hcju/isabelleps huggingface dataset:
par_INF_distrib: fixes X :: "'d::type set"   and c :: "'a::refinement_lattice"   and d :: "'d::type \<Rightarrow> 'a::refinement_lattice" assumes "(X::'d::type set) \<noteq> {}" shows "(c::'a::refinement_lattice) \<parallel> \<Sqinter> ((d::'d::type \<Rightarrow> 'a::refinement_lattice) ` (X::'d::type set)) = (\<Sqinter>x::'d::type\<in>X. c \<parallel> d x)"
flrestriction_def: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<bullet> A = set {[a, b]\<^sub>\<circ> |a b. a \<in>\<^sub>\<circ> A \<and> [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
domain_eq_zeroprimeideal: shows "domain R = primeideal {\<zero>} R"
irrefl_p: shows "irreflp happened_immediately_before\<^sup>+\<^sup>+"
minus_mult_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring plus zero minus uminus times" shows "uminus (times a b) = times a (uminus b)"
the_NF_step: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> A" shows "the_NF A a = the_NF A b"
iteration_prod_unfold: fixes y :: "'a"   and x :: "'a" shows "(1::'a) + y \<cdot> (x \<cdot> y)\<^sup>\<infinity> \<cdot> x = (y \<cdot> x)\<^sup>\<infinity>"
F2set1_bd: fixes x :: "('d, 'e, 'f) F2" shows "|F2set1 x| \<le>o IFbd"
CD_2: shows "D \<in> f \<turnstile> \<C> \<union> g \<turnstile> \<C>"
domintros_13: fixes v :: "char list option"   and va :: "exp0"   and vc :: "exp0" shows "is_value_dom (exp0.Let v va vc)"
mor_def: fixes B1 :: "'b set"   and B2 :: "'c set"   and s1 :: "('d, 'b, 'c) F1 \<Rightarrow> 'b"   and s2 :: "('d, 'b, 'c) F2 \<Rightarrow> 'c"   and B1' :: "'e set"   and B2' :: "'f set"   and s1' :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2' :: "('d, 'e, 'f) F2 \<Rightarrow> 'f"   and f :: "'b \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'f" shows "mor B1 B2 s1 s2 B1' B2' s1' s2' f g = (((\<forall>a\<in>B1. f a \<in> B1') \<and> (\<forall>a\<in>B2. g a \<in> B2')) \<and> (\<forall>z\<in>F1in UNIV B1 B2. f (s1 z) = s1' (F1map id f g z)) \<and> (\<forall>z\<in>F2in UNIV B1 B2. g (s2 z) = s2' (F2map id f g z)))"
i2: shows "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
ffact_monomial: fixes n :: "nat"   and a :: "'a" shows "ffact n a = (\<Sum>k = 0..n. (- (1::'a)) ^ (n - k) * of_nat (stirling n k) * a ^ k)"
R_def: fixes p :: "'a"   and q :: "'a" shows "R p q \<equiv> Sup {x. d p \<cdot> x \<le> x \<cdot> d q}"
ldropWhile_def: fixes P :: "'c \<Rightarrow> bool" shows "ldropWhile P \<equiv> llist.fixp_fun (\<lambda>ldropWhile xs. case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then ldropWhile xs' else xs)"
the_\<nu>_props_2: shows "local.iso w.the_\<nu>"
Trg_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "Trg (Dom t) = Trg t"
pdevs_apply_Pdevs_add_slist: fixes xs :: "(nat, 'd) slist"   and ys :: "(nat, 'd) slist"   and i :: "nat" shows "pdevs_apply (Pdevs (add_slist xs ys)) i = pdevs_apply (Pdevs xs) i + pdevs_apply (Pdevs ys) i"
\<R>\<^sub>1_dma_eq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "dma mem\<^sub>1 = dma mem\<^sub>2"
seq_monosub: fixes s :: "nat \<Rightarrow> 'b" shows "\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. s (f n))"
fin_CANDS_aux: fixes M :: "trm"   and x :: "name"   and B :: "ty" assumes "\<not> fin M x"   and "x:M \<in> NEGn B (\<parallel><B>\<parallel>)" shows "x:M \<in> AXIOMSn B \<or> x:M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
simps_5: fixes vs :: "'b list"   and bs :: "'b list"   and a :: "tm" shows "Ifm vs bs (Eq a) = (Itm vs bs a = (0::'b))"
compare_EQ_dest: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare"   and "compare\<cdot>x\<cdot>y = EQ" shows "x = y"
slp_of_fas: fixes fas :: "floatarith list"   and xs :: "real list" shows "take (length fas) (interpret_slp (slp_of_fas fas) xs) = interpret_floatariths fas xs"
induct: fixes x1 :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x2 :: "OBJ"   and x3 :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and P :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set \<Rightarrow> OBJ \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<Rightarrow> bool" assumes "x1 \<rhd> x2 : x3"   and "\<And>G. P G Skip (\<lambda>s t. t = s)"   and "\<And>G x e. P G (Assign x e) (\<lambda>s t. t = (update (fst s) x (evalE e (fst s)), snd s))"   and "\<And>G x C. P G (New x C) (\<lambda>s t. \<exists>l. l \<notin> Dom (snd s) \<and> t = (update (fst s) x (RVal (Loc l)), (l, C, []) # snd s))"   and "\<And>G x y F. P G (Get x y F) (\<lambda>s t. \<exists>l C Flds v. fst s y = RVal (Loc l) \<and> lookup (snd s) l = Some (C, Flds) \<and> lookup Flds F = Some v \<and> t = (update (fst s) x v, snd s))"   and "\<And>G x F e. P G (Put x F e) (\<lambda>s t. \<exists>l C Flds. fst s x = RVal (Loc l) \<and> lookup (snd s) l = Some (C, Flds) \<and> t = (fst s, (l, C, (F, evalE e (fst s)) # Flds) # snd s))"   and "\<And>G c A d B. \<lbrakk>G \<rhd> c : A; P G c A; G \<rhd> d : B; P G d B\<rbrakk> \<Longrightarrow> P G (Comp c d) (\<lambda>s t. \<exists>r. A s r \<and> B r t)"   and "\<And>G c A d B b. \<lbrakk>G \<rhd> c : A; P G c A; G \<rhd> d : B; P G d B\<rbrakk> \<Longrightarrow> P G (Iff b c d) (\<lambda>s t. (evalB b (fst s) \<longrightarrow> A s t) \<and> (\<not> evalB b (fst s) \<longrightarrow> B s t))"   and "\<And>G c B b A. \<lbrakk>G \<rhd> c : B; P G c B; \<forall>s. \<not> evalB b (fst s) \<longrightarrow> A s s; \<forall>s r t. evalB b (fst s) \<longrightarrow> B s r \<longrightarrow> A r t \<longrightarrow> A s t\<rbrakk> \<Longrightarrow> P G (While b c) (\<lambda>s t. A s t \<and> \<not> evalB b (fst t))"   and "\<And>A G. \<lbrakk>({A} \<union> G) \<rhd> body : A; P ({A} \<union> G) body A\<rbrakk> \<Longrightarrow> P G Call A"   and "\<And>A G. A \<in> G \<Longrightarrow> P G Call A"   and "\<And>G c A B. \<lbrakk>G \<rhd> c : A; P G c A; \<forall>s t. A s t \<longrightarrow> B s t\<rbrakk> \<Longrightarrow> P G c B" shows "P x1 x2 x3"
nat_le_Suc_less_imp: fixes x :: "nat"   and y :: "nat" assumes "x < y" shows "x \<le> y - Suc 0"
ordstNS_ordst_trans: fixes X :: "char list"   and Y :: "char list"   and Z :: "char list" assumes "ordstNS X Y"   and "ordst Y Z" shows "ordst X Z"
perm_sym: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs <~~> ys" shows "ys <~~> xs"
fact_group_trivial_iff: assumes "finite (carrier G)" shows "(carrier (G Mod H) = {\<one>\<^bsub>G Mod H\<^esub>}) = (H = carrier G)"
addsI: fixes a :: "'a"   and b :: "'a"   and k :: "'a" assumes "a = b + k" shows "b adds a"
trace_def: fixes sig :: "'f signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
36: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> y \<squnion> z \<sqinter> (u \<sqinter> y) = (x \<squnion> z \<sqinter> u) \<sqinter> y"
1: shows "match \<Delta> s s1 vl1 a ou s' vl'"
distinct1_rotate: fixes xs :: "'c list" shows "distinct (rotate1 xs) = distinct xs"
cp_def: fixes P :: "'b com option"   and s :: "'b" shows "cp P s \<equiv> {l. l ! 0 = (P, s) \<and> l \<in> cptn}"
multrow_carrier: fixes mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "nat"   and a :: "'c"   and A :: "'c mat"   and n :: "nat"   and nc :: "nat" shows "(mat_multrow_gen mul k a A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
cases: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" assumes "a1 \<rhd> a2 \<approx>\<^sub>w a3" obtains \<Psi> :: "'b" and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" where "a1 = \<Psi>" and "a2 = P" and "a3 = Q" and "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<weakenBisim> Q" and "\<Psi> \<rhd> P \<leadsto>\<^sub>w<weakenBisim> Q" and "\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<approx>\<^sub>w Q" and "\<Psi> \<rhd> Q \<approx>\<^sub>w P"
A: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> r"   and "(y, x) \<in> r" shows "x = y"
is_projection_on_plus: fixes A :: "'b set"   and B :: "'b set"   and \<pi>A :: "'b \<Rightarrow> 'b"   and \<pi>B :: "'b \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> is_orthogonal x y"   and "closed_csubspace A"   and "closed_csubspace B"   and "is_projection_on \<pi>A A"   and "is_projection_on \<pi>B B" shows "is_projection_on (\<lambda>x. \<pi>A x + \<pi>B x) (A +\<^sub>M B)"
fmlookup_add: fixes m :: "('d, 'c) fmap"   and n :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (m ++\<^sub>f n) x = (if x |\<in>| fmdom n then fmlookup n x else fmlookup m x)"
indep_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "indep_in \<E> X" shows "X \<subseteq> \<E>"
smult_power: fixes a :: "'b"   and p :: "'b poly"   and n :: "nat" shows "smult a p ^ n = smult (a ^ n) (p ^ n)"
mirror_\<delta>: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs)" shows "\<delta> (mirror p) = \<delta> p"
t: shows "finite t \<and> g C1_differentiable_on s - t"
g2: shows "vec_of_poly_n (q * u) n + vec_of_poly_n (smult k r) n = g"
vconn_implies_hpar_4: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "trg\<^sub>D g = trg\<^sub>D f"
cong_0: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c zero = (b = c)"
less_eq_real_def: fixes x :: "real"   and y :: "real" shows "(x \<le> y) = (x < y \<or> x = y)"
tr_s_newI: fixes var :: "char list"   and T :: "x \<Rightarrow> x option"   and var' :: "char list"   and ctx :: "ctx"   and cl :: "cl" assumes "(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'" shows "tr_s T (s_new var ctx cl) (s_new var' ctx cl)"
set_bit_word_of_int: fixes x :: "int"   and n :: "nat"   and b :: "bool" shows "set_bit_class.set_bit (word_of_int x) n b = word_of_int (bin_sc n b x)"
indicator_times: fixes A :: "'d set"   and B :: "'e set"   and x :: "'d \<times> 'e" shows "indicator (A \<times> B) x = indicator A (fst x) * indicator B (snd x)"
product_sigma_finite_def: fixes M :: "'c \<Rightarrow> 'd measure" shows "product_sigma_finite M \<equiv> \<forall>i. sigma_finite_measure (M i)"
SUP_insert: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` insert a A) = sup (f a) (Sup (f ` A))"
finished_no_pending: fixes v :: "'v" assumes "v \<in> dom (finished s)" shows "pending s `` {v} = {}"
finite_dimensionE': fixes K :: "'a set"   and E :: "'a set" assumes "Hom.finite_dimension K E" obtains n :: "nat" where "Hom.dimension n K E"
fst: fixes P :: "'a trm"   and A :: "'a trm"   and B :: "'a trm" assumes "P \<noteq> SimplyTyped.Pair A B"   and "NF P" shows "NF (Fst P)"
ub1: shows "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> x < ub1"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_uminus" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> vrat_uminus). -\<^sub>\<rat> a = b"
is_singleton_altdef: fixes A :: "'c set" shows "is_singleton A = (card A = 1)"
ran: shows "ran E \<union> set (blocks_types es) \<subseteq> types P"
assms_8: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r)|\<^sub>r"
list_orderE: fixes ops :: "('b \<times> 'b option) list"   and x :: "'b"   and y :: "'b" assumes "list_order ops x y" shows "\<exists>xs ys zs. interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
nonzero_mult_divide_mult_cancel_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
"1_4": fixes i :: "nat" shows "X i \<in> sets (Pi\<^sub>M (J i) M)"
inter_same_diff: fixes A :: "'a set"   and S :: "'a set" shows "A \<inter> (A - S) = A - S"
axioms_1: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "tolerance A equiv" shows "symmetric A equiv"
Cinfinite_cexp: fixes q :: "('c \<times> 'c) set"   and r :: "('d \<times> 'd) set" assumes "ctwo \<le>o q"   and "Cinfinite r" shows "Cinfinite (q ^c r)"
insertion_fst_tmf_polys: fixes e :: "nat \<Rightarrow> real"   and z :: "float interval list" shows "insertion e (map_poly real_of_float (fst (tmf_polys z))) = (\<Sum>i<length z. e 0 ^ i * real_of_float (mid (z ! i)))"
fls_const_uminus: fixes x :: "'a" shows "fls_const (- x) = - fls_const x"
split_Vertex2_eq_0I: fixes y :: "'v vertex"   and x :: "'v"   and f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "\<And>z. y \<noteq> Edge z x" shows "split f (y, Vertex x) = 0"
pred_const_less: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and c :: "'c" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{c<..} \<in> sets N" shows "Measurable.pred M (\<lambda>x. c < f x)"
is_rewritable_ConsD: fixes b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t" assumes "is_rewritable (b # bs) p u"   and "u \<prec>\<^sub>t lt b" shows "is_rewritable bs p u"
partition_on_remove_singleton: fixes A :: "'b set"   and P :: "'b set set"   and X :: "'b set"   and A' :: "'b set" assumes "partition_on A P"   and "X \<in> P"   and "A - X = A'" shows "partition_on A' (P - {X})"
space_pair_measure: fixes A :: "'a measure"   and B :: "'b measure" shows "space (A \<Otimes>\<^sub>M B) = space A \<times> space B"
nrows_spvec: fixes x :: "(nat \<times> 'a) list" shows "nrows (sparse_row_vector x) \<le> Suc 0"
binary_def: fixes a :: "'c"   and b :: "'c" shows "binary a b = (\<lambda>x. b)(0 := a)"
"case": fixes f :: "char list \<Rightarrow> cl \<Rightarrow> fd list \<Rightarrow> meth_def list \<Rightarrow> 'a"   and x1 :: "char list"   and x2 :: "cl"   and x3 :: "fd list"   and x4 :: "meth_def list" shows "(case cld_def x1 x2 x3 x4 of cld_def x xa xb xc \<Rightarrow> f x xa xb xc) = f x1 x2 x3 x4"
fold_Option_bind_eq_Some_at_each_point_Some': assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (zip xs ys @ [(x, y)]) (Some subs) = Some subs'" obtains point :: "'c" where "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (zip xs ys) (Some subs) = Some point" and "fold (\<lambda>(x, y) option. Option.bind option (f x y)) [(x, y)] (Some point) = Some subs'"
plus_assoc: fixes A :: "'b tensor"   and ds :: "nat list"   and B :: "'b tensor"   and C :: "'b tensor" assumes "dims A = ds"   and "dims B = ds"   and "dims C = ds" shows "A + B + C = A + (B + C)"
hom_prod_mset: fixes X :: "'a poly poly multiset" shows "poly_y_x (\<Prod>\<^sub># X) = \<Prod>\<^sub># (image_mset poly_y_x X)"
vec_lambda_beta: fixes g :: "'b \<Rightarrow> 'a"   and i :: "'b" shows "vec_lambda g $h i = g i"
le_real_sqrt_sumsq: fixes x :: "real"   and y :: "real" shows "x \<le> sqrt (x * x + y * y)"
list_interval_def: fixes l :: "'a list"   and u :: "'a list" shows "list_interval l u = {x. list_all2 (\<le>) l x \<and> list_all2 (\<le>) x u}"
simps_4: fixes u :: "'b sort_key extended"   and l :: "'b sort_key extended"   and i :: "'b" assumes "u \<le> l" shows "\<Psi> (l, u) i = undefined"
share_all_until_volatile_write_RMW_commute: fixes i :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and a :: "nat"   and t :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d \<times> 'e"   and p' :: "'b"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "'c"   and \<O>' :: "'d \<times> 'e"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, RMW a t (D, f) cond ret A L R W # is, \<theta>, [], \<D>, \<O>)" shows "share_all_until_volatile_write (ls[i := (p', is, \<theta>', [], \<D>', \<O>')]) S = share_all_until_volatile_write ls S"
scaleR_add_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "real"   and x :: "'a"   and y :: "'a" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (plus x y) = plus (scaleR a x) (scaleR a y)"
open_contains_cball: fixes S :: "'a set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
HF1_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e HF1"
mon_c_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and c :: "'b list multiset" shows "mon_c fg c \<equiv> \<Union> {mon_s fg s |s. s \<in># c}"
a_omega: fixes x :: "'a"   and y :: "'a" shows "ad (\<Omega> x y) = ad y +  |x\<rangle> y"
wfp_on_inv_image_betw: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "wfp_on P B" shows "wfp_on (inv_image_betw P f A B) A"
arbitrary_union_of_alt: fixes Q :: "'c set \<Rightarrow> bool"   and S :: "'c set" shows "(arbitrary union_of Q) S = (\<forall>x\<in>S. \<exists>U. Q U \<and> x \<in> U \<and> U \<subseteq> S)"
monotone_directed_image: fixes I :: "'b set"   and ir :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "monotone_on I ir r f"   and "directed I ir" shows "directed (f ` I) r"
finite_IN: fixes A :: "'a set" shows "(\<Sum>\<^sup>+ y\<in>A. f (y, x)) \<noteq> \<top>"
csrp_eq_coset: fixes I :: "'a set"   and x :: "'a set" assumes "ideal R I"   and "x \<in> carrier (R /\<^sub>r I)" shows "csrp_fn R I x \<uplus>\<^bsub>R\<^esub> I = x"
Ann_1: shows "\<turnstile>\<^sub>! (r \<^bold>\<longleftrightarrow>\<^sub>! reduce r)"
subset_UNIV: fixes A :: "'e set" shows "A \<subseteq> UNIV"
lastAssertedLiteralIsUniq: fixes literal :: "Literal"   and literalList :: "Literal list"   and valuation :: "Literal list"   and literal' :: "Literal" assumes "isLastAssertedLiteral literal literalList valuation"   and "isLastAssertedLiteral literal' literalList valuation" shows "literal = literal'"
fold_insert_remove: fixes A :: "'a\<^sub>m set"   and z :: "'a\<^sub>m option"   and x :: "'a\<^sub>m" assumes "finite A" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z (insert x A) = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z (A - {x}) \<bind> m x"
mono_2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and P' :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<diamond>P) \<sigma>"   and "\<And>\<sigma>. P \<sigma> \<Longrightarrow> P' \<sigma>" shows "(\<diamond>P') \<sigma>"
esing_bot: fixes x :: "'c" shows "esing x\<cdot>\<bottom> = \<bottom>"
long_dividesI: fixes b :: "'a list list"   and r :: "'a list list"   and p :: "'a list list"   and q :: "'a list list" assumes "b \<in> carrier (poly_ring (K [X]))"   and "r \<in> carrier (poly_ring (K [X]))"   and "p = q \<otimes>\<^bsub>poly_ring (K [X])\<^esub> b \<oplus>\<^bsub>poly_ring (K [X])\<^esub> r"   and "r = [] \<or> degree r < degree q" shows "UP.long_divides p q (b, r)"
assms_2: shows "H = (\<Union>h\<in>set hs. G + {h})"
filter_mset_add_mset: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and A :: "'c multiset" shows "filter_mset P (add_mset x A) = (if P x then add_mset x (filter_mset P A) else filter_mset P A)"
mem_carrier: fixes x :: "'a" assumes "x \<in> derived G (carrier G)" shows "x \<in> carrier G"
assms_1: shows "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"
step_elim_cases_1: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (Skip, s) \<rightarrow> u" shows "P"
linear_inj_imp_surj: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f" shows "surj f"
suffix_nth: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "x [k..] n = x (k + n)"
pointermap_empty_def: shows "pointermap_empty \<equiv> hm_new \<bind> (\<lambda>hm. arl_empty \<bind> (\<lambda>arl. return \<lparr>entriesi = arl, getentryi = hm\<rparr>))"
preserves_inv: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "fst (C.VV.inv f) \<star>\<^sub>C snd (C.VV.inv f) = C.inv (fst f \<star>\<^sub>C snd f)"
C_verifies_PInitRes: fixes MsgPInitRes :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and Chall_C :: "nat"   and Chall_M :: "nat"   and P :: "agent"   and EKj :: "nat"   and M :: "agent"   and evs :: "event list" assumes "MsgPInitRes = \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Nonce Chall_M, cert P EKj onlyEnc (priSK RCA)\<rbrace>"   and "Crypt (priSK M) (Hash MsgPInitRes) \<in> parts (knows Spy evs)"   and "evs \<in> set_pur"   and "M \<notin> bad" shows "\<exists>j trans. Notes M \<lbrace>Number LID_M, Agent P, trans\<rbrace> \<in> set evs \<and> P = PG j \<and> EKj = pubEK P"