hcju/isabelleps huggingface dataset:
axioms_2: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2" shows "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
differing_vars_subst: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "differing_vars mem\<^sub>1 mem\<^sub>2 \<subseteq> dom \<sigma>" shows "subst \<sigma> mem\<^sub>1 = subst \<sigma> mem\<^sub>2"
transfer_action_pnet_closed: fixes s :: "'s net_state"   and a :: "'m node_action"   and s' :: "'s net_state"   and n :: "net_tree" assumes "(s, a, s') \<in> automaton.trans (closed (pnet np n))" obtains a' :: "'m node_action" where "(s, a', s') \<in> automaton.trans (pnet np n)" and "\<And>\<sigma> \<zeta> \<sigma>' \<zeta>'. \<lbrakk>((\<sigma>, \<zeta>), a', \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n); \<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>' j = \<sigma> j\<rbrakk> \<Longrightarrow> ((\<sigma>, \<zeta>), a, \<sigma>', \<zeta>') \<in> automaton.trans (oclosed (opnet onp n))"
recv_marker_other_channels_not_shrinking: fixes ev :: "('p, 's, 'm) event"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and c :: "('p, 's, 'm) configuration"   and c' :: "('p, 's, 'm) configuration"   and i' :: "nat" assumes "ev = RecvMarker i p q"   and "c \<turnstile> ev \<mapsto> c'" shows "(length (msgs c i') \<le> length (msgs c' i')) = (i \<noteq> i')"
table_X: shows "table (args_n args) (args_R args) X"
int_vertices_simps_2: fixes vv :: "'a \<times> 'b \<times> 'a"   and p :: "('a \<times> 'b \<times> 'a) list" shows "int_vertices (vv # p) = insert (fst vv) (int_vertices p)"
test_test_compl: fixes x :: "'a" assumes "is_test x" shows "is_test (tc x)"
Setcompr_eq_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "{f x |x. x \<in> A} = f ` A"
intros_1: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and packet_tagger :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and state_update :: "'c \<Rightarrow> final_decision \<Rightarrow> 'd \<Rightarrow> 'c"   and \<sigma>\<^sub>0 :: "'c"   and built_in_chain :: "char list"   and default_policy :: "action"   and ps :: "'d list" shows "semantics_stateful_packet_tagging \<Gamma> \<gamma> packet_tagger state_update \<sigma>\<^sub>0 (built_in_chain, default_policy) ps [] \<sigma>\<^sub>0"
create_get: fixes x :: "'b \<Longrightarrow> 'c"   and v :: "'b" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (create\<^bsub>x\<^esub> v) = v"
assms_3: shows "has_density M (embed_measure N f) \<delta>"
smcf_dghm_def: fixes \<CC> :: "V" shows "smcf_dghm \<CC> = [\<CC>\<lparr>ObjMap\<rparr>, \<CC>\<lparr>ArrMap\<rparr>, smc_dg (\<CC>\<lparr>HomDom\<rparr>), smc_dg (\<CC>\<lparr>HomCod\<rparr>)]\<^sub>\<circ>"
less_numeral_extra_1: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
restrict_map_def: fixes m :: "'a \<Rightarrow> 'b option"   and A :: "'a set" shows "m |` A = (\<lambda>x. if x \<in> A then m x else None)"
emp_not_basis: fixes x :: "'b list"   and G :: "'b list set" assumes "x \<in> \<BB> G" shows "x \<noteq> \<epsilon>"
rat_step_then_int_poly_step: fixes q :: "int poly"   and a1 :: "int poly"   and a2 :: "int poly"   and b2 :: "int"   and c1 :: "int poly"   and c2 :: "int poly" assumes "q \<noteq> 0"   and "(of_int_poly a1, of_int_poly a2) = (div_rat_poly_step (of_int_poly q) \<circ> rat_of_int) b2 (of_int_poly c1, of_int_poly c2)" shows "Some (a1, a2) = div_int_poly_step q b2 (Some (c1, c2))"
ccw_translate3_minus: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and x :: "real" assumes "det3 0 a b \<noteq> 0"   and "x < 1"   and "ccw 0 a (b - x *\<^sub>R a)" shows "ccw 0 a b"
knows_composition2_s: fixes m :: "Secrets"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "sKS m \<notin> specKeysSecrets P"   and "sKS m \<notin> specKeysSecrets Q"   and "knows Q [sE m]"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "knows PQ [sE m]"
b: fixes k :: "complex" assumes "cmod k < 1" shows "P (blaschke k)"
listset_singletonI: fixes a :: "'b"   and A :: "'b set"   and ys :: "'b list" assumes "a \<in> A"   and "ys = [a]" shows "ys \<in> listset [A]"
free_basis_dec_morph: fixes u :: "'b list"   and G :: "'b list set"   and v :: "'b list" assumes "u \<in> \<langle>G\<rangle>\<^sub>F"   and "v \<in> \<langle>G\<rangle>\<^sub>F" shows "Dec \<BB>\<^sub>F G u \<cdot> v = (Dec \<BB>\<^sub>F G u) \<cdot> (Dec \<BB>\<^sub>F G v)"
deadlock_free_def: fixes P :: "'b process" shows "deadlock_free P \<equiv> DF UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
Low_source_Exit_edge: shows "\<exists>a. valid_edge a \<and> sourcenode a = (_Low_) \<and> targetnode a = (_Exit_) \<and> kind a = (\<lambda>s. True)\<^sub>\<surd>"
rep_eq: fixes x :: "'b set" shows "Rep_ttree (many_among x) = {xs. set xs \<subseteq> x}"
inf_sup_distrib2: fixes y :: "'b upred"   and z :: "'b upred"   and x :: "'b upred" shows "((y \<or> z) \<and> x) = (y \<and> x \<or> z \<and> x)"
SN_imp_wf: fixes r :: "('b \<times> 'b) set" assumes "SN r" shows "wf (r\<inverse>)"
gen_dfs_simps_2: fixes succs :: "'e \<Rightarrow> 'e list"   and ins :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d"   and x :: "'e"   and xs :: "'e list" shows "gen_dfs succs ins memb S (x # xs) = (if memb x S then gen_dfs succs ins memb S xs else gen_dfs succs ins memb (ins x S) (succs x @ xs))"
ocircline_set_ounit_circle: shows "ocircline_set ounit_circle = circline_set unit_circle"
\<Psi>eq: shows "\<Psi>\<^sub>P\<^sub>Q = (p \<bullet> \<Psi>\<^sub>P) \<otimes> (p \<bullet> \<Psi>\<^sub>Q)"
assms: fixes n :: "nat" shows "X n \<in> M \<rightarrow>\<^sub>M N"
support_upd_subset: fixes z :: "'e"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" shows "support z A (f(x := y)) \<subseteq> support z A f \<union> {x}"
space_closed: fixes M :: "'d measure" shows "sets M \<subseteq> Pow (space M)"
can_step_def: fixes lstep :: "'b \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and l :: "'b"   and s :: "'d" shows "system.can_step lstep l s \<equiv> \<exists>s'. lstep l s s'"
diag_block_mat_singleton: fixes A :: "'b mat" shows "diag_block_mat [A] = A"
R1: shows "\<forall>x y. ((x, y) \<in> r) = (\<exists>u. p x y u = 0)"
fps_dvd_iff: fixes f :: "'b fps"   and g :: "'b fps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "(f dvd g) = (subdegree f \<le> subdegree g)"
c_K5_not_comb_planar: shows "\<not> comb_planar (with_proj c_K5)"
lunit'_simps_1: fixes f :: "'a" assumes "C.ide f" shows "C.arr \<l>\<^sub>C\<^sup>-\<^sup>1[f]"
cong_diff_nat: fixes a :: "nat"   and b :: "nat"   and m :: "nat"   and c :: "nat"   and d :: "nat" assumes "[a = b] (mod m)"   and "[c = d] (mod m)"   and "c \<le> a"   and "d \<le> b" shows "[a - c = b - d] (mod m)"
fun_upd_def: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'d" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
circ_tang_vec_opposite_orient: fixes \<mu> :: "complex"   and E :: "complex"   and p :: "bool" shows "circ_tang_vec \<mu> E p = - circ_tang_vec \<mu> E (\<not> p)"
revcast_rev_ucast: fixes w :: "'b word" shows "revcast (word_reverse w) = word_reverse (ucast w)"
all_snapshotted: shows "\<forall>p. ps c' p \<noteq> None"
sum_list_distinct_filter: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> int" assumes "distinct xs"   and "distinct ys"   and "set ys \<subseteq> set xs"   and "\<And>x. x \<in> set xs - set ys \<Longrightarrow> f x = 0" shows "sum_list (map f xs) = sum_list (map f ys)"
freshSets_7: fixes x :: "name"   and X :: "name set"   and C :: "'d" shows "insert x X \<sharp>* C = (x \<sharp> C \<and> X \<sharp>* C)"
finite_ordertype_le_card: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "finite A"   and "wf r"   and "trans r" shows "ordertype A r \<le> ord_of_nat (card A)"
compatible_different_no_read: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var" assumes "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"   and "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"   and "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "i \<sqsubset> length cms\<^sub>1"   and "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "doesnt_read (fst (cms\<^sub>1 ! i)) x \<and> doesnt_read (fst (cms\<^sub>2 ! i)) x"
Red_F_\<G>_equiv_def: fixes N :: "'f set" shows "local.Red_F_\<G> N = {C. \<forall>Di\<in>\<G>_F C. Di \<in> Red_F_G (\<G>_Fset N) \<or> (\<exists>E\<in>N - local.Red_F_\<G> N. Prec_F_g Di E C \<and> Di \<in> \<G>_F E)}"
suffix_second: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "second (s |\<^sub>s n) = s (Suc n)"
Act_eqvt: fixes p :: "perm"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c) formula" shows "p \<bullet> Formula.Act \<alpha> x = Formula.Act (p \<bullet> \<alpha>) (p \<bullet> x)"
finite_imp_finite_component_Keys: fixes F :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "finite F" shows "finite (component_of_term ` Keys F)"
fresh_tyenv_None: fixes x :: "Syntax.var"   and \<Gamma> :: "(Syntax.var, ty) fmap" shows "atom x \<sharp> \<Gamma> = (\<Gamma> $$ x = None)"
merkle_interface_def: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" shows "merkle_interface h bo m \<equiv> ((\<forall>a b. (h a = h b) = (\<exists>ab. m a b = Some ab)) \<and> (\<forall>a. m a a = Some a)) \<and> (\<forall>a b. m a b = m b a) \<and> (\<forall>a b c. m a b \<bind> m c = m b c \<bind> m a) \<and> (\<forall>a b. bo a b = (m a b = Some b))"
plus_flow_outside: fixes e :: "'k \<times> 'k"   and G :: "('k, 'l) graph_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "e \<notin> \<^bold>E\<^bsub>G\<^esub>" shows "(f \<oplus>\<^bsub>G\<^esub> g) e = 0"
unit_simps_5: fixes a :: "'a" assumes "B.obj a" shows "B.cod \<i>[a] = a"
order_zmset_exists_foundation_neg: fixes M :: "'a zmultiset"   and t :: "'a" assumes "zcount M t < 0" shows "\<exists>s\<le>t. zcount M s < 0 \<and> (\<forall>u. zcount M u < 0 \<longrightarrow> \<not> u < s)"
scale_eq_0_iff: fixes a :: "real"   and x :: "'c" shows "(a *\<^sub>R x = (0::'c)) = (a = 0 \<or> x = (0::'c))"
open_imp_locally_connected: fixes S :: "'c set" assumes "open S" shows "locally connected S"
llast_linfinite: fixes xs :: "'c llist" assumes "\<not> lfinite xs" shows "llast xs = undefined"
less_minimum: fixes P :: "'a \<Rightarrow> bool"   and d :: "'a" assumes "minimum_elem D {x \<in> carrier D. P x} d" shows "\<forall>a. a \<prec> d \<and> a \<in> carrier D \<longrightarrow> \<not> P a"
simps_2: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "postQ (x ::= a) s = s[a/x]"
reduce_nested_mpoly_prod: fixes p1 :: "'b mpoly mpoly"   and p2 :: "'b mpoly mpoly" shows "reduce_nested_mpoly (p1 * p2) = reduce_nested_mpoly p1 * reduce_nested_mpoly p2"
circ_separate_4: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> x * x\<^sup>\<circ> * ((1::'a) \<squnion> y)" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * y\<^sup>\<circ>"
dvd_triv_right: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a (times b a)"
rep_eq: fixes x :: "'a freeword"   and xa :: "'a freeword" shows "freeword (x + xa) = prappend_signed_list (freeword x) (freeword xa)"
euclidean_representation_sum: fixes f :: "'b \<Rightarrow> real"   and b :: "'b" shows "((\<Sum>i\<in>Basis. f i *\<^sub>R i) = b) = (\<forall>i\<in>Basis. f i = b \<bullet> i)"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" shows "subtrees_mset \<langle>l, x, r\<rangle> = add_mset \<langle>l, x, r\<rangle> (subtrees_mset l + subtrees_mset r)"
of_int_floor_cancel: fixes x :: "'c" shows "(of_int \<lfloor>x\<rfloor> = x) = (\<exists>n. x = of_int n)"
sub_Hd_HdE: fixes \<zeta> :: "('a, 'b) hd"   and \<xi> :: "('a, 'b) hd" assumes "sub (Hd \<zeta>) (Hd \<xi>)" obtains "\<xi> = \<zeta>"
Tensor_right_id: fixes xs :: "int poly fract list list" shows "xs \<otimes> [[1]] = xs"
new_character_data_types_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
geq: shows "g ` (S - {0::'a}) = T - {0::'a}"
negchecks_model_def: fixes \<I> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and D :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set"   and X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "negchecks_model \<I> D X F G \<equiv> \<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> list_ex (\<lambda>f. fst f \<cdot> \<delta> \<circ>\<^sub>s \<I> \<noteq> snd f \<cdot> \<delta> \<circ>\<^sub>s \<I>) F \<or> list_ex (\<lambda>f. f \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> D) G"
permute_swap_cancel: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "atom"   and b :: "atom"   and x :: "'a" assumes "class.pt permute" shows "permute (a \<rightleftharpoons> b) (permute (a \<rightleftharpoons> b) x) = x"
MkIde_Dom': fixes a :: "('o, 'a) arr" assumes "ide a" shows "MkIde (Dom a) = a"
blocksRedsFinal: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and l'' :: "char list \<Rightarrow> val option" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "distinct Vs"   and "P \<turnstile> \<langle>e,(h, l(Vs [\<mapsto>] vs))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"   and "final e'"   and "l'' = override_on l' l (set Vs)" shows "P \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l'')\<rangle>"
continuous_on_Icc_at_rightD: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on {a..b} f"   and "a < b" shows "(f \<longlongrightarrow> f a) (at_right a)"
subset_imp_leadsETo: fixes A :: "'d set"   and B :: "'d set"   and F :: "'d program"   and CC :: "'d set set" assumes "A \<subseteq> B" shows "F \<in> A leadsTo[CC] B"
lc3_aux: shows "{l. length l = 3 \<and> valid l} = {[B, B, B], [R, R, R]}"
chain_imageI: fixes le_a :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Y :: "'g set"   and le_b :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" assumes "Complete_Partial_Order.chain le_a Y"   and "\<And>x y. \<lbrakk>x \<in> Y; y \<in> Y; le_a x y\<rbrakk> \<Longrightarrow> le_b (f x) (f y)" shows "Complete_Partial_Order.chain le_b (f ` Y)"
S2_and'_def: fixes B :: "bool \<times> bool"   and A :: "bool \<times> bool"   and s2 :: "bool" shows "S2_and' B A s2 = (let (a1, a2) = A; (b1, b2) = B in S2_14_OT B s2 \<bind> (\<lambda>V. let s1 = s2 \<oplus> (a1 \<oplus> b1 \<and> a2 \<oplus> b2) in return_spmf ((B, V), s1, s2)))"
mgu_gives_subst_idem: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<theta>" shows "subst_idem \<theta>"
succs_def: fixes rel :: "(nat \<times> nat) set"   and x :: "nat" shows "succs rel x = {y. (x, y) \<in> rel}"
divides_def: fixes p1 :: "'a list"   and p2 :: "'a list" shows "p1 divides p2 = (\<exists>q. poly p2 = poly (p1 *** q))"
sum_encode_inverse: fixes x :: "nat + nat" shows "sum_decode (sum_encode x) = x"
rel_ipurge_def: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d" shows "rel_ipurge P I D u \<equiv> {(xs, ys). xs \<in> traces P \<and> ys \<in> traces P \<and> ipurge_tr_rev I D u xs = ipurge_tr_rev I D u ys}"
abs_nat_pp_inverse: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat"   and t :: "(nat, nat) pp" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "rep_nat_pp (abs_nat_pp t) = t"
nn_integral_restrict_space: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = set_nn_integral M \<Omega> f"
Quotient_dual_ord: shows "Quotient (=) to_dual_ord of_dual_ord cr_dual_ord"
prems: fixes x :: "'c" assumes "x \<in> results_gpv \<I> gpv" shows "f x = g x"
prev_in_basics_1: fixes A :: "'c set"   and a :: "'c"   and b :: "'c"   and u :: "'c" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b"   and "u \<in> {a<..b}" shows "prev_in A u \<in> A"
possible_assignments_for_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c" shows "possible_assignments_for \<Psi> v \<equiv> map (Pair v) (the (range_of \<Psi> v))"
fixp_induct: fixes P :: "(('c stream \<Rightarrow> bool) \<Rightarrow> 'c stream \<Rightarrow> enat) \<Rightarrow> bool" assumes "lfp.admissible (\<lambda>scount. P (curry scount))"   and "P (\<lambda>scount P. \<Squnion> {})"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> P (\<lambda>a b. if a b then eSuc (\<omega> a (stl b)) else \<omega> a (stl b))" shows "P scount"
list_to_poly_def: fixes R :: "('c, 'd) ring_scheme"   and as :: "'c list"   and n :: "nat" shows "list_to_poly R as n = (if n < length as then as ! n else \<zero>\<^bsub>R\<^esub>)"
context_le_def: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and \<Gamma>' :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.context_le \<Gamma> \<Gamma>' \<equiv> dom \<Gamma> = dom \<Gamma>' \<and> (\<forall>x\<in>dom \<Gamma>. the (\<Gamma> x) \<sqsubseteq> the (\<Gamma>' x))"
finite_Field: fixes r :: "('a \<times> 'a) set" assumes "finite r" shows "finite (Field r)"
that: fixes x' :: "'u" assumes "max s P = Some x'"   and "x' \<in> \<alpha> s"   and "P x'"   and "\<forall>x\<in>\<alpha> s. P x \<longrightarrow> x \<le> x'" shows "thesis"
snd_snd_snd_foldl_echelon_form_of_column_k_det_iarrays: fixes n :: "'c"   and A :: "'c iarray iarray"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and k :: "nat" shows "snd (snd (snd (foldl echelon_form_of_column_k_det_iarrays (n, A, 0, bezout) [0..<k]))) = bezout"
homotopic_loops_eq: fixes p :: "real \<Rightarrow> 'a"   and s :: "'a set"   and q :: "real \<Rightarrow> 'a" assumes "path p"   and "path_image p \<subseteq> s"   and "pathfinish p = pathstart p"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t" shows "homotopic_loops s p q"
ABV: fixes b :: "'a" assumes "b \<in> B" shows "(b, a) \<in> r"
i_in_Red_I_N: shows "\<iota> \<in> Red_I_\<G> N"
sorted_aux0: shows "sorted_wrt (\<lambda>x y. fst y < fst x) aux0"
binomial_symmetric: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "n choose k = n choose (n - k)"
ring_seq_smult_closed: fixes R :: "('a, 'b) ring_scheme"   and s :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "U_function_ring R"   and "s \<in> carrier R\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier R" shows "a \<odot>\<^bsub>R\<^bsup>\<omega>\<^esup>\<^esub> s \<in> carrier R\<^bsup>\<omega>\<^esup>"
wf_RP_combined_relation: shows "wf (natLess <*lex*> (natLess <*lex*> natLess <*lex*> natLess) <*lex*> natLess <*lex*> natLess <*lex*> natLess)"
insert_4: fixes x :: "'a" assumes "x \<in> insert x__ F" shows "0 < f x"
assms: shows "\<forall>T. gap i T \<le> gap i t"
rebalance_last_tree_inorder: fixes t :: "'a btree"   and sub :: "'a btree"   and ts :: "('a btree \<times> 'a) list"   and list :: "('a btree \<times> 'a) list"   and sep :: "'a"   and k :: "nat" assumes "height t = height sub"   and "ts = list @ [(sub, sep)]" shows "inorder (rebalance_last_tree k ts t) = inorder (Node ts t)"
all_long_pers: fixes w :: "'b list"   and n :: "nat" assumes "w \<noteq> \<epsilon>"   and "\<^bold>|w\<^bold>| \<le> n" shows "periodN w n"
sel_1: fixes x1 :: "nat"   and x2 :: "v store_v list"   and x3 :: "'a ffi_state"   and x4 :: "tid_or_exn set"   and x5 :: "char list list set" shows "clock (make_state x1 x2 x3 x4 x5) = x1"
prems_2: shows "oid \<notin> set (map fst (xs @ [x]))"
path_rev: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" shows "path g u (revp p) v = path g v p u"
intro: fixes S :: "'a set"   and ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>p q r t. \<lbrakk>distinct4 p q r t; in4 S p q r t; ccw t q r; ccw p t r; ccw p q t\<rbrakk> \<Longrightarrow> ccw p q r" shows "ccw_system4_axioms ccw S"
utp_weak_partial_order: fixes T :: "'a hrel \<Rightarrow> 'a hrel" shows "weak_partial_order (utp_order T)"
Busemann_function_minus_infinity_imp_convergent: fixes xi :: "'c Gromov_completion"   and u :: "'b \<Rightarrow> 'c"   and x :: "'c"   and F :: "'b filter" assumes "((\<lambda>xa. ereal (Busemann_function_at xi (u xa) x)) \<longlongrightarrow> - \<infinity>) F" shows "((\<lambda>n. to_Gromov_completion (u n)) \<longlongrightarrow> xi) F"