hcju/isabelleps huggingface dataset:
suc_singl_pred: fixes a :: "'a"   and a' :: "'a" assumes "a \<in> Field r"   and "local.aboveS a \<noteq> {}"   and "(a', local.suc {a}) \<in> r"   and "a' \<noteq> local.suc {a}" shows "a' = a \<or> (a', a) \<in> r"
table_of_mapkey_SomeD: fixes C :: "'c"   and t :: "('b \<times> 'a) list"   and k :: "'b"   and D :: "'c"   and x :: "'a" assumes "table_of (map (\<lambda>(k, x). ((k, C), x)) t) (k, D) = Some x" shows "C = D \<and> table_of t k = Some x"
distinct_signs_hyp: shows "distinct (snd (snd (reduce_system p (qs, M_mat signs subsets, subsets, signs))))"
fmrel_fmdom_eq: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('c, 'a) fmap"   and y :: "('c, 'b) fmap" assumes "fmrel P x y" shows "fmdom x = fmdom y"
ignores_old_varsD: fixes P :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool"   and mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list"   and mdss' :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" assumes "ignores_old_vars P"   and "length mdss = length mdss'"   and "length mdss' = length cms"   and "map (\<lambda>x m. x m \<inter> - range var\<^sub>C_of) mdss = map (\<lambda>x m. x m \<inter> - range var\<^sub>C_of) mdss'" shows "P mdss = P mdss'"
lq_rev_rq: fixes v :: "'b list"   and u :: "'b list" shows "rev v\<inverse>\<^sup>>rev u = rev (u\<^sup><\<inverse>v )"
isUbD: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "isUb R S x"   and "y \<in> S" shows "y \<le> x"
timedelayed_e2: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<delta>\<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and K\<^sub>3 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "digraph G"   and "pseudo_graph G" shows "graph G"
take_all_iff: fixes n :: "nat"   and xs :: "'c list" shows "(take n xs = xs) = (length xs \<le> n)"
trace_callee_eq_def: fixes callee1 :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and callee2 :: "'h \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'h) spmf"   and A :: "'f set"   and p :: "'e spmf"   and q :: "'h spmf" shows "trace_callee_eq callee1 callee2 A p q = (\<forall>xs. set xs \<subseteq> A \<times> UNIV \<longrightarrow> (\<forall>x\<in>A. trace_callee callee1 p xs x = trace_callee callee2 q xs x))"
part_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n)" shows "part (BrnNFT cl dl) (UNpart1 cl dl P)"
not_white_marked: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "\<not> white r s"   and "obj_at (\<lambda>s. True) r s" shows "marked r s"
image_empty: fixes f :: "'d \<Rightarrow> 'c" shows "bot_pres (\<P> f)"
prob_subset_dom_subset: fixes PROB1 :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and PROB2 :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "PROB1 \<subseteq> PROB2" shows "prob_dom PROB1 \<subseteq> prob_dom PROB2"
cr_envT_stateTD: fixes s :: "'f"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m1 :: "('f, 'g) envT"   and m2 :: "('f, 'h) stateT" assumes "cr_envT_stateT s M m1 m2" shows "M (run_env m1 s) (run_state m2 s)"
UniqueSucState_Target_ChiPlus_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and T :: "'d"   and U :: "'d"   and V :: "'d"   and W :: "'d" assumes "TS \<in> HPT ST"   and "(S, T) \<in> ChiRel (HA ST)"   and "(S, U) \<in> ChiRel (HA ST)"   and "V \<in> Target TS"   and "(V, W) \<in> ChiRel (HA ST)"   and "T \<notin> ChiStar (HA ST) `` Source TS"   and "(W, U) \<in> (Restr (ChiRel (HA ST)) (HAInitStates (HA ST)))\<^sup>+"   and "T \<in> Conf ST"   and "(S, U) \<in> ChiRel (HA ST)" shows "T = U"
pivot_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and s :: "('a, 'b) state" shows "pivot_code x\<^sub>i x\<^sub>j s \<equiv> \<T>_update (pivot_tableau_code x\<^sub>i x\<^sub>j (\<T> s)) s"
first_pd_no_icd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> n pd\<rightarrow> \<pi> 0"   and "\<forall>l<n. \<pi> l \<noteq> \<pi> n" shows "\<forall>l. \<not> n icd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
a_export'': fixes x :: "'a"   and y :: "'a" shows "ad (ad x \<cdot> y) = d x + ad y"
ObjY: shows "Y \<in> obj\<^bsub>C\<^esub>"
P_conforms: shows "path_conforms_with_strategy p P (override_on \<sigma>' \<sigma> (A - W))"
rquot_accepting: fixes q :: "nat" assumes "dfa_is_node (rquot A n) q" shows "dfa_accepting (rquot A n) q = (\<exists>m. dfa_accepting A (dfa_steps A q (zeros m n)))"
topspace_def: fixes T :: "'b topology" shows "topspace T = \<Union> {S. openin T S}"
exclude_one_is_pseudominimal_in_below_in: fixes w :: "'w"   and T :: "'w set"   and s :: "'w" assumes "w \<in> W"   and "T \<in> Pow S"   and "s \<in> S - T" shows "dual_order.pseudominimal_in (dual_order.below_in \<P> (w +o \<langle>T\<rangle>)) (w +o \<langle>S - {s}\<rangle>)"
sccs_verts_conv_scc_of: shows "sccs_verts = scc_of ` pverts (list_digraph G)"
ancestors_h2: shows "h2 \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors_h2"
ls: shows "ls = take n ls @ [lls] @ drop (Suc n) ls"
CX5h_irc: fixes h :: "H2" shows "irc (CX5h h)"
16: shows "irreflexive (prim_P w v e\<^sup>T * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
test_absorb_1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot" shows "sup x (inf x y) = x"
alternate_Suc: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and k :: "nat" shows "alternate f g (Suc k) = (if even k then f else g) \<circ> alternate f g k"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and f3 :: "'a"   and agent :: "agent" shows "P (case agent of Server \<Rightarrow> f1 | Friend x \<Rightarrow> f2 x | Spy \<Rightarrow> f3) = ((agent = Server \<longrightarrow> P f1) \<and> (\<forall>x2. agent = Friend x2 \<longrightarrow> P (f2 x2)) \<and> (agent = Spy \<longrightarrow> P f3))"
rel_fp64_def: shows "rel_fp64 \<equiv> \<lambda>x y. x = float_of_fp64 y"
WHILEI_body_trimono: fixes I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" shows "trimono (WHILEI_body (\<bind>) RETURN I b f)"
b4: fixes a :: "name"   and s2 :: "lam"   and s1 :: "lam" assumes "a \<sharp> s2" shows " App (Lam [a].s1) s2 \<longrightarrow>\<^sub>\<beta> s1[a::=s2]"
rs: shows "(\<sigma>, s) \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) (otherwith S {i} (oarrivemsg I)) (other U {i})"
rev_exhaust2: fixes xs :: "'a list" obtains "xs = []"   | ys :: "'a list" and y :: "'a" where "xs = ys @ [y]"
vimage_vconst_on_vempty: fixes A :: "V"   and B :: "V"   and c :: "V" assumes "vdisjnt A B" shows "vconst_on A c `\<^sub>\<circ> B = 0"
BinOp: fixes Env :: "env"   and B :: "lname set"   and e1 :: "expr"   and E1 :: "assigned"   and e2 :: "expr"   and A :: "assigned"   and binop :: "binop" assumes "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1"   and "Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A"   and "binop \<noteq> CondAnd"   and "binop \<noteq> CondOr" shows "Env\<turnstile> B \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<guillemotright> A"
rep_eq: fixes x :: "'b ttree"   and xa :: "'b ttree" shows "Rep_ttree (x \<otimes>\<otimes> xa) = \<Union> {xs \<otimes> ys |xs ys. xs \<in> Rep_ttree x \<and> ys \<in> Rep_ttree xa}"
inv: shows "ChamberComplexIsomorphism Y X (the_inv_into (\<Union> X) f)"
assms_1: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<diamond>Q) s)) \<sigma>"
list_adds: fixes t :: "'b tree"   and n :: "nat"   and xs :: "'b list" assumes "braun t"   and "size t = n" shows "list (adds xs n t) = list t @ xs"
nub_body'''_def: shows "nub_body''' \<equiv> \<Lambda> f r. case nextR\<cdot>r of Nothing \<Rightarrow> lnil | Just\<cdot>(x, xs) \<Rightarrow> x :@ f\<cdot>(filterR\<cdot>x\<cdot>xs)"
img_is_subalgebra: fixes K :: "'a list set"   and V :: "'a list set" assumes "K \<subseteq> carrier (K [X])"   and "subalgebra K V (K [X])" shows "subalgebra (rupture_surj A P ` K) (rupture_surj A P ` V) (Rupt A P)"
rk2_consistent_traj_set: fixes h :: "real"   and t :: "real"   and u :: "real"   and T :: "real set"   and X :: "'e set"   and x :: "real \<Rightarrow> 'e"   and f :: "real \<times> 'e \<Rightarrow> 'e"   and g' :: "real \<times> 'e \<Rightarrow> real \<times> 'e \<Rightarrow> 'e"   and f' :: "real \<times> 'e \<Rightarrow> (real \<times> 'e) \<Rightarrow>\<^sub>L 'e"   and g'' :: "real \<times> 'e \<Rightarrow> real \<times> 'e \<Rightarrow> (real \<times> 'e) \<Rightarrow>\<^sub>L 'e"   and f'' :: "real \<times> 'e \<Rightarrow> (real \<times> 'e) \<Rightarrow>\<^sub>L (real \<times> 'e) \<Rightarrow>\<^sub>L 'e"   and p :: "real"   and R :: "'e set" assumes "0 \<le> h"   and "t + h \<le> u"   and "{t..u} \<subseteq> T"   and "X \<noteq> {}"   and "convex X"   and "\<And>s. s \<in> {t..u} \<Longrightarrow> (x has_vector_derivative f (s, x s)) (at s within {t..u})"   and "\<And>tx. tx \<in> T \<times> X \<Longrightarrow> (f has_derivative g' tx) (at tx)"   and "\<And>tx. tx \<in> T \<times> X \<Longrightarrow> (f' has_derivative g'' tx) (at tx)"   and "\<And>tx. tx \<in> T \<times> X \<Longrightarrow> g' tx = blinfun_apply (f' tx)"   and "\<And>tx. tx \<in> T \<times> X \<Longrightarrow> g'' tx = blinfun_apply (f'' tx)"   and "bounded (f'' ` (T \<times> X))"   and "\<And>s. s \<in> {t..u} \<Longrightarrow> x s \<in> X"   and "0 < p"   and "p \<le> 1"   and "x t + (h * p) *\<^sub>R f (t, x t) \<in> X"   and "convex R"   and "closed R"   and "\<And>s1 s2. \<lbrakk>0 \<le> s1; s1 \<le> 1; 0 \<le> s2; s2 \<le> 1\<rbrakk> \<Longrightarrow> discrete_evolution (rk2_increment p (\<lambda>t x. f (t, x))) (t + h) t (x t) + heun_remainder1 x f f' f'' t h s1 - heun_remainder2 p x f f'' t h s2 \<in> R" shows "x (t + h) \<in> R"
simps_1: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat" shows "Graph.isPath c u [] v = (u = v)"
assms: shows "[a, a]\<^sub>\<circ> \<in>\<^sub>\<circ> fid_on A"
c2dc_principal: shows "c2dc (principal_char G) = principal_dchar n"
p\<^sub>1_simps: shows "trg t\<^sub>0u\<^sub>1.p\<^sub>1 = src t\<^sub>0"
distinct_1: fixes x21 :: "color"   and x22 :: "('b, 'c) rbt"   and x23 :: "'b"   and x24 :: "'c"   and x25 :: "('b, 'c) rbt" shows "rbt.Empty \<noteq> Branch x21 x22 x23 x24 x25"
leg1_simps_3: shows "src (tab\<^sub>1 s) = src (tab\<^sub>0 s)"
dvd_GcdD: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "x dvd Gcd A"   and "y \<in> A" shows "x dvd y"
filter_insort_triv: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "\<not> P x" shows "filter P (linorder.insort_key less_eq f x xs) = filter P xs"
greek_less_singleton: fixes a :: "accent \<times> 'a"   and b :: "accent \<times> 'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> letter_less r" shows "([a], [b]) \<in> greek_less r"
monom_pow: fixes v :: "nat"   and n0 :: "nat"   and a :: "'b"   and n :: "nat" shows "monom (Poly_Mapping.single v n0) a ^ n = monom (Poly_Mapping.single v (n0 * n)) (a ^ n)"
elims: fixes x :: "'a ifex"   and xa :: "'a"   and xb :: "bool"   and y :: "'a ifex" assumes "restrict x xa xb = y" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" and var :: "'a" and val :: "bool" where "x = IF v t e" and "xa = var" and "xb = val" and "y = (let rt = restrict t var val; re = restrict e var val in if v = var then if val then rt else re else IF v rt re)"   | uu_ :: "'a" and uv_ :: "bool" where "x = Trueif" and "xa = uu___" and "xb = uv___" and "y = Trueif"   | uu_ :: "'a" and uv_ :: "bool" where "x = Falseif" and "xa = uu_" and "xb = uv_" and "y = Falseif"
locked_in_different: fixes s :: "v_state"   and r1 :: "nat"   and v :: "val"   and r2 :: "nat"   and w :: "val" assumes "s \<in> Vinv2"   and "locked_in s r1 v"   and "locked_in s r2 w"   and "r1 < r2" shows "v = w"
le_cINF_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and u :: "'c multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_below (f ` A)" shows "(u \<subseteq># Inf (f ` A)) = (\<forall>x\<in>A. u \<subseteq># f x)"
downclosed_set_iff: fixes X :: "'a set" shows "downclosed_set X = (\<forall>x\<in>X. \<forall>y\<le>x. y \<in> X)"
carrier_is_subcring: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "subcring (carrier R) R"
psen_en: fixes s :: "'state"   and p :: "'process" assumes "s \<in> nodes" shows "pac p \<inter> {a. en a s} \<subseteq> psen p s"
retract_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "retract G \<equiv> galois_connection G \<and> retract_axioms G"
topological_space_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool" shows "topological_space S is_open \<equiv> (is_open S \<and> is_open {}) \<and> (\<forall>U. is_open U \<longrightarrow> U \<subseteq> S) \<and> (\<forall>U V. is_open U \<longrightarrow> is_open V \<longrightarrow> is_open (U \<inter> V)) \<and> (\<forall>F. (\<forall>x. x \<in> F \<longrightarrow> is_open x) \<longrightarrow> is_open (\<Union>x\<in>F. x))"
map_lasso_def: fixes f :: "'e \<Rightarrow> 'd"   and L :: "('e, 'f) lasso_scheme" shows "map_lasso f L \<equiv> \<lparr>lasso_reach = map f (lasso_reach L), lasso_va = f (lasso_va L), lasso_cysfx = map f (lasso_cysfx L)\<rparr>"
dcache_mod_def: fixes c :: "word20 \<times> word12"   and val :: "word8"   and state :: "'b sparc_state" shows "dcache_mod c val state \<equiv> state\<lparr>cache := cache state\<lparr>dcache := dcache (cache state)(c \<mapsto> val)\<rparr>\<rparr>"
set_iterator_weaken_R: fixes iti :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R"   and "\<And>x y. \<lbrakk>x \<in> S0; y \<in> S0; R x y\<rbrakk> \<Longrightarrow> R' x y" shows "set_iterator_genord iti S0 R'"
0: shows "sum b (Outside' X a) \<le> sum b (Outside' (X \<union> {i}) a \<union> {i} \<times> {\<Union> (a `` (X \<union> {i}))})"
swap_id_idempotent: fixes a :: "'a"   and b :: "'a" shows "a \<rightleftharpoons>\<^sub>F b \<circ> a \<rightleftharpoons>\<^sub>F b = id"
summands_valid: fixes A :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "A \<subseteq> carrier_vec n" shows "\<forall>v\<in>A. a v \<cdot>\<^sub>v v \<in> carrier_vec n"
SUP_upper2: fixes A :: "'c set"   and U\<^sub>2 :: "'c set"   and u :: "'al"   and f :: "'c \<Rightarrow> 'al"   and i :: "'c" assumes "A \<subseteq> U\<^sub>2"   and "u \<in> U"   and "\<forall>x\<in>U\<^sub>2. f x \<in> U"   and "i \<in> A"   and "u \<le>\<^sub>o\<^sub>w f i" shows "u \<le>\<^sub>o\<^sub>w \<Squnion>\<^sub>o\<^sub>w (f ` A)"
abs2_strict_1: shows "abs2\<cdot>\<bottom> = \<bottom>"
sum_pos2: fixes I :: "'b set"   and i :: "'b"   and f :: "'b \<Rightarrow> 'c multiset" assumes "finite I"   and "i \<in> I"   and "{#} \<subset># f i"   and "\<And>i. i \<in> I \<Longrightarrow> {#} \<subseteq># f i" shows "{#} \<subset># comm_monoid_add.sum (+) {#} f I"
relpow_code_def: shows "Transitive_Closure.relpow = compow"
ldrop_def: shows "ldrop \<equiv> curry (llist.fixp_fun (\<lambda>ldrop (n, xs). case n of 0 \<Rightarrow> xs | eSuc n' \<Rightarrow> case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> curry ldrop n' xs'))"
assms_1: shows "P (nsqn\<^sub>r (dsn, dsk, val, hops, nhip))"
intro: fixes get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "get_dom_component = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component get_root_node to_tree_order"   and "is_strongly_dom_component_safe = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe get_root_node to_tree_order"   and "is_weakly_dom_component_safe = l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe get_root_node to_tree_order" shows "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node"
ts''_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [x] : ts'' _> ts'"
exp_suc: fixes b :: "nat"   and a :: "'a" shows "a \<up> Suc b = a \<up> b @ [a]"
dvd: fixes j :: "nat" assumes "2 \<le> j"   and "j < i" shows "\<not> j dvd n"
disc_2: fixes x21 :: "'a"   and x22 :: "'a llist" shows "\<not> lnull (LCons x21 x22)"
Rep_cpx_sqr_mat_inverse: fixes x :: "cpx_sqr_mat" shows "Abs_cpx_sqr_mat (Rep_cpx_sqr_mat x) = x"
card_of_lists: fixes A :: "'b set" shows "|A| \<le>o |lists A|"
red_rtranclp_0_in_pmdl: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* p 0" shows "p \<in> local.punit.pmdl F"
swap_lassocr_inverse: fixes \<I>1 :: "('h, 'k) \<I>"   and \<I>2 :: "('i, 'l) \<I>"   and \<I>3 :: "('j, 'm) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3),\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3) \<turnstile>\<^sub>C swap_lassocr \<odot> swap_lassocr \<sim> 1\<^sub>C"
L\<^sub>2_p: shows "L\<^sub>2 \<subseteq> C\<^sub>2 \<and> L\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<eta> = L\<^sub>2' \<and> C\<^sub>2 - L\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<eta> = C\<^sub>2' - L\<^sub>2'"
simps_10: fixes \<C> :: "t_context"   and t1 :: "t"   and t2 :: "t"   and sx :: "sx option"   and ts :: "checker_type" shows "check_single \<C> (Cvtop t1 Reinterpret t2 sx) ts = (if t1 \<noteq> t2 \<and> t_length t1 = t_length t2 \<and> sx = None then type_update ts [TSome t2] (Type [t1]) else Bot)"
d_wfc_equals_dwfc: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" assumes "deterministic P" shows "weakly_future_consistent P I D R = d_weakly_future_consistent P I D R"
compatAbsVsubst_def: fixes xs :: "'varSort"   and s :: "'c"   and f :: "'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set \<Rightarrow> 'g" shows "compatAbsVsubst xs s f \<equiv> \<forall>x X x' X'. (\<forall>y. X #[y // x]_xs = X' #[y // x']_xs) \<longrightarrow> f x X = f x' X'"
wf_ps_triple_def: fixes T :: "(node \<rightharpoonup>\<^sub>f topbot) \<times> (node + node fset \<times> command_gadget \<times> node fset) \<times> (node \<rightharpoonup>\<^sub>f topbot)" shows "wf_ps_triple T = (case snd3 T of Inl v \<Rightarrow> \<exists>\<sigma>. v |\<notin>| fmdom \<sigma> \<and> fst3 T = [ {|v|} |=> Top ] ++\<^sub>f \<sigma> \<and> thd3 T = [ {|v|} |=> Bot ] ++\<^sub>f \<sigma> | Inr e \<Rightarrow> \<exists>\<sigma>. fst3 e |\<union>| thd3 e |\<inter>| fmdom \<sigma> = {||} \<and> fst3 T = [ fst3 e |=> Bot ] ++\<^sub>f \<sigma> \<and> thd3 T = [ thd3 e |=> Top ] ++\<^sub>f \<sigma>)"
cod_char: fixes f :: "'c \<times> 'c \<times> 'c" shows "C.VVV.cod f = (if C.VVV.arr f then C.VxVxV.cod f else C.VxVxV.null)"
simps: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b" shows "r\<^sup>*\<^sup>* a1 a2 = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>*\<^sup>* a b \<and> r b c))"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<exists>!n. \<forall>f. C n f = n \<and> C f n = n" shows "partial_magma C"
NTDeta: shows "NTDom \<eta> = YFtor C @@ X"
intensional_simps_13: fixes P :: "'b \<Rightarrow> bool" shows "P \<longrightarrow> P \<equiv> #True"
sectionE: fixes f :: "'a" assumes "C.section f" obtains g :: "'a" where "C.ide (g \<cdot>\<^sub>C f)"
asymp_equiv_eventually_zeros: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. (f x = (0::'b)) = (g x = (0::'b))"
SLC_join: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r'' :: "'r"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and \<tau>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<sigma>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and v :: "('r, 'l, 'v) val"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r''))])"   and "s\<^sub>2 = (s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r'' := None)"   and "s\<^sub>1 r'' = Some (\<sigma>', \<tau>', VE v)"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
ring_iso_imp_img_ring: fixes h :: "'c \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "h \<in> ring_iso (S\<lparr>zero := h \<zero>\<rparr>) S" shows "ring (S\<lparr>zero := h \<zero>\<^bsub>S\<lparr>zero := h \<zero>\<rparr>\<^esub>\<rparr>)"
expand: fixes list :: "'a list"   and list' :: "'a list" assumes "(list = []) = (list' = [])"   and "\<lbrakk>list \<noteq> []; list' \<noteq> []\<rbrakk> \<Longrightarrow> hd list = hd list' \<and> tl list = tl list'" shows "list = list'"
branch_prob_altdef: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "branch_prob \<Y> \<rho> = \<integral>\<^sup>+ x. \<delta> (merge V V' (x, \<rho>)) \<partial>state_measure V \<Gamma>"
g_step_is_pr: shows "g_step \<in> PrimRec2"
le_less: fixes y :: "'a list"   and x :: "'a list" shows "(y \<le>lex x) = (y <lex x \<or> x = y)"