hcju/isabelleps huggingface dataset:
Mapping_RBT_impl_of: fixes t :: "('b, 'c) mapping_rbt" shows "Mapping_RBT (mapping_rbt.impl_of t) = t"
intersect'_language: fixes A :: "('a, 'b) nbta"   and B :: "('a, 'c) nbta" shows "NGBTA.language (intersect' A B) = NBTA.language A \<inter> NBTA.language B"
apply_operations_def: fixes es :: "('msgid \<times> 'op) list" shows "hb.apply_operations es \<equiv> foldl (\<rhd>) Some (map interp_msg es)"
ex_crossing_path: fixes Q :: "'a set" assumes "Q \<in> \<P>" shows "\<exists>R\<in>\<P>. R \<noteq> Q \<and> (\<exists>c. c \<in> R \<and> c \<in> Q)"
wf_def: fixes r :: "('b \<times> 'b) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
catch_Jump: fixes G :: "prog"   and j :: "jump"   and s :: "st"   and tn :: "qtname" shows "\<not> G,(Some (Jump j), s)\<turnstile>catch tn"
remove_vertex_V: fixes G :: "('a, 'b) Graph_scheme"   and x :: "'a" assumes "Digraph G" shows "V\<^bsub>Digraph.remove_vertex G x\<^esub> = V\<^bsub>G\<^esub> - {x}"
wf_update_matchF: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and ne :: "nat"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and mrs :: "mregex list"   and nt :: "nat"   and rels :: "event_data option list set list" assumes "wf_matchF_aux \<sigma> V n R I r aux ne 0"   and "safe_regex Futu Strict r"   and "to_mregex r = (mr, \<phi>s)"   and "mrs = sorted_list_of_set (LPDs mr)"   and "nt = \<tau> \<sigma> (ne + length aux)"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length aux) \<phi>)) \<phi>s rels" shows "wf_matchF_aux \<sigma> V n R I r (update_matchF n I mr mrs rels nt aux) ne 0"
Collect_mono: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
bounded_linear_Pair: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. (f x, g x))"
H_k_dim: fixes k :: "nat" assumes "k < n" shows "H_k k \<in> carrier_mat (2 ^ Suc k) (2 ^ Suc k)"
irrefl_gt: fixes xs :: "'a list" shows "\<not> ext gt xs xs"
binomial_ring: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "(a + b) ^ n = (\<Sum>k\<le>n. of_nat (n choose k) * a ^ k * b ^ (n - k))"
beukers_integral2_conv_int_combination: obtains A :: "int" and B :: "int" where "0 < A" and "local.beukers_integral2 = real_of_int A * Re (zeta 3) + real_of_int B / real (Lcm {1..n} ^ 3)"
path_inf_priorities_def: fixes P :: "'a llist" shows "G'.path_inf_priorities P \<equiv> {k. \<forall>n. k \<in> lset (ldropn n (lmap \<omega>\<^bsub>G'\<^esub> P))}"
cases: fixes a1 :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and a2 :: "'thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action"   and a3 :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" assumes "h.mthr.if.redT a1 a2 a3" obtains t :: "'thread_id" and x :: "status \<times> 'x" and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" and ta :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action" and x' :: "status \<times> 'x" and m' :: "'heap" and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "a1 = sa__" and "a2 = (tb__, ta)" and "a3 = s'a__" and "h.mthr.if.r_syntax tb__ x__ (shr sa__) ta x' m'" and "thr sa__ tb__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "h.mthr.if.actions_ok sa__ tb__ ta" and "redT_upd sa__ tb__ ta x' m' s'a__"   | s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" and t :: "'thread_id" and x :: "status \<times> 'x" and n :: "'addr" and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" and ln :: "'addr \<Rightarrow>f nat" where "a1 = s" and "a2 = (t, K$ [], [], [], [], [], (map NormalAction \<circ> convert_RA) ln)" and "a3 = s'" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
surj_std_dual: assumes "subspace S"   and "finite B" shows "dual_space S \<subseteq> local.std_dual ` S"
dijkstra_loop_invar_and_empty: shows "case dijkstra_loop of (Q, V) \<Rightarrow> D_invar_impl Q V \<and> Q_is_empty Q"
rel_fun_eq_rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(=) ===> R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
frechet_derivative: fixes x :: "'a" shows "frechet_derivative f (at x) = f"
games_lfp_nonempty: shows "Opair Empty Empty \<in> games_lfp"
nzi: shows "(1 - \<i> * z) * inverse (1 + \<i> * z) \<noteq> 0"
monoid_axioms: shows "Group_Theory.monoid carrier_direct_lim add_rel \<lfloor>V, \<zero>\<^bsub>V\<^esub>\<rfloor>"
history_consistent_hd_prog: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p' :: "'p"   and xs :: "'p memref list"   and p :: "'p" assumes "history_consistent \<theta> p' xs" shows "history_consistent \<theta> (hd_prog p xs) xs"
m_ovsmfidiesi: shows "m O (ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>) \<subseteq> b \<union> s \<union> m"
saturated_binary_rule_def: fixes Rule :: "'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> 'a Literal set \<Rightarrow> bool"   and S :: "'a Literal set set" shows "saturated_binary_rule Rule S \<equiv> \<forall>P1 P2 C. P1 \<in> S \<and> P2 \<in> S \<and> Rule P1 P2 C \<longrightarrow> redundant C S"
eq_iff: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" shows "(a = b) = (a \<sqsubseteq> b \<and> b \<sqsubseteq> a)"
closure_of: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of S = topspace X \<inter> (S \<union> X derived_set_of S)"
cases: fixes x :: "init_call_status" obtains C :: "char list" and Cs :: "char list list" where "x = Calling C Cs__"   | Cs :: "char list list" and a :: "nat" where "x = Throwing Csa__ a"   | Cs :: "char list list" where "x = Called Cs"   | "x = No_ics"
stutter_equiv_forw_subst: fixes w\<^sub>1 :: "nat \<Rightarrow> 'a"   and w\<^sub>2 :: "nat \<Rightarrow> 'a"   and w\<^sub>3 :: "nat \<Rightarrow> 'a" assumes "w\<^sub>1 = w\<^sub>2"   and "w\<^sub>2 \<approx> w\<^sub>3" shows "w\<^sub>1 \<approx> w\<^sub>3"
ordinal_exp_oLog_le: fixes x :: "ordinal"   and b :: "ordinal" assumes "0 < x"   and "oSuc 0 < b" shows "b ** oLog b x \<le> x"
l_5: fixes x :: "'a" shows "(SUP i. l i x) = a * f x + g x"
ego_other2_def: fixes t :: "real" shows "ego_other2 t = other.s t - u t"
assms_2: shows "pred_stream (\<lambda>u. u \<in> V) xs"
cong_simp: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B =simp=> g x = h x" shows "sum g A = sum h B"
nonempty_connected_components_of: fixes C :: "'b set"   and X :: "'b topology" assumes "C \<in> connected_components_of X" shows "C \<noteq> {}"
simps: fixes q :: "'b queue" shows "list q = front_list q @ rear_list q"
reduced_row_echelon_form_def': fixes A :: "(('a, 'b) vec, 'c) vec" shows "reduced_row_echelon_form A = ((\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (1::'a)) \<and> (\<forall>i. i < i + (1::'c) \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row (i + (1::'c)) A \<longrightarrow> (LEAST k. A $ i $ k \<noteq> (0::'a)) < (LEAST k. A $ (i + (1::'c)) $ k \<noteq> (0::'a))) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (0::'a))))"
ok_SKIP2: fixes F :: "'b program" shows "F ok \<bottom>"
lst: shows "lst (fst \<sigma>3', snd \<sigma>3' @ snd \<sigma>3'') = lst (fst \<tau>3', snd \<tau>3' @ snd \<tau>3'')"
eventually_bot: fixes P :: "'a \<Rightarrow> bool" shows "eventually P bot"
continuous_Sigma: fixes y :: "'b \<Rightarrow> 'c"   and T :: "'b set"   and X :: "'b \<Rightarrow> 'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "y \<in> Pi T X"   and "continuous_on (Sigma T X) (\<lambda>(t, x). f t x)"   and "continuous_on T y" shows "continuous_on T (\<lambda>x. f x (y x))"
m2r_succ_def: fixes s :: "'d \<Rightarrow> 'd option set" shows "m2r_succ s \<equiv> \<lambda>x. case x of None \<Rightarrow> {} | Some a \<Rightarrow> s a"
h_diff_le_h_plus: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_diff i e \<le> h_plus i e"
simps_1: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "sym.implInsec A B M = \<langle>InsecTag, Messages.Agent A, Messages.Agent B, M\<rangle>"
left_total_rel_resource: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "left_unique A"   and "right_total A"   and "left_total B" shows "left_total (rel_resource A B)"
elims_2: fixes x :: "('b \<times> nat) tree" assumes "wbt x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> nat) tree" and uu_ :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (uu_, n), r\<rangle>" and "n = size l + size r + 1 \<and> balanced1 l r \<and> balanced1 r l \<and> wbt l \<and> wbt r"
NUM_def: shows "NUM \<equiv> {U. \<exists>\<psi>\<in>\<R>\<^sup>2. \<forall>f\<in>U. \<exists>i. \<psi> i = f}"
assms_3: shows "e = Unop_i t iop \<or> e = Unop_f t fop \<or> e = Testop t testop \<or> e = Cvtop t2 cvtop t sx"
encode_problem_parallel_complete_iii_b: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and C :: "sat_plan_variable literal set"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "is_valid_problem_strips \<Pi>"   and "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"   and "C \<in> cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) (length \<pi>))"   and "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"   and "\<forall>l op. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<le> l \<and> l < length \<pi> \<longrightarrow> \<not> \<A> (Operator l (index (\<Pi>\<^sub>\<O>) op))"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)" shows "clause_semantics \<A> C"
comm_monoid_mset_def: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k" shows "comm_monoid_mset f z \<equiv> comm_monoid f z"
c_join_a_eq_e: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "c_aux a b c \<squnion> a_aux a b c = e_aux a b c"
assm: shows "consec i j \<and> consec (i \<squnion> j) k"
set_shadow_root_is_l_set_shadow_root: shows "l_set_shadow_root ShadowRootClass.type_wf set_shadow_root set_shadow_root_locs"
composition_series_triv_group: shows "(carrier G = {\<one>\<^bsub>G\<^esub>}) = (\<HH> = [{\<one>\<^bsub>G\<^esub>}])"
converged_cd_same_suc: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<forall>k k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k' \<and> l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k \<longrightarrow> \<pi> (Suc k) = \<pi>' (Suc k')"   and "l < m"   and "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'" shows "\<exists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
nhop_update_unk_val: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and hops :: "nat"   and ip :: "nat" shows "the (nhop (update rt dip (dsn, unk, val, hops, ip)) dip) = ip"
l_set_shadow_root_get_disconnected_nodes_axioms: shows "l_set_shadow_root_get_disconnected_nodes set_shadow_root_locs get_disconnected_nodes_locs"
in_car_final1: fixes s1 :: "'e \<Rightarrow> ('d, 'e, 'f) F1"   and s2 :: "'f \<Rightarrow> ('d, 'e, 'f) F2"   and x :: "'e" shows "(Equiv_Relations.proj (lsbis1 carT1 carT2 strT1 strT2) \<circ> beh1 s1 s2) x \<in> car_final1"
assms_1: shows "\<forall>x y. x - y = x \<sqinter> - y"
sign_vec_semantics: fixes fml :: "nat fml"   and ls :: "rat poly list"   and x :: "real" assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length ls" shows "lookup_sem fml (map (\<lambda>p. rpoly p x) ls) = lookup_sem fml (sign_vec ls x)"
dist_eqs_1: fixes a' :: "'a"   and slist' :: "[:'a:]" shows "[::] \<noteq> a' :# slist'"
FG_proj_mult_leftdelta: fixes g :: "'g"   and r :: "'a"   and x :: "('a, 'g) aezfun" assumes "g \<in> G" shows "FG_proj (r \<delta>\<delta> g * x) = r \<delta>\<delta> g * FG_proj x"
line_in: fixes x :: "real" assumes "x \<in> {0..1}" shows "b + x *\<^sub>R (a - b) \<in> D"
delete_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
finite_gpv_try: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and gpv' :: "('f, 'd, 'e) gpv" shows "finite_gpv \<I> (TRY gpv ELSE gpv') = (finite_gpv \<I> gpv \<and> (colossless_gpv \<I> gpv \<or> finite_gpv \<I> gpv'))"
sigma_algebra_trivial: fixes \<Omega> :: "'a set" shows "sigma_algebra \<Omega> {{}, \<Omega>}"
assm_bound_funct_OT_12: fixes \<sigma> :: "bool"   and b1 :: "bool"   and b0 :: "bool"   and D :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf"   and HCP_ad :: "real" assumes "local.etp.HCP_adv \<A> \<sigma> (if \<sigma> then b1 else b0) D \<le> HCP_ad" shows "\<bar>spmf (funct_OT_12 (b0, b1) \<sigma> \<bind> (\<lambda>(out1, out2). local.etp.HCP_game \<A> \<sigma> out2 D)) True - 1 / 2\<bar> \<le> HCP_ad"
rsplit_def_auxs_1: fixes n :: "nat"   and w :: "nat \<times> int" shows "bin_rsplit n w = bin_rsplit_aux n (fst w) (snd w) []"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_ring_hom hom"   and "surjective hom" shows "ring_isom hom"
a1_leadsto_b1: shows "\<turnstile> \<box>($PsiInv \<and> [(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2)) \<and> SF(N1)_(x, y, sem, pc1, pc2) \<and> \<box>SF(N2)_(x, y, sem, pc1, pc2) \<longrightarrow> (pc1 = #a \<leadsto> pc1 = #b)"
gpv'_def: shows "gpv' = Generative_Probabilistic_Value.Done x"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "arr (src f) = arr f"
iszero_def: 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 z :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.iszero zero z = (z = zero)"
f3: shows "s \<otimes> s' \<otimes> s'' = s \<otimes> (s' \<otimes> s'')"
absorb2: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># b" shows "a \<union># b = b"
lunit_simps_3: fixes f :: "'a" assumes "ide f" shows "trg \<l>[f] = trg f"
complete_ht_new: shows "<emp> hm_new <is_hashtable (replicate (def_hashmap_size TYPE('c)) [])>"
allNetsDistinct_def: fixes p :: "('a set set, 'b) Combinators list" shows "allNetsDistinct p = (\<forall>a b. a \<noteq> b \<and> a \<in> set (net_list p) \<and> b \<in> set (net_list p) \<longrightarrow> netsDistinct a b)"
rt_fresh_as_nsqnr: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "nsqn\<^sub>r (the (rt2 dip)) = nsqn\<^sub>r (the (rt1 dip))"
except_id_iff: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" shows "(except p S = p) = (keys p \<inter> S = {})"
peek_and_Normal: fixes P :: "'b assn"   and p :: "state \<Rightarrow> bool" shows "(Normal P \<and>. p) = Normal (P \<and>. p)"
rbt_greater_simps_1: fixes k :: "'a" shows "(k \<guillemotleft>| Empty) = True"
neg_true_until_always: fixes p :: "'a" shows "- (\<top> until - p) = \<box> p"
tab\<^sub>0_def: fixes r :: "'a" shows "tab\<^sub>0 r \<equiv> SOME f. \<exists>g. isomorphic_rep (VVV.iso_class_rep \<lbrakk>r\<rbrakk>) f g"
test_def: shows "test \<equiv> cs_empty \<bind> (\<lambda>l. cs_append ''a'' l \<bind> (\<lambda>l. cs_append ''b'' l \<bind> (\<lambda>l. cs_append ''c'' l \<bind> (\<lambda>l. cs_prepend ''0'' l \<bind> (\<lambda>l. cs_rotate l \<bind> (\<lambda>l. cs_pop l \<bind> (\<lambda>(v1, l). cs_pop l \<bind> (\<lambda>(v2, l). cs_pop l \<bind> (\<lambda>(v3, l). cs_pop l \<bind> (\<lambda>(v4, l). return [v1, v2, v3, v4]))))))))))"
cut_greater_Int: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>> t = A \<down>> t \<inter> B \<down>> t"
deriv: fixes x :: "real" assumes "x \<in> {real_of_int a<..<real_of_int b} - A" shows "(f has_vector_derivative f' x) (at x)"
union_assoc: fixes M :: "'d multiset"   and N :: "'d multiset"   and K :: "'d multiset" shows "M + N + K = M + (N + K)"
1: fixes n :: "nat" shows "\<Union> (S ` {..n}) \<in> lmeasurable"
int_of_uint32_0_iff: fixes n :: "uint32" shows "(int_of_uint32 n = 0) = (n = 0)"
gR_rel_mult: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "D x y" shows "partial_semigroup.gR_rel times D x (times x y)"
assms_3: shows "B\<^sub>1 \<subseteq> B\<^sub>2"
op_dghm_components_2: fixes \<FF> :: "V" shows "op_dghm \<FF>\<lparr>ArrMap\<rparr> = \<FF>\<lparr>ArrMap\<rparr>"
chamber_retraction2: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>" shows "f ` C = C"
assms_3: fixes bs :: "'a list" assumes "bs \<in> lists A"   and "sum_list bs = a" shows "order.greater_eq (length bs) (length as)"
lprod_single: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set" assumes "(a, b) \<in> R" shows "([a], [b]) \<in> lprod R"
mem_val_alt_10_mem_equal_1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 ((pa AND 68719476732) + 1) s1 = mem_val_alt 10 ((pa AND 68719476732) + 1) s2"
sgn1_pos_eq_one: fixes x :: "nat" assumes "0 < sgn1 x" shows "sgn1 x = 1"
NewInitOOMReds: fixes s :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "\<nexists>sfs. shp s C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,s,False\<rangle> \<rightarrow>* \<langle>Val v',(h', l', sh'),b'\<rangle>"   and "new_Addr h' = None"   and "is_class P C" shows "P \<turnstile> \<langle>new C,s,False\<rangle> \<rightarrow>* \<langle>THROW OutOfMemory,(h', l', sh'),False\<rangle>"
mat1_mult_right: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and nr :: "nat"   and nc :: "nat"   and m :: "'c list list" assumes "class.semiring_1 one times plus zero"   and "mat nr nc m" shows "mat_multI zero plus times nr m (mat1I zero one nc) = m"
list_of_vec_append: fixes v :: "'a vec"   and w :: "'a vec" shows "list_of_vec (v @\<^sub>v w) = list_of_vec v @ list_of_vec w"
Lp_space_def: fixes p :: "ennreal"   and M :: "'a measure" shows "\<LL> p M = (if p = 0 then quasinorm_of (1, \<lambda>f. if f \<in> borel_measurable M then 0 else \<infinity>) else if p < \<infinity> then quasinorm_of (if p < 1 then 2 powr (1 / enn2real p - 1) else 1, \<lambda>f. if f \<in> borel_measurable M \<and> integrable M (\<lambda>x. \<bar>f x\<bar> powr enn2real p) then ennreal ((LINT x|M. \<bar>f x\<bar> powr enn2real p) powr (1 / enn2real p)) else \<infinity>) else quasinorm_of (1, \<lambda>f. if f \<in> borel_measurable M then e2ennreal (esssup M (\<lambda>x. ereal \<bar>f x\<bar>)) else \<infinity>))"
stable_sort_key_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sk :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c list \<Rightarrow> 'c list" assumes "class.linorder less_eq less" shows "linorder.stable_sort_key sk = (\<forall>f xs k. filter (\<lambda>y. f y = k) (sk f xs) = filter (\<lambda>y. f y = k) xs)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) \<mu>"
nth_append_singl: fixes i :: "nat"   and al :: "'a list"   and a :: "'a" assumes "i < length al" shows "(al @ [a]) ! i = al ! i"
lincomb_list_as_lincomb: fixes vs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "set vs \<subseteq> carrier_vec n" shows "lincomb_list c vs = lincomb (mk_coeff vs c) (set vs)"
rep_inject: fixes x :: "('b, 'c) mapping"   and y :: "('b, 'c) mapping" shows "(rep x = rep y) = (x = y)"
fermat_theorem_mod_ring: fixes a :: "'a mod_ring" shows "a ^ CARD('a) = a"
tinstT_iff_ex_raw_match': fixes T1 :: "typ"   and T2 :: "typ" shows "tinstT T1 T2 = (\<exists>subs. raw_match' T2 T1 Map.empty = Some subs)"
start_fresh_variable_fresh: fixes cs :: "('a \<times> QDelta ns_constraint) list" shows "\<forall>var\<in>vars_constraints (map snd cs). var < start_fresh_variable cs"
dickson_lessI: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'c \<Rightarrow> nat"   and s :: "'c"   and m :: "nat"   and t :: "'c" assumes "gd_powerprod ord ord_strict"   and "d s \<le> m"   and "d t \<le> m"   and "ord_strict s t" shows "gd_powerprod.dickson_less ord_strict d m s t"
inj_hom_preserves_independent: fixes Us :: "'a list" assumes "inj_on h (R.Span K Us)"   and "R.independent K Us" shows "S.independent (h ` K) (map h Us)"