hcju/isabelleps huggingface dataset:
continuous_le_on_closure: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real"   and x :: "'a"   and a :: "real" assumes "continuous_on (closure s) f"   and "x \<in> closure s"   and "\<And>x. x \<in> s \<Longrightarrow> f x \<le> a" shows "f x \<le> a"
w_def: shows "x1 < w \<and> w < x2 \<and> poly (pderiv (prod_list qs)) w = 0"
rel_simps_27: fixes n :: "num" shows "(0::'c) \<le> numeral n"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
not_eq_order_implies_strict: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
pr_comp1_3': fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec1'"   and "pr_conv_3_to_1 g \<in> PrimRec1'" shows "pr_conv_3_to_1 (\<lambda>x y z. f (g x y z)) \<in> PrimRec1'"
\<pi>_upper_bound: fixes x :: "real" assumes "2 \<le> x" shows "\<pi> x < 3 * (exp (- 1) + ln 2) * x / ln x"
Int_insert_right: fixes A :: "'e set"   and a :: "'e"   and B :: "'e set" shows "A \<inter> insert a B = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
finite_imp_compactin: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X"   and "finite S" shows "compactin X S"
assms_3: shows "\<not> VS\<^sub>1.dependent B"
gfp_dual_var: fixes f :: "'b \<Rightarrow> 'b" shows "\<partial> (gfp f) = lfp (\<partial>\<^sub>F f)"
a: shows "\<integral>\<^sup>+ x. ennreal (f x) * nn_cond_exp M F gp x \<partial>M < \<infinity>"
reads_restr_suc: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "'n"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n" shows "suc n \<sigma> = suc n \<sigma>'"
the_def: fixes option :: "'a option" shows "the option = (case option of \<lfloor>x2\<rfloor> \<Rightarrow> x2)"
fvi_Suc: fixes x :: "nat"   and b :: "nat"   and \<phi> :: "Formula.formula" shows "(x \<in> Formula.fvi (Suc b) \<phi>) = (Suc x \<in> Formula.fvi b \<phi>)"
sequentially_imp_eventually_within: fixes "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.first_countable_topology open"   and "\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> filterlim f (topological_space.nhds open a) sequentially \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))" shows "eventually P (topological_space.at_within open a s)"
combine_collapse: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "<s|s> = s"
inject: fixes x1 :: "'a"   and x2 :: "'a tree list"   and y1 :: "'a"   and y2 :: "'a tree list" shows "(NODE x1 x2 = NODE y1 y2) = (x1 = y1 \<and> x2 = y2)"
exists_next_if_can_occur: fixes ev :: "('p, 's, 'm) event"   and c :: "('p, 's, 'm) configuration" assumes "can_occur ev c" shows "\<exists>c'. c \<turnstile> ev \<mapsto> c'"
going_to_principal: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'c set" shows "f going_to principal A within B = principal (f -` A \<inter> B)"
IH_3: fixes x3 :: "(bexp \<times> int) list"   and x :: "nat list"   and xa :: "nat"   and pc' :: "nat" assumes "pc < array_length ins"   and "array_get ins pc = ChoiceI x3"   and "x = concat (map (\<lambda>(b, i). if bval b s then [nat (int (pc + 1) + i)] else []) x3)"   and "x \<noteq> []"   and "xa \<in> set x"   and "pc < xa"   and "xa < pc_bound ins"   and "pc' \<in> set (exec' ins s xa)" shows "pc' < pc_bound ins"
collapse: fixes list :: "'b list" assumes "list \<noteq> \<epsilon>" shows "hd list # tl list = list"
finite_Range: fixes r :: "('b \<times> 'c) set" assumes "finite r" shows "finite (Range r)"
cPar1B: fixes a :: "subject"   and x :: "name"   and P' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> C" shows "F C (P \<parallel> !P) (a\<guillemotleft>x\<guillemotright> \<prec> P' \<parallel> !P)"
coplanar_pseudo_trans: fixes P :: "'p"   and Q :: "'p"   and R :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Col P Q R"   and "Coplanar P Q R A"   and "Coplanar P Q R B"   and "Coplanar P Q R C"   and "Coplanar P Q R D" shows "Coplanar A B C D"
vector_to_cblinfun_cblinfun_apply: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and \<psi> :: "'c" shows "vector_to_cblinfun (A *\<^sub>V \<psi>) = A o\<^sub>C\<^sub>L vector_to_cblinfun \<psi>"
p2rel_def: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "p2rel P \<equiv> {(x, y). P x y}"
sum_right: fixes x :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and g :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "x *\<^sub>V sum g S = (\<Sum>i\<in>S. x *\<^sub>V g i)"
pelims: fixes x :: "key"   and y :: "key" assumes "invKey x = y"   and "invKey_dom x" obtains A :: "agent" where "x = pubK A__" and "y = priK A__" and "invKey_dom (pubK A__)"   | A :: "agent" where "x = priK A" and "y = pubK A" and "invKey_dom (priK A)"   | v :: "fresh_t" where "x = sesK v" and "y = sesK v" and "invKey_dom (sesK v)"   | va :: "agent" where "x = shrK va" and "y = shrK va" and "invKey_dom (shrK va)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_add_hom hom"   and "zero_hom_0 hom" shows "monoid_add_hom_0 hom"
Cons_3: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
OrdP_linear_imp: fixes x :: "tm"   and y :: "tm" shows "{} \<turnstile> OrdP x IMP OrdP y IMP x IN y OR x EQ y OR y IN x"
ts_in_type: shows "ts_rep \<in> {ts. (\<forall>c. fst (snd ts) c \<sqinter> fst (snd (snd ts)) c = bot) \<and> (\<forall>c. 1 \<le> |fst (snd ts) c|) \<and> (\<forall>c. |fst (snd ts) c| \<le> 2) \<and> (\<forall>c. |fst (snd (snd ts)) c| \<le> 1) \<and> (\<forall>c. |fst (snd ts) c| + |fst (snd (snd ts)) c| \<le> 2) \<and> (\<forall>c. fst (snd (snd ts)) c \<noteq> bot \<longrightarrow> (\<exists>n. Rep_nat_int (fst (snd ts) c) \<union> Rep_nat_int (fst (snd (snd ts)) c) = {n, n + 1})) \<and> (\<forall>c. 0 < fst (snd (snd (snd (snd ts)))) c) \<and> (\<forall>c. 0 < snd (snd (snd (snd (snd ts)))) c)}"
p': shows "p = (\<Sum>i = 0..<dim_row A. v $ i \<cdot> row_to_poly (pps_to_list S) (row A i))"
lm158: fixes X :: "'b set" shows "finite X = (X \<in> range set)"
it_OK: shows "map_iterator (iti m) (\<alpha> m)"
nth_append_simps_1: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "i < length xs" shows "(xs @ ys) ! i = xs ! i"
fsafe_And_simps: fixes p :: "('a, 'b, 'c) formula"   and q :: "('a, 'b, 'c) formula" shows "fsafe (p && q) = (fsafe p \<and> fsafe q)"
C: shows "x # xs \<simeq> {ys, zs, P}"
fls_plus_const: fixes x :: "'b"   and y :: "'b" shows "fls_const x + fls_const y = fls_const (x + y)"
\<sigma>_cong: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and t :: "MIR.num"   and t' :: "MIR.num"   and c :: "int" assumes "iszlfm p (a # bs)"   and "Inum (a # bs) t = Inum (a # bs) t'" shows "Ifm (a # bs) (\<sigma> p c t) = Ifm (a # bs) (\<sigma> p c t')"
prems_4: shows "ts ! i = (p, is, \<theta> |` (dom \<theta> - read_tmps ((r # sb) @ sb')), x, \<D>, \<O>, \<R>)"
exec_rem_effless_as_proj_eq_exec_as_proj: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and vs :: "'c set" shows "exec_plan s (as_proj (rem_effectless_act as) vs) = exec_plan s (as_proj as vs)"
typing_cond_prot_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and \<P> :: "('d strand_label \<times> ('a, 'b) strand_step) list set" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2" shows "labeled_typed_model.typing_cond_prot arity Ana \<Gamma> \<P> \<equiv> labeled_typed_model.wf\<^sub>l\<^sub>s\<^sub>t\<^sub>s \<P> \<and> typed_model.tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> (\<Union> (trms\<^sub>l\<^sub>s\<^sub>t ` \<P>)) \<and> (\<forall>t\<in>\<Union> (trms\<^sub>l\<^sub>s\<^sub>t ` \<P>). intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t) \<and> (\<forall>\<A>\<in>\<P>. list_all (typed_model.tfr\<^sub>s\<^sub>t\<^sub>p \<Gamma>) (unlabel \<A>)) \<and> typed_model.Ana_invar_subst Ana (\<Union> (ik\<^sub>s\<^sub>t ` unlabel ` \<P>) \<union> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` unlabel ` \<P>))"
carrier_class: fixes x :: "'b" shows "x \<in> carrier cring_class_ops"
less_eq_imp_interp_subseteq_Interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<le> D" shows "interp C \<subseteq> Interp D"
get_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "h \<turnstile> ok get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| document_ptr_kinds h"
nat_leSuc0_conv: fixes n :: "nat" shows "(n \<le> Suc 0) = (n = 0 \<or> n = Suc 0)"
intros_27: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (global \<C>)"   and "tg_t (global \<C> ! i) = t" shows "\<C> \<turnstile> [Get_global i] : [] _> [t]"
distinct_1: fixes x1 :: "'c"   and x2 :: "'d" shows "Inl x1 \<noteq> Inr x2"
simps_1: fixes x2 :: "'b"   and y2 :: "'b" shows "(Ref x2 = Ref y2) = (x2 = y2)"
IH: fixes i :: "nat"   and "is" :: "nat list" assumes "i < hd (dims A)"   and "is \<lhd> dims (subtensor A i)" shows "lookup (subtensor A i) is \<in> set (vec (subtensor A i))"
fixespointwise_comp: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'b" assumes "fixespointwise f A"   and "fixespointwise g A" shows "fixespointwise (g \<circ> f) A"
set_intros_2: fixes y :: "'b"   and x22 :: "'b list"   and x21 :: "'b" assumes "y \<in> ran\<^sub>l x22" shows "y \<in> ran\<^sub>l (x21 # x22)"
simps_1: fixes uu :: "'e"   and u :: "'d \<Rightarrow> 'e" shows "[[]\<rightarrow>uu]u = u"
MOST_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>\<^sub>\<infinity>x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "\<forall>\<^sub>\<infinity>x. Q x"
value_eq_true: fixes a :: "value option"   and b :: "value option" shows "(value_eq a b = true) = (\<exists>x y. a = Some x \<and> b = Some y \<and> x = y)"
fps_inverse_zero': assumes "inverse (0::'b) = (0::'b)" shows "inverse 0 = 0"
P_no_deadend_v0: assumes "\<not> lnull (ltl P)" shows "\<not> deadend v"
iMOD_pred_insert_conv: fixes m :: "nat"   and r :: "nat" assumes "m \<le> r" shows "insert (r - m) [ r, mod m ] = [ r - m, mod m ]"
span_explicit': fixes b :: "'a set" shows "cspan b = {\<Sum>v | f v \<noteq> 0. f v *\<^sub>C v |f. finite {v. f v \<noteq> 0} \<and> (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> b)}"
distinct_fst_alist_of: fixes q :: "('c, 'd) pq" shows "distinct (map fst (alist_of q))"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "web_flow \<Gamma> f = (current \<Gamma> f \<and> (\<forall>x. x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub> \<longrightarrow> x \<notin> A \<Gamma> \<longrightarrow> x \<notin> B \<Gamma> \<longrightarrow> KIR f x))"
simps_7: fixes v :: "'d strand_label"   and vb :: "'f list"   and vc :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and vd :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (v, \<forall>vb\<langle>\<or>\<noteq>: vc \<or>\<notin>: vd\<rangle>) = (v, \<forall>vb\<langle>\<or>\<noteq>: vc \<or>\<notin>: vd\<rangle>)"
quicksort_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" shows "WB_Sort.quicksort R h = (\<lambda>(lo, hi, xs0). REC\<^sub>T (\<lambda>f (lo, hi, xs). ASSERT (lo \<le> hi \<and> hi < length xs \<and> mset xs = mset xs0) \<bind> (\<lambda>_. SPEC (uncurry (partition_spec R h xs lo hi)) \<bind> (\<lambda>(xs, p). ASSERT (mset xs = mset xs0) \<bind> (\<lambda>_. (if p - 1 \<le> lo then RETURN xs else f (lo, p - 1, xs)) \<bind> (\<lambda>xs. ASSERT (mset xs = mset xs0) \<bind> (\<lambda>_. if hi \<le> p + 1 then RETURN xs else f (p + 1, hi, xs))))))) (lo, hi, xs0))"
simps_1: fixes committed :: "nat set"   and justifying_exec :: "('b \<times> ('a, 'b) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and action_translation :: "nat \<Rightarrow> nat"   and more :: "'c" shows "action_translation \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, action_translation = action_translation, \<dots> = more\<rparr> = action_translation"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(edge_rev M x = edge_rev M y) = (x = y)"
eq: shows "(\<lambda>x. Blinfun (\<lambda>a. dterm_sem (extendf I x) \<theta>\<^sub>1 \<nu> * extendf_deriv I i \<theta>\<^sub>2 \<nu> x a + extendf_deriv I i \<theta>\<^sub>1 \<nu> x a * dterm_sem (extendf I x) \<theta>\<^sub>2 \<nu>)) = (\<lambda>x. dterm_sem (extendf I x) \<theta>\<^sub>1 \<nu> *\<^sub>R Blinfun (extendf_deriv I i \<theta>\<^sub>2 \<nu> x) + dterm_sem (extendf I x) \<theta>\<^sub>2 \<nu> *\<^sub>R Blinfun (extendf_deriv I i \<theta>\<^sub>1 \<nu> x))"
subspace_scale: fixes S :: "'b set"   and x :: "'b"   and c :: "complex" assumes "csubspace S"   and "x \<in> S" shows "c *\<^sub>C x \<in> S"
Max_in: fixes A :: "'t set" assumes "finite A"   and "A \<noteq> {}" shows "ord_term_lin.Max A \<in> A"
sg_2: shows "subgraph (S j) (S (max i j))"
sel_9: fixes x41 :: "'c list"   and x42 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "bvars\<^sub>s\<^sub>t\<^sub>p \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t = x41"
F_subset_imp_T_subset: fixes P :: "'a process"   and Q :: "'a process" assumes "F P \<subseteq> F Q" shows "T P \<subseteq> T Q"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "ty list"   and a2 :: "val list"   and a3 :: "val list" assumes "P \<turnstile> a1 Casts a2 to a3 " obtains "a1 = []" and "a2 = []" and "a3 = []"   | T :: "ty" and v :: "val" and v' :: "val" and Ts :: "ty list" and vs :: "val list" and vs' :: "val list" where "a1 = T # Ts" and "a2 = v # vs" and "a3 = v' # vs'" and "P \<turnstile> T casts v to v' " and "P \<turnstile> Ts Casts vs to vs' "
B: fixes ys :: "'a list"   and zs :: "'a list" shows "xs \<cong> {ys, zs, P} \<longrightarrow> map f xs \<cong> {map f ys, map f zs, \<lambda>w ws. P (inv f w) (map (inv f) ws)}"
intro: fixes P :: "'c \<Rightarrow> bool"   and E :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>P a; E a b\<rbrakk> \<Longrightarrow> P b" shows "Graph_Invariant E P"
distinct_22: fixes name1' :: "name"   and name2' :: "name"   and pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "name1'<name2'>.pi' \<noteq> name1{name2}.pi"
set_of_elts: fixes x :: "V" shows "set (elts x) = x"
no_trailing_unfold: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "no_trailing P xs = (xs \<noteq> [] \<longrightarrow> \<not> P (last xs))"
is_empty_set: fixes xs :: "'c list" shows "Set.is_empty (set xs) = List.null xs"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>\<epsilon> f : FG.map a \<rightarrow>\<^sub>C C.map b\<guillemotright>"
l: shows "\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e"
series_comparison_complex: fixes g :: "nat \<Rightarrow> complex"   and N :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "summable g"   and "\<And>n. g n \<in> \<real>"   and "\<And>n. 0 \<le> Re (g n)"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> cmod (g n)" shows "summable f"
ex_conj_sylow_group: fixes H :: "'a set"   and b :: "nat"   and P :: "'a set" assumes "H \<in> subgroups_of_size (p ^ b)"   and "P \<in> subgroups_of_size (p ^ a)" obtains g :: "'a" where "g \<in> carrier G" and "H \<subseteq> g <# (P #> inv g)"
lens_comp_right_id: fixes X :: "'d \<Longrightarrow> 'e" shows "X ;\<^sub>L \<Sigma> = X"
minCB: fixes x :: "'a" assumes "x \<in> C"   and "x \<notin> B" shows "bc \<le> x \<and> (x \<le> bc \<longrightarrow> bc = x)"
map_poly_comm_semiring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_comm_semiring_hom hom \<equiv> comm_semiring_hom hom"
rel_intros_2: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x2 :: "'d"   and y2 :: "'e" assumes "R x2 y2" shows "rel_option R (Some x2) (Some y2)"
left_idem: fixes a :: "'e"   and b :: "'e" shows "sup a (sup a b) = sup a b"
same_togetherTr: fixes A :: "'b set"   and H :: "'a set"   and n :: "nat" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "\<forall>s f. s \<in> {j. j \<le> n} \<rightarrow> A \<and> f \<in> {j. j \<le> n} \<rightarrow> H \<longrightarrow> (\<exists>t\<in>{j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} \<rightarrow> A. \<exists>g\<in>{j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} \<rightarrow> f ` {j. j \<le> n}. surj_to g {j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} (f ` {j. j \<le> n}) \<and> \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n = \<Sigma>\<^sub>e M (\<lambda>k. t k \<cdot>\<^sub>s g k) (card (f ` {j. j \<le> n}) - Suc 0))"
assms_2: shows "\<forall>x y. le x y \<or> le y x"
Says_K6: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and A :: "agent"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
domains_subsume_globally_def: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "\<S>\<^sub>G s = (\<forall>r ls. s r = Some ls \<longrightarrow> \<S> ls)"
rev_sub_closed: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "rev_compose R q p \<in> carrier (UP R)"
r_def: shows "r \<equiv> if rs = {} then (x + b) / 2 else (Min rs + x) / 2"
independent_backwards_3: fixes K :: "'a set"   and u :: "'a"   and Us :: "'a list" assumes "local.independent K (u # Us)" shows "u \<in> carrier R"
smult_conv_mult: fixes s :: "'b"   and p :: "'b mpoly" shows "MPoly_Type.smult s p = MPoly_Type.monom 0 s * p"
prems_2: shows "a \<in> all_shared (x # sb) \<union> all_unshared (x # sb) \<union> X"
sel_2: fixes x1 :: "((('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) list"   and x2 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x3 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x4 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x5 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x6 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_receive (Transaction x1 x2 x3 x4 x5 x6) = x2"
distinct_set_replace: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list" assumes "distinct xs" shows "set (replace x ys xs) = (if x \<in> set xs then set xs - {x} \<union> set ys else set xs)"
hyps_3: fixes A :: "char list set"   and A' :: "char list set" assumes "D A c A'"   and "A \<subseteq> dom s" shows "\<exists>t. s' = Some t \<and> A' \<subseteq> dom t"
wens_Int_eq: fixes T :: "'b set"   and B :: "'b set"   and F :: "'b program"   and act :: "('b \<times> 'b) set" assumes "T - B \<subseteq> awp F T"   and "act \<in> Acts F" shows "T \<inter> wens F act B = T \<inter> wens F act (T \<inter> B)"
integral_sum: fixes I :: "'d set"   and M :: "'e measure"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "LINT x|M. (\<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
inv_one: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G" shows "inv\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>G\<^esub>"
restrict_def: fixes ioa :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and actns :: "'d set" shows "restrict ioa actns \<equiv> (restrict_asig (asig_of ioa) actns, starts_of ioa, trans_of ioa)"
rbt_insert_with_key_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c"   and t :: "('a, 'c) rbt" shows "rbt_insert_with_key f k v t = paint B (rbt_ins f k v t)"
f_id: fixes f :: "int poly" assumes "f \<in># fs" shows "Mp f = f"
finite_total_preorder_on_axioms: shows "finite_total_preorder_on (inv f -` carrier) (map_relation (inv f) le)"
gauss_int_periodic: shows "periodic_arithmetic (\<lambda>x. gauss_sum_int (int x)) n"
abs_eq: fixes x :: "circline_mat" shows "calc_ideal_points (abs_circline x) = abs_complex_homo ` calc_ideal_points_clmat_hcoords x"
tensor_preserves_ide: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "ide (MC.tensor a b)"
lincomb_replicate_left: fixes r :: "'r"   and ms :: "'m list"   and k :: "nat" assumes "r \<in> R"   and "set ms \<subseteq> M" shows "replicate k r \<bullet>\<cdot> ms = r \<cdot> (\<Sum>m\<leftarrow>take k ms. m)"
rel_intros_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x21 :: "'b"   and y21 :: "'c"   and x22 :: "'b list"   and y22 :: "'c list" assumes "R x21 y21"   and "list_all2 R x22 y22" shows "list_all2 R (x21 # x22) (y21 # y22)"
set_update_with: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and al :: "('a, 'b) alist" shows "AssocList.set (update_with v k f al) = AssocList.set al - {k} \<times> UNIV \<union> {(k, f (case DAList.lookup al k of None \<Rightarrow> v | Some v \<Rightarrow> v))}"
Cd_range': fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Cd d X" shows "x \<in> X \<inter> Field (Pd d)"
fst_snd_flip: fixes xy :: "'a \<times> 'b" shows "cc xy = (snd \<circ>\<circ> case_prod) (\<lambda>x y. (y, x)) xy"
rel_interior_rel_interior: fixes S :: "'b set" assumes "convex S" shows "rel_interior (rel_interior S) = rel_interior S"
crename_ax2: fixes N :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and c :: "coname" assumes "N[a\<turnstile>c>b] = Ax x c" shows "\<exists>d. N = Ax x d"
graph_Entry_axioms_def: fixes \<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" shows "graph_Entry_axioms \<alpha>n invar inEdges' Entry \<equiv> (\<forall>g. Entry g \<in> set (\<alpha>n g)) \<and> (\<forall>g. invar g \<longrightarrow> graph_path_base.inEdges inEdges' g (Entry g) = []) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> invar g \<longrightarrow> (\<exists>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns n))"
degree_pCons_eq: fixes p :: "'c poly"   and a :: "'c" assumes "p \<noteq> 0" shows "degree (pCons a p) = Suc (degree p)"