hcju/isabelleps huggingface dataset:
awalk_verts_append2: fixes u :: "'a"   and p :: "'b list"   and q :: "'b list"   and v :: "'a" assumes "awalk u (p @ q) v" shows "awalk_verts u (p @ q) = butlast (awalk_verts u p) @ awalk_verts (awlast u p) q"
fast_filter_def: fixes a :: "nat list"   and b :: "nat list" shows "fast_filter a b = filter (post_cond a b) (map (\<lambda>(x, y). (fst x, fst y)) (tl (c12_generate_check a b a b)))"
openin_inject: fixes x :: "'a topology"   and y :: "'a topology" shows "(openin x = openin y) = (x = y)"
assms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
similar_mat_four_block_0_0: fixes A1 :: "'a mat"   and B1 :: "'a mat"   and A2 :: "'a mat"   and B2 :: "'a mat"   and n :: "nat"   and m :: "nat" assumes "similar_mat A1 B1"   and "similar_mat A2 B2"   and "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat m m" shows "similar_mat (four_block_mat A1 (0\<^sub>m n m) (0\<^sub>m m n) A2) (four_block_mat B1 (0\<^sub>m n m) (0\<^sub>m m n) B2)"
lexord_eq_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "lexord_eq_fusion g1 g2 s1 s2 = lexordp_eq (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
ras_shrink_def: fixes s :: "'c array \<times> nat" shows "ras_shrink s \<equiv> let (a, n) = s; a = if 128 * n \<le> array_length a \<and> 4 < n then array_shrink a n else a in (a, n)"
FDTs: shows "map_of FDTs (F, D) = \<lfloor>(b, T)\<rfloor>"
ring_hom_cring_axioms_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "ring_hom_cring_axioms R S h \<equiv> h \<in> ring_hom R S"
mult_le_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m \<le> k * n) = (0 < k \<longrightarrow> m \<le> n)"
members_of_only_horiz_div_line_integrable': fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "only_horizontal_division one_chain two_chain_typeII"   and "(k, \<gamma>) \<in> one_chain"   and "(k, \<gamma>) \<in> one_chain"   and "finite two_chain_typeII"   and "\<forall>two_cube\<in>two_chain_typeII. valid_two_cube two_cube" shows "line_integral_exists F {j} \<gamma>"
assms_3: shows "g \<turnstile> n-ns\<rightarrow>m"
widen_antisym: fixes m :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and T :: "ty"   and U :: "ty" assumes "wf_prog m P"   and "subtype P T U"   and "subtype P U T" shows "T = U"
fold_set_fold_remdups: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and y :: "'h"   and xs :: "'g list" assumes "comp_fun_commute f" shows "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
intro: fixes M :: "'d \<Rightarrow> 'e measure" assumes "\<And>i. sigma_finite_measure (M i)" shows "product_sigma_finite M"
ss: shows "{#b1, b2#} \<subseteq># filter_mset ((\<subseteq>) {x, y}) \<B>"
goodInp_def: fixes inp :: "'c \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "goodInp inp \<equiv> liftAll good inp \<and> |{i. inp i \<noteq> None}| <o |UNIV|"
infi: fixes x__ :: "'a"   and x'__ :: "'b"   and x :: "'a"   and x' :: "'b" assumes "(x__, x'__) \<in> A"   and "(x, x') \<in> A" shows "(infi x__ x, inf x'__ x') \<in> A"
da_a: fixes x :: "'a" shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> (a x) = a x"
floor_real_def: fixes x :: "real" shows "\<lfloor>x\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
rel_intros_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" assumes "R x2 y2" shows "rel_option R \<lfloor>x2\<rfloor> \<lfloor>y2\<rfloor>"
powr_diff: fixes w :: "'b"   and z1 :: "'b"   and z2 :: "'b" shows "w powr (z1 - z2) = w powr z1 / w powr z2"
sum_diff1_nat: fixes f :: "'b \<Rightarrow> nat"   and A :: "'b set"   and a :: "'b" shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
foldl_flip_Cons_append: fixes ys :: "['b]"   and xs :: "['b]" shows "foldl\<cdot>(flip\<cdot>Cons)\<cdot>ys\<cdot>xs = foldl\<cdot>(flip\<cdot>Cons)\<cdot>[]\<cdot>xs ++ ys"
5: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G"   and "g \<notin> M"   and "q'' g \<odot> g \<noteq> 0" shows "lt (q'' g \<odot> g) \<prec>\<^sub>t v"
simps_54: fixes x6 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type" shows "OrderedSet x6 \<noteq> Collection x4"
game2'_def: shows "game2' = (\<lambda>\<A>. key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (oracle_encrypt2' key b \<oplus>\<^sub>O oracle_decrypt2 key) \<A> (False, Map_empty) \<bind> (\<lambda>(b', bad, D). return_spmf (b = b', bad)))))"
"3_2": shows "fold (count cs) u (count cs (shd w) k) = l"
finite_freeVarsFL: fixes gamma :: "formula list" shows "finite (freeVarsFL gamma)"
surj_plus: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.ab_group_add plus zero minus uminus" shows "surj (plus a)"
awalk_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "S.awalk u p v \<equiv> u \<in> pverts (mk_symmetric G) \<and> set p \<subseteq> parcs (mk_symmetric G) \<and> pcas u p v"
nm_F: fixes y :: "'a"   and N' :: "'a stream measure"   and M' :: "'a stream measure" assumes "y \<in> set_pmf (p N' M')"   and "(N', M') \<in> F" shows "(n N' M' y, m N' M' y) \<in> F"
assm: shows "\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> |lan va| = 1 \<and> (\<forall>c. \<parallel>len va ts c\<parallel> = 0 \<or> restrict va (clm ts) c = bot \<and> restrict va (res ts) c = bot)) \<and> 0 < \<parallel>ext u\<parallel> \<and> |lan u| = 1 \<and> (\<forall>c. \<parallel>len u ts c\<parallel> = 0 \<or> restrict u (clm ts) c = bot \<and> restrict u (res ts) c = bot)"
matroid_ax_3: fixes X :: "Points set"   and Y :: "Points set" shows "rk (X \<union> Y) + rk (X \<inter> Y) \<le> rk X + rk Y"
AxiomR: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and Q :: "('fun, 'var) Term.term \<Rightarrow> bool" assumes "t \<in> M" shows "\<langle>M;Q\<rangle> \<turnstile>\<^sub>r t"
blinfun_minus_comp_distrib: fixes a :: "'d \<Rightarrow>\<^sub>L 'c"   and b :: "'d \<Rightarrow>\<^sub>L 'c"   and c :: "'b \<Rightarrow>\<^sub>L 'd" shows "a - b o\<^sub>L c = (a o\<^sub>L c) - (b o\<^sub>L c)"
igSubstIGAbs_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstIGAbs MOD \<equiv> \<forall>ys y Y xs x s X. isInBar (xs, s) \<and> igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
rest: shows "Strat = take n Strat @ [Strat ! n] @ rest"
diff_fun_axioms: shows "diff_fun k (charts_submanifold W') f'"
enc_reflects_binary_pred_iff_indRelR_reflects_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "enc_reflects_binary_pred Pred = rel_reflects_binary_pred indRelR Pred"
NB_Crypt_imp_Alice_msg: fixes A :: "agent"   and NB :: "nat"   and evs :: "event list" assumes "Crypt (shrK A) (Nonce NB) \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> woolam" shows "\<exists>B. Says A B (Crypt (shrK A) (Nonce NB)) \<in> set evs"
safe_distance_3r_def: shows "safe_distance_3r = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / 2 / a\<^sub>e - v\<^sub>o * \<delta> - 1 / 2 * a\<^sub>o * \<delta>\<^sup>2"
get_dom_component_no_overlap: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_component :: "(_) 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"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'" shows "set c \<inter> set c' = {} \<or> c = c'"
intro: fixes permute :: "'c perm \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x. permute 0 x = x"   and "\<And>p q x. permute (p + q) x = permute p (permute q x)" shows "permute permute"
list_all2_nthD2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length ys" shows "P (xs ! p) (ys ! p)"
Inf_eq_PInfty: fixes S :: "ereal set" shows "(Inf S = \<infinity>) = (S = {} \<or> S = {\<infinity>})"
infinite_super: fixes S :: "'e set"   and T :: "'e set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
neq_def: fixes c :: "int"   and t :: "MIR.num" shows "neq c t = (if c = 0 then NEq t else if 0 < c then NEq (CN 0 c t) else NEq (CN 0 (- c) (Neg t)))"
mod_mod_power: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ min m n"
const_mem_subst_cases: fixes c :: "'a"   and \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and M :: "('a, 'c) Term.term set" assumes "Fun c [] \<in> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>" shows "Fun c [] \<in> M \<or> Fun c [] \<in> \<theta> ` fv\<^sub>s\<^sub>e\<^sub>t M"
coinduct: fixes X :: "'transition stream \<Rightarrow> 'state \<Rightarrow> bool"   and x1 :: "'transition stream"   and x2 :: "'state" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>a p r. x1 = a ## r \<and> x2 = p \<and> enabled a p \<and> (X r (execute a p) \<or> run r (execute a p))" shows "run x1 x2"
Class_widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list" shows "P \<turnstile> Class C \<le> Class D = P \<turnstile> C \<preceq>\<^sup>* D"
coboundedI2: fixes c :: "'d multiset"   and b :: "'d multiset"   and a :: "'d multiset" assumes "c \<subseteq># b" shows "c \<subseteq># a \<union># b"
redT_preserves: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" assumes "lifting_wf final r P"   and "multithreaded_base.redT final r convert_RA s (t, ta) s'"   and "ts_ok P (thr s) (shr s)" shows "ts_ok P (thr s') (shr s')"
symmetry_preserves_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and M :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "M Midpoint A A'"   and "M Midpoint B B'"   and "M Midpoint C C'" shows "A B C CongA A' B' C'"
permute_fun_def: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'd" shows "p \<bullet> f = (\<lambda>x. p \<bullet> f (- p \<bullet> x))"
reflect_poly_irreducible: fixes p :: "int poly"   and x :: "'a" assumes "irreducible p"   and "p represents x"   and "x \<noteq> (0::'a)" shows "irreducible (reflect_poly p)"
analytic_on_inverse: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f analytic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" shows "(\<lambda>z. inverse (f z)) analytic_on S"
order_less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y < z" shows "x < z"
homologous_rel_singular_chain: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S a b" shows "singular_chain p X a = singular_chain p X b"
set_of_regions_spec: fixes R :: "('c \<Rightarrow> real) set"   and v :: "'c \<Rightarrow> real"   and R' :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>"   and "v \<in> R"   and "R' \<in> Succ \<R> R" shows "\<exists>t\<ge>0. [v \<oplus> t]\<^sub>\<R> = R'"
ASum_Suc: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" shows "ASum f (Suc n) = ASum f n + f (Suc n)"
hash_blinded_simp: fixes h' :: "'b \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a"   and st :: "'c rose_tree" shows "hash_tree h' (blind_source_tree h st) = hash_source_tree h st"
sup_least: fixes y :: "'c \<times> 'd"   and x :: "'c \<times> 'd"   and z :: "'c \<times> 'd" assumes "less_eq_prod y x"   and "less_eq_prod z x" shows "less_eq_prod (sup_prod y z) x"
P''Trans: shows "\<Psi> \<rhd> P'' \<longmapsto> \<alpha> \<prec> P'"
waiting_switch_independent: shows "\<forall>n s. waiting (cswitch n s) = waiting s"
factorize_int_last_nz_poly_def: fixes f :: "int poly" shows "local.factorize_int_last_nz_poly f = (let df = degree f in if df = 0 then (coeff f 0, []) else if df = 1 then (content f, [(primitive_part f, 0)]) else local.internal_int_poly_factorization f)"
check_tableau_index_valid: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "index_valid as (check s) = index_valid as s"
quotient_is_ring: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R" shows "ring (R Quot I)"
t_2: fixes x :: "nat" shows "eval t [x] = t' x"
vcg_conj: fixes I :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and c :: "('answer, 'location, 'question, 'state) com"   and Q :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and R :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<lbrace>I\<rbrace> c \<lbrace>Q\<rbrace>"   and "\<lbrace>I\<rbrace> c \<lbrace>R\<rbrace>" shows "\<lbrace>I\<rbrace> c \<lbrace>\<lambda>s. Q s \<and> R s\<rbrace>"
Cod_comp: fixes f :: "('obj, 'obj) concrete_category.arr"   and g :: "('obj, 'obj) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Cod (g \<cdot> f) = Cod g"
lowner_le_keep_under_measurement: fixes M :: "complex mat"   and n :: "nat"   and A :: "complex mat"   and B :: "complex mat" assumes "M \<in> carrier_mat n n"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A \<le>\<^sub>L B" shows "adjoint M * A * M \<le>\<^sub>L adjoint M * B * M"
right_unique_rel_resource: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A"   and "right_unique B" shows "right_unique (rel_resource A B)"
S3Hist: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> HNext rmhist p \<and> $S3 rmhist p \<and> unchanged r p \<longrightarrow> unchanged (rmhist!p)"
th0: fixes x :: "nat \<Rightarrow> 'b vec"   and y :: "nat \<Rightarrow> 'b vec" shows "mat\<^sub>r n n (\<lambda>i. if i \<in> {0..<n} then x i else y i) = mat\<^sub>r n n x"
qs1_in: shows "qs1 \<in> listset (s # ss)"
simps_2: fixes s1 :: "'d stream"   and s2 :: "'d stream" shows "stl (sinterleave s1 s2) = sinterleave s2 (stl s1)"
int_of_uint8_def: shows "int_of_uint8 \<equiv> map_fun Rep_uint8 id uint"
ln_bound: fixes x :: "real" assumes "0 < x" shows "ln x \<le> x"
convergent_prod_imp_convergent: fixes f :: "nat \<Rightarrow> 'e" assumes "convergent_prod f" shows "convergent (\<lambda>n. prod f {..n})"
arr_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g)"
irreducible\<^sub>d_factorization_exists_1: fixes p :: "'b poly" assumes "0 < degree p" shows "\<exists>fs. fs \<noteq> [] \<and> (\<forall>f\<in>set fs. irreducible\<^sub>d f \<and> degree f \<le> degree p) \<and> p = prod_list fs"
i_def: shows "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
less_linear: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec>\<^bsub>D\<^esub> b \<or> a = b \<or> b \<prec>\<^bsub>D\<^esub> a"
s_prod_test_aux1: fixes S :: "('a \<times> 'a set) set"   and a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set" assumes "S \<subseteq> 1\<^sub>\<sigma>"   and "(a, A) \<in> R \<cdot> S" shows "(a, A) \<in> R \<and> (\<forall>a\<in>A. (a, {a}) \<in> S)"
sgn_euclidean_space_def: fixes x :: "'a euclidean_space" shows "sgn x = x /\<^sub>R norm x"
AbstAtomicP_Eq: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstAtomicP v i (Q_Eq x y) (Q_Eq x' y')"
is_strongly_scdom_component_safe_def: shows "Core_DOM_SC_DOM_Components.is_strongly_scdom_component_safe \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_scdom_component_safe Core_DOM_Functions.get_owner_document get_disconnected_nodes Core_DOM_Functions.to_tree_order"
quality_increases_rreq_rrep_props: fixes \<sigma> :: "'a \<Rightarrow> state"   and sip :: "'a"   and \<sigma>' :: "'a \<Rightarrow> state"   and sn :: "nat"   and ip :: "nat"   and hops :: "nat" assumes "quality_increases (\<sigma> sip) (\<sigma>' sip)"   and "1 \<le> sn"   and "ip \<in> kD (rt (\<sigma> sip)) \<and> sn \<le> nsqn (rt (\<sigma> sip)) ip \<and> (nsqn (rt (\<sigma> sip)) ip = sn \<longrightarrow> the (dhops (rt (\<sigma> sip)) ip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) ip) = Aodv_Basic.inv)" shows "ip \<in> kD (rt (\<sigma>' sip)) \<and> sn \<le> nsqn (rt (\<sigma>' sip)) ip \<and> (nsqn (rt (\<sigma>' sip)) ip = sn \<longrightarrow> the (dhops (rt (\<sigma>' sip)) ip) \<le> hops \<or> the (flag (rt (\<sigma>' sip)) ip) = Aodv_Basic.inv)"
thm_cont_e_1: fixes v :: "i" shows "[\<^bold>\<diamond>(\<^bold>\<exists>x. \<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>) in v]"
imageE: fixes b :: "'a::type"   and f :: "'b::type \<Rightarrow> 'a::type"   and A :: "'b::type set" assumes "(b::'a::type) \<in> (f::'b::type \<Rightarrow> 'a::type) ` (A::'b::type set)" obtains x :: "'b::type" where "(b::'a::type) = (f::'b::type \<Rightarrow> 'a::type) (x::'b::type)" and "(x::'b::type) \<in> (A::'b::type set)"
simps_1: fixes p :: "real mpoly" shows "aNeg (Less p) = Leq (- p)"
hyps_3: assumes "P h'"   and "final h''"   and "tame h''" shows "h' [f']\<rightarrow>* h''"
cut_greater_ge_inext_conv: fixes t :: "nat"   and I :: "nat set" assumes "t \<in> I"   and "t \<noteq> Max I \<or> infinite I" shows "I \<down>> t = I \<down>\<ge> inext t I"
algebraic_uminus: fixes x :: "'b" assumes "algebraic x" shows "algebraic (- x)"
finite_measure_Union: fixes A :: "'b set"   and M :: "'b pmf"   and B :: "'b set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M"   and "A \<inter> B = {}" shows "measure_pmf.prob M (A \<union> B) = measure_pmf.prob M A + measure_pmf.prob M B"
newton_coefficients_def: shows "local.newton_coefficients = map hd (local.newton_coefficients_main (rev fs) (rev xs))"
braun_indices1: fixes t :: "'b tree" shows "0 \<notin> braun_indices t"
Seq: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and c\<^sub>1 :: "('f, 'e, 'g) com"   and R :: "'f set"   and A :: "'f set"   and c\<^sub>2 :: "('f, 'e, 'g) com"   and Q :: "'f set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 R,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Seq c\<^sub>1 c\<^sub>2 Q,A"
delta_pos_neg_least_2: fixes rm :: "('am \<times> 'bm) admS \<times> ('av \<times> 'bv) admS"   and rp :: "('am \<times> 'bm) admS \<times> ('av \<times> 'bv) admS" assumes "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "delta_pos \<le> rp"
distinct: fixes i :: "nat" shows "p i = 0 \<or> p i = 1"
assms: shows "unifier\<^sub>t\<^sub>s \<sigma> (literal_to_term ` L)"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "ide a"   and "ide a'"   and "\<guillemotleft>g : \<I> \<otimes> a' \<rightarrow> \<I> \<otimes> a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> \<I> \<otimes> f = g"
mult_unit_dvd_iff': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd (1::'a)" shows "(a * b dvd c) = (b dvd c)"
matrix_equality: shows "mat_of_rows n (map of_int_hom.vec_hom fs) = gs1.M (length (map of_int_hom.vec_hom fs)) * mat_of_rows n (map gs1.gso [0..<length (map of_int_hom.vec_hom fs)])"
retraction_map_snd: fixes X :: "'b topology"   and Y :: "'c topology" shows "retraction_map (prod_topology X Y) Y snd = (topspace X = {} \<longrightarrow> topspace Y = {})"
eq: shows "rev ([b] \<cdot> [a] \<^sup>@ m) = [a] \<^sup>@ m \<cdot> [b]"
prism_suml_def: shows "Inl\<^sub>\<triangle> = \<lparr>prism_match = \<lambda>v. case v of Inl x \<Rightarrow> Some x | Inr x \<Rightarrow> Map.empty x, prism_build = Inl\<rparr>"
assms: shows "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>"
intros: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"