hcju/isabelleps huggingface dataset:
insert_into_member_list_equivalence: fixes Sets :: "'b set list"   and new_el :: "'b"   and S :: "'b set" assumes "distinct Sets" shows "set (insert_into_member_list new_el Sets S) = insert_into_member new_el (set Sets) S"
inst_ifp_refl: shows "\<forall>u. rifp u u"
valid_weak_tau_modality_iff_tau_steps: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and x :: "('a, 'd, 'b) formula" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "nominal_ts.valid satisfies transition P (indexed_weak_nominal_ts.weak_tau_modality \<tau> x) = (\<exists>n. nominal_ts.valid satisfies transition P (indexed_weak_nominal_ts.tau_steps \<tau> x n))"
loop_stepE: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "B\<^sub>1 \<noteq> {}"   and "u \<in> V \<inter> S"   and "real c < W B\<^sub>1 + w u"   and "real c < W B\<^sub>2 + w u" shows "inv\<^sub>3 (P\<^sub>1 \<union> wrap B\<^sub>1) (P\<^sub>2 \<union> wrap B\<^sub>2) {} {u} (V - {u})"
asubGroup_def: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set" shows "A +> H = b_ag A \<guillemotright> H "
I_lift_nnf_qe: fixes qe :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "'a list" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)"   and "\<And>xs \<phi>. nqfree \<phi> \<Longrightarrow> DLO.I (qe \<phi>) xs = (\<exists>x. DLO.I \<phi> (x # xs))" shows "DLO.I (DLO.lift_nnf_qe qe \<phi>) xs = DLO.I \<phi> xs"
ennreal_inverse_eq_top_iff: fixes a :: "ennreal" shows "(inverse a = \<top>) = (a = 0)"
between_nextElem: fixes vs :: "'a list"   and u :: "'a"   and v :: "'a" assumes "pre_between vs u v" shows "between vs u (nextElem vs (hd vs) v) = between vs u v @ [v]"
int_s: shows "0 < s \<and> s \<le> t"
to_nat_mono': fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less_eq a b" shows "mod_type.to_nat Rep a \<le> mod_type.to_nat Rep b"
eventually_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<diamond>F \<equiv> \<not> \<box>\<not> F"
clamp_in_interval: fixes a :: "'b"   and b :: "'b"   and x :: "'b" assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" shows "clamp a b x \<in> cbox a b"
add_right: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'d"   and b :: "'e"   and b' :: "'e" assumes "bounded_sesquilinear prod" shows "prod a (b + b') = prod a b + prod a b'"
llast_def: shows "llast \<equiv> \<mu> llast. \<Lambda> v. Fixrec.run\<cdot>(match_lcons\<cdot>v\<cdot>(\<Lambda> x yys. Fixrec.succeed\<cdot>(case yys of lnil \<Rightarrow> x | y :@ ys \<Rightarrow> llast\<cdot>yys)))"
alpha_Tree_rel_relcomp_trivialI: fixes x :: "('b, 'c, 'd) Tree"   and y :: "'e"   and R :: "(('b, 'c, 'd) Tree \<times> 'e) set" assumes "(x, y) \<in> R" shows "(x, y) \<in> Validity.alpha_Tree_rel O R"
xy: fixes x :: "real"   and y :: "real" assumes "x < y"   and "{r \<in> \<rat>. r < x} = {r \<in> \<rat>. r < y}" shows "x = y"
existence_ivl_initial_time: fixes x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "t0 \<in> existence_ivl t0 x0"
A5: shows "\<lfloor>\<P> (\<lambda>x w. \<forall>xa. xa x w \<and> (\<forall>xb. (xb x \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<forall>x. (existsAt x \<^bold>\<rightarrow> xa x \<^bold>\<rightarrow> xb x) v)) w) \<longrightarrow> (\<^bold>\<box>mexistsActB xa) w)\<rfloor>"
invar_inserts_tries: fixes t :: "('c, 'd list) trie"   and key :: "'d \<Rightarrow> 'c list"   and xs :: "'d list" assumes "invar_trie t" shows "invar_trie (inserts_tries key xs t)"
map_converter_def: fixes a :: "'u \<Rightarrow> 'q"   and b :: "'r \<Rightarrow> 'v"   and out :: "'s \<Rightarrow> 'w"   and inn :: "'x \<Rightarrow> 't" shows "map_converter a b out inn \<equiv> corec_converter (\<lambda>conv. map_gpv (map_prod b Inr) out \<circ> map_gpv' id id inn \<circ> run_converter conv \<circ> a)"
cond_unit_F: fixes P :: "('a, 'b) uexpr"   and Q :: "('a, 'b) uexpr" shows "P \<triangleleft> false \<triangleright> Q = Q"
MergeGuardsI_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P merge_guards c Q,A"
rd_assign_is_action: fixes s :: "'d" shows "R (true \<turnstile> \<lambda>(A, A'). ref A' = ref A \<and> tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = s) \<in> {p. is_CSP_process p}"
lex_ord_pair_EqD: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option"   and k :: "'a" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "lex_ord_pair_tc f xs ys = Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "f k (lookup_pair_tc xs k) (lookup_pair_tc ys k) = Some Eq"
CFG_equal_Use_equal_call: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r' :: "'ret"   and p' :: "'pname"   and fs' :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p'\<^esub>fs'"   and "sourcenode a = sourcenode a'"   and "pred (kind a) s"   and "pred (kind a') s'"   and "snd (hd s) = snd (hd s')"   and "length s = length s'"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V" shows "a = a'"
measure_def: fixes M :: "'b measure"   and A :: "'b set" shows "Sigma_Algebra.measure M A = enn2real (emeasure M A)"
Inf_lower: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and x :: "'g"   and A :: "'g set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq (Inf A) x"
1: shows "B - rat_poly_times (rat_poly_times A A) A = B - A ^ 3"
properties_for_sort_key: fixes ys :: "'c list"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'a" assumes "mset ys = mset xs"   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"   and "sorted (map f ys)" shows "sort_key f xs = ys"
fds_eqI: fixes f :: "'a fds"   and g :: "'a fds" assumes "\<And>n. 0 < n \<Longrightarrow> fds_nth f n = fds_nth g n" shows "f = g"
of_nat_single: shows "of_nat = single (0::'b) \<circ> of_nat"
101: fixes v :: "nat \<times> 'a" shows "finite (greachable A w (graph A w (2 * card (nodes A))) v)"
update_unsat_id: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and c :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<U> (update x c s) = \<U> s"
half_chamber_system_disjoint_union_2: shows "f \<turnstile> folding_g.\<C> \<inter> (g \<turnstile> folding_g.\<C>) = {}"
exec_appendL_if: fixes P' :: "instr list"   and i :: "int"   and P :: "instr list"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and j :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list"   and i' :: "int" assumes "size P' \<le> i"   and "P \<turnstile> (i - size P', s, stk) \<rightarrow>* (j, s', stk')"   and "i' = size P' + j" shows "P' @ P \<turnstile> (i, s, stk) \<rightarrow>* (i', s', stk')"
odd_even_cases: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "\<And>n. P (Suc (2 * n))"   and "\<And>n. P (2 * n + 2)" shows "P n"
rewriteR_comp_comp: fixes g :: "'c \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'c"   and r :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd" assumes "g \<circ> h = r" shows "f \<circ> g \<circ> h = f \<circ> r"
addN'_conv: fixes ga :: "'g"   and n :: "'node"   and m :: "'var \<Rightarrow> 'node set option" shows "addN' ga n m = (\<lambda>v. case m v of None \<Rightarrow> if v \<in> set (sorted_list_of_set (uses ga n)) then Some {n} else None | Some N \<Rightarrow> if v \<in> set (sorted_list_of_set (uses ga n)) then Some (insert n N) else Some N)"
monotone_comp: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and g :: "'f \<Rightarrow> 'g"   and ordc :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" assumes "monotone orda ordb g"   and "monotone ordb ordc f" shows "monotone orda ordc (f \<circ> g)"
add_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ab_semigroup_add plus" shows "plus a b = plus b a"
cases: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" assumes "element_type a1 a2" obtains \<tau> :: "'a OCL_Types.type" where "a1 = Collection \<tau>__" and "a2 = \<tau>__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Set \<tau>'__" and "a2 = \<tau>'__"   | \<tau> :: "'a OCL_Types.type" where "a1 = OrderedSet \<tau>''__" and "a2 = \<tau>''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Bag \<tau>'''__" and "a2 = \<tau>'''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Sequence \<tau>" and "a2 = \<tau>"
monoE: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr"   and f' :: "('a, 'a set) Maps.arr" assumes "Maps.mono g"   and "Maps.seq g f"   and "Maps.seq g f'"   and "g \<odot> f = g \<odot> f'" shows "f' = f"
assms_3: shows "fst (rep_nat_term u) = 0"
flatten_sigma_1_sf_subset: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "sf (\<phi>[M]\<^sub>\<Sigma>\<^sub>1) \<subseteq> (\<Union>\<phi>\<in>sf \<phi>. sf (\<phi>[M]\<^sub>\<Sigma>\<^sub>1))"
val_stuck: fixes e :: "exp" assumes "isval e" shows "stuck e"
DBM_val_bounded_neg_cycle: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and i :: "nat"   and xs :: "nat list" assumes "u \<turnstile>\<^bsub>v,n\<^esub> M"   and "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "len M i i xs < \<one>"   and "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)" shows "False"
exec_gpv_oracle_bisim_bad: fixes bad2 :: "'a \<Rightarrow> bool"   and s2 :: "'a"   and X_bad :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s1 :: "'b"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and bad1 :: "'b \<Rightarrow> bool"   and oracle1 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and oracle2 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('e, 'c, 'd) gpv"   and R :: "'e \<times> 'b \<Rightarrow> 'e \<times> 'a \<Rightarrow> bool" assumes "if bad2 s2 then X_bad s1 s2 else X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>s2. bad2 s2 \<Longrightarrow> callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"   and "\<And>s1. bad1 s1 \<Longrightarrow> callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"   and "\<And>s1 x. bad1 s1 \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. bad2 s2 \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "lossless_gpv \<I> gpv"   and "\<And>s1. \<I> \<turnstile>c oracle1 s1 \<surd>"   and "\<And>s2. \<I> \<turnstile>c oracle2 s2 \<surd>"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>a s1 b s2. \<lbrakk>bad1 s1 = bad2 s2; \<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2; bad2 s2 \<Longrightarrow> X_bad s1 s2\<rbrakk> \<Longrightarrow> R (a, s1) (b, s2)" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document get_parent get_parent_locs remove_child remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs adopt_node adopt_node_locs DocumentClass.known_ptr DocumentClass.type_wf get_child_nodes get_child_nodes_locs DocumentClass.known_ptrs set_child_nodes set_child_nodes_locs remove"
pullback_simplex_like: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "finite P"   and "ordering.simplex_like less_eq' less' (f ` P)" shows "ordering.simplex_like less_eq less P"
domintros: fixes data :: "nat \<times> nat \<times> 'd"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "\<And>ac ad bb ae bc af ag bd xe ah be. \<lbrakk>data = (ac, ad, bb); ps \<noteq> []; (ae, bc) = compl gs bs (ps -- sel gs bs ps (ad, bb)) (sel gs bs ps (ad, bb)) (ad, bb); \<not> ac \<le> count_const_lt_components (fst (compl gs bs (ps -- sel gs bs ps (ad, bb)) (sel gs bs ps (ad, bb)) (ad, bb))); (af, ag, bd) = add_indices (compl gs bs (ps -- sel gs bs ps (ad, bb)) (sel gs bs ps (ad, bb)) (ad, bb)) (ad, bb); (xe, ah, be) = add_indices (compl gs bs (ps -- sel gs bs ps (ad, bb)) (sel gs bs ps (ad, bb)) (ad, bb)) (ad, bb)\<rbrakk> \<Longrightarrow> gb_schema_aux_dom ((ac - count_const_lt_components (fst (compl gs bs (ps -- sel gs bs ps (ad, bb)) (sel gs bs ps (ad, bb)) (ad, bb))), ah, be), ab gs bs xe (ah, be), ap gs bs (ps -- sel gs bs ps (ad, bb)) xe (ah, be))" shows "gb_schema_aux_dom (data, bs, ps)"
empty: fixes g :: "'d \<Rightarrow> 'a" shows "F g {} = \<^bold>1"
augment_with_path_def: fixes p :: "(nat \<times> nat) list" shows "f.augment_with_path p \<equiv> f.augment (f.augmentingFlow p)"
simps_4: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "nqfree (And \<phi>\<^sub>1 \<phi>\<^sub>2) = (nqfree \<phi>\<^sub>1 \<and> nqfree \<phi>\<^sub>2)"
commute: fixes a :: "'y"   and b :: "'y" shows "a + b = b + a"
is_midpoint_id: fixes A :: "'p"   and B :: "'p" assumes "A Midpoint A B" shows "A = B"
wf\<^sub>s\<^sub>s\<^sub>t_vars_mono: fixes V :: "'c set"   and S :: "('d, 'c) stateful_strand_step list"   and W :: "'c set" assumes "wf'\<^sub>s\<^sub>s\<^sub>t V S" shows "wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> W) S"
coboundedI2: fixes c :: "'c"   and b :: "'c"   and a :: "'c" assumes "c \<le> b" shows "c \<le> a + b"
B: shows "\<exists>u\<in>sinks_aux I D U (xs @ [x]). \<exists>v\<in>V. (u, v) \<in> I"
fSup_distr_var: fixes F :: "('c \<Rightarrow> 'b) set"   and g :: "'a \<Rightarrow> 'c" shows "\<Squnion> F \<circ> g = (\<Squnion>f\<in>F. f \<circ> g)"
apply_rsp': fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and x :: "'d"   and y :: "'e" assumes "rel_fun R1 R2 f g"   and "R1 x y" shows "R2 (f x) (g y)"
duh: fixes A :: "'a set"   and B :: "'a set" assumes "A \<inter> B = {}" shows "A \<subseteq> - B"
HSem_restr_cong: fixes h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> 'Value"   and \<rho>' :: "Vars.var \<Rightarrow> 'Value" assumes "\<rho> f|` (- domA h) = \<rho>' f|` (- domA h)" shows "\<lbrace>h\<rbrace>\<rho> = \<lbrace>h\<rbrace>\<rho>'"
UnifierD_1: fixes f :: "'a"   and X :: "('a, 'b) Term.term list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and g :: "'a"   and Y :: "('a, 'b) Term.term list" assumes "Unifier \<sigma> (Fun f X) (Fun g Y)" shows "f = g"
\<R>\<^sub>3_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and 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" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "\<forall>x. (dma x = High \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v\<^sub>1 v\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v\<^sub>1)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v\<^sub>2)\<rangle>) \<in> \<R>\<^sub>3 \<Gamma>')) \<and> (dma x = Low \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v)\<rangle>) \<in> \<R>\<^sub>3 \<Gamma>'))"
UNlift_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and II :: "nat set" shows "UNlift cl dl P F II \<equiv> local.shift dl (pickT cl P II) (F (pickT cl P II) (back cl (pickT cl P II) II))"
Issues_def: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" shows "(A Issues B with X on evs) = (\<exists>Y. Says A B Y \<in> set evs \<and> X \<in> parts {Y} \<and> X \<notin> parts (knows Spy (takeWhile (\<lambda>z. z \<noteq> Says A B Y) (rev evs))))"
crel_vs_bind_ignore: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "('mem, 'e) state"   and S :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'f"   and d :: "('mem, 'g) state" assumes "crel_vs R a b"   and "crel_vs S c d" shows "crel_vs R a (d \<bind> (\<lambda>_. b))"
nn: fixes x :: "'b" assumes "x \<in> space MX" shows "0 \<le> f x"
id_act_1: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P \<^bold>\<equiv> \<^bold>\<A>(x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
simps_13: fixes x21 :: "char list"   and x22 :: "aexp"   and x31 :: "acom"   and x32 :: "acom" shows "x21 ::= x22 \<noteq> x31;; x32"
pchoose_assoc: fixes p :: "real"   and r :: "real"   and s :: "real"   and q :: "real"   and m :: "'m"   and m' :: "'m"   and m'' :: "'m" assumes "min 1 (max 0 p) = min 1 (max 0 r) * min 1 (max 0 s)"   and "1 - min 1 (max 0 s) = (1 - min 1 (max 0 p)) * (1 - min 1 (max 0 q))" shows "m \<lhd> p \<rhd> (m' \<lhd> q \<rhd> m'') = (m \<lhd> r \<rhd> m') \<lhd> s \<rhd> m''"
sumhr_split_add: fixes n :: "nat star"   and p :: "nat star"   and f :: "nat \<Rightarrow> real" assumes "n < p" shows "sumhr (0, n, f) + sumhr (n, p, f) = sumhr (0, p, f)"
induct: fixes P :: "'c Tree \<Rightarrow> bool"   and a0 :: "'c Tree" assumes "P E"   and "\<And>v. P (T v E E)"   and "\<And>v va vb vc. (\<not> val (T va vb vc) \<le> v \<Longrightarrow> P (set_val (T va vb vc) v)) \<Longrightarrow> P (T v (T va vb vc) E)"   and "\<And>v va vb vc. (\<not> val (T va vb vc) \<le> v \<Longrightarrow> P (set_val (T va vb vc) v)) \<Longrightarrow> P (T v E (T va vb vc))"   and "\<And>v va vb vc vd ve vf. \<lbrakk>\<lbrakk>val (T vd ve vf) \<le> val (T va vb vc); \<not> val (T va vb vc) \<le> v\<rbrakk> \<Longrightarrow> P (set_val (T va vb vc) v); \<lbrakk>\<not> val (T vd ve vf) \<le> val (T va vb vc); \<not> val (T vd ve vf) \<le> v\<rbrakk> \<Longrightarrow> P (set_val (T vd ve vf) v)\<rbrakk> \<Longrightarrow> P (T v (T va vb vc) (T vd ve vf))"   and "\<And>v vd ve vf va vb vc. \<lbrakk>\<lbrakk>val (T va vb vc) \<le> val (T vd ve vf); \<not> val (T vd ve vf) \<le> v\<rbrakk> \<Longrightarrow> P (set_val (T vd ve vf) v); \<lbrakk>\<not> val (T va vb vc) \<le> val (T vd ve vf); \<not> val (T va vb vc) \<le> v\<rbrakk> \<Longrightarrow> P (set_val (T va vb vc) v)\<rbrakk> \<Longrightarrow> P (T v (T vd ve vf) (T va vb vc))" shows "P a0"
nonpos_linorder_cases: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less_eq a zero"   and "less_eq b zero" obtains "less a b"   | "a = b"   | "less b a"
type_subst_rename: fixes Y :: "tyvrs"   and T :: "ty"   and X :: "tyvrs"   and U :: "ty" assumes "Y \<sharp> T" shows "([(Y, X)] \<bullet> T)[Y \<mapsto> U]\<^sub>\<tau> = T[X \<mapsto> U]\<^sub>\<tau>"
indexed_fmember_unique: fixes x :: "'b"   and i :: "nat"   and s :: "'b fset"   and y :: "'b"   and j :: "nat" assumes "x |\<in>|\<^bsub>i\<^esub> s"   and "y |\<in>|\<^bsub>j\<^esub> s" shows "(x = y) = (i = j)"
healthy_wp_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)" shows "healthy (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
resTrans_1: fixes x :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "<\<nu>x>x{b}.P \<longmapsto> Rs" shows "False"
assms: shows "Act \<alpha>1 x1 = Act \<alpha>2 x2"
verts3_subdivide: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b" assumes "pair_sym_digraph G"   and "e \<in> parcs G"   and "w \<notin> pverts G" shows "verts3 (with_proj (subdivide G e w)) = verts3 (with_proj G)"
elims: fixes x :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and y :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat" assumes "eval_step_mmuaux x = y" obtains tp :: "nat" and tss :: "nat queue" and len :: "nat" and maskL :: "bool list" and maskR :: "bool list" and a1_map :: "('a option list, nat) mapping" and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping" and "done" :: "'a option list set list" and done_length :: "nat" where "x = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)" and "y = (case safe_hd tss of (Some ts, tss') \<Rightarrow> case Mapping.lookup a2_map (tp - len) of Some m \<Rightarrow> let m = Mapping.filter (\<lambda>_. ts_tp_lt' ts (tp - len)) m; T = Mapping.keys m; a2_map = Mapping.update (tp - len + 1) (case Mapping.lookup a2_map (tp - len + 1) of Some x \<Rightarrow> Mapping.combine max_tstp m x) a2_map; a2_map = Mapping.delete (tp - len) a2_map in (tp, tl_queue tss', len - 1, maskL, maskR, a1_map, a2_map, T # done, done_length + 1))"
0: shows "smap fst (konigDown t S) = srs @- steps"