hcju/isabelleps huggingface dataset:
seqp_groupcastTE: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>i\<^sub>p\<^sub>s :: "'e \<Rightarrow> nat set"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "a = groupcast (s\<^sub>i\<^sub>p\<^sub>s \<xi>) (s\<^sub>m\<^sub>s\<^sub>g \<xi>)" and "\<xi>' = \<xi>" and "q = p"
has_codomain_iff_arr: fixes f :: "('obj, 'obj) concrete_category.arr" shows "(codomains f \<noteq> {}) = arr f"
irreducible\<^sub>d_via_tests: fixes ua :: "int poly"   and vsa :: "int poly list"   and da :: "nat" assumes "0 < degree ua"   and "comm_monoid_mult_class.coprime (lead_coeff ua) p"   and "unique_factorization_m ua (lead_coeff ua, mset vsa)"   and "p.square_free_m ua"   and "\<And>v. v \<in> set vsa \<Longrightarrow> Mp v = v"   and "size (mset vsa) < da + da"   and "\<And>ws. \<lbrakk>ws \<subseteq># mset vsa; ws \<noteq> {#}; size ws < da\<rbrakk> \<Longrightarrow> test_dvd ua ws" shows "irreducible\<^sub>d ua"
to_mini_is_mini: fixes F :: "'a formula" shows "is_mini_formula (to_mini_formula F)"
disjoint_equi: fixes ts :: "'b set list" assumes "disjoint_list_rec ts" shows "disjoint (set ts)"
hyps_2: fixes T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list" assumes "u = Fun (Fu f) T" shows "\<exists>S. length S = length T \<and> t = Fun (Fu f) S"
assms_5: shows "h \<turnstile> get_elements_by_tag_name ptr name \<rightarrow>\<^sub>h h'"
cyclic: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "ccw' p q r" shows "ccw' q r p"
assms_2: shows "x \<in> \<lfloor>U, s\<rfloor>"
simps: fixes A :: "'a list"   and sz :: "nat" shows "array_shrink (Array A) sz = (if length A < sz then undefined else Array (take sz A))"
msubset_mset_set_iff: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "(mset_set A \<subseteq># mset_set B) = (A \<subseteq> B)"
gr_implies_not_zero: fixes m :: "'k multiset"   and n :: "'k multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
cod_inv: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.cod (VoV.inv f) = VoV.dom f"
lta_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "C B A LtA D E F"
hom_inverse: fixes x :: "real" shows "of_real (inverse x) = inverse (of_real x)"
Pi_iff: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
assms_2: shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"
lookup_single: fixes k :: "'c"   and v :: "'b"   and k' :: "'c" shows "lookup (monomial v k) k' = (v when k = k')"
eventually_le_le: fixes P :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> 'b"   and net :: "'a filter"   and R :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in net. P x \<le> Q x"   and "\<forall>\<^sub>F x in net. Q x \<le> R x" shows "\<forall>\<^sub>F x in net. P x \<le> R x"
rep_eq: fixes x :: "('a, 'b) \<I>" shows "outs_\<I> x = {out. Rep_\<I> x out \<noteq> {}}"
Bisimulation_Invariants_axioms: shows "Bisimulation_Invariants A B (\<sim>) PA QA PB QB"
that: assumes "arr_Set \<alpha> S"   and "S\<lparr>ArrDom\<rparr> = A"   and "S\<lparr>ArrCod\<rparr> = B" shows "thesis"
BPd_BCh_mechanism_domain: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh BPd BCh"
intro: fixes A :: "V"   and f :: "V" assumes "nop_bij A (2\<^sub>\<nat>) f" shows "binop_bij A f"
k: shows "(spr_simAbs (fst k), snd k) \<in> (\<Union>a. SPRdet.sim_equiv_class a ` SPR.jkbpC) \<times> UNIV"
nn_integral_0_iff_AE: fixes u :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (AE x in M. u x = 0)"
ereal_eq_minus_iff: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" shows "(x = z - y) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and> (y = - \<infinity> \<longrightarrow> x = \<infinity>) \<and> (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = - \<infinity>))"
nonempty: shows "valid_ENC n ` FO \<noteq> {}"
CR_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "CR_on r A = (\<forall>a\<in>A. \<forall>b c. (a, b) \<in> r\<^sup>* \<and> (a, c) \<in> r\<^sup>* \<longrightarrow> (b, c) \<in> r\<^sup>\<down>)"
dominates_unsnoc': fixes g :: "'g"   and n :: "'node"   and m :: "'node"   and m' :: "'node"   and ms :: "'node list" assumes "invar g"   and "dominates g n m"   and "g \<turnstile> m'-ms\<rightarrow>m"   and "\<forall>x\<in>set (tl ms). x \<noteq> n" shows "dominates g n m'"
ideD_1: fixes a :: "'d" assumes "D.ide a" shows "D.arr a"
yoneda_functor_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b" shows "yoneda_functor C S \<phi> \<equiv> (category C \<and> dual_category C \<and> product_category (dual_category.comp C) C) \<and> (replete_set_category S \<and> hom_functor C S \<phi>) \<and> functor_category (dual_category.comp C) S \<and> curried_functor' (dual_category.comp C) C S (hom_functor.map C S \<phi>)"
le_diff_conv2: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "(i \<le> j - k) = (i + k \<le> j)"
out_degree_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a" assumes "digraph_isomorphism hom"   and "u \<in> verts G" shows "out_degree (app_iso hom G) (iso_verts hom u) = out_degree G u"
Tree\<^sub>\<alpha>_free_4: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and f :: "'d"   and \<phi> :: "'b" shows "Not\<^sub>\<alpha> t\<^sub>\<alpha> \<noteq> Pred\<^sub>\<alpha> f \<phi>"
that: fixes ys' :: "'b list" assumes "ys' \<in> listset xs"   and "ys = map f ys'" shows "thesis"
character_data_ptr_casts_commute3: fixes node_ptr :: "(_) node_ptr" assumes "is_character_data_ptr_kind node_ptr" shows "cast (the (cast node_ptr)) = node_ptr"
simps_29: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x41 ;; x42 \<noteq> x31@[x32]"
assms: shows "x \<rightarrow>\<^sup>*\<^bsub>G\<^esub> y"
enat_add1_eq: fixes x :: "nat"   and y :: "enat"   and z :: "enat" shows "(enat x + y = enat x + z) = (y = z)"
find_closest_pair_code_eq: fixes \<delta> :: "real"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and \<delta>' :: "int"   and C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list"   and \<Delta>' :: "int"   and C\<^sub>0' :: "int \<times> int"   and C\<^sub>1' :: "int \<times> int" assumes "\<delta> = dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"   and "\<delta>' = dist_code c\<^sub>0 c\<^sub>1"   and "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps"   and "(\<Delta>', C\<^sub>0', C\<^sub>1') = find_closest_pair_code (\<delta>', c\<^sub>0, c\<^sub>1) ps" shows "C\<^sub>0 = C\<^sub>0' \<and> C\<^sub>1 = C\<^sub>1'"
notEmpty: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list" assumes "execution trans sends start exec trace" shows "1 \<le> length exec"
comm_semiring_0_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.comm_semiring_0 plus zero times \<equiv> class.comm_monoid_add plus zero \<and> class.comm_semiring plus times \<and> class.mult_zero times zero"
transD: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "trans r"   and "x \<succeq>[r] y"   and "y \<succeq>[r] z" shows "x \<succeq>[r] z"
iT_Div_cut_greater: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" assumes "0 < k" shows "(I \<oslash> k) \<down>> t = I \<down>> (t * k + (k - Suc 0)) \<oslash> k"
WT_converter_of_resource: fixes \<I> :: "('i, 'j) \<I>"   and res :: "('i, 'j) resource"   and \<I>' :: "('k, 'l) \<I>" assumes "\<I> \<turnstile>res res \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C converter_of_resource res \<surd>"
wf_cdecl_def: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> \<not> P \<turnstile> D \<preceq>\<^sup>* C \<and> (\<forall>m\<in>set ms. wf_overriding P D m)) \<and> (C = Thread \<longrightarrow> (\<exists>m. (run, [], Void, m) \<in> set ms))"
relabel_pres_dis_invar: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set" assumes "discharge_invar c s t u fo lo no f l n"   and "Network.relabel_precond c t f l u" shows "discharge_invar c s t u fo lo no f (Network.relabel_effect c f l u) (n(u := Graph.adjacent_nodes c u))"
Greatest_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "order.Greatest less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq y x))"
u_nonneg: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a"   and n :: "nat" shows "0 \<le> MC_syntax.u K x y n"
is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C I\<^sub>C.map I\<^sub>C.cmp"
d\<^sub>2: fixes a :: "'a" assumes "alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}"   and "a \<in> alphabet t\<^sub>2" shows "depth (Node w t\<^sub>1 t\<^sub>2) a = depth t\<^sub>2 a + 1"
pid_gstep_succE: fixes gc' :: "('c, 'ls, 'gs) pid_global_config"   and gc :: "('c, 'ls, 'gs) pid_global_config" assumes "gc' \<in> pid_gstep_succ gc" obtains pid :: "nat" and a :: "'a" and c' :: "'c" and ls' :: "'ls" and gs' :: "'gs" where "pid_valid gc pid" and "cstep (cmd_of_pid gc pid) a c'" and "en (ls_of_pid gc pid, pid_global_config.state gc) a" and "ex (ls_of_pid gc pid, pid_global_config.state gc) a = (ls', gs')" and "gc' = \<lparr>pid_global_config.processes = (pid_global_config.processes gc)[pid := \<lparr>command = c', state = ls'\<rparr>], state = gs'\<rparr>"
zero_order_1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
log_le: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "1 < a"   and "0 < x"   and "x \<le> y" shows "log a x \<le> log a y"
lvivl_rel_br: shows "lvivl_rel = br (\<lambda>(x, y). set_of_ivl (eucl_of_list x, eucl_of_list y)) (\<lambda>(x, y). length x = DIM('a) \<and> length y = DIM('a))"
intro: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "is_tm_dghm_axioms \<alpha> \<FF>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
assms_1: shows "N1 \<subseteq>\<^sub>N N"
ldropn_iterates: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "ldropn n (iterates f x) = iterates f ((f ^^ n) x)"
rep_eq: fixes x :: "'e" shows "generator (replicate_prod x) = replicate_raw x"
closed_levelset: fixes s :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "continuous_on UNIV s" shows "closed {x. s x = c}"
is_kth_obs_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat" shows "IFC_def.is_kth_obs prob \<pi> k i = (card (IFC_def.obs_ids prob \<pi> \<inter> {..<i}) = k \<and> IFC_def.att prob (\<pi> i) \<noteq> None)"
not_Some_eq: fixes x :: "'d option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
arl_get_def: shows "arl_get \<equiv> \<lambda>(a, n). Array.nth a"
gen_model_rels: fixes w' :: "'e"   and w'' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and w :: "'e"   and a :: "'f" assumes "(w', w'') \<in> relations (gen_model M w) a" shows "(w', w'') \<in> relations M a"
jj_2: shows "\<not> ps (S t j) p \<noteq> None"
irreducible\<^sub>d_def_0: fixes f :: "'a poly" shows "irreducible\<^sub>d f = (degree f \<noteq> 0 \<and> (\<forall>g h. degree g \<noteq> 0 \<longrightarrow> degree h \<noteq> 0 \<longrightarrow> f \<noteq> g * h))"
lm089: fixes f :: "('d \<times> 'c) set"   and X :: "'d set" shows "Range f - f `` X \<subseteq> Range (f outside X)"
Unfold2: fixes wa :: "'a set list" shows "{\<psi>. G_depth \<psi> \<le> G_depth (af\<^sub>G \<phi> wa)} = {\<psi>. G_depth \<psi> \<le> G_depth \<phi>}"
leq_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f \<sqsubseteq> g \<equiv> \<forall>x. f x \<sqsubseteq> g x"
conjugate_exponent_ennrealI_3: fixes p :: "ennreal"   and q :: "ennreal" assumes "1 / p + 1 / q = 1" shows "p = conjugate_exponent q"
m2: shows "mC_C theta c c2 s s2 P2 F2"
simObs: shows "\<forall>a ec t. t \<in> jkbpC \<and> simAbs ec = sim_equiv_class a t \<longrightarrow> simObs a ec = envObs a (tLast t)"
cong: fixes I :: "('a \<times> 'b) spmf"   and range :: "'a \<Rightarrow> 'c set"   and B :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "ETP_base.protocol I range B F F\<^sub>i\<^sub>n\<^sub>v \<equiv> ETP_base.protocol I range B F F\<^sub>i\<^sub>n\<^sub>v"
list_of_oalist_of_list_id: fixes xs :: "('a \<times> 'd) list \<times> 'o" assumes "oalist_inv xs" shows "list_of_oalistz (oalist_of_listz xs) = xs"
simps: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'d" shows "(r OO s) a1 a2 = (\<exists>a b c. a1 = a \<and> a2 = c \<and> r a b \<and> s b c)"
resChainEqvt: fixes perm :: "(name \<times> name) list"   and xvec :: "name list"   and P :: "('d, 'e, 'f) psi" shows "perm \<bullet> \<lparr>\<nu>*xvec\<rparr>P = \<lparr>\<nu>*perm \<bullet> xvec\<rparr>perm \<bullet> P"
listsp_infI: fixes A :: "'g \<Rightarrow> bool"   and l :: "'g list"   and B :: "'g \<Rightarrow> bool" assumes "listsp A l"   and "listsp B l" shows "listsp (A \<sqinter> B) l"
real_root_pos2: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 \<le> x" shows "root n (x ^ n) = x"
simps_3: fixes v :: "'a list"   and vb :: "'a list"   and vc :: "'a list list" shows "local.c_merge_all (v # vb # vc) = local.c_merge_pairs (v # vb # vc) + local.c_merge_all (merge_pairs key (v # vb # vc))"
lexord_eq_alt_def1: fixes a :: "String.literal list"   and b :: "String.literal list" shows "(a \<le> b) = lexord_eq a b"
\<A>1d_1: shows "\<A>1 = to_st (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>1d)"
case_2: fixes f1 :: "int \<Rightarrow> int list \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and x21 :: "int"   and x22 :: "int"   and x23 :: "int list" shows "(case Dvd x21 x22 x23 of Le x xa \<Rightarrow> f1 x xa | Dvd x xa xb \<Rightarrow> f2 x xa xb | NDvd x xa xb \<Rightarrow> f3 x xa xb) = f2 x21 x22 x23"
summable_diff: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "summable f"   and "summable g" shows "summable (\<lambda>n. f n - g n)"
simps_2: fixes l :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree"   and x :: "'c \<times> 'd"   and uu :: "'e"   and mkp :: "'c \<times> 'd"   and r :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" shows "invpst \<langle>l, (x, uu, mkp), r\<rangle> = (invpst l \<and> invpst r \<and> is_min2 mkp (set (Tree2.inorder l @ x # Tree2.inorder r)))"
1: shows "((a \<otimes> \<l>[b \<otimes> c]) \<cdot> (a \<otimes> \<a>[\<I>, b, c])) \<cdot> \<a>[a, \<I> \<otimes> b, c] \<cdot> (\<a>[a, \<I>, b] \<otimes> c) = (a \<otimes> \<l>[b] \<otimes> c) \<cdot> \<a>[a, \<I> \<otimes> b, c] \<cdot> (\<a>[a, \<I>, b] \<otimes> c)"
times_S: fixes x :: "'a"   and n :: "nat" shows "x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
filter_empty_every_not: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "(filter P l = []) = list_all (\<lambda>x. \<not> P x) l"
mpoly_map_vars_monom_single: fixes f :: "nat \<Rightarrow> nat"   and i :: "nat"   and n :: "nat"   and c :: "'b" assumes "bij f" shows "mpoly_map_vars f (MPoly_Type.monom (Poly_Mapping.single i n) c) = MPoly_Type.monom (Poly_Mapping.single (f i) n) c"
higher_pderiv_monom_eq_zero: fixes n :: "nat"   and m :: "nat"   and c :: "'a" assumes "n + 1 < m" shows "(pderiv ^^ m) (Polynomial.monom c n) = 0"
Least1I: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.order less_eq less"   and "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y)" shows "P (ord.Least less_eq P)"
hyps: assumes "x \<in> set ys"   and "xs \<in> set (subseqs zs)" shows "x # xs \<in> set (subseqs (ys @ zs))"
drop_tl: fixes n :: "nat"   and xs :: "'a list" shows "drop n (tl xs) = tl (drop n xs)"
hyps_2: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []" shows "let state' = exhaustiveUnitPropagate (applyUnitPropagate state') in getConflictFlag state' \<or> getQ state' = []"
epiE: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and g' :: "('a, 'a set) Maps.arr" assumes "Maps.epi f"   and "Maps.seq g f"   and "Maps.seq g' f"   and "g \<odot> f = g' \<odot> f" shows "g = g'"
Zp_defs_5: shows "(\<oplus>) = padic_add p"
hyps_2: shows "(\<Union>a\<in>toplevel_summands r2. toplevel_summands \<guillemotleft>deriv as \<guillemotleft>a\<guillemotright>\<guillemotright>) = toplevel_summands \<guillemotleft>deriv as \<guillemotleft>r2\<guillemotright>\<guillemotright>"
push_def: fixes x :: "('d \<times> 'e) \<times> 'f" shows "push x \<equiv> (fst (fst x), snd (fst x), snd x)"
partition_on_the_part_mem: fixes A :: "'a set"   and P :: "'a set set"   and x :: "'a" assumes "partition_on A P"   and "x \<in> A" shows "(THE X. x \<in> X \<and> X \<in> P) \<in> P"
sinh_minus: fixes x :: "'a" shows "sinh (- x) = - sinh x"
cod_eqI': fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "a \<in> CC.codomains f" shows "a = CC.cod f"
closed_def: fixes "open" :: "'d set \<Rightarrow> bool"   and S :: "'d set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
continuous_map_const: fixes X :: "'b topology"   and Y :: "'c topology"   and C :: "'c" shows "continuous_map X Y (\<lambda>x. C) = (topspace X = {} \<or> C \<in> topspace Y)"
munionM_comp_fun_commute: fixes return :: "'d multiset \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d multiset \<Rightarrow> 'e) \<Rightarrow> 'e" assumes "monad_commute return bind" shows "comp_fun_commute (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A + B))))"
is_rewrite_ordD3: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and a :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and b :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "is_rewrite_ord rword" obtains "rword a b"   | "\<not> rword a b" and "rword b a"
i_take_Suc_Cons: fixes n :: "nat"   and x :: "'d"   and f :: "nat \<Rightarrow> 'd" shows "([x] \<frown> f) \<Down> Suc n = x # f \<Down> n"
step_z_complete1: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g"   and l' :: "'e"   and u' :: "'f \<Rightarrow> 'g" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>" shows "\<exists>Z. A \<turnstile> \<langle>l, {u}\<rangle> \<leadsto> \<langle>l', Z\<rangle> \<and> u' \<in> Z"
A_def: shows "A \<equiv> {x \<in> space M. limsup (\<lambda>n. ereal (birkhoff_sum f n x)) = \<infinity>}"
Domain_totalize_act: fixes act :: "('b \<times> 'b) set" shows "Domain (totalize_act act) = UNIV"
accept\<^sub>G\<^sub>R_I: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and F :: "('b \<times> 'c \<times> 'b) set"   and \<I> :: "('b \<times> 'c \<times> 'b) set set"   and w :: "nat \<Rightarrow> 'c"   and \<alpha> :: "(('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set set) set" assumes "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, \<I>) w"   and "(F, \<I>) \<in> \<alpha>" shows "accept\<^sub>G\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w"
IH: fixes c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "(singular_subdivision p ^^ n) c = frag_extend (\<lambda>f. chain_map p f ((simplicial_subdivision p ^^ n) (frag_of (restrict id (standard_simplex p))))) c"
Key_unique_SesKey: fixes K :: "nat"   and SesKey :: "nat"   and B :: "agent"   and T :: "msg"   and evs :: "event list"   and K' :: "nat"   and B' :: "agent"   and T' :: "msg" assumes "Crypt K \<lbrace>Key SesKey, Agent B, T\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt K' \<lbrace>Key SesKey, Agent B', T'\<rbrace> \<in> parts (knows Spy evs)"   and "Key SesKey \<notin> analz (knows Spy evs)"   and "evs \<in> kerbV" shows "K = K' \<and> B = B' \<and> T = T'"
WT_fail_resource: fixes \<I> :: "('h, 'i) \<I>" shows "\<I> \<turnstile>res fail_resource \<surd>"
takeWhile_filter': fixes xs :: "(nat \<times> 'a) list"   and nt :: "nat"   and c :: "nat" assumes "sorted (map fst xs)"   and "\<forall>t\<in>fst ` set xs. t \<le> nt" shows "takeWhile (\<lambda>(t, X). c \<le> nt - t) xs = filter (\<lambda>(t, X). c \<le> nt - t) xs"
B: shows "B \<in> f \<turnstile> folding.\<C>"
tendsto_uminus_nhds: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.topological_group_add minus plus zero uminus open" shows "filterlim uminus (topological_space.nhds open (uminus a)) (topological_space.nhds open a)"
x0_in_carrier: shows "x0 \<in> carrier \<G>"
ex_cm_imp_conds_1: fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "\<exists>c'. next_change_multiplicity' c c' loc t n" shows "\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc) \<and> t' \<le> t"
list_permutes_def: fixes xs :: "('b \<times> 'b) list"   and A :: "'b set" shows "list_permutes xs A = (set (map fst xs) \<subseteq> A \<and> set (map snd xs) = set (map fst xs) \<and> distinct (map fst xs) \<and> distinct (map snd xs))"
coplanar_perm_3: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar A C D B"