: "parity" assumes "P Even Even"   and "P Odd Odd"   and "P Even Odd"   and "P Odd Even"   and "\<And>y. P Either y"   and "P Even Either"   and "P Odd Either" shows "P a0 a1"
upper: fixes t :: "real" assumes "t \<in> T"   and "t0 < t" shows "f t (w t) < w' t"
Hermite_def: fixes associates :: "'d set"   and residues :: "'d \<Rightarrow> 'd set"   and A :: "(('d, 'e) vec, 'f) vec" shows "Hermite associates residues A = (Complete_set_non_associates associates \<and> Complete_set_residues residues \<and> echelon_form A \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'d)) \<in> associates) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j<i. A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'d)) \<in> residues (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'d))))))"
and_assoc: fixes a :: "bool"   and b :: "bool"   and c :: "bool" shows "(a \<and> b \<and> c) = ((a \<and> b) \<and> c)"
sup_left_commute: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x + (y + z) = y + (x + z)"
ftf_syn32: fixes u :: "'a event list"   and u1 :: "'a event list"   and t :: "'a event list"   and t1 :: "'a event list"   and s :: "'a event list"   and A :: "'a set" assumes "u = u1 @ [tick] \<and> t = t1 @ [tick] \<and> s setinterleaves ((t, u), insert tick (ev ` A))" shows "\<exists>s1. s1 setinterleaves ((t1, u1), insert tick (ev ` A)) \<and> s = s1 @ [tick]"
Seq_MtransC_invert: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and d' :: "('test, 'atom) com"   and t' :: "'state" assumes "(c1 ;; c2, s) \<rightarrow>*c (d', t')" obtains c1' :: "('test, 'atom) com" where "(c1, s) \<rightarrow>*c (c1', t')" and "d' = c1' ;; c2"   | s' :: "'state" where "(c1, s) \<rightarrow>*t s'" and "(c2, s') \<rightarrow>*c (d', t')"
vunion_complement: fixes J :: "V"   and I :: "V" assumes "J \<subseteq>\<^sub>\<circ> I" shows "I -\<^sub>\<circ> J \<union>\<^sub>\<circ> J = I"
oalist_inv_list_of_oalist: fixes x :: "('a, 'b) oalist_ntm" shows "ko_ntm.oalist_inv (list_of_oalist_ntm x)"
binomial_symmetric_plus: fixes n :: "nat"   and k :: "nat" shows "n + k choose n = n + k choose k"
InitNonObjectReds: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and r :: "fdecl list \<times> (char list list \<times> char list exp) mdecl list"   and sh' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "C \<noteq> Object"   and "class P C = \<lfloor>(D, r)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Processing))"   and "P \<turnstile> \<langle>INIT C' (D # C # Cs,False) \<leftarrow> e,(h, l, sh'),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
lub_sum_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "class.idempotent_semiring one times sup less_eq less bot" shows "(\<forall>k. less_eq (f k) x) = less_eq (comm_monoid_add.sum sup bot f {k. True}) x"
orderI: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a = a \<inter># b" shows "a \<subseteq># b"
surjective: fixes r :: "('a, 'b, 'c, 'd) configuration_scheme" shows "r = \<lparr>states = states r, msgs = msgs r, process_snapshot = ps r, channel_snapshot = cs r, \<dots> = more r\<rparr>"
ucast_ucast_id: fixes x :: "'b word" assumes "LENGTH('b) < LENGTH('c)" shows "UCAST('c \<rightarrow> 'b) (UCAST('b \<rightarrow> 'c) x) = x"
full_pmdl_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "nat set" shows "(p \<in> full_pmdl K) = (component_of_term ` keys p \<subseteq> K)"
sel_2: fixes x1 :: "'a"   and x2 :: "'a" shows "high (Interval x1 x2) = x2"
assms_3: shows "x \<in> MPLS_s PROB s"
transpD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
convex_on_alt: fixes C :: "'a set"   and f :: "'a \<Rightarrow> real" shows "convex_on C f = (\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
character_double_dual_iso: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "finite_comm_group G"   and "x \<in> carrier G" shows "character (Characters G) (finite_comm_group.double_dual_iso G x)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f0 :: "'a"   and f1 :: "'a" assumes "partial_magma.arr C f0 \<and> partial_magma.arr C f1 \<and> partial_magma.cod C f0 = partial_magma.cod C f1" shows "cospan_diagram_axioms C f0 f1"
left_spec_def: fixes j :: "nat" shows "local.left_spec j = (if \<exists>i<j. xs ! i < xs ! j then Some (GREATEST i. i < j \<and> xs ! i < xs ! j) else None)"
card_seteq: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
fbd_ifbd_inv2_inv: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "(bd\<^sub>\<F> \<circ> bd\<^sup>-\<^sub>\<F>) \<phi> = \<phi>" shows "Sup_pres \<phi>"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a ; (b + c) = a ; b + a ; c"
g'': fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "g'' tx = ($) (f'' tx)"
span_eq: fixes S :: "'a set"   and T :: "'a set" shows "(span S = span T) = (S \<subseteq> span T \<and> T \<subseteq> span S)"
normalize_quot_proj_1: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "fst (normalize_quot (a, b)) = a div (gcd a b * unit_factor b)"
simps: fixes i :: "nat"   and t :: "nat"   and e :: "nat"   and s :: "bddi" shows "iteci i t e s = lowest_topsci [i, t, e] s \<bind> (\<lambda>lt. case lt of None \<Rightarrow> case_ifexici (return (t, s)) (return (e, s)) (\<lambda>_ _ _. raise STR ''Cannot happen'') i s | Some a \<Rightarrow> restrict_topci i a True s \<bind> (\<lambda>ti. restrict_topci t a True s \<bind> (\<lambda>tt. restrict_topci e a True s \<bind> (\<lambda>te. restrict_topci i a False s \<bind> (\<lambda>fi. restrict_topci t a False s \<bind> (\<lambda>ft. restrict_topci e a False s \<bind> (\<lambda>fe. iteci ti tt te s \<bind> (\<lambda>(tb, s'). iteci fi ft fe s' \<bind> (\<lambda>(fb, s''). ifci a tb fb s'')))))))))"
complex_unimodular_polar: fixes z :: "complex" assumes "cmod z = 1" obtains t :: "real" where "0 \<le> t" and "t < 2 * pi" and "z = Complex (cos t) (sin t)"
cases: fixes t :: "nat"   and a1 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and a2 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "blstep t a1 a2" obtains c :: "nat" and n :: "nat" and ts :: "nat \<Rightarrow> thread" and c' :: "nat" and n' :: "nat" and s' :: "thread" where "a1 = (c, n, ts)" and "a2 = (c', n', ts(t := s'))" and "t < N" and "bstep_sng (c, n, ts t) (c', n', s')"
hyps: assumes "trace init t final"   and "\<not> ps (S t i) p \<noteq> None"   and "channel cid = Some (q, p)" shows "cs (S t i) cid = cs init cid"
resComm: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x><\<nu>y>P \<sim>\<^sup>s\<^sub>l <\<nu>y><\<nu>x>P"
mset_merge: fixes xs :: "'a list"   and ys :: "'a list" shows "mset (local.merge xs ys) = mset xs + mset ys"
Un_Plus_bij_betw: fixes A :: "'a set"   and B :: "'a set" assumes "A \<inter> B = {}" shows "\<exists>f. bij_betw f (A \<union> B) (A <+> B)"
rel_spmf_mono_strong: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h spmf"   and g :: "'i spmf"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>A x y; x \<in> set_spmf f; y \<in> set_spmf g\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
StrongEq_L_trans: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b"   and z :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "\<tau> \<Turnstile> x \<triangleq> y"   and "\<tau> \<Turnstile> y \<triangleq> z" shows "\<tau> \<Turnstile> x \<triangleq> z"
simps: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" shows "not_final_thread s t = ((\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x) \<or> (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> ln \<noteq> no_wait_locks) \<or> (\<exists>x w ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> wset s t = \<lfloor>w\<rfloor>))"
iso_lunit: fixes f :: "'a" assumes "ide f" shows "local.iso \<l>[f]"
d: shows "ball c d \<inter> affine hull S \<subseteq> S"
1: shows "(Ze1, Re1) = poincare_circle_euclidean (of_complex z1) r1"
Cd_simps_1: fixes d :: "'d" shows "Cd d {} = {}"
simps: fixes C :: "'b literal set"   and S :: "'b trm set"   and \<sigma> :: "('b \<times> 'b trm) list" shows "subst_ecl (Ecl C S) \<sigma> = Ecl (subst_cl C \<sigma>) {t \<lhd> \<sigma> |t. t \<in> S}"
coeff_const: fixes a :: "'a"   and i :: "nat" shows "poly.coeff [:a:] i = (if i = 0 then a else (0::'a))"
wellformed_complete_item_derives: fixes x :: "item" assumes "wellformed_item x"   and "is_complete x" shows "derives [item_nonterminal x] (item_\<alpha> x)"
UN_insert: fixes B :: "'e \<Rightarrow> 'd set"   and a :: "'e"   and A :: "'e set" shows "\<Union> (B ` insert a A) = B a \<union> \<Union> (B ` A)"
int_pow_eq_id: fixes x :: "'c"   and i :: "int" assumes "x \<in> carrier H" shows "(x [^]\<^bsub>H\<^esub> i = \<one>\<^bsub>H\<^esub>) = (int (H.ord x) dvd i)"
imp_nfoldli_def: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool Heap"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and s :: "'c" shows "imp_nfoldli l c f s \<equiv> heap.fixp_fun (\<lambda>D (l, s). case l of [] \<Rightarrow> return s | x # ls \<Rightarrow> c s \<bind> (\<lambda>b. if b then f x s \<bind> (\<lambda>s'. D (ls, s')) else return s)) (l, s)"
AD: shows "A \<inter> D \<in> null_sets lborel"
surjective: fixes r :: "'a ConvSchema_scheme" shows "r = \<lparr>cLengthF = cLengthF r, cMassF = cMassF r, cTimeF = cTimeF r, cCurrentF = cCurrentF r, cTemperatureF = cTemperatureF r, cAmountF = cAmountF r, cIntensityF = cIntensityF r, \<dots> = ConvSchema.more r\<rparr>"
int_eq': shows "(fst (bezw x q) * int x + snd (bezw x q) * int q) mod int q = int (1 mod q)"
j: shows "atom j \<sharp> (s, s', i, x, y, y')"
compile_mn_correct': fixes B :: "abc_inst list"   and arity :: "nat"   and fap :: "abc_inst list"   and ft :: "nat"   and fft :: "nat"   and xs :: "nat list"   and f :: "recf"   and r :: "nat"   and far :: "nat"   and anything :: "nat list" assumes "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"   and "ft = max (Suc arity) fft"   and "length xs = arity"   and "terminate f (xs @ [r])"   and "\<And>anything. {\<lambda>nl. nl = xs @ r # 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fft - Suc far) @ anything}"   and "\<forall>i<r. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"   and "\<forall>i<r. 0 < rec_exec f (xs @ [i])"   and "rec_exec f (xs @ [r]) = 0"   and "rec_ci f = (fap, far, fft)" shows "{\<lambda>nl. nl = xs @ 0 \<up> (max (Suc arity) fft - arity) @ anything} fap @ B {\<lambda>nl. nl = xs @ rec_exec (Mn arity f) xs # 0 \<up> (max (Suc arity) fft - Suc arity) @ anything}"
simps_1: fixes i :: "nat"   and \<Gamma> :: "nat \<Rightarrow> db"   and k :: "nat" shows "closing_subst (DBVar i) \<Gamma> k = (if k \<le> i then \<Gamma> (i - k) else DBVar i)"
id_nec4_1: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[x\<^sup>P \<^bold>\<noteq>\<^sub>E y\<^sup>P \<^bold>\<equiv> \<^bold>\<box>x\<^sup>P \<^bold>\<noteq>\<^sub>E y\<^sup>P in v]"
assms_5: shows "integrable M (\<lambda>x. \<bar>g x\<bar> powr p)"
abs_eq: shows "One_pdevs = Abs_pdev One_pdevs_raw"
Fvars_empty_ground: fixes t :: "trm" assumes "Fvars t = {}" shows "ground t"
assms_1: shows "ps \<in> sterms \<Gamma> (\<Gamma> pn)"
dgrad_set_le_Un: fixes d :: "'c \<Rightarrow> nat"   and S :: "'c set"   and T :: "'c set"   and U :: "'c set" shows "dgrad_set_le d (S \<union> T) U = (dgrad_set_le d S U \<and> dgrad_set_le d T U)"
ptrm_alpha_equiv_transp: shows "transp (\<approx>)"
Mp_factorization_m: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" assumes "poly_mod_2 m" shows "poly_mod.factorization_m m (poly_mod.Mp m f) cfs = poly_mod.factorization_m m f cfs"
epi_group_mul_eq: shows "((\<lambda>(x, y). x \<otimes> y) \<in> epi (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (A <#> B = carrier G \<and> (\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = y \<otimes> x))"
fsts: shows "set (map fst fs) \<subseteq> set ftrs"
rationalizes_on_f_range_on_V_axiom_on_2: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set"   and r :: "('b \<times> 'b) set" assumes "rationalizes_on A f r" shows "V_axiom_on A f"
normalize_mult_normalize_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize (times a (normalize b)) = normalize (times a b)"
size_eq_0_iff_empty: fixes M :: "'c multiset" shows "(size M = 0) = (M = {#})"
cod_char: fixes f :: "'a" shows "Right.cod f = (if Right.arr f then cod f else null)"
Path_snoc: fixes f :: "'b \<Rightarrow> 'b ref"   and a :: "'b"   and q :: "'b ref"   and p :: "'b ref"   and as :: "'b list" assumes "Path (f(a := q)) p as (Ref a)" shows "Path (f(a := q)) p (as @ [a]) q"
intros: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y" shows "Wellfounded.accp r x"
word_bw_comms_3: fixes x :: "'b word"   and y :: "'b word" shows "x XOR y = y XOR x"
last_state_reachable: fixes A :: "('b, 'c) ioa"   and e :: "'b \<times> ('c \<times> 'b) list" assumes "is_exec_of A e" shows "reachable A (last_state e)"
XSvar: shows "|XS| <o |UNIV| \<or> finite XS"