hcju/isabelleps huggingface dataset:
is_store_renaming_def: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and \<sigma>' :: "'e \<Rightarrow> ('d, 'e, 'f) val option" shows "is_store_renaming \<alpha> \<beta> \<sigma> \<sigma>' \<equiv> \<forall>l. case \<sigma> l of None \<Rightarrow> \<sigma>' (\<beta> l) = None | Some v \<Rightarrow> \<sigma>' (\<beta> l) = Some (\<R>\<^sub>V \<alpha> \<beta> v)"
splitAt_is_sublist2R: fixes ram :: "'a"   and vs :: "'a list" assumes "ram \<in> set vs" shows "is_sublist (ram # snd (splitAt ram vs)) vs"
of_int_eq_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and w :: "int"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus w = ring_1.of_int one plus zero minus z) = (w = z)"
indRelRSTPO_impl_SRel_and_TRel_reflect_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelRSTPO SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
collision_def: fixes time_set :: "real set" shows "delayed_safe_distance.collision time_set \<equiv> \<exists>t\<in>time_set. ego2.s t = delayed_safe_distance.other.s t"
repl_succ: fixes n :: "'a"   and x :: "'b" shows "repl (succ n) x = x # repl n x"
unit_simps_3: shows "cod \<eta>\<^sub>0 = d\<^sub>0 \<star> e\<^sub>0"
sorted_nth_strict_mono: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "sorted xs"   and "distinct xs"   and "i < j"   and "j < length xs" shows "xs ! i < xs ! j"
star_contr: fixes X :: "'a set"   and Y :: "'a set" shows "X\<^sup>\<star> \<cdot> Y = (\<Union>n. X ^ n \<cdot> Y)"
minus: fixes x :: "'a" shows "- x *s x = - (x *s x)"
two_comparisons_into_case_order_8: fixes x :: "'a"   and y :: "'a"   and Q :: "'e"   and P :: "'e"   and R :: "'e" shows "(if lt x y then Q else if y = x then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
LeftDerivation_drop: shows "LeftDerivation (drop (Suc i) \<alpha>) (take (n - m) F) (drop (Suc j) \<beta>)"
zero_hom_0_axioms: shows "zero_hom_0 ((*) x_y)"
less_or_eq_imp_le: fixes m :: "nat"   and n :: "nat" assumes "m < n \<or> m = n" shows "m \<le> n"
less_add_same_cancel2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus b a) = less zero b"
fg_state_change: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and n :: "nat"   and p :: "'f" assumes "fg_run A rho HOs SHOs coords"   and "round (rho (Suc n)) p = round (rho n) p" shows "state (rho (Suc n)) p = state (rho n) p"
rev_finite_subset: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "order.greater_eq B A" shows "finite A"
1: shows "commutative_square f g (\<p>\<^sub>1[f, g] \<cdot> h) (\<p>\<^sub>0[f, g] \<cdot> h)"
arr_Par_ArrDom_in_Vset: shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
subset_nodes_inpow: fixes xs :: "nat list" assumes "Sorted_Less.sorted (rev xs)"   and "set xs \<subseteq> set nodes" shows "xs \<in> (rev \<circ> sorted_list_of_set) ` Pow (set nodes)"
elims: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "'edge list" assumes "upd_cs x xa = y" obtains cs :: "'edge list" where "x = csa__" and "xa = []" and "y = csa__"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> upd_cs (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> upd_cs cs as | c' # cs' \<Rightarrow> upd_cs cs' as | _ \<Rightarrow> upd_cs cs as)"
monofunI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" shows "monofun f"
subst_apply_fv_subset: fixes t :: "('d, 'c) Term.term"   and V :: "'c set"   and \<delta> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "fv t \<subseteq> V" shows "fv (t \<cdot> \<delta>) \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)"
comp_simp: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<times> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.arr C1 (C1 (fst g) (fst f))"   and "partial_magma.arr C2 (C2 (snd g) (snd f))" shows "product_category.comp C1 C2 g f = (C1 (fst g) (fst f), C2 (snd g) (snd f))"
trans_to_dterms: fixes \<xi> :: "'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"   and "wellformed \<Gamma>" shows "\<forall>r\<in>sterms \<Gamma> q. r \<in> dterms \<Gamma> p"
simps: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "scount P \<omega> = (if P \<omega> then eSuc (scount P (stl \<omega>)) else scount P (stl \<omega>))"
norm_pre_ex_rule: fixes P :: "'a \<Rightarrow> assn"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" assumes "\<And>x. <P x> f <Q>" shows "<\<exists>\<^sub>Ax. P x> f <Q>"
assms: shows "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'"
elliptic_imp_not_parabolic_loxodromic_2: fixes f :: "'b \<Rightarrow> 'b" assumes "elliptic_isometry f" shows "\<not> loxodromic_isometry f"
labelAtOut_def: fixes v :: "'vertex"   and p :: "('form, 'var) out_port" shows "labelAtOut v p = subst (inst v) (freshen (vidx v) (labelsOut (nodeOf v) p))"
G_reach_conv: fixes v :: "'a"   and w :: "'a" shows "(v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w) = (v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w \<or> v = u \<and> w = u)"
left_tree_initial': fixes s :: "'a"   and t :: "'a" assumes "s \<in> V"   and "t \<in> V"   and "s \<noteq> t" shows "t \<notin> left_tree s t"
Object_fields: fixes P :: "'a prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and C :: "char list"   and F :: "char list" assumes "P \<turnstile> Object has_fields FDTs"   and "C \<noteq> Object" shows "map_of FDTs (F, C) = None"
vector_def: fixes l :: "'b list" shows "vector l = (\<chi>i. foldr (\<lambda>x f n. (f (n + (1::'c)))(n := x)) l (\<lambda>n x. 0::'b) (1::'c) i)"
inject_3: fixes x51 :: "'b mrexp2"   and x52 :: "'b mrexp2"   and x53 :: "bool"   and x54 :: "bool"   and y51 :: "'b mrexp2"   and y52 :: "'b mrexp2"   and y53 :: "bool"   and y54 :: "bool" shows "(Times2 x51 x52 x53 x54 = Times2 y51 y52 y53 y54) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53 \<and> x54 = y54)"
"4_1": fixes t :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>t\<rangle>) = fv t"
assms_3: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes document_ptr :: "(_) document_ptr"   and uu :: "unit" shows "local.CD.a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr uu = Heap_Error_Monad.bind document_ptr_kinds_M (\<lambda>document_ptrs. if document_ptr \<in> set document_ptrs then return document_ptr else error SegmentationFault)"
max_diff_distrib_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "max x y - z = max (x - z) (y - z)"
change_respecting_dom_unique: fixes c :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set"   and g :: "('Var \<Rightarrow> 'Val option) \<Rightarrow> 'Var \<Rightarrow> 'Val option" assumes "change_respecting \<langle>c, mds, mem\<rangle> \<langle>c', mds', mem'\<rangle> X g" shows "\<exists>d. \<forall>f. dom f = X \<longrightarrow> d = dom (g f)"
metric_CauchyI: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
succ_is_dep: fixes m :: "('a, 'b) insert_message"   and i :: "'a" shows "S m = \<lbrakk>i\<rbrakk> \<longrightarrow> i \<in> deps (Insert m)"
prj_simps_13: shows "C.dom gfg.Prj\<^sub>1 = Chn (g \<star> f \<star> g)"
poly_power_zero_iff: fixes p :: "'b poly"   and n :: "nat"   and x :: "'b" shows "(poly (p ^ n) x = (0::'b)) = (n \<noteq> 0 \<and> poly p x = (0::'b))"
mult_assoc2: fixes V :: "'b set"   and x :: "'b"   and a :: "real"   and b :: "real" assumes "vectorspace V"   and "x \<in> V" shows "a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
pullback_iso_self: fixes f :: "'a" assumes "local.iso f" shows "\<p>\<^sub>0[f, f] = \<p>\<^sub>1[f, f]"
ordLeq_def2: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" shows "(r \<le>o s) = (Well_order r \<and> Well_order s \<and> (\<exists>f. \<forall>a\<in>Field r. f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)))"
scc_of_G_tail: fixes u :: "'a" assumes "u \<in> G.scc_of (tail G a)" shows "H.scc_of u = H.scc_of (tail G a) \<or> H.scc_of u = H.scc_of (head G a)"
assms_2: shows "\<forall>r\<in>range f \<inter> space N. \<exists>A\<in>sets N. range f \<inter> A = {r}"
intro: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "df_system s\<^sub>0 lstep"   and "fair_run_axioms s\<^sub>0 lstep s" shows "fair_run s\<^sub>0 lstep s"
zero_eq_add_iff_both_eq_0: fixes x :: "'k multiset"   and y :: "'k multiset" shows "({#} = x + y) = (x = {#} \<and> y = {#})"
nonzero_insert: fixes xs :: "'a list"   and p :: "pexpr"   and ps :: "pexpr list" shows "nonzero xs (List.insert p ps) = (peval xs p \<noteq> \<zero> \<and> nonzero xs ps)"
pw_eq_on_def: fixes S :: "'b set"   and w :: "'c \<Rightarrow> 'b set"   and w' :: "'c \<Rightarrow> 'b set" shows "pw_eq_on S w w' \<equiv> \<forall>i. w i \<inter> S = w' i \<inter> S"
set_sorted_list_of_set: fixes A :: "'a set" assumes "finite A" shows "set (sorted_list_of_set A) = A"
a2: fixes x :: "'x"   and f :: "'x \<Rightarrow> 'a nres" assumes "\<And>x'. \<lbrakk>pre x'; (x', x) \<in> V\<rbrakk> \<Longrightarrow> f x' \<le> (if nofail (f x') then M x' else FAIL)"   and "pre x"   and "REC\<^sub>T body = f" shows "body f x \<le> (if nofail (body f x) then M x else FAIL)"
nn_integral_subalgebra: fixes f :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'c measure" assumes "f \<in> borel_measurable N"   and "sets N \<subseteq> sets M"   and "space N = space M"   and "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" shows "integral\<^sup>N N f = integral\<^sup>N M f"
nodesI_2: fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph" assumes "(u, v) \<in> edges g" shows "v \<in> nodes g"
prob_space_stream_space: fixes M :: "'b measure" assumes "prob_space M" shows "prob_space (stream_space M)"
residue_primroot_Carmichael: fixes n :: "nat"   and g :: "nat" assumes "residue_primroot n g" shows "Carmichael n = totient n"
length_rotate1: fixes xs :: "'b list" shows "length (rotate1 xs) = length xs"
coeff_mult_C: fixes a :: "'b poly"   and x :: "'b"   and n :: "nat" shows "coeff (a * C x) n = coeff a n * x"
subgraph_def: fixes G :: "('c, 'd) graph"   and H :: "('c, 'd) graph" shows "subgraph G H \<equiv> nodes G = nodes H \<and> edges G \<subseteq> edges H"
starfun: fixes f :: "'b \<Rightarrow> 'a"   and X :: "nat \<Rightarrow> 'b" shows "(*f* f) (star_n X) = star_n (\<lambda>n. f (X n))"
assms: shows "f \<in>\<^sub>\<circ> vproduct (set {a, b}) A"
tObsC_tLast: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "tLast (tObsC t) = (envObsC (es (tLast t)), pubActs (tLast t))"
lless_eq: fixes x :: "int"   and y :: "int" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
fr_Send7: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat"   and j :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n"   and "j < n"   and "nReturn k t = []" shows "nSend j t = []"
dec_word_enc: fixes w :: "'a list"   and I :: "(nat + nat set) list" shows "dec_word (enc (w, I)) = w"
pgwt_ground: fixes t :: "('fun, 'var) Term.term" assumes "public_ground_wf_term t" shows "fv t = {}"
countableE_infinite: fixes S :: "'b set" assumes "countable S"   and "infinite S" obtains e :: "'b \<Rightarrow> nat" where "bij_betw e S UNIV"
frontier_real_atMost: fixes a :: "real" shows "frontier {..a} = {a}"
sym_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(\<not> x \<bowtie> y) = (\<not> y \<bowtie> x)"
fls_lr_inverse_starting0_2: fixes g :: "'a fls" shows "fls_right_inverse g (0::'a) = 0"
neg: fixes x :: "'a" assumes "x \<in> S" shows "negligible (U x)"
basechambers_half_chamber_systems_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "g ` C0 \<in> g \<turnstile> ChamberComplex.chamber_system X"
clean_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "clean A w V \<equiv> {v \<in> V. infinite (greachable A w V v)}"
dotprod_le_right: fixes v :: "nat list"   and w :: "nat list"   and b :: "nat list" assumes "v \<le>\<^sub>v w"   and "length b = length w" shows "b \<bullet> v \<le> b \<bullet> w"
coboundedI2: fixes c :: "'t"   and b :: "'t"   and a :: "'t" assumes "c \<preceq>\<^sub>t b" shows "c \<preceq>\<^sub>t ord_term_lin.max a b"
O_simps_2: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "\<not> \<gamma> trn" shows "O (trn # tr) = O tr"
dvd_productE: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "p dvd a * b" obtains x :: "'a" and y :: "'a" where "p = x * y" and "x dvd a" and "y dvd b"
nths_Cons: fixes x :: "'g"   and l :: "'g list"   and A :: "nat set" shows "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"
std_apart_apart: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set" shows "vars\<^sub>l\<^sub>s (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''1'' @ x))) \<inter> vars\<^sub>l\<^sub>s (C\<^sub>2 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''2'' @ x))) = {}"
sum_eq_0_iff: fixes F :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "finite F" shows "(comm_monoid_add.sum (+) {#} f F = {#}) = (\<forall>a\<in>F. f a = {#})"
inverse_arrowsI: fixes g :: "'a2"   and f :: "'a2" assumes "C2.ide (g \<cdot>\<^sub>2 f)"   and "C2.ide (f \<cdot>\<^sub>2 g)" shows "C2.inverse_arrows f g"
intros_2: fixes x :: "'a"   and y :: "'a"   and ns :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "(x, y) \<in> ns"   and "(X, Y) \<in> multpw ns" shows "(add_mset x X, add_mset y Y) \<in> multpw ns"
eq_trans: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<chi> :: "'b ltln" assumes "\<phi> \<sim>\<^sub>P \<psi>"   and "\<psi> \<sim>\<^sub>P \<chi>" shows "\<phi> \<sim>\<^sub>P \<chi>"
rel_Grp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "rel_option (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_option x \<subseteq> A} (map_option f)"
insert_Diff_if: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(x \<triangleright> A) - B = (if x \<in> B then A - B else x \<triangleright> A - B)"
cong: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "comm_monoid_add.sum plus zero \<equiv> comm_monoid_add.sum plus zero"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes element_ptr :: "(_) element_ptr"   and element :: "('o, 'm, 'n, 'q, 's option) RElement_scheme" shows "put element_ptr element = put (cast element_ptr) (cast element)"
vec_index_vec_of_list: fixes xs :: "'b list"   and i :: "nat" shows "vec_of_list xs $ i = xs ! i"
is_projection_on_cadjoint: fixes \<pi> :: "'a \<Rightarrow> 'a"   and M :: "'a set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "\<pi>\<^sup>\<dagger> = \<pi>"
injective_inv: fixes t :: "'a"   and e :: "'a" assumes "injective t"   and "e * t\<^sup>T = bot"   and "arc e" shows "injective (t \<squnion> e)"
sel_2: fixes x1 :: "le_rel"   and x2 :: "linear_poly"   and x3 :: "'a" shows "lec_poly (Le_Constraint x1 x2 x3) = x2"
not_eq_outside: fixes a :: "'c"   and A :: "'c set" assumes "a \<notin> A" shows "\<forall>b\<in>A. b \<noteq> a"
less_eq_nmultiset_def: fixes X :: "'a nmultiset"   and Y :: "'a nmultiset" shows "(X \<le> Y) = (X < Y \<or> X = Y)"
progress_simps_2: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and p :: "String.literal"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Let p \<phi> \<psi>) j = Monitor.progress \<sigma> (P(p \<mapsto> Monitor.progress \<sigma> P \<phi> j)) \<psi> j"
ffact_Suc_rev: fixes n :: "nat"   and m :: "'a" shows "ffact (Suc n) m = (m - of_nat n) * ffact n m"
rbt_max_opt_Branch: fixes t2 :: "('b, unit) RBT_Impl.rbt"   and c :: "color"   and t1 :: "('b, unit) RBT_Impl.rbt"   and k :: "'b" assumes "t2 \<noteq> rbt.Empty" shows "rbt_max_opt (Branch c t1 k () t2) = rbt_max_opt t2"
downset_appendE: fixes xss :: "'b list set"   and xs :: "'b list"   and ys :: "'b list" assumes "downset xss"   and "xs @ ys \<in> xss" shows "xs \<in> xss"
cprod_def: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" shows "r1 *c r2 = |Field r1 \<times> Field r2|"
weak_reduction_bisimulation_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_bisimulation (Rel\<^sup>+) Cal"
su2_3: shows "su2 \<in> existence_ivl0 x"
real_sqrt_gt_1_iff: fixes y :: "real" shows "(1 < sqrt y) = (1 < y)"
run_one_step_basic_get_local_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x14 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Get_local x14) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
assms_5: shows "snd (rep_nat_term u) = snd (rep_nat_term v)"
extremum_unique: fixes a :: "'d upred" shows "(a \<sqsubseteq> true) = (a = true)"
cring_simprules_26: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "cring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
PRECOND_tag_def: fixes P :: "'b" shows "PRECOND_tag P \<equiv> P"
hyps: shows "\<Psi> \<rhd> \<tau>.P \<longmapsto> \<tau> \<prec> P'"
a_set_child_nodes_tups_def: shows "a_set_child_nodes_tups \<equiv> [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, set_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
has_density_cong: fixes N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_density M N f = has_density M N g"
nxt_stable: fixes b :: "nat"   and b' :: "nat" assumes "nxt b \<down>= b'" shows "\<forall>x<e_length b. e_nth b x = e_nth b' x"
wf_wm_imp_basic: fixes M :: "object atom formula set" assumes "wf_world_model M" shows "wm_basic M"
th2: shows "Inum (x # bs) (CN 0 (fst (zsplit0 t)) (snd (zsplit0 t))) = Inum (x # bs) t \<and> numbound0 (snd (zsplit0 t))"
wf_trm_subtermeq: fixes t :: "('fun, 'b) Term.term"   and s :: "('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "s \<sqsubseteq> t" shows "wf\<^sub>t\<^sub>r\<^sub>m s"
lin_indep: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
IH: fixes i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list" assumes "P @ P' \<turnstile> (i, s, stk) \<rightarrow>^k (n, s', stk')"   and "size P \<le> i"   and "exits P' \<subseteq> {0..}" shows "P' \<turnstile> (i - size P, s, stk) \<rightarrow>^k (n - size P, s', stk')"
pell_mul_eq_trivial_nat_iff: fixes D :: "nat"   and z1 :: "nat \<times> nat"   and z2 :: "nat \<times> nat" assumes "pell D" shows "(pell.pell_mul D z1 z2 = (Suc 0, 0)) = (z1 = (Suc 0, 0) \<and> z2 = (Suc 0, 0))"
\<omega>_in_vhom: shows "S.in_hom \<i>[a] (a \<star> a) a"
countable_basis: fixes x :: "'b" obtains A :: "nat \<Rightarrow> 'b set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> x"
arr_mset_remove_last: fixes l :: "int"   and r :: "int"   and arr :: "int \<Rightarrow> 'a" assumes "l \<le> r" shows "arr_mset arr l r = arr_mset arr l (r + 1) - {#arr r#}"
Vr_integral: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "Idomain (Vr K v)"
cring_simprules_27: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
measurable_count_space_const: fixes c :: "'b"   and M :: "'a measure" shows "(\<lambda>x. c) \<in> M \<rightarrow>\<^sub>M count_space UNIV"