hcju/isabelleps huggingface dataset:
arr_to_hfun_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr_to_hfun f = \<lbrace>XY \<^bold>\<in> ide_to_hf (local.dom f) * ide_to_hf (cod f). hsnd XY = DOWN (Fun f (UP (hfst XY)))\<rbrace>"
IH_1: fixes n :: "nat"   and l :: "nat list" assumes "ackloop_dom (ack m n # l)" shows "ackloop_dom (n # m # l)"
intros_3: fixes u :: "'c \<Rightarrow> 'b"   and r :: "'c"   and d :: "'b" assumes "u r < d" shows "dbm_entry_val u (Some r) None (Lt d)"
islE: fixes x :: "'c + 'd" assumes "isl x" obtains l :: "'c" where "x = Inl l"
assms_6: shows "has_greatest_postfixpoint (h \<circ> g)"
emeasure_eq: fixes A :: "nat set set set" shows "emeasure P A = ennreal (if A \<subseteq> Pow S_edges then \<Sum>edges\<in>A. p ^ card edges * (1 - p) ^ card (S_edges - edges) else 0)"
le_fin_member': fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "traces.le_fin ind [a] (u @ v)"   and "a \<in> set u" shows "traces.le_fin ind [a] u"
lift2_le_ub: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and u :: "'b" assumes "semilat (err A, Err.le r, lift2 f)"   and "x \<in> A"   and "y \<in> A"   and "x \<squnion>\<^bsub>f\<^esub> y = OK z"   and "u \<in> A"   and "x \<sqsubseteq>\<^bsub>r\<^esub> u"   and "y \<sqsubseteq>\<^bsub>r\<^esub> u" shows "z \<sqsubseteq>\<^bsub>r\<^esub> u"
inc_seq_greater: fixes f :: "nat \<Rightarrow> nat"   and N :: "nat" assumes "\<forall>i. f i < f (Suc i)" shows "\<exists>i. N < f i"
rel_interior: fixes S :: "'a set" shows "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
diff_less_imp_less_add: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j - k < i" shows "j < i + k"
homeomorphic_maps_swap: fixes X :: "'c topology"   and Y :: "'d topology" shows "homeomorphic_maps (prod_topology X Y) (prod_topology Y X) (\<lambda>(x, y). (y, x)) (\<lambda>(y, x). (x, y))"
t_wcode_main_def: shows "t_wcode_main = t_wcode_main_first_part @ shift t_twice 12 @ [(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]"
summands_equal: fixes a :: "rat"   and b :: "rat"   and c :: "rat"   and d :: "rat" assumes "a = b"   and "c = d" shows "a + c = b + d"
mem_simps_2: fixes c :: "'c" shows "(c \<in> {}) = False"
matrix_matrix_mult_def: fixes m :: "(('b, 'e) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and m' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $h i $h k * m' $h k $h j"
cont_case_prod_applyI: fixes luba :: "('b \<times> 'c) set \<Rightarrow> 'b \<times> 'c"   and orda :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'e" assumes "cont luba orda lubb ordb (\<lambda>(a, b). f a b y)" shows "cont luba orda lubb ordb (\<lambda>x. (case x of (x, xa) \<Rightarrow> f x xa) y)"
cr_alist_def: shows "cr_alist \<equiv> \<lambda>x y. x = alist.impl_of y"
eval_fps_0: fixes z :: "'a" shows "eval_fps 0 z = (0::'a)"
lunit_naturality: fixes \<mu> :: "'b" assumes "arr \<mu>" shows "\<mu> \<cdot> \<l>[local.dom \<mu>] = \<l>[cod \<mu>] \<cdot> (trg \<mu> \<star> \<mu>)"
ee_factorsD: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "essentially_equal G as bs"   and "factors G as a"   and "factors G bs b"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim> b"
equivalence_map_def: fixes f :: "'d" shows "D.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. D.internal_equivalence f g \<eta> \<epsilon>"
natLeq_on_Card_order: fixes n :: "nat" shows "Card_order (natLeq_on n)"
le_Integer_bottom_iff: fixes x :: "Integer"   and y :: "Integer" shows "(le\<cdot>x\<cdot>y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
ideD_3: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.VV.cod a = a"
nonzero_mult_div_cancel_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "b \<noteq> zero" shows "divide (times a b) b = a"
set_mset_Union_mset: fixes MM :: "'k multiset multiset" shows "set_mset (\<Sum>\<^sub># MM) = \<Union> (set_mset ` set_mset MM)"
dominated_convergence_1: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and h :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>k. f k integrable_on S"   and "h integrable_on S"   and "\<And>k x. x \<in> S \<Longrightarrow> norm (f k x) \<le> h x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x" shows "g integrable_on S"
non_neg_mat_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "non_neg_mat A \<equiv> \<forall>a\<in>elements_mat_h A. (0::'b) \<le> a"
simps_1: fixes uu :: "'c \<Rightarrow> 'b dres" shows "dFAIL \<bind> uu = dFAIL"
sum_list_sum_nth: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'c list" assumes "class.monoid_add plus zero" shows "sum_list xs = sum ((!) xs) {0..<length xs}"
exists_m: shows "\<exists>m. A $ m $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> (GREATEST m. \<not> is_zero_row_upt_k m k A) + (1::'rows) \<le> m"
soundness: fixes idx :: "'n"   and r :: "('a, 'k) aformula"   and s :: "('a, 'k) aformula" assumes "bounded.check_eqv idx r s" shows "rel_language (=) (language\<^sub>b idx r) (language\<^sub>b idx s)"
map: fixes f :: "'d \<Rightarrow> 'b"   and x1 :: "'d"   and x2 :: "'d stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
ex1E: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'b" where "P x" and "\<forall>y. P y \<longrightarrow> y = x"
wffp_imp_appendL: fixes sl1 :: "('St, 'U, 'C) state list"   and sl2 :: "('St, 'U, 'C) state list" assumes "wffp (sl1 @ sl2)"   and "sl1 \<noteq> []" shows "wffp sl1"
hcomplex_of_hypreal_pow: fixes x :: "real star"   and n :: "nat" shows "hcomplex_of_hypreal (x ^ n) = hcomplex_of_hypreal x ^ n"
associated_polynomials_iff: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p \<sim>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q = (\<exists>k\<in>K - {\<zero>\<^bsub>K [X]\<^esub>}. p = [k] \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q)"
Gs: shows "mat_of_rows n (map gs1.gso [0..<m]) \<in> carrier_mat m n"
ring_tOp_commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r y = y \<cdot>\<^sub>r x"
fps_divide_nth_base: fixes g :: "'a fps"   and f :: "'a fps" assumes "subdegree g \<le> subdegree f" shows "fps_nth (f / g) (subdegree f - subdegree g) = fps_nth f (subdegree f) * inverse (fps_nth g (subdegree g))"
is_functor_if_value_is_ide: assumes "C.ide a" shows "functor (\<cdot>\<^sub>J) (\<cdot>) A.map"
HT_partial_mods_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and mods :: "char list set"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT_partial_mods \<pi> mods P c Q \<equiv> HT_partial \<pi> P c (\<lambda>s\<^sub>0 s. Q s\<^sub>0 s \<and> modifies mods s s\<^sub>0)"
filterlim_power_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and n :: "nat" assumes "filterlim f at_infinity F"   and "0 < n" shows "LIM x F. f x ^ n :> at_infinity"
lincomb_units: fixes i :: "nat"   and a :: "'a vec \<Rightarrow> 'a" assumes "i < n" shows "lincomb a (set (unit_vecs n)) $ i = a (unit_vec n i)"
tendsto_def: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter" shows "(f \<longlongrightarrow> l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
less_imp_le: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "ko.lt ko x y" shows "ko.le ko x y"
tab_simps_2: shows "src \<rho> = src (src w)"
upset_set_downset_set_dual: shows "(`) \<partial> \<circ> \<Up> = \<Down> \<circ> (`) \<partial>"
bounded_hashcode_nat_bounds: fixes def_hashmap_size :: "'a itself \<Rightarrow> nat"   and n :: "nat"   and hashcode :: "'a \<Rightarrow> uint32"   and a :: "'a" assumes "class.hashable def_hashmap_size"   and "1 < n" shows "hashable.bounded_hashcode_nat hashcode n a < n"
bind_left_fail_SE': fixes P :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "fail\<^sub>S\<^sub>E ;- P = fail\<^sub>S\<^sub>E"
keys_subset_Keys: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and F :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "f \<in> F" shows "keys f \<subseteq> Keys F"
initial: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme" assumes "p = initial A" shows "p \<in> nodes A"
mod_add_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) (modulo b c)) c = modulo (plus a b) c"
R\<^sub>C_ofD: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>1\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>2\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C' :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>2\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "secure_refinement \<R>\<^sub>A \<R> P"   and "(\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C', mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> R\<^sub>C_of \<R>\<^sub>A \<R> P" shows "(\<exists>c\<^sub>1\<^sub>A c\<^sub>2\<^sub>A. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A) \<in> \<R>\<^sub>A) \<and> mds\<^sub>C' = mds\<^sub>C \<and> conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C \<and> (\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C', mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> P"
\<epsilon>_def: shows "\<epsilon> = Min (insert 1 {(u y - u x) / (f x - f y) |x y. x \<prec>[le] y \<and> f y < f x}) / 2"
symmetric_mpoly_sum: fixes B :: "'a set"   and A :: "nat set"   and f :: "'a \<Rightarrow> 'b mpoly" assumes "\<And>i. i \<in> B \<Longrightarrow> symmetric_mpoly A (f i)" shows "symmetric_mpoly A (sum f B)"
abs_eq: fixes xa :: "perm"   and x :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "xa \<bullet> FL_Formula.abs_Tree\<^sub>\<alpha> x = FL_Formula.abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
amd_delete_def: fixes dflt :: "'b"   and k :: "nat"   and a :: "'b array" shows "amd_delete dflt k a \<equiv> array_set_dyn dflt a k dflt"
assms_2: shows "gen_division s (cubeImage ` two_chain_typeII)"
intro: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping" assumes "Incremental_NS_Constraint_Ops init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc"   and "To_ns to_ns from_ns" shows "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns"
InconsistencyAutomatic: shows "False \<^bold>\<leftarrow> (\<exists>x. \<^bold>\<not> (E x))"
on_circline_cmat_cvec_circline_equation: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex"   and z1 :: "complex"   and z2 :: "complex" shows "on_circline_cmat_cvec (A, B, C, D) (z1, z2) = circline_equation A B C D z1 z2"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_same a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "vars_mset s \<subseteq># vars_mset t" and "wt t = wt s" and "head t = head s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
refine_IdD: fixes c :: "'c nres"   and a :: "'c nres" assumes "c \<le> \<Down> Id a" shows "c \<le> a"
is_glb_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and s1 :: "'e"   and s2 :: "'e" shows "pre_RDR.is_glb \<delta> s s1 s2 \<equiv> pre_RDR.is_lb \<delta> s s1 s2 \<and> (\<forall>s'. pre_RDR.is_lb \<delta> s' s1 s2 \<longrightarrow> pre_RDR.less_eq \<delta> s' s)"
heap_is_wellformed_children_distinct: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
yoneda_lemma_fixed_e_axioms: shows "yoneda_lemma_fixed_e (\<cdot>) (\<cdot>\<^sub>S) \<phi> F a (\<E> \<tau>)"
eq_strict_2: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and y :: "'a" assumes "class.Eq_strict eq" shows "eq\<cdot>\<bottom>\<cdot>y = \<bottom>"
sccs_verts_def: shows "H.sccs_verts = {S. S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> u)}"
If_cases: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "finite A" shows "(\<Prod>x\<in>A. if P x then h x else g x) = prod h (A \<inter> {x. P x}) * prod g (A \<inter> - {x. P x})"
in_atms_of_minusD: fixes x :: "'b"   and A :: "'b literal multiset"   and B :: "'b literal multiset" assumes "x \<in> atms_of (A - B)" shows "x \<in> atms_of A"
simps_5: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r \<phi> and\<^sub>r \<psi> = (\<xi> \<Turnstile>\<^sub>r \<phi> \<and> \<xi> \<Turnstile>\<^sub>r \<psi>)"
Subring_zero_ring_zero: fixes S :: "('a, 'd) Ring_scheme" assumes "Subring R S" shows "\<zero>\<^bsub>S\<^esub> = \<zero>"
runit_agreement: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.runit a = \<r>[a]"
tr_trms_subset: fixes A' :: "('fun, 'var) strand_step list"   and A :: "('fun, 'var) stateful_strand_step list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "A' \<in> set (tr A D)" shows "trms\<^sub>s\<^sub>t A' \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` set D"
assms: shows "\<exists>s2'. (if \<mu>1 s1' s1 then trsys2.silent_moves else trsys2.silent_movet) s2 s2' \<and> s1' \<approx> s2'"
simps: fixes a1 :: "'a"   and a2 :: "'c"   and r :: "('a \<times> 'd) set"   and s :: "('d \<times> 'c) set" shows "((a1, a2) \<in> r O s) = (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r \<and> (b, c) \<in> s)"
star_def: fixes P :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool" shows "star P Q = (\<lambda>h. \<exists>h1 h2. h = h1 ++ h2 \<and> h1 \<bottom> h2 \<and> P h1 \<and> Q h2)"
two_pers: fixes w :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "w \<le>p u\<^sup>\<omega>"   and "w \<le>p v\<^sup>\<omega>"   and "\<^bold>|u\<^bold>| + \<^bold>|v\<^bold>| \<le> \<^bold>|w\<^bold>|" shows "u \<cdot> v = v \<cdot> u"
Y: fixes \<rho> :: "nat \<rightarrow> ValD" shows "\<lbrakk>Ycomb\<rbrakk>\<rho> = ValF\<cdot>fixD"
pr1: shows "\<guillemotleft>pr1 a b : local.prod a b \<rightarrow> a\<guillemotright>"
simps_1: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "QE_dnf opt step (And \<phi>\<^sub>1 \<phi>\<^sub>2) = and (QE_dnf opt step \<phi>\<^sub>1) (QE_dnf opt step \<phi>\<^sub>2)"
monom_eq_const_iff: fixes c :: "'a"   and n :: "nat"   and d :: "'a" shows "(monom c n = [:d:]) = (c = d \<and> (c = (0::'a) \<or> n = 0))"
le_fin_reflp: fixes w :: "'item list" shows "w \<preceq>\<^sub>F w"
rel_inject: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'f"   and y :: "'g" shows "rel_nondetT' R1 R2 (NondetT x) (NondetT y) = R2 x y"
simps_20: fixes vb :: "'b"   and vc :: "value list"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (In vb vc) (Gt v va) = False"
mk_minsky_copy2: fixes u :: "nat"   and v :: "nat"   and w :: "nat" assumes "u \<noteq> v"   and "u \<noteq> w"   and "v \<noteq> w" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = u then 0 else if x = v then vs u else if x = w then vs u else vs x))"
prod_vwb_lens: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'g" assumes "vwb_lens X"   and "vwb_lens Y" shows "vwb_lens (X \<times> Y)"
AA_E: fixes X :: "'a set" assumes "X \<in> AA" obtains "X \<subseteq> A" and "f ` X \<subseteq> X" and "\<And>Y s. \<lbrakk>Y \<subseteq> X; extreme_bound A (\<sqsubseteq>) Y s\<rbrakk> \<Longrightarrow> s \<in> X"
pred: shows "is_pred_syz (new_syz_sigs ss bs p) (sig_of_pair p)"
find_crossing_edges_on_path: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "path g u p v"   and "P u"   and "P v" obtains "\<forall>(u, v)\<in>set p. P u \<and> P v"   | u\<^sub>1 :: "'b" and v\<^sub>1 :: "'b" and v\<^sub>2 :: "'b" and u\<^sub>2 :: "'b" and p\<^sub>1 :: "('b \<times> 'b) list" and p\<^sub>2 :: "('b \<times> 'b) list" and p\<^sub>3 :: "('b \<times> 'b) list" where "p = p\<^sub>1 @ [(u\<^sub>1, v\<^sub>1)] @ p\<^sub>2 @ [(u\<^sub>2, v\<^sub>2)] @ p\<^sub>3" and "P u\<^sub>1" and "\<not> P v\<^sub>1" and "\<not> P u\<^sub>2" and "P v\<^sub>2"
inf2D2: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c" assumes "(A \<sqinter> B) x y" shows "B x y"
le_max_image_of_finite: fixes f :: "'c \<Rightarrow> 'b"   and i :: "'c" shows "f i \<le> Max {x. \<exists>i. x = f i}"
system_needed_asses_def: fixes PROB :: "(('a, 'b) fmap \<times> 'c) set"   and s :: "('a, 'b) fmap" shows "system_needed_asses PROB s \<equiv> fmrestrict_set (system_needed_vars PROB s) s"
inj_on_UnI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f A"   and "inj_on f B"   and "f ` (A - B) \<inter> f ` (B - A) = {}" shows "inj_on f (A \<union> B)"
propex\<^sub>2: fixes F :: "\<Pi>\<^sub>2" shows "\<exists>r. Some r = Semantics.d\<^sub>2 F"
redT_updWs_def: fixes t :: "'b" shows "redT_updWs t = rtrancl3p (redT_updW t)"
Fourier_sum_offset_Dirichlet_kernel: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) = (LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) / pi"
tabulation_data_axioms: shows "tabulation_data (\<cdot>) (\<star>) \<a> \<i> src trg r' ((\<phi> \<star> f) \<cdot> \<rho>) f g"
disjoint_unionD2: fixes A :: "'b set set"   and B :: "'b set set" assumes "disjoint (A \<union> B)" shows "disjoint B"
Inequality: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S"   and "ineq_model \<I> X F" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (S @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
foo_sig_def: shows "foo_sig = \<lparr>name = foo, parTs = [Class Base]\<rparr>"
RPDs_Times: fixes r :: "mregex"   and s :: "mregex" shows "RPDs (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesL r (RPDs s) \<union> RPDs r"
assms_5: shows "u \<notin> circline_set x_axis"
tl_obtain_elem: fixes xs :: "'c list" assumes "xs \<noteq> []"   and "tl xs = []" obtains e :: "'c" where "xs = [e]"
hdb_cons_subset_nodes: shows "set (hd b # a ! hd b) \<subseteq> set nodes"
g'_def: shows "g' = \<^bold>g [^] x"
direct_decompE: fixes A :: "'c set"   and ss :: "'c set list"   and a :: "'c" assumes "direct_decomp A ss"   and "a \<in> A" obtains qs :: "'c list" where "qs \<in> listset ss" and "a = sum_list qs"
inst_step_atomicity: shows "\<forall>s a. rcurrent (rstep s a) = rcurrent s"
packing_covering_imp_balance: fixes V :: "'b set"   and B :: "'b set multiset"   and k :: "int"   and t :: "int"   and \<Lambda>\<^sub>t :: "int" assumes "t_packing_design V B k t \<Lambda>\<^sub>t"   and "t_covering_design V B k t \<Lambda>\<^sub>t" shows "t_wise_balance V B t \<Lambda>\<^sub>t"
flow_in_domain: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> Y.existence_ivl t0 x0" shows "Y.flow t0 x0 t \<in> Y"
mcont_times_ennreal2: fixes x :: "ennreal" shows "mcont Sup (\<le>) Sup (\<le>) (\<lambda>y. y * x)"
return_bind: fixes x :: "unit"   and f :: "unit \<Rightarrow> 'm" shows "bind (return x) f = f x"
continuous_on_subset: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and t :: "'c set" assumes "continuous_on s f"   and "t \<subseteq> s" shows "continuous_on t f"