hcju/isabelleps huggingface dataset:
valof_alt: fixes x :: "('a, 'b) IEEE.float" shows "valof x = (if IEEE.exponent x = 0 then if sign x = 0 then valofd x else - valofd x else if sign x = 0 then valofn x else - valofn x)"
isomorphic_refl: fixes G :: "nat set \<times> nat set set" assumes "uwellformed G" shows "G \<simeq> G"
decseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'c" shows "decseq f = (\<forall>n. f (Suc n) \<le> f n)"
singletonize_L4Ports_src: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and pts :: "ipt_l4_ports"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "match_list (\<beta>, \<alpha>) (map (Match \<circ> Src_Ports) (singletonize_L4Ports pts)) a p = matches (\<beta>, \<alpha>) (Match (Src_Ports pts)) a p"
sym: fixes s :: "'c event list"   and X :: "'c event set"   and t :: "'c event list" shows "setinterleaving (s, X, t) = setinterleaving (t, X, s)"
i_append_imp_iprefix: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "xs \<sqsubseteq> xs \<frown> f"
mult_left: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" shows "(\<lambda>x. h x * f x) \<in> L F (\<lambda>x. h x * g x)"
KM_map_def: shows "KM_map \<equiv> \<Lambda> f. cfun_map\<cdot>(cfun_map\<cdot>f\<cdot>ID)\<cdot>ID"
simps: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and p :: "'e pmf"   and q :: "'f pmf" shows "rel_pmf R p q = (\<exists>pq. (\<forall>x xa. (x, xa) \<in> set_pmf pq \<longrightarrow> R x xa) \<and> map_pmf fst pq = p \<and> map_pmf snd pq = q)"
get_min_in: fixes h :: "'b tree" assumes "h \<noteq> \<langle>\<rangle>" shows "get_min h \<in> set_tree h"
ipurge_tr_aux_foldr_subset: fixes U :: "'e set"   and V :: "'e set"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "U \<subseteq> V" shows "ipurge_tr_aux I D U (ipurge_tr_aux_foldr I D P V xs) = ipurge_tr_aux_foldr I D P V xs"
assms: shows "{#} \<in> N_of_state (Liminf_wstate Sts) \<union> P_of_state (Liminf_wstate Sts) \<union> Q_of_state (Liminf_wstate Sts)"
wp_conj: fixes P :: "('b, 'c) urel"   and q :: "'c upred"   and r :: "'c upred" shows "P wp (q \<and> r) = (P wp q \<and> P wp r)"
assms_1: shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> Q \<longmapsto> Rs"
assms_1: shows "is_ground_atm (A \<cdot> \<sigma>)"
list_assn_aux_append2: fixes l2 :: "'a list"   and l2' :: "'b list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l1 :: "'a list"   and l1' :: "'b list" assumes "length l2 = length l2'" shows "list_assn P (l1 @ l2) (l1' @ l2') = list_assn P l1 l1' * list_assn P l2 l2'"
complex_homo_coords_eq_equivp: shows "equivp (\<approx>)"
Ts: shows "P \<turnstile> rev (take n ST) [\<le>] Ts"
feq: fixes z :: "complex" assumes "z \<in> S" shows "f z = cos (g z)"
prems_1: shows "a \<le> mod_type_class.from_nat i"
assms_2: shows "redundancy_criterion \<Gamma> Rf Ri"
simple: shows "simple_group (G\<lparr>carrier := \<GG> ! (i - 1 + 1)\<rparr> Mod \<GG> ! (i - 1))"
emeasure_eq_measure: fixes M :: "'c measure"   and A :: "'c set" assumes "finite_measure M" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
gSubstGAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstGAbs MOD \<equiv> \<forall>ys y Y' Y xs x s X' X. isInBar (xs, s) \<and> wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> (xs \<noteq> ys \<or> x \<noteq> y) \<and> fresh xs x Y' \<and> gFresh MOD xs x Y' Y \<longrightarrow> gSubstAbs MOD ys Y' Y y (Abs xs x X') (gAbs MOD xs x X' X) = gAbs MOD xs x (X' #[Y' / y]_ys) (gSubst MOD ys Y' Y y X' X)"
pick_card: fixes I :: "nat set"   and i :: "nat" shows "pick I (card {a \<in> I. a < i}) = (LEAST a. a \<in> I \<and> i \<le> a)"
inj_zero_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_zero_hom hom \<equiv> zero_hom hom \<and> injective hom"
Uk1: shows "uk \<in> carrier (residue_ring (p ^ k)) \<and> uk * p ^ n = x (n + k)"
additive_subgroup_a_kernel: shows "additive_subgroup (a_kernel G H h) G"
epi_right_invertible: fixes g :: "'a \<Rightarrow> 'b"   and H :: "('a, 'c) monoid_scheme"   and G :: "('b, 'd) monoid_scheme"   and f :: "'b \<Rightarrow> 'a" assumes "g \<in> hom H G"   and "f \<in> carrier G \<rightarrow> carrier H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> g (f x) = x" shows "g \<in> epi H G"
int_dvd_int_iff: fixes m :: "nat"   and n :: "nat" shows "(int m dvd int n) = (m dvd n)"
small_big_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> o[F](g1)"   and "f2 \<in> O[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> o[F](\<lambda>x. g1 x * g2 x)"
simps_46: fixes x7 :: "'b ptrm"   and x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "PSnd x7 \<noteq> PPair x51 x52"
liftAll_lift_ext: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and inp :: "'c \<Rightarrow> 'b option" shows "liftAll (\<lambda>x. f x = g x) inp = (lift f inp = lift g inp)"
blocks_with_x_partition: fixes x :: "'a" assumes "x \<in> \<V>" shows "partition_on (\<V> - {x}) {b - {x} |b. b \<in># \<B> \<and> x \<in> b}"
trnr\<^sub>\<epsilon>_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c"   and u :: "'c"   and \<nu> :: "'c" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_bicategory.trnr\<^sub>\<epsilon> V H \<a> \<i> src f g \<epsilon> u \<nu> \<equiv> V (bicategory.runit V H \<a> \<i> src u) (V (H u \<epsilon>) (V (\<a> u f g) (H \<nu> g)))"
extendRuleNonEmpty: fixes \<Gamma> :: "('c, 'd) form multiset"   and \<Delta> :: "('c, 'd) form multiset"   and r :: "('c, 'd) sequent list \<times> ('c, 'd) sequent" assumes "\<not> (\<Gamma> = \<Empt> \<and> \<Delta> = \<Empt>)" shows "extendRule ( \<Gamma> \<Rightarrow>* \<Delta>) r \<noteq> r"
rsub_def: fixes interval_tree :: "interval_tree" shows "rsub interval_tree = (case interval_tree of Node x21 x22 x23 x24 \<Rightarrow> x24)"
elims_1: fixes x :: "ty"   and y :: "bool" assumes "is_Array x = y" obtains T :: "ty" where "x = T\<lfloor>\<rceil>" and "y"   | "x = Void" and "\<not> y"   | "x = Boolean" and "\<not> y"   | "x = Integer" and "\<not> y"   | "x = NT" and "\<not> y"   | v :: "String.literal" where "x = Class v" and "\<not> y"
set_take_drop_i_le_j: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list" assumes "i \<le> j" shows "set xs = set (take j xs) \<union> set (drop i xs)"
fun_type_inv_wf: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<Gamma> t = Fun f T"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "arity f = length T"
in_enumerate_iff_nth_default_eq: fixes x :: "'b"   and dflt :: "'b"   and n :: "nat"   and xs :: "'b list" assumes "x \<noteq> dflt" shows "((n, x) \<in> set (enumerate 0 xs)) = (nth_default dflt xs n = x)"
left_neutral: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and a :: "'k" assumes "monoid f z" shows "f z a = a"
simps: fixes b :: "bool" shows "the_Bool (Bool b) = b"
SPEC_eq_is_RETURN_2: fixes y :: "'b" shows "SPEC (\<lambda>x. x = y) = RETURN y"
isDiagonal_diagonal_to_Smith: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "isDiagonal A" shows "isDiagonal (diagonal_to_Smith A bezout)"
aless_imp_le: fixes x :: "ant"   and y :: "ant" assumes "x < y" shows "x \<le> y"
essentially_surjective: fixes b :: "'c" assumes "C.ide b" shows "\<exists>a. D.ide a \<and> C.isomorphic (G a) b"
CFG_intra_edge_transfer_uses_only_Use: fixes a :: "'a"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and s' :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "valid_edge a"   and "\<forall>V\<in>Use (sourcenode a). local.CFG.state_val s V = local.CFG.state_val s' V"   and "intra_kind (kind a)"   and "local.CFG.pred (kind a) s"   and "local.CFG.pred (kind a) s'" shows "\<forall>V\<in>Def (sourcenode a). local.CFG.state_val (local.CFG.transfer (kind a) s) V = local.CFG.state_val (local.CFG.transfer (kind a) s') V"
magQ_one: shows "\<lbrakk>1\<rbrakk>\<^sub>Q = (1::'e)"
i_in_red_inf: shows "to_F \<iota> \<in> no_labels.Red_I_\<G> (fst ` (N1p \<union> Mp))"
Arg_cnj_eq_inverse: fixes z :: "complex" assumes "z \<noteq> 0" shows "Arg (cnj z) = Arg (inverse z)"
seq_app_second: fixes s :: "'a"   and \<sigma> :: "nat \<Rightarrow> 'a" shows "(s ## \<sigma>) 1 = \<sigma> 0"
dghm_def: shows "\<GG> = [\<GG>\<lparr>ObjMap\<rparr>, \<GG>\<lparr>ArrMap\<rparr>, \<GG>\<lparr>HomDom\<rparr>, \<GG>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
norm_triangle_half_l: fixes x :: "'a"   and y :: "'a"   and e :: "real"   and x' :: "'a" assumes "norm (x - y) < e / 2"   and "norm (x' - y) < e / 2" shows "norm (x - x') < e"
valid_init_msaux: fixes L :: "nat set"   and R :: "nat set"   and I :: "\<I>"   and n :: "nat"   and pos :: "bool" assumes "L \<subseteq> R" shows "valid_msaux (init_args I n L R pos) 0 (init_msaux (init_args I n L R pos)) []"
iMOD_sub1: fixes r :: "nat"   and k :: "nat"   and m :: "nat" assumes "r \<le> k" shows "k \<ominus> [ r, mod m ] = [ (k - r) mod m, mod m, (k - r) div m ]"
power2_minus: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) 2 = power.power one times a 2"
i_join_i_take: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and I :: "nat set" shows "f \<Down> n \<Join>\<^sub> I = f \<Join>\<^sub> I \<Down> card (I \<down>< n)"
shiftl_shiftr3: fixes b :: "nat"   and c :: "nat"   and a :: "'a word" assumes "b \<le> c" shows "a << b >> c = (a >> c - b) && mask (size a - c)"
MsgC0: shows "msgs c0 = (\<lambda>m. if \<exists>p. m = <p, inM False> then 1 else 0)"
absorb_iff1: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<cdot> b = a)"
card_of_ordIso_finite: fixes A :: "'b set"   and B :: "'c set" assumes "|A| =o |B|" shows "finite A = finite B"
gt_ex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
preserves_reflects_arr: fixes f :: "'a" shows "Left_a.arr (H\<^sub>L a f) = Left_a.arr f"
assms_4: shows "consistent_signs_at_roots p qs \<subseteq> set signs"
sgn_greater: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sgn :: "'k \<Rightarrow> 'k"   and abs :: "'k \<Rightarrow> 'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and a :: "'k" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (sgn a) = less zero a"
enres_lift_def: fixes m :: "'e nres" shows "enres_lift m \<equiv> m \<bind> (\<lambda>x. RETURN (Inr x))"
x1: shows "mod_type_class.from_nat y = mod_type_class.from_nat x"
sum_succ_eqpoll: fixes a :: "V"   and b :: "V" shows "elts (ZFC_in_HOL.succ a \<Uplus> b) \<approx> elts (ZFC_in_HOL.succ (a \<Uplus> b))"
alpha_Tree_equivp: shows "equivp (=\<^sub>\<alpha>)"
SbisL_length: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list" assumes "SbisL cl dl" shows "length cl = length dl"
Trivial_Aheap_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "Trivial_Aheap \<Gamma> e = (\<Lambda> a. (\<lambda>x. up\<cdot>0) f|` domA \<Gamma>)"
vert_f21: shows "vertices f21 = a # vs2 @ v # ws"
to_N: fixes s :: "'a" assumes "s \<in> S" shows "\<exists>t\<in>N. (s, t) \<in> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*"
are_operator_effects_consistent_def: fixes op\<^sub>1 :: "('c, 'd) strips_operator_scheme"   and op\<^sub>2 :: "('c, 'e) strips_operator_scheme" shows "STRIPS_Semantics.are_operator_effects_consistent op\<^sub>1 op\<^sub>2 \<equiv> let add\<^sub>1 = add_effects_of op\<^sub>1; add\<^sub>2 = add_effects_of op\<^sub>2; del\<^sub>1 = delete_effects_of op\<^sub>1; del\<^sub>2 = delete_effects_of op\<^sub>2 in \<not> list_ex (\<lambda>v. list_ex ((=) v) del\<^sub>2) add\<^sub>1 \<and> \<not> list_ex (\<lambda>v. list_ex ((=) v) add\<^sub>2) del\<^sub>1"
sub_branch_def: fixes f :: "'e \<Rightarrow> 'd"   and blocks :: "(('c, 'e) fm list \<times> 'e) list" shows "sub_branch f blocks \<equiv> map (sub_block f) blocks"
assms_10: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
cyclic_mapI: fixes xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "cyclic xs ys" shows "cyclic (map f xs) (map f ys)"
monom_mult_one_left: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult (1::'c) (0::'b) p = p"
hfset_hinsert: fixes a :: "hf"   and b :: "hf" shows "hfset (b \<triangleleft> a) = insert a (hfset b)"
pivot_fun_def: fixes A :: "'b mat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat" shows "pivot_fun A f nc \<equiv> let nr = dim_row A in \<forall>i<nr. f i \<le> nc \<and> (f i < nc \<longrightarrow> A $$ (i, f i) = (1::'b) \<and> (\<forall>i'<nr. i' \<noteq> i \<longrightarrow> A $$ (i', f i) = (0::'b))) \<and> (\<forall>j<f i. A $$ (i, j) = (0::'b)) \<and> (Suc i < nr \<longrightarrow> f i < f (Suc i) \<or> f (Suc i) = nc)"
time_transl: fixes C :: "Nielson_VCG.acom" shows "Nielson_VCGi.time (transl C) = Nielson_VCG.time C"
is_projection_on_def: fixes \<pi> :: "'a \<Rightarrow> 'a"   and M :: "'a set" shows "is_projection_on \<pi> M = (\<forall>h. is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) (\<pi> h))"
rtb_rbl_ariths_1: fixes w :: "'a word"   and ys :: "bool list" assumes "rev (to_bl w) = ys" shows "rev (to_bl (word_succ w)) = rbl_succ ys"
pivot_funI: fixes A :: "'b mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat" assumes "dim_row A = nr"   and "\<And>i. i < nr \<Longrightarrow> f i \<le> nc"   and "\<And>i j. \<lbrakk>i < nr; j < f i\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'b)"   and "\<And>i. \<lbrakk>i < nr; Suc i < nr\<rbrakk> \<Longrightarrow> f i < f (Suc i) \<or> f (Suc i) = nc"   and "\<And>i. \<lbrakk>i < nr; f i < nc\<rbrakk> \<Longrightarrow> A $$ (i, f i) = (1::'b)"   and "\<And>i i'. \<lbrakk>i < nr; f i < nc; i' < nr; i' \<noteq> i\<rbrakk> \<Longrightarrow> A $$ (i', f i) = (0::'b)" shows "pivot_fun A f nc"
IrrGSubspace_iso_constituent: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and W :: "'c set"   and Us :: "'c set list" assumes "FinGroupRepresentation G smult V"   and "V \<noteq> 0"   and "W \<subseteq> V"   and "W \<noteq> 0"   and "IrrFinGroupRepresentation G smult W"   and "\<forall>U\<in>set Us. IrrFinGroupRepresentation G smult U"   and "0 \<notin> set Us"   and "V = (\<Oplus>U\<leftarrow>Us. U)" shows "\<exists>U\<in>set Us. \<exists>T. FGModuleIso G smult W smult T U"
igSubstIPresIGWls_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstIPresIGWls MOD \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> igWls MOD s (igSubst MOD ys Y y X)"
rank_Sup: fixes a :: "V" shows "rank a = (SUP y\<in>elts a. ZFC_in_HOL.succ (rank y))"
x_not_j: shows "x \<noteq> mod_type_class.to_nat j"
token_succeeds_alt_def: fixes x :: "nat" shows "\<MM>.token_succeeds x = (\<forall>\<^sub>\<infinity>n. \<MM>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
foldr_map: fixes g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'a" shows "foldr g (map f xs) a = foldr (g \<circ> f) xs a"
linear_irreducible\<^sub>d: fixes p :: "'a poly" assumes "degree p = 1" shows "irreducible\<^sub>d p"
dflt_size_aux_g0: shows "0 < dflt_size_aux"
\<ff>_strict_borderI: fixes y :: "nat"   and s :: "'a list"   and i :: "nat"   and j :: "nat" assumes "y = \<ff> s (i - 1)"   and "strict_border (take (i - 1) s) (take (j - 1) s)" shows "strict_border (take (y - 1) s) (take (j - 1) s)"
ChamberComplexEndomorphism_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" shows "ChamberComplexEndomorphism X f \<equiv> ChamberComplexMorphism X X f \<and> ChamberComplexEndomorphism_axioms X f"
scaling_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "scaling t \<equiv> \<forall>P c x. sound P \<and> 0 \<le> c \<longrightarrow> c * t P x = t (\<lambda>x. c * P x) x"
roifex_set_var_subtree_1: fixes b :: "'a ifex"   and v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and ass :: "'a \<Rightarrow> bool" assumes "ro_ifex b"   and "b = IF v t e" shows "val_ifex b (ass(v := True)) = val_ifex t ass"
PiE_E: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "f \<in> Pi\<^sub>E A B" obtains "x \<in> A" and "f x \<in> B x"   | "x \<notin> A" and "f x = undefined"
IllegalThreadState_def: shows "IllegalThreadState = STR ''java/lang/IllegalThreadStateException''"
square_part_pos: fixes n :: "nat" assumes "0 < n" shows "0 < square_part n"
no_await_trancl_impl: fixes ctx :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and ctx' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "ctx \<leadsto>\<^sub>w\<^sup>+ ctx'" shows "no_await (fst (fst ctx)) \<longrightarrow> no_await (fst (fst ctx'))"
\<R>_def: shows "\<R> \<equiv> \<lambda>\<Gamma> \<S> P. {(xa, x). \<R>p \<Gamma> \<S> P xa x}"
mult_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m = k * n) = (m = n \<or> k = 0)"
inf_sup_aci_7: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "sup_prod x (sup_prod y z) = sup_prod y (sup_prod x z)"
addmod32: fixes a :: "bit list"   and b :: "bit list" shows "addmod32 a b = rev (select (rev (nat_to_bv (bv_to_nat a + bv_to_nat b))) 0 31)"
freshResidualChain3_1: fixes xvec :: "name list"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" assumes "xvec \<sharp>* (\<alpha> \<prec> P)"   and "xvec \<sharp>* bn \<alpha>" shows "xvec \<sharp>* \<alpha>"
slice_kind_Return: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f" shows "slice_kind S a = \<lambda>cf. True\<hookleftarrow>\<^bsub>p\<^esub>\<lambda>cf cf'. cf'"
wf: shows "wf_matchF_invar \<sigma> V n R I r st (ne + length aux)"
guardedFrameSupp_1: fixes P :: "('a, 'b, 'c) psi"   and x :: "name" assumes "guarded P" shows "x \<sharp> extractFrame P"
vertex_in_separator_1: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and s :: "'g"   and t :: "'g"   and v :: "'e" assumes "TreeDecomposition G T bag"   and "s \<rightarrow>\<^bsub>T\<^esub> t"   and "v \<in> TreeDecomposition.left_part T bag s t"   and "v \<in> TreeDecomposition.left_part T bag t s" shows "v \<in> bag s"
assms: shows "NodeS i s R \<in> init (pnet np p)"
comm_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "comm_monoid f z" shows "f a z = a"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "Adj.C.arr f"   and "Adj.C.dom f = a" shows "f \<cdot>\<^sub>C a = f"
unique_rootD_2: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x" shows "real_of_1 plr \<le> real_of_rat (rai_ub plr)"
boundedE: fixes A :: "'a multiset set"   and x :: "'a multiset" assumes "finite A"   and "A \<noteq> {}"   and "subset_mset.Sup_fin A \<subseteq># x" obtains "\<And>a. a \<in> A \<Longrightarrow> a \<subseteq># x"
induct: fixes P :: "Ferrack.num \<Rightarrow> bool"   and a0 :: "Ferrack.num" assumes "\<And>i. P (C i)"   and "\<And>n c t. P t \<Longrightarrow> P (CN n c t)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)" shows "P a0"
assms_4: shows "r\<lparr>a'\<rparr> = b'"
sum: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b"   and k :: "'a"   and v :: "'b" assumes "P f"   and "P g"   and "g = Poly_Mapping.single k v"   and "k \<notin> keys f" shows "P (f + g)"
prec_split2_aux: fixes A :: "assn"   and B :: "assn" shows "A * B * true \<Longrightarrow>\<^sub>A B * true"
r: fixes i :: "'a" assumes "i \<in> K" shows "0 < r i"
psiFreshSet_9: fixes X :: "name set"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "X \<sharp>* \<rparr>N.P = (X \<sharp>* N \<and> X \<sharp>* P)"
preserves_arr: fixes f :: "'a" assumes "S.arr f" shows "S.arr (L f)"
assms_1: fixes n :: "nat" shows "F n \<in> borel_measurable M"
intPF_def: fixes p :: "'psym"   and al :: "univ list" shows "intPF p al \<equiv> intP p (map2 fromFull (parOf p) al)"
size_20: fixes x91 :: "'b ltln"   and x92 :: "'b ltln" shows "size (x91 R\<^sub>n x92) = size x91 + size x92 + Suc 0"
ladder_cut_prefix_def: fixes d :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_cut_prefix d L = (ladder_shift_j d L)[0 := (ladder_n L 0, ladder_j L 0 - d, ladder_i L 0 - d)]"
mult_smult_right: fixes p :: "'b poly"   and a :: "'b"   and q :: "'b poly" shows "p * Polynomial.smult a q = Polynomial.smult a (p * q)"
RP_step_imp_derive_step: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and St' :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set" assumes "St \<leadsto>RP St'" shows "lclss_of_state St \<rhd>L lclss_of_state St'"
atU_single_top: fixes U :: "'a set"   and u :: "'a"   and r :: "'a list" shows "atU U {#u # r#} = (u \<in> U)"
nat_less_diff_conv: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j - k" shows "i + k < j"
mult_carrier_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B \<in> carrier_mat nr nc"
A: shows "domain (R\<lparr>carrier := H\<rparr>)"
simps_1: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_aux A [] bezout = A"
map_oo: fixes f :: "'c \<rightarrow> 'b"   and g :: "'d \<rightarrow> 'c"   and xs :: "['d]" shows "map\<cdot>f\<cdot>(map\<cdot>g\<cdot>xs) = map\<cdot>(f oo g)\<cdot>xs"
distinct_12: fixes x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list" shows "Label x41 x42 x43 \<noteq> Trap"
map_gpv_parametric': fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and A' :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and C' :: "'j \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'n \<Rightarrow> bool" shows "rel_fun (rel_fun A A') (rel_fun (rel_fun C C') (rel_fun (rel_gpv'' A C R) (rel_gpv'' A' C' R))) map_gpv map_gpv"
reduce_basis_cost_2: shows "cost (reduce_basis_cost fs_init) \<le> initial_gso_cost + body_cost * num_loops"