hcju/isabelleps huggingface dataset:
unlabel_ik\<^sub>s\<^sub>t_is_rcv_set: fixes A :: "('f \<times> ('d, 'e) strand_step) list" shows "ik\<^sub>s\<^sub>t (unlabel A) = {uu_. \<exists>l t. uu_ = t \<and> (l, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set A}"
fds_perzeta_one_half: shows "fds_perzeta (1 / 2) = fds ((^) (- 1))"
lexmax_singleton: fixes r :: "('b \<times> 'b) set"   and \<alpha> :: "'b" shows "r|[\<alpha>]| = {#\<alpha>#}"
meth: shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"
seqr_left_unit: fixes P :: "('d, 'e) urel" shows "II ;; P = P"
cost_swapSyms_le: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "b \<in> alphabet t"   and "freq t a \<le> freq t b"   and "depth t a \<le> depth t b" shows "cost (swapSyms t a b) \<le> cost t"
induced_id_inj: shows "inj_on (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) id) (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>)"
AE_iff_null_sets: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "complete_measure M" shows "almost_everywhere M P = ({x \<in> space M. \<not> P x} \<in> null_sets M)"
Abs_rename_set: fixes p :: "perm"   and bs :: "atom set"   and x :: "'a" assumes "(p \<bullet> bs) \<sharp>* x" shows "\<exists>q. [bs]set. x = [p \<bullet> bs]set. q \<bullet> x \<and> q \<bullet> bs = p \<bullet> bs"
tendsto_add_ereal_MInf: fixes y :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "y \<noteq> \<infinity>"   and "(f \<longlongrightarrow> - \<infinity>) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> - \<infinity>) F"
convex_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<And>t x y. \<lbrakk>0 < t; t < 1; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" shows "convex_on A f"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>x xs y ys. \<lbrakk>P xs (y # ys); P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
val_bind_tm: fixes m :: "'b tm"   and f :: "'b \<Rightarrow> 'a tm" shows "Time_Monad.val (m \<bind> f) = (let x = Time_Monad.val m in Time_Monad.val (f x))"
step: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
Class_is_Left_Coset: fixes g :: "'a" assumes "g \<in> R" shows "additive.Class g = g +| I"
ChiPlus_ChiRel: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(S, T) \<in> ChiRel A" shows "(S, T) \<in> ChiPlus A"
and_P1_security: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "((bool \<times> bool) \<times> bool \<times> 'v_14_OT1) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "and_secret_sharing.adv_P1 n m1 m2 D \<le> adv_14_OT n"
index_row_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "Matrix.row A i $v j = A $$ (i, j)"
simps: fixes ws :: "'a list"   and Z :: "'a set" shows "local.rtrancl_while_invariant (ws, Z) = (x \<in> Z \<and> set ws \<subseteq> Z \<and> distinct ws \<and> {(x, y). y \<in> set (f x)} `` (Z - set ws) \<subseteq> Z \<and> Z \<subseteq> {(x, y). y \<in> set (f x)}\<^sup>* `` {x} \<and> Ball (Z - set ws) p)"
assign: fixes x :: "'Var"   and e :: "'AExp"   and mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simplep (x \<leftarrow> e) mem Stop (mem(x := ev\<^sub>A mem e))"
succ_list_it_correct: fixes g :: "'G"   and v :: "'V" assumes "invar g" shows "set_iterator (succ_list_it g v) (succ (\<alpha> g) v)"
set_less_finite_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B" shows "ord.set_less less_eq A B = ord.set_less_aux less_eq A B"
subcategory_axioms: shows "subcategory VxV.comp (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu> \<noteq> null)"
type_definition_unit: shows "type_definition Rep_unit Abs_unit {True}"
dg_prod_ObjE: fixes a :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>" obtains "vsv a" and "\<D>\<^sub>\<circ> a = I" and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> a\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>"
combSlice_Return_parent_node: fixes n' :: "'node SDG_node"   and S :: "'node SDG_node set"   and n'' :: "'node SDG_node"   and p :: "'pname"   and n :: "'node SDG_node" assumes "n' \<in> S"   and "n'' s-p\<rightarrow>\<^bsub>ret\<^esub> CFG_node (parent_node n')"   and "n \<in> sum_SDG_slice2 n'" shows "n \<in> combine_SDG_slices S"
halt_steps2: fixes A :: "abc_inst list"   and st :: "nat"   and lm :: "nat list"   and stp :: "nat" assumes "length A \<le> st" shows "abc_steps_l (st, lm) A stp = (st, lm)"
assms_2: fixes x :: "'a" assumes "(3::'a) \<le> x" shows "f x = g x"
simps_1: fixes P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and i :: "'b"   and j :: "'c" shows "row_add_iterate_PA (P, A) 0 i j = (if i = (0::'b) then (P, A) else (row_add P (0::'b) i (- A $ (0::'b) $ j), row_add A (0::'b) i (- A $ (0::'b) $ j)))"
runit_simps_1: fixes f :: "'b" assumes "D.ide f" shows "D.arr \<r>\<^sub>D[f]"
rep_eq: fixes x :: "mini_alg_unique" shows "real_of_u x = real_of (Rep_mini_alg_unique x)"
boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<preceq> b \<sqinter> c" obtains "a \<preceq> b" and "a \<preceq> c"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) replete_setcat.comp (\<lambda>_. meta_adjunction.inC) (\<lambda>_. meta_adjunction.inD) F G \<phi> \<psi> (meta_adjunction.\<eta> (\<cdot>\<^sub>D) F \<phi>) (meta_adjunction.\<epsilon> (\<cdot>\<^sub>C) F G \<psi>) (transformation_by_components.map (product_category.comp G\<chi>.Cop.comp (\<cdot>\<^sub>C)) replete_setcat.comp (hom_functor.map (\<cdot>\<^sub>C) replete_setcat.comp (\<lambda>_. meta_adjunction.inC) \<circ> product_functor.map G\<chi>.Cop.comp (\<cdot>\<^sub>C) (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>C) (dual_functor.map F) C.map) (meta_adjunction.\<Phi>o (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<phi>)) (inverse_transformation.map (product_category.comp G\<chi>.Cop.comp (\<cdot>\<^sub>C)) replete_setcat.comp (hom_functor.map (\<cdot>\<^sub>D) replete_setcat.comp (\<lambda>_. meta_adjunction.inD) \<circ> product_functor.map G\<chi>.Cop.comp (\<cdot>\<^sub>C) G\<chi>.Cop.comp (\<cdot>\<^sub>D) G\<chi>.Cop.map G) (transformation_by_components.map (product_category.comp G\<chi>.Cop.comp (\<cdot>\<^sub>C)) replete_setcat.comp (hom_functor.map (\<cdot>\<^sub>C) replete_setcat.comp (\<lambda>_. meta_adjunction.inC) \<circ> product_functor.map G\<chi>.Cop.comp (\<cdot>\<^sub>C) (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>C) (dual_functor.map F) C.map) (meta_adjunction.\<Phi>o (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<phi>)))"
psimps_2: fixes M :: "'b trm"   and N :: "'b trm"   and c :: "'b" assumes "unify_dom (M \<cdot> N, Const c)" shows "unify (M \<cdot> N) (Const c) = None"
subset_singletonD: fixes A :: "'b set"   and x :: "'b" assumes "order.greater_eq {x} A" shows "A = {} \<or> A = {x}"
suffix_conc_snd: fixes u :: "'a list"   and n :: "nat"   and v :: "nat \<Rightarrow> 'a" assumes "length u \<le> n" shows "suffix n (u \<frown> v) = suffix (n - length u) v"
foldr_map: fixes g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and a :: "'b" shows "foldr g (list.map f xs) a = foldr (g \<circ> f) xs a"
Bnf: fixes x :: "complex" assumes "x \<in> convex hull {a, b, c}" shows "cmod (f x) \<le> B"
hyps: shows "(gval (foldr gAnd G (Bc True)) s = true) = (\<forall>g\<in>set G. gval g s = true)"
face_cycle_set_def: fixes a :: "'b" shows "GM.face_cycle_set a = orbit GM.face_cycle_succ a"
length_eq_card_columns: fixes A :: "(('c, 'd) vec, 'e) vec" shows "IArray.length (matrix_to_iarray A !! 0) = CARD('d)"
d_min: shows "\<forall>DDa. set_mset DDa \<subseteq> N \<and> set_mset DDa \<union> CC \<Turnstile> {E} \<and> (\<forall>D'\<in>#DDa. D' < D) \<longrightarrow> DD \<le> DDa"
rb_aux_inv2_no_zero_red: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "hom_grading dgrad"   and "is_regular_sequence fs"   and "rb_aux_inv2 (bs, ss, p # ps)"   and "\<not> sig_crit bs (new_syz_sigs ss bs p) p" shows "rep_list (sig_trd bs (poly_of_pair p)) \<noteq> 0"
intro: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "monoid G composition unit"   and "Group_Theory.group_axioms G composition unit" shows "group G composition unit"
simps_2: fixes p :: "'c"   and Tl :: "'a trm list" shows "gA (Pr p Tl) = Pr (Oldp p) (map gT Tl)"
Follows_LeadsTo: fixes F :: "'b program"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and k :: "'c" assumes "F \<in> f Fols g" shows "F \<in> {s. k \<le> g s} \<longmapsto>w {s. k \<le> f s}"
assms_4: fixes s :: "'a" assumes "s \<in> D"   and "V s = 0" shows "V' s (f s) < 0"
flatten2_im: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "\<Sum> ((\<lambda>y. \<Sum> ((\<lambda>x. f x y) ` A)) ` B) = \<Sum> ((\<lambda>p. f (fst p) (snd p)) ` (A \<times> B))"
gen_mono_on_subset: fixes T :: "'b set"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "T \<subseteq> S"   and "gen_mono_on f R S" shows "gen_mono_on f R T"
next_ge: fixes "next" :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "fresh next" shows "x < next x"
Red1AAccNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and i :: "'addr val"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<lfloor>Val i\<rceil>,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
space_eq: fixes n :: "'a measure" assumes "n \<in> M" shows "space n = space m"
deg_zero: shows "deg Zp \<zero>\<^bsub>UP Zp\<^esub> = 0"
par: shows "par (\<Lambda> a b c (local.comp (eval b c) (HF'.prod h b))) h"
induct: fixes P :: "'b hp list \<Rightarrow> bool"   and a0 :: "'b hp list" assumes "P []"   and "\<And>x hsl hsr. \<lbrakk>P hsl; P hsr\<rbrakk> \<Longrightarrow> P (Hp x hsl # hsr)" shows "P a0"
conjunctive_def: shows "Comp.conjunctive = {x. \<forall>y z. x \<circ> inf y z = inf (x \<circ> y) (x \<circ> z)}"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "\<F>C.arr (\<F>C.inclusion_of_generators f)"
isVal_transform: fixes a :: "Arity"   and e :: "exp" shows "isVal (ccTransform a e) = isVal e"
synthetic_div_unique: fixes p :: "'a poly"   and c :: "'a"   and q :: "'a poly"   and r :: "'a" assumes "p + smult c q = pCons r q" shows "r = poly p c \<and> q = synthetic_div p c"
decE_Nil: fixes n :: "nat"   and k :: "nat" shows "\<down>\<^sub>e n k [] = []"
Healthy_intro: fixes H :: "'d upred \<Rightarrow> 'd upred"   and P :: "'d upred" assumes "H P = P" shows "P is H"
of_int_eq_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   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)"
fixp_above_inside: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "a \<in> Field leq" shows "bourbaki_witt_fixpoint.fixp_above lub leq f a = lub (bourbaki_witt_fixpoint.iterates_above lub leq f a)"
length_step: fixes s :: "'a list"   and r :: "'a"   and as :: "nat \<times> nat list" shows "length (step s r as) = length s"
min_less_iff_disj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.linorder less_eq less" shows "less (ord.min less_eq x y) z = (less x z \<or> less y z)"
init_satisfies_tableau: fixes t :: "(nat \<times> linear_poly) list" shows "\<langle>\<V> (init t)\<rangle> \<Turnstile>\<^sub>t t"
left_idem: fixes a :: "'c"   and b :: "'c" shows "sup a (sup a b) = sup a b"
min: fixes y :: "'t" assumes "y \<prec>\<^sub>t v" shows "y \<notin> {v. v \<prec>\<^sub>t u \<and> d (pp_of_term v) \<le> dgrad_max d \<and> component_of_term v < length fs \<and> \<not> is_RB_in d rword G v}"
map_return_pmf: fixes f :: "'s \<Rightarrow> 'r"   and x :: "'s" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
that: fixes T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list" assumes "(\<Gamma>, e, S) \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> (\<Delta>, z, S)" shows "thesis"
order_subst2: fixes a :: "'g"   and b :: "'g"   and f :: "'g \<Rightarrow> 'h"   and c :: "'h" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
nsub_RI_not_init: fixes e :: "'a exp"   and C :: "char list" assumes "\<not> sub_RI e" shows "not_init C e"
A1_def: shows "A1 = mat_of_row (Matrix.row A 0)"
E\<^sub>i\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i" shows "E (x \<cdot> y) \<^bold>\<leftarrow> dom x \<cong> cod y \<^bold>\<and> E (cod y)"
wlp_is_weakest_liberal_precondition: fixes dims :: "nat list"   and S :: "com"   and P :: "complex mat" assumes "state_sig.well_com dims S"   and "state_sig.is_quantum_predicate dims P" shows "state_sig.is_weakest_liberal_precondition dims (state_sig.wlp dims S P) S P"
par: shows "par (local.comp (pr0 (cod f) (cod g)) (tuple f g)) g"
Sup_atLeast: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (ord.atLeast less_eq x) = top"
Cauchy_Schwarz_ineq2: fixes x :: "'a"   and y :: "'a" shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
trivial_relative_homology_group_topspace: fixes p :: "int"   and X :: "'a topology" shows "trivial_group (relative_homology_group p X (topspace X))"
fst_snd_Gauss_Jordan_column_k_PA_eq: fixes P :: "(('b, 'd) vec, 'd) vec"   and i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "fst (snd (Gauss_Jordan_column_k_PA (P, i, A) k)) = fst (Gauss_Jordan_column_k (i, A) k)"
intro: fixes V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>B :: "'d \<Rightarrow> 'd"   and src\<^sub>B :: "'d \<Rightarrow> 'd"   and trg\<^sub>B :: "'d \<Rightarrow> 'd" assumes "bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "op_bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
components_are_iso: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "D.iso (cmp a)"
step: fixes \<phi>' :: "'val"   and g :: "'g"   and \<phi> :: "'val"   and \<phi>'' :: "'val" assumes "\<phi>' \<in> reachable g \<phi>"   and "phiArg g \<phi>' \<phi>''"   and "unnecessaryPhi g \<phi>''" shows "\<phi>'' \<in> reachable g \<phi>"
pred_sets2: fixes A :: "'c set"   and N :: "'c measure"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "A \<in> sets N"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "Measurable.pred M (\<lambda>x. f x \<in> A)"
rep_eq: fixes x :: "('d, 'e) \<I>" shows "responses_\<I> x = Rep_\<I> x"
select_convs_1: fixes src :: "'d"   and tgt :: "'d"   and more :: "'e" shows "src \<lparr>src = src, tgt = tgt, \<dots> = more\<rparr> = src"
dep3_def: fixes Q :: "'a set"   and R :: "'a set"   and S :: "'a set" shows "dep3 Q R S \<equiv> \<exists>x. dep3_event Q R S x"
assms_2: assumes "p \<noteq> (0::'a)"   and "\<not> p dvd (1::'a)"   and "\<And>a b. p = a * b \<Longrightarrow> a dvd (1::'a) \<or> b dvd (1::'a)" shows "thesis"
p: shows "\<down>\<^sub>t (c1, s) = p"
W_eq: fixes i :: "nat" assumes "i \<in> N" shows "domain (\<psi> i) = W i"
strong_release_or_left_distrib: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi>\<^sub>1 :: "'b ltln"   and \<phi>\<^sub>2 :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n (\<phi>\<^sub>1 or\<^sub>n \<phi>\<^sub>2) M\<^sub>n \<psi> = w \<Turnstile>\<^sub>n (\<phi>\<^sub>1 M\<^sub>n \<psi>) or\<^sub>n (\<phi>\<^sub>2 M\<^sub>n \<psi>)"
S_intersections_in_S_1: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q"   and "proj2_abs p \<in> hyp2" shows "S_intersection1 p q \<in> S"
STCalWB_reachesBarbST: fixes P :: "('procS, 'procT) Proc"   and a :: "'barbs" shows "P\<Down><STCalWB SWB TWB>a = ((\<exists>S. S \<in>S P \<and> S\<Down><SWB>a) \<or> (\<exists>T. T \<in>T P \<and> T\<Down><TWB>a))"
Dom_memb: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "arr f"   and "t \<in> f" shows "Dom t = DOM f"
max_fun_diff_in: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rt.FINFUNC"   and "g \<in> rt.FINFUNC" shows "t.max_fun_diff f g \<in> {a \<in> Field t. f a \<noteq> g a}"
local_time_incseq: fixes A :: "'a set"   and x :: "'a" shows "incseq (\<lambda>n. local_time A n x)"
l8_20_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and C' :: "'a"   and D :: "'a"   and A :: "'a"   and C :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong P C' D"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A C' C"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B D C"   and "B \<noteq> C" shows "A \<noteq> P"
ring_one: fixes R :: "('h, 'i) Ring_scheme" assumes "Ring R" shows "1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R"
map_ident: fixes p :: "'a spmf" shows "map_spmf (\<lambda>x. x) p = p"
induct: fixes x1 :: "char list"   and x2 :: "char list"   and P :: "'a prog"   and Pa :: "char list \<Rightarrow> char list \<Rightarrow> bool" assumes "P \<turnstile> x1 \<prec>\<^sup>1 x2"   and "\<And>C D rest. \<lbrakk>class P C = \<lfloor>(D, rest)\<rfloor>; C \<noteq> Object\<rbrakk> \<Longrightarrow> Pa C D" shows "Pa x1 x2"
inj_on_opp_chambers'': fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> \<C> - f \<turnstile> \<C>"   and "D \<in> \<C> - f \<turnstile> \<C>"   and "f ` C = f ` D" shows "C = D"
filterlim_at_infinity_imp_filterlim_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "filterlim f at_infinity F"   and "\<forall>\<^sub>F x in F. 0 < f x" shows "filterlim f at_top F"
convergent_limsup_cl: fixes X :: "nat \<Rightarrow> 'c" assumes "convergent X" shows "limsup X = lim X"
NSLIM_D: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and L :: "'d"   and x :: "'c star" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"   and "x \<noteq> star_of a"   and "x \<approx> star_of a" shows "(*f* f) x \<approx> star_of L"
AVar: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and T :: "ty"   and i :: "expr" assumes "E,dt\<Turnstile>e\<Colon>-T.[]"   and "E,dt\<Turnstile>i\<Colon>-PrimT Integer" shows "E,dt\<Turnstile>e.[i]\<Colon>=T"
downset_substitute: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" shows "downset (Collect (substitute' f T t))"
ln_diff_le_inverse: fixes x :: "real" assumes "1 \<le> x" shows "ln (x + 1) - ln x < 1 / x"
streams_stream_space: fixes M :: "'b measure" shows "streams (space M) \<in> sets (stream_space M)"
prod_rel_sv: fixes R1 :: "('a \<times> 'b) set"   and R2 :: "('c \<times> 'd) set" assumes "single_valued R1"   and "single_valued R2" shows "single_valued (R1 \<times>\<^sub>r R2)"
nn_integral_multc: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * c \<partial>M = integral\<^sup>N M f * c"
frameResFreshChain: fixes xvec :: "name list"   and F :: "'b frame" assumes "xvec \<sharp>* F" shows "\<lparr>\<nu>*xvec\<rparr>F \<simeq>\<^sub>F F"
poly_pos_geq: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x\<ge>a. 0 < poly p x) = (p \<noteq> 0 \<and> poly_inf p = 1 \<and> poly p a \<noteq> 0 \<and> count_roots_above p a = 0)"
inject_1: fixes x11 :: "nat"   and x12 :: "'a"   and x13 :: "nat set"   and y11 :: "nat"   and y12 :: "'a"   and y13 :: "nat set" shows "(NodeS x11 x12 x13 = NodeS y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
alphaAbs_sym: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and A' :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "A $= A'" shows "A' $= A"
mimg_module: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "R module mimg\<^bsub>R M,N\<^esub> f"
impl_adjoin: fixes P :: "'a upred"   and Q :: "'a upred"   and R :: "'a upred" shows "((P \<Rightarrow> Q) \<and> R) = ((P \<and> R \<Rightarrow> Q \<and> R) \<and> R)"
extremum_strict: fixes a :: "'a word" shows "\<not> - 1 < a"
Joints_aforms_of_ivlsD2: fixes zs :: "real list"   and xs :: "real list"   and ys :: "real list" assumes "zs \<in> Joints (aforms_of_ivls xs ys)"   and "list_all2 (\<le>) xs ys" shows "list_all2 (\<le>) zs ys"
fv_subst_eq: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var" shows "fv e[y::=x] = fv e - {y} \<union> (if y \<in> fv e then {x} else {})"
disc_71: fixes x5 :: "protocol" shows "\<not> is_Dst_Ports (Prot x5)"
vfsequence_axioms: shows "vfsequence xs\<^sub>2"
hasty_env_def: fixes ve :: "ExVar \<Rightarrow> Val option"   and te :: "ExVar \<Rightarrow> Ty option" shows "ve hastyenv te  \<equiv> dom ve = dom te \<and> (\<forall>x. x \<in> dom ve \<longrightarrow> the (ve x) hasty the (te x))"
summable_zero_power': fixes f :: "nat \<Rightarrow> 'b" shows "summable (\<lambda>n. f n * (0::'b) ^ n)"
\<alpha>edges_irrefl: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"   and "wgraph_by_map.invar M_lookup M_invar g" shows "irrefl (wgraph_by_map.\<alpha>edges_aux M_lookup g)"
safe_state_def: fixes x :: "'a"   and y :: "'b" shows "safe_state x y \<equiv> True"
SeqCompositionality: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and c1 :: "com"   and c2 :: "com" assumes "nonInterference \<Gamma> c1"   and "nonInterference \<Gamma> c2" shows "nonInterference \<Gamma> (c1;; c2)"
finite_dimensional_real_vector_space_pair_1_on_def: fixes S1 :: "'c set"   and S2 :: "'d set"   and basis :: "'c set" shows "finite_dimensional_real_vector_space_pair_1_on S1 S2 basis \<equiv> finite_dimensional_real_vector_space_on S1 basis \<and> real_vector_space_on S2"
p_residue_ring_car_memI: fixes m :: "int"   and k :: "nat" assumes "0 \<le> m"   and "m < p ^ k" shows "m \<in> carrier (residue_ring (p ^ k))"
l'_def: fixes x :: "'a" shows "l' x = (if x \<in> I then l x else f x)"