hcju/isabelleps huggingface dataset:
Prim_Invar2_ctd_Presentation_Loc_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" shows "Prim_Invar2_ctd_Presentation_Loc w g r Q \<pi> \<equiv> Prim2.prim_invar2_ctd w g r Q \<pi>"
xivp_2: shows "Y\<^sub>1 t\<^sub>0 = s"
sub_conjI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P Q. \<lbrakk>sound P; sound Q\<rbrakk> \<Longrightarrow> t P && t Q \<tturnstile> t (P && Q)" shows "sub_conj t"
ce_rel_lm_20: fixes x :: "nat"   and r :: "(nat \<times> nat) set" shows "((c_fst x, c_snd x) \<in> r) = (x \<in> ce_rel_to_set r)"
g: shows "\<guillemotleft>g : trg\<^sub>C f \<rightarrow>\<^sub>C src\<^sub>C f\<guillemotright> \<and> C.ide g \<and> D.isomorphic (F g) g'"
crit_pair_cbelow_same: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'e"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m" shows "crit_pair_cbelow_on d m F p p"
PartialMeetContraction_axioms: shows "PartialMeetContraction Cn \<gamma>\<^sub>A\<^sub>G\<^sub>M"
path_trans: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and l3 :: "'vtx"   and ys :: "('vtx \<times> 'lbl \<times> 'vtx) list" assumes "path l1 l2 xs"   and "path l2 l3 ys" shows "path l1 l3 (xs @ ys)"
subgroup_incl: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I (G\<lparr>carrier := S\<rparr>)"   and "subgroup J (G\<lparr>carrier := S\<rparr>)"   and "I \<subseteq> J" shows "subgroup I (G\<lparr>carrier := S, carrier := J\<rparr>)"
distinct_9: shows "control_var\<^sub>C \<noteq> reg\<^sub>C"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
preserves_arr: fixes f :: "'s" assumes "S.arr f" shows "S'.arr (\<Phi> f)"
rel_setI: fixes A :: "'b set"   and B :: "'c set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
nat_dvd_not_less: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m < n" shows "\<not> n dvd m"
prod_pow: fixes p :: "'b"   and n :: "nat" shows "(\<Prod>i = 0..<n. p) = p ^ n"
less_plus_elim: fixes i :: "nat"   and n1 :: "nat"   and n2 :: "nat" assumes "i < n1 + n2" obtains "i < n1"   | i2 :: "nat" where "i2 < n2" and "i = n1 + i2"
domintros_3: fixes uw :: "(char list, char list) id0" shows "is_value_dom (Var uw)"
assert_opt_def: fixes v :: "'c option" shows "assert_opt v \<equiv> case v of None \<Rightarrow> fail | Some v \<Rightarrow> return v"
filter_mset_eq_conv: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c multiset"   and N :: "'c multiset" shows "(filter_mset P M = N) = (N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a))"
ind': assumes "\<forall>g\<in>set (take n gs). terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))" shows "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
abs_eq_iff: fixes x :: "(11, 52) IEEE.float"   and y :: "(11, 52) IEEE.float" shows "(abs_double x = abs_double y) = (IEEE.is_nan x \<and> IEEE.is_nan y \<or> x = y)"
gf_U'_tendsto_U': fixes x :: "'s"   and y :: "'s" assumes "recurrent x"   and "(x, y) \<in> acc" shows "((\<lambda>z. ennreal (gf_U' x y z)) \<longlongrightarrow> U' x y) (at_left 1)"
BB: fixes i :: "nat" assumes "i < n" shows "(B * B) $$ (i, i) = (B $$ (i, i))\<^sup>2"
HPhase1or2ReadElse_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d r"   and "HInv4c s q" shows "HInv4c s' q"
l_get_dom_component_get_element_by_id\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_element_by_id\<^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_dom_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_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
ipoly_nth_root: fixes p :: "int poly"   and x :: "'b" shows "ipoly (local.poly_nth_root p) x = ipoly p (x ^ n)"
right_diff_distrib: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b - c) = a * b - a * c"
filt_equiv_sets: fixes F :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure"   and N :: "'d measure" assumes "filt_equiv F M N" shows "sets M = sets N"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Cod t) (Cod t) (Cod t)"
optimize_primitive_univ_correct_matchexpr: fixes \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and m :: "'b common_primitive match_expr" shows "matches (common_matcher, \<alpha>) m = matches (common_matcher, \<alpha>) (optimize_primitive_univ m)"
eventually_subst: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<forall>\<^sub>F n in F. P n = Q n" shows "eventually P F = eventually Q F"
finite_paramst_1: fixes t :: "'a FOL_Fitting.term" shows "finite (paramst t)"
sup_idem: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "class.semilattice_sup sup less_eq less" shows "sup x x = x"
fp_def: shows "fp = (\<lambda>x. max (f x) 0)"
not_is_red_empty: fixes f :: "'t \<Rightarrow>\<^sub>0 'c" shows "\<not> is_red {} f"
ifte_def: fixes a :: "'c"   and x :: "'c"   and y :: "'c" shows "ifte a x y = a \<sqinter> x \<squnion> - a \<sqinter> y"
rec_enclen_def: shows "rec_enclen = CN (rec_max1 (CN rec_not [CN rec_inside [recf.Id 2 1, CN rec_predecessor [recf.Id 2 0]]])) [recf.Id 1 0, recf.Id 1 0]"
mult_ge1_mono_1: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "(1::'b) \<le> b" shows "a \<le> a * b"
map_undefined: fixes a :: "'a" assumes "a \<notin> G" shows "contained.Class a = undefined"
assms_2: assumes "\<phi> \<pi>l"   and "\<psi> \<pi>l" shows "\<chi>"
assms_8: shows "\<not> is_sig_red (\<prec>\<^sub>t) (=) G p"
append: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and xs :: "'e list"   and ys :: "'e list" assumes "monoid_list f z" shows "monoid_list.F f z (xs @ ys) = f (monoid_list.F f z xs) (monoid_list.F f z ys)"
subtract_too_much: fixes I :: "\<I>"   and i :: "nat" assumes "(case right I of enat n \<Rightarrow> n | \<infinity> \<Rightarrow> left I) < i" shows "subtract i I = subtract (case right I of enat n \<Rightarrow> n | \<infinity> \<Rightarrow> left I) I"
lm088: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "Outside' X R \<subseteq> R"
aexp_induct_separate_V_cases: fixes P :: "vname aexp \<Rightarrow> bool"   and a :: "vname aexp" assumes "\<And>x. P (L x)"   and "\<And>x. P (V (I x))"   and "\<And>x. P (V (R x))"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Plus x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Minus x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Times x1a x2a)" shows "P a"
list_ball_nth: fixes n :: "nat"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "n < length xs"   and "\<forall>x\<in>set xs. P x" shows "P (xs ! n)"
is_sentence_cons: fixes x :: "symbol"   and xs :: "symbol list" shows "is_sentence (x # xs) = (is_symbol x \<and> is_sentence xs)"
r2s_inj: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R \<subseteq> Id"   and "S \<subseteq> Id"   and "r2s R = r2s S" shows "R = S"
projection_on_subset: fixes Y :: "'b set"   and X :: "'b set"   and l :: "'b list" assumes "Y \<subseteq> X"   and "l \<upharpoonleft> X = []" shows "l \<upharpoonleft> Y = []"
projl_def: fixes sum :: "'c + 'd" shows "projl sum = (case sum of Inl x1 \<Rightarrow> x1)"
game3_def: shows "game3 = (\<lambda>f x y. coin_spmf \<bind> (\<lambda>b. exec_gpv hash_oracle'' (\<A>1 (\<^bold>g [^] x)) ({}, Map.empty) \<bind> (\<lambda>(((msg1, msg2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains msg1 msg2) \<bind> (\<lambda>_. f b msg1 msg2 (spmf_of_set (nlists UNIV len_plain)) \<bind> (\<lambda>h'. let cipher = (\<^bold>g [^] y, h') in exec_gpv hash_oracle'' (\<A>2 cipher \<sigma>) (s, s_h) \<bind> (\<lambda>(guess, s', s_h'). return_spmf (guess = b, \<^bold>g [^] (x * y) \<in> s')))) ELSE (coin_spmf \<bind> (\<lambda>b. return_spmf (b, \<^bold>g [^] (x * y) \<in> s))))))"
fconst_on_vintersection: fixes A :: "V"   and B :: "V"   and c :: "V" shows "fconst_on (A \<inter>\<^sub>\<circ> B) c = fconst_on A c \<inter>\<^sub>\<circ> fconst_on B c"
2: shows "coreflexive ((v \<sqinter> - F * e * top \<sqinter> top * e\<^sup>T * F) * top * (v \<sqinter> - F * e * top \<sqinter> top * e\<^sup>T * F)\<^sup>T)"
foundation1: fixes \<tau> :: "'a state \<times> 'a state" shows "\<tau> \<Turnstile> true"
ctr_simps_UNIV_def: shows "{x. True} = UNIV"
equivD1: fixes x :: "'a"   and y :: "'a" assumes "x \<approx> y" shows "x \<le> y"
int_right_shift_def: fixes x :: "int \<Rightarrow> 'b" shows "int_right_shift x = (\<lambda>i. x (i - 1))"
Cd_Xd: fixes x :: "'x"   and d :: "'d"   and X :: "'x set" assumes "x \<in> Cd d X" shows "Xd x = d"
float_distinct_8: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_normal a \<and> is_denormal a)"
terminates_rbt_entries_generator: shows "terminates (rbt_has_next, rbt_entries_next)"
path_SeqSecond: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node"   and c\<^sub>1 :: "cmd" assumes "While_CFG.path prog n as n'"   and "n \<noteq> (_Entry_)"   and "as \<noteq> []" shows "While_CFG.path (c\<^sub>1;; prog) (n \<oplus> #:c\<^sub>1) (as \<oplus>s #:c\<^sub>1) (n' \<oplus> #:c\<^sub>1)"
h2b: shows "filter (\<lambda>y. y < dim_col A) (map snd (pivot_positions A)) = map snd (pivot_positions A)"
subcocycle_avg_ereal_birkhoff: fixes u :: "'a \<Rightarrow> real" assumes "integrable M u" shows "subcocycle_avg_ereal (birkhoff_sum u) = ereal (integral\<^sup>L M u)"
simps: fixes a :: "'a set"   and A :: "'a set set" shows "(a \<in> tiling A) = (a = {} \<or> (\<exists>a t. a = a \<union> t \<and> a \<in> A \<and> t \<in> tiling A \<and> a \<inter> t = {}))"
h: shows "\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))"
region_unique: fixes v :: "'a \<Rightarrow> real"   and R :: "('a \<Rightarrow> real) set" assumes "v \<in> R"   and "R \<in> \<R>" shows "[v]\<^sub>\<R> = R"
step_thread_eq_None_conv: fixes update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" shows "(step_thread update_state s t = None) = (t \<notin> active_threads s)"
bitlen_Float: fixes m :: "int"   and e :: "int" shows "bitlen \<bar>mantissa (Float m e)\<bar> + exponent (Float m e) = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
pointermap_sane_def: fixes m :: "('b, 'c) pointermap_scheme" shows "pointermap_sane m \<equiv> distinct (entries m) \<and> (\<forall>n\<in>{..<length (entries m)}. getentry m (entries m ! n) = Some n) \<and> (\<forall>p i. getentry m p = Some i \<longrightarrow> entries m ! i = p \<and> i < length (entries m))"
totient_even: fixes n :: "nat" assumes "2 < n" shows "even (totient n)"
s_def: shows "s \<equiv> \<lambda>b. if findr b \<down>= e_length b then Some 0 else Some (Suc (the (findr b)))"
skew_invar: fixes t :: "('b \<times> nat) tree" assumes "invar t" shows "skew t = t"
iter_state_unfold: fixes f :: "'a \<Rightarrow> ('d, 'e) state"   and x :: "'a" shows "iter_state f x = (if cnt x then f x \<bind> (\<lambda>_. iter_state f (nxt x)) else State_Monad.return ())"
isodistinct_remisodups: fixes Us :: "('a, 'g) aezfun set list"   and V :: "('a, 'g) aezfun set"   and W :: "('a, 'g) aezfun set" assumes "\<forall>U\<in>set Us. FGModule G (*) U"   and "V \<in> set (remisodups Us)"   and "W \<in> set (remisodups Us)"   and "V \<noteq> W" shows "\<nexists>T. FGModuleIso G (*) V (*) T W"
c_2: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "stateful_strand_step.is_Receive stateful_strand_step" shows "receive\<langle>the_msg stateful_strand_step\<rangle> = stateful_strand_step"
lam: shows "int (size (filter_mset ((\<subseteq>) ps) \<B>)) = \<Lambda>"
assms_1: shows "v \<succeq>[\<R>] u \<and> (u, v) \<notin> \<R>"
plus_uexpr_def: fixes u :: "('a, 'b) uexpr"   and v :: "('a, 'b) uexpr" shows "u + v = bop (+) u v"
inducts: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and x :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "reachablep A I x"   and "\<And>s. s \<in> init A \<Longrightarrow> P s"   and "\<And>s a s'. \<lbrakk>reachablep A I s; P s; (s, a, s') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P s'" shows "P x"
div_by_0: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a zero = zero"
induced_vector_def: fixes v :: "'v" shows "induced_vector v \<equiv> if v \<in> V then \<lambda>y. if y \<in> FH then BaseRep.FG_proj y \<cdot> v else (0::'v) else 0"
welltypedE_1: fixes x :: "char list \<times> Unary_PCF.type"   and T :: "Unary_PCF.type" assumes "\<langle>x\<rangle> ::: T" obtains xa :: "char list" where "x = (xa, T)"
rel_symp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symp R" shows "symp (list_all2 R)"
fps_conv_radius_divide: fixes g :: "complex fps"   and f :: "complex fps"   and r :: "ereal" assumes "subdegree g \<le> subdegree f"   and "0 < fps_conv_radius f"   and "0 < fps_conv_radius g"   and "0 < r"   and "\<And>z. \<lbrakk>z \<in> eball 0 r; z \<noteq> 0\<rbrakk> \<Longrightarrow> eval_fps g z \<noteq> 0" shows "Min {r, fps_conv_radius f, fps_conv_radius g} \<le> fps_conv_radius (f / g)"
Sup_SUP_eq: fixes S :: "('b \<Rightarrow> bool) set" shows "Sup S = (\<lambda>x. x \<in> \<Union> (Collect ` S))"
index_permute_list: fixes \<pi> :: "nat \<Rightarrow> nat"   and xs :: "'b list"   and x :: "'b" assumes "\<pi> permutes {..<length xs}"   and "distinct xs"   and "x \<in> set xs" shows "index (permute_list \<pi> xs) x = inv \<pi> (index xs x)"
crename_ImpR: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and c :: "coname"   and N :: "trm"   and d :: "coname" assumes "R[a\<turnstile>c>b] = ImpR x.<c>.N d"   and "c \<sharp> (R, a, b)"   and "d \<sharp> (a, b)"   and "x \<sharp> R" shows "\<exists>N'. R = ImpR x.<c>.N' d \<and> N'[a\<turnstile>c>b] = N"
lock_ok_las'_into_lock_on_las: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and las :: "'a lock_actions" assumes "lock_ok_las' ls t las"   and "\<And>l. l \<in> collect_locks' las \<Longrightarrow> may_lock (ls $ l) t" shows "lock_ok_las ls t las"
oarrivemsg_simps_4: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and i1 :: "nat"   and i2 :: "nat" shows "oarrivemsg P \<sigma> (connect(i1, i2)) = True"
Sup_level: fixes L :: "level set" shows "\<Squnion> L \<equiv> if Hi \<in> L then Hi else Lo"
coeff_nonzero_dim_vec_non_zero_1: fixes v :: "rat vec"   and i :: "nat" assumes "Abstract_Linear_Poly.coeff (vec_to_lpoly v) i \<noteq> 0" shows "v $ i \<noteq> 0"
path_connectedin_singleton: fixes X :: "'b topology"   and a :: "'b" shows "path_connectedin X {a} = (a \<in> topspace X)"
rell_FGco_def: fixes L1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool" shows "rell_FGco L1 = rel_FGco L1 (=) (=) (=) (=) (=) (=) (=) (=) (=) (=)"
l_coset_eq_set_mult: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and H :: "'e set" shows "x <#\<^bsub>G\<^esub> H = {x} <#>\<^bsub>G\<^esub> H"
Bex_set_replicate: fixes n :: "nat"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>set (replicate n a). P x) = (P a \<and> n \<noteq> 0)"
sublist_skip: fixes l1 :: "'a list"   and h :: "'a"   and l1' :: "'a list"   and l2 :: "'a list"   and l2a :: "'a list"   and l2b :: "'a list" assumes "l1 = h # l1'"   and "l2 = l2a @ [h] @ l2b"   and "subseq l1 l2"   and "\<not> ListMem h l2a" shows "subseq l1 (h # l2b)"
pq_order_normal_series: fixes q :: "nat"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := H\<rparr>))"   and "order (G\<lparr>carrier := H\<rparr>) = q * p"   and "prime p"   and "prime q"   and "p < q" shows "normal_series (G\<lparr>carrier := H\<rparr>) [{\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>}, THE H. H \<in> Hgroup.subgroups_of_size q, carrier (G\<lparr>carrier := H\<rparr>)]"
has_derivative_mult: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. f x * g x) has_derivative (\<lambda>h. f x * g' h + f' h * g x)) (at x within s)"
cases: fixes a :: "(coname \<times> ty) list" assumes "validc a" obtains "aa__ = []"   | a :: "coname" and \<Delta> :: "(coname \<times> ty) list" and T :: "ty" where "aa__ = (a, T) # \<Delta>" and "a \<sharp> \<Delta>" and "validc \<Delta>"
simps_1: fixes i :: "nat"   and s :: "'d"   and R :: "nat set" shows "net_ips (NodeS i s R) = {i}"
assms_3: assumes "P = \<bottom>" shows "Q \<noteq> FF"
superposition_def: fixes P1 :: "'a eclause"   and P2 :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" shows "superposition P1 P2 C \<sigma> k C' = (\<exists>L t s u v M p Cl_P1 Cl_P2 Cl_C polarity t' u' L' trms_C. L \<in> Cl_P1 \<and> M \<in> Cl_P2 \<and> eligible_literal L P1 \<sigma> \<and> eligible_literal M P2 \<sigma> \<and> variable_disjoint P1 P2 \<and> Cl_P1 = cl_ecl P1 \<and> Cl_P2 = cl_ecl P2 \<and> \<not> is_a_variable u' \<and> allowed_redex u' P1 \<sigma> \<and> trms_C = get_trms Cl_C (dom_trms Cl_C (subst_set (trms_ecl P1 \<union> trms_ecl P2 \<union> {r. \<exists>q. (q, p) \<in> pos_ord P1 t \<and> subterm t q r}) \<sigma>)) k \<and> C = Ecl Cl_C trms_C \<and> orient_lit_inst M u v pos \<sigma> \<and> orient_lit_inst L t s polarity \<sigma> \<and> u \<lhd> \<sigma> \<noteq> v \<lhd> \<sigma> \<and> subterm t p u' \<and> ck_unifier u' u \<sigma> k \<and> replace_subterm t p v t' \<and> (k = FirstOrder \<or> (subst_lit M \<sigma>, subst_lit L \<sigma>) \<in> lit_ord) \<and> (k = FirstOrder \<or> strictly_maximal_literal P2 M \<sigma>) \<and> L' = mk_lit polarity (Eq t' s) \<and> Cl_C = subst_cl C' \<sigma> \<and> C' = Cl_P1 - {L} \<union> (Cl_P2 - {M} \<union> {L'}))"
rtos_i_mod_r_2: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0" shows "rtos r s i mod r = i div s"
isodefl_fmapU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'a \<rightarrow> udom\<cdot>'a"   and t :: "udom defl" assumes "class.prefunctor fmapU" shows "isodefl (fmapU\<cdot>(cast\<cdot>t)) (TC('a)\<cdot>t)"
lemma_2_4_6: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a l\<rightarrow> b \<le> (c l\<rightarrow> a) l\<rightarrow> c l\<rightarrow> b"
accepts_Times_iff: fixes fsm1 :: "'b fsm"   and fsm2 :: "'b fsm"   and xs :: "'b list" shows "accepts (Times fsm1 fsm2) xs = (accepts fsm1 xs \<and> accepts fsm2 xs)"
graphD_aggressive: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
ext_inject: fixes zc :: "int"   and more :: "'a"   and zc' :: "int"   and more' :: "'a" shows "(\<lparr>zc = zc, \<dots> = more\<rparr> = \<lparr>zc = zc', \<dots> = more'\<rparr>) = (zc = zc' \<and> more = more')"
distinct_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
ceiling_one: shows "\<lceil>1::'b\<rceil> = 1"
inja: fixes A :: "nat set"   and B :: "nat set" assumes "A \<subseteq> {0..<m} \<union> {0..<Suc r}"   and "B \<subseteq> {0..<m} \<union> {0..<Suc r}"   and "A \<inter> B = {}" shows "a ` A \<inter> a ` B = {}"
mgu_complete: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "mgu s t = None" shows "unifiers {(s, t)} = {}"
IH: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updTs' ts TAS) (upd_invs I P TAS)"
F2: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "F 2 = map_poly to_fract G2"
max_bag_card_in_bag_cards: fixes G :: "('a, 'b) Graph_scheme"   and T :: "('c, 'd) Graph_scheme"   and bag :: "'c \<Rightarrow> 'a set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> \<noteq> {}" shows "TreeDecomposition.max_bag_card T bag \<in> {card (bag t) |t. t \<in> V\<^bsub>T\<^esub>}"
represents_root_poly_1: fixes p :: "int poly"   and x :: "complex" assumes "poly (complex_of_int_poly p) x = 0"   and "p \<noteq> 0" shows "root_poly_Re p represents Re x"
cmp_combine_paramD: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D1 :: "'b set"   and cmp1 :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res"   and D2 :: "'b set"   and cmp2 :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res"   and R :: "('a \<times> 'b) set" assumes "(cmp, cmp_combine D1 cmp1 D2 cmp2) \<in> R \<rightarrow> R \<rightarrow> comp_res_rel"   and "Range R \<subseteq> D1" shows "(cmp, cmp1) \<in> R \<rightarrow> R \<rightarrow> comp_res_rel"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "P TBase"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)"   and "\<And>ty. P ty \<Longrightarrow> P (T ty)" shows "P ty"
abs_eq: fixes x :: "'b" shows "Pred\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (tPred x)"
constant_imageI: fixes f :: "'a \<Rightarrow> 'c" assumes "\<exists>c. \<forall>x\<in>space M. f x = c" shows "\<forall>x\<in>space M. f x = constant_image f"
Cons_prefix_Cons: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
gen_gba_impl_rel_eext_def_internal: fixes Rm :: "('g \<times> 'j) set"   and Rl :: "('f \<times> ('h \<Rightarrow> 'i \<Rightarrow> bool)) set" shows "gen_gba_impl_rel_eext Rm Rl \<equiv> {(\<lparr>gbai_L = Li, \<dots> = mi\<rparr>, \<lparr>gba_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
ANR_imp_locally_path_connected: fixes S :: "'b set" assumes "ANR S" shows "locally path_connected S"
diff_inclusion: shows "diff k (charts_submanifold S) charts (\<lambda>x. x)"
rr_smaller: shows "1 + length r - length d - 1 = 1 + length (rev (tl (minus_poly_rev_list (map ((*) lc) (rev r)) (map ((*) (hd (rev r))) (rev d))))) - length d"