hcju/isabelleps huggingface dataset:
compose_actions_Cons: fixes f :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option"   and s :: "predicate \<times> object list \<Rightarrow> bool"   and s' :: "predicate \<times> object list \<Rightarrow> bool"   and fs :: "((predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option) list" assumes "f s = Some s'" shows "compose_actions (f # fs) s = compose_actions fs s'"
sub0T: shows "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0::'a}) \<subseteq> sphere (0::'a) 1 \<inter> T"
nat_pow_mult: fixes x :: "nat \<times> nat"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier S" shows "x [^]\<^bsub>S\<^esub> n \<otimes>\<^bsub>S\<^esub> x [^]\<^bsub>S\<^esub> m = x [^]\<^bsub>S\<^esub> (n + m)"
c6: assumes "\<pi>\<^sub>2 r \<le> sqn rt ip"   and "\<pi>\<^sub>3 r = kno"   and "sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val" shows "P (rt(ip \<mapsto> the (rt ip)))"
that: fixes B :: "real" assumes "\<And>h z. \<lbrakk>h \<in> Y; z \<in> K\<rbrakk> \<Longrightarrow> cmod (h z) \<le> B" shows "thesis"
move_trans: fixes ts :: "traffic"   and ts' :: "traffic"   and ts'' :: "traffic"   and v :: "view" shows "ts \<^bold>\<Rightarrow> ts' \<and> ts' \<^bold>\<Rightarrow> ts'' \<longrightarrow> move ts' ts'' (move ts ts' v) = move ts ts'' v"
s_of_ns_def: fixes ns :: "(nat \<times> form) list" shows "s_of_ns ns = map snd ns"
matrix_mul_rid: fixes A :: "(('b, 'c) vec, 'd) vec" shows "A ** mat (1::'b) = A"
B: shows "x * \<top> \<sqinter> x ^ o = x"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (\<tau>'\<sigma>'.map f) = A.arr f"
canon_ang_2: fixes \<alpha> :: "real" shows "\<downharpoonright>\<alpha>\<downharpoonleft> \<le> pi"
iT_inext_3: fixes n :: "nat"   and t :: "nat"   and d :: "nat" assumes "n \<le> t"   and "t < n + d" shows "inext t [n\<dots>,d] = Suc t"
has_ring_isomorphism: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF>1 :: "'d set \<Rightarrow> 'e set"   and \<rho>1 :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str1 :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str1 :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str1 :: "'d set \<Rightarrow> 'e"   and one_str1 :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d"   and \<FF>2 :: "'d set \<Rightarrow> 'e set"   and \<rho>2 :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and add_str2 :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str2 :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str2 :: "'d set \<Rightarrow> 'e"   and one_str2 :: "'d set \<Rightarrow> 'e"   and U :: "'d set" assumes "iso_stalks S is_open \<FF>1 \<rho>1 b add_str1 mult_str1 zero_str1 one_str1 I x \<FF>2 \<rho>2 add_str2 mult_str2 zero_str2 one_str2"   and "U \<in> I" shows "ring_isomorphism (identity (stalk.carrier_stalk is_open \<FF>1 \<rho>1 x)) (stalk.carrier_stalk is_open \<FF>1 \<rho>1 x) (stalk.add_stalk \<FF>1 \<rho>1 add_str1 I) (stalk.mult_stalk \<FF>1 \<rho>1 mult_str1 I) (stalk.zero_stalk \<FF>1 \<rho>1 zero_str1 I U) (stalk.one_stalk \<FF>1 \<rho>1 one_str1 I U) (stalk.carrier_stalk is_open \<FF>2 \<rho>2 x) (stalk.add_stalk \<FF>2 \<rho>2 add_str2 I) (stalk.mult_stalk \<FF>2 \<rho>2 mult_str2 I) (stalk.zero_stalk \<FF>2 \<rho>2 zero_str2 I U) (stalk.one_stalk \<FF>2 \<rho>2 one_str2 I U)"
assms_3: shows "subpath (add_edge g e) v es (tgt e)"
succs_Cons: fixes x :: "instr"   and xs :: "instr list"   and n :: "int" shows "succs (x # xs) n = isuccs x n \<union> succs xs (1 + n)"
r_inv: fixes x :: "'b" assumes "x \<in> carrier (G i)" shows "x \<otimes>\<^bsub>G i\<^esub> inv\<^bsub>G i\<^esub> x = \<one>\<^bsub>G i\<^esub>"
abelian_monoid_def: fixes G :: "('e, 'f) ring_scheme" shows "abelian_monoid G \<equiv> Group.comm_monoid (add_monoid G)"
size_split_min: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and t :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "t \<noteq> \<langle>\<rangle>" shows "size t = Suc (size (snd (WBT.split_min \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 t)))"
ass_function_euclidean_def: fixes p :: "'a" shows "ass_function_euclidean p = normalize p"
EF_lem000: fixes f :: "'a set" shows "EF f = f \<union> EX' (EF f)"
true_lit_iff: fixes I :: "'b set"   and L :: "'b literal" shows "(I \<Turnstile>l L) = (\<exists>A. L = Pos A \<and> A \<in> I \<or> L = Neg A \<and> A \<notin> I)"
hcomp_isomorphic_ide: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f g"   and "Span.ide h"   and "Span.src f = Span.trg h" shows "Span.isomorphic (f \<circ> h) (g \<circ> h)"
distinct_insert: fixes x :: "'b"   and xs :: "'b list" shows "distinct (List.insert x xs) = distinct xs"
delete_eq: fixes k :: "'a" shows "AList.delete k = filter (\<lambda>(k', uu_). k \<noteq> k')"
flow0_image_UNIV: fixes x :: "'a"   and t :: "real"   and S :: "real set" assumes "existence_ivl0 x = UNIV" shows "flow0 (flow0 x t) ` S = flow0 x ` (\<lambda>s. s + t) ` S"
Least_equality: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y" shows "ord.Least less_eq P = x"
newton_seq_closed: fixes m :: "nat" shows "ns m \<in> carrier Zp"
take_cols_subsets_og_cols: fixes subsets :: "nat list list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list" assumes "length subsets = length signs"   and "invertible_mat (M_mat signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs" shows "set (take_indices (cols (M_mat signs subsets)) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) \<subseteq> set (cols (M_mat signs subsets))"
triangle_equiv_form_1: shows "((\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]) = (\<l>[f] \<cdot> (\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[f] = f)"
aadd_less_mono_z: fixes x :: "ant"   and y :: "ant"   and z :: "int" assumes "x < y" shows "x + ant z < y + ant z"
HEndPhase0_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2b s"   and "HEndPhase0 s s' p" shows "Inv2b s'"
ext: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. f x = g x" shows "f = g"
iteration_unfoldr_distr: fixes y :: "'a"   and x :: "'a" shows "y + x\<^sup>\<infinity> \<cdot> x \<cdot> y = x\<^sup>\<infinity> \<cdot> y"
is_pot_ordD2: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order"   and u :: "('a, 'b) pp \<times> nat"   and v :: "('a, 'b) pp \<times> nat" assumes "is_pot_ord_pprod to"   and "le_of_nat_term_order to u v" shows "snd u \<le> snd v"
types_preserved_local_const: fixes n :: "nat"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[Local n i vs es]\<rparr> \<leadsto> \<lparr>es\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : ts _> ts'"   and "const_list es" shows "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'"
inject: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree"   and y21 :: "'b tree"   and y22 :: "'b"   and y23 :: "'b tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
inf_le1: fixes x :: "'a"   and y :: "'a" shows "inf x y \<le> x"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)" shows "H \<subseteq> carrier (G\<lparr>carrier := H__\<rparr>)"
Icc_eq_Icc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and h :: "'b"   and l' :: "'b"   and h' :: "'b" assumes "class.order less_eq less" shows "(ord.atLeastAtMost less_eq l h = ord.atLeastAtMost less_eq l' h') = (l = l' \<and> h = h' \<or> \<not> less_eq l h \<and> \<not> less_eq l' h')"
hyps: fixes nc :: "nat" assumes "mat nr nc m" shows "mat nc nr (Matrix_Legacy.transpose nr m)"
exec_step_ind_Invokestatic_NoMethod: fixes D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and n :: "nat"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "\<nexists>Ts T m D b. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "exec_step_ind (StepI (Invokestatic C M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NoSuchMethodError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
lens_indep_get: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'f" assumes "x \<bowtie> y" shows "get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"
elem_in_partition: fixes x :: "'b"   and A :: "'b set"   and P :: "'b set set" assumes "x \<in> A"   and "P partitions A" obtains X :: "'b set" where "x \<in> X" and "X \<in> P"
keys_except: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "keys (except p S) = keys p - S"
\<G>\<F>_subfrmlsn: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> \<phi> w \<subseteq> subfrmlsn \<phi>"
fls_lr_inverse_X_inv_1: fixes x :: "'a" shows "fls_left_inverse fls_X_inv x = fls_shift (- 1) (fls_const x)"
share_owns_shared: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "dom (share sb \<S>) \<subseteq> \<O> \<union> dom \<S>"
elts_\<omega>: shows "elts \<omega> = {\<alpha>. \<exists>n. \<alpha> = ord_of_nat n}"
discI_2: fixes llist :: "'b llist"   and x21 :: "'b"   and x22 :: "'b llist" assumes "llist = LCons x21 x22" shows "\<not> lnull llist"
dom_insert: shows "subst_domain (\<sigma>1(v := t)) = insert v (subst_domain \<sigma>1)"
linear_poly_root: fixes a :: "'a"   and as :: "'a list" assumes "a \<in> set as" shows "poly (\<Prod>a\<leftarrow>as. [:- a, 1::'a:]) a = (0::'a)"
select_convs_2: fixes ta_initial :: "'a set"   and ta_rules :: "('a, 'b) ta_rule set"   and more :: "'c" shows "ta_rules \<lparr>ta_initial = ta_initial, ta_rules = ta_rules, \<dots> = more\<rparr> = ta_rules"
unifiable_imp_empty: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "unifiable (set_mset E)" shows "(E, {#}) \<in> unif\<^sup>*"
1: shows "purgeIdle sl \<noteq> [] \<and> hd (purgeIdle sl) = hd sl"
ary_s_ge_x: shows "arity_var x \<le> arity s"
Inf_set_def: fixes A :: "'a set set" shows "\<Inter> A = {x. \<Sqinter> ((\<in>) x ` A)}"
RTranCl_conv: fixes g :: "'a"   and h :: "'a"   and succs :: "'a \<Rightarrow> 'a list" shows "(g [succs]\<rightarrow>* h) = ((g, h) \<in> (Rel succs)\<^sup>*)"
equiintegrable_halfspace_restrictions_ge: fixes F :: "('a \<Rightarrow> 'b) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "F equiintegrable_on cbox a b"   and "f \<in> F"   and "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm (f x)" shows "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<le> x \<bullet> i then h x else (0::'b)}) equiintegrable_on cbox a b"
choose_good: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and v :: "'c" assumes "WellOrderedStrategies G S p good r"   and "v \<in> S" shows "WellOrderedStrategies.choose good r v \<in> good v"
weak_equivalence_implies_weak_bisimilarity: fixes P :: "'state"   and Q :: "'state" assumes "P \<equiv>\<cdot> Q" shows "P \<approx>\<cdot> Q"
has_bochner_integral_add: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and g :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "has_bochner_integral M f x"   and "has_bochner_integral M g y" shows "has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
restriction_res_leq_two: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "|restrict v (res ts) c| \<le> 2"
of_nat_Suc: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc m) = plus one (semiring_1.of_nat one plus zero m)"
eqvtI_1: fixes P :: "pi"   and Rs :: "residual"   and perm :: "(name \<times> name) list" assumes "P \<Longrightarrow>\<^sub>l Rs" shows "perm \<bullet> P \<Longrightarrow>\<^sub>l perm \<bullet> Rs"
bij_betw_def: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and B :: "'h set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"
advantage_obsf_distinguisher: fixes \<A> :: "('y, 'z) distinguisher"   and ideal_resource :: "('y, 'z) resource"   and real_resource :: "('y, 'z) resource" shows "advantage (obsf_distinguisher \<A>) (obsf_resource ideal_resource) (obsf_resource real_resource) = advantage \<A> ideal_resource real_resource"
assms_4: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "P y" shows "P x"
proj_unl_cons_3: fixes l :: "'e"   and a :: "'d"   and A :: "('e strand_label \<times> 'd) list" shows "proj_unl l ((\<star>, a) # A) = a # proj_unl l A"
augm_func_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and A :: "'d set"   and m :: "nat"   and g :: "nat \<Rightarrow> 'd"   and B :: "'d set" shows "augm_func n f A m g B = (\<lambda>i\<in>{j. j \<le> n + m}. if i \<le> n then f i else if Suc n \<le> i \<and> i \<le> n + m then g (sliden (Suc n) i) else undefined)"
cSup_eq_Max: fixes X :: "'a set" assumes "finite X"   and "X \<noteq> {}" shows "\<Squnion> X = Max X"
d2: shows "poly (prod_list (retrieve_polys qs xa)) x < 0 \<longrightarrow> prod_list (map ((!) sign) xa) = - 1"
is_path_undir_sym: fixes v :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_path_undir G v p v'" shows "is_path_undir G v' (rev (map (\<lambda>(u, w, u'). (u', w, u)) p)) v"
compl_le_swap2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<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 y :: "'c"   and x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq (uminus y) x" shows "less_eq (uminus x) y"
case_map_option: fixes g :: "'a"   and h :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c option" shows "(case map_option f x of None \<Rightarrow> g | Some x \<Rightarrow> h x) = (case x of None \<Rightarrow> g | Some x \<Rightarrow> (h \<circ> f) x)"
neg_equal_0_iff_equal: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "(uminus a = zero) = (a = zero)"
inv_character_inv_character: fixes \<chi> :: "'a \<Rightarrow> complex" shows "inv_character (inv_character \<chi>) = \<chi>"
assms_4: shows "hX (\<phi> Pd' Ch' ds) h = hX (\<phi> Pd' (Ch'(h := Ch'')) ds) h"
not_Some_eq: fixes x :: "'c option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
f_strict: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "is_conjugation f g" shows "f 0 = 0"
case_eq_if: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g"   and list :: "'h list" shows "(case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (if list = [] then f1 else f2 (hd list) (tl list))"
IH: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and w :: "'a \<Rightarrow> nat"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "\<forall>x y. x \<le> g x y \<and> y \<le> g x y" shows "w x \<le> wlp R w g f x l"
Rep_mat_impl_cases: fixes y :: "nat \<times> nat \<times> 'b iarray iarray" assumes "y \<in> {(nr, nc, m). IArray.length m = nr \<and> IArray.all (\<lambda>r. IArray.length r = nc) m}" obtains x :: "'b mat_impl" where "y = Rep_mat_impl x"
inRS2: fixes x :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "x \<notin> set p" shows "x \<notin> set (removeShadowRules2 p)"
cos_arctan_not_zero: fixes x :: "real" shows "cos (arctan x) \<noteq> 0"
square_simulation: fixes f :: "state \<Rightarrow> 'a"   and B :: "state \<times> state \<Rightarrow> bool"   and g :: "state \<Rightarrow> 'b"   and A :: "state \<times> state \<Rightarrow> bool" assumes "\<turnstile> unchanged f \<and> \<not> B \<longrightarrow> unchanged g"   and "\<turnstile> A \<and> \<not> unchanged g \<longrightarrow> B" shows "\<turnstile> [A]_f \<longrightarrow> [B]_g"
cmf: shows "continuous_map (subtopology (Euclidean_space n) U) (Euclidean_space n) f"
add_mset: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and N :: "'b multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (add_mset x N) = f x (comm_monoid_mset.F f z N)"
octo_of_real_sel_7: fixes x :: "real" shows "Im6 (octo_of_real x) = 0"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<sim>\<^sup>s P"
lambda_system_eq: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "algebra \<Omega> M" shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x\<in>M. f (x \<inter> l) + f (x - l) = f x}"
quasi_isometry_onD_4: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_on X f" shows "0 \<le> C"
floor_eq_iff: fixes x :: "'a"   and a :: "int" shows "(\<lfloor>x\<rfloor> = a) = (of_int a \<le> x \<and> x < of_int a + (1::'a))"
nn_integral_bind: fixes f :: "'b \<Rightarrow> ennreal"   and B :: "'b measure"   and N :: "'c \<Rightarrow> 'b measure"   and M :: "'c measure" assumes "f \<in> borel_measurable B"   and "N \<in> M \<rightarrow>\<^sub>M subprob_algebra B" shows "integral\<^sup>N (M \<bind> N) f = \<integral>\<^sup>+ x. integral\<^sup>N (N x) f \<partial>M"
mono_Mprefix: fixes A :: "'b set" shows "monofun (Mprefix A)"
sum_divisors_coprime_mult: fixes a :: "nat"   and b :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "coprime a b" shows "(\<Sum>d | d dvd a * b. f d) = (\<Sum>r | r dvd a. \<Sum>s | s dvd b. f (r * s))"
subst_no_occs: fixes v :: "'b"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" assumes "Var v \<notin> subterms t" shows "t \<cdot> Var(v := s) = t"
check_nsc_ok: fixes nsc :: "('i \<times> 'a ns_constraint) list"   and J :: "'i set"   and s :: "'s"   and s' :: "'s" assumes "invariant_nsc nsc J s"   and "check_nsc s = Inr s'" shows "checked_nsc nsc J s'"
AC_def: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "a \<Squnion> b \<equiv> \<lambda>ab Q s. max (a ab Q s) (b ab Q s)"
to_nat_on_surj: fixes A :: "'c set"   and n :: "nat" assumes "countable A"   and "infinite A" shows "\<exists>a\<in>A. to_nat_on A a = n"
execute_parallel_operator_no_effect_if: fixes ops :: "'b strips_operator list"   and v :: "'b"   and s :: "'b \<Rightarrow> bool option" assumes "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "execute_parallel_operator s ops v = s v"
blaschke_0_id: shows "blaschke 0 = id_moebius"
readable: shows "\<forall>asset\<in>stocks Mkt. borel_adapt_stoch_proc G (prices Mkt asset)"
A2: shows "nat_to_sch (sch_to_nat sch2) = sch2"
ev_introR: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "\<phi> \<pi>l" shows "ev \<phi> \<pi>l"
mod_less_divisor: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "m mod n < n"
rec_ci_SucSuc_n: fixes g :: "recf"   and gap :: "abc_inst list"   and gar :: "nat"   and gft :: "nat"   and x :: "nat"   and xs :: "nat list"   and n :: "nat"   and f :: "recf"   and y :: "nat" assumes "rec_ci g = (gap, gar, gft)"   and "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"   and "length xs = n"   and "Suc y \<le> x" shows "gar = Suc (Suc n)"
mono_def: fixes f :: "'a \<Rightarrow> 'd" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
ideal_eq_right_ideal: fixes I :: "'a set" shows "ideal I = right_ideal I"
interior_greaterThanLessThan_real: fixes a :: "real"   and b :: "real" shows "interior {a<..<b} = {a<..<b}"
eventually_nhds_x_imp_x: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "class.topological_space open"   and "eventually P (topological_space.nhds open x)" shows "P x"
inf_compl_bot_left2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<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 x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (inf (uminus x) y) = bot"
infs_all: fixes w :: "'b stream"   and P :: "'b \<Rightarrow> bool" assumes "Ball (sset w) P" shows "infs P w"
ls_copy_map_of: fixes ls :: "('a \<times> 'b) list list"   and ld :: "('a \<times> 'b) list list"   and n :: "nat" assumes "ht_hash ls"   and "ht_distinct ls"   and "ht_hash ld"   and "ht_distinct ld"   and "1 < length ld"   and "n \<le> length ls" shows "map_of (concat (ls_copy n ls ld)) = map_of (concat ld) ++ map_of (concat (take n ls))"
open_iff: fixes t :: "'a set" shows "(t open) = (t = {} \<or> t = carrier)"
abs_eq: fixes x :: "'d" shows "blinfun_inner_left x = Blinfun (\<lambda>y. y \<bullet> x)"
List_hd_not_in_tl: fixes h :: "'c \<Rightarrow> 'c"   and a :: "'c"   and as :: "'c list" assumes "List h (h a) as" shows "a \<notin> set as"
to_nat_the_solve_consistent_rref: fixes A :: "(('d, 'e) vec, 'f) vec"   and j :: "'e" assumes "reduced_row_echelon_form A"   and "\<exists>i. A $ i $ j = (1::'d) \<and> j = (LEAST n. A $ i $ n \<noteq> (0::'d))" shows "mod_type_class.to_nat (THE i. A $ i $ j = (1::'d)) = least_non_zero_position_of_vector (column_iarray (mod_type_class.to_nat j) (matrix_to_iarray A))"