hcju/isabelleps huggingface dataset:
equivalent_objects_def: fixes a :: "'d"   and b :: "'d" shows "D.equivalent_objects a b \<equiv> \<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright> \<and> D.equivalence_map f"
frestrictionE: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A" obtains a :: "V" and b :: "V" where "x = [a, b]\<^sub>\<circ>" and "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> A" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
top_left_normalize: fixes a :: "'a" shows "lcm (1::'a) a = normalize a"
constr_fp_uniq: fixes F :: "'a upred \<Rightarrow> 'a upred"   and E :: "nat \<Rightarrow> 'a upred"   and C :: "'a upred" assumes "constr F E"   and "Monotonic F"   and "\<Or>range E = C" shows "(C \<and> \<mu> F) = (C \<and> \<nu> F)"
not_wf: fixes j :: "int" assumes "j < i" shows "\<bar>ifac (j - 1)\<bar> < \<bar>ifac j\<bar>"
map_S_id0: shows "map_S id id id id id id = id"
unit_simps_4: fixes a :: "'a" assumes "C.obj a" shows "C.dom \<i>\<^sub>C[a] = a \<star>\<^sub>C a"
semilattice_axioms: shows "semilattice (\<squnion>)"
thetaSDone_def: shows "thetaSDone \<equiv> {(Done, Done)}"
BA: shows "\<forall>a\<in>B \<inter> A. (\<lambda>a\<in>A. if a \<in> B then f (generate G {a}) else \<one>) a = \<one>"
orbit_def: fixes x :: "'b" shows "orbit x = {y. \<exists>g\<in>carrier G. y = g \<odot> x}"
singleton: fixes F :: "'b filter"   and a :: "'b" assumes "freeultrafilter F" shows "\<not> (\<forall>\<^sub>F x in F. x = a)"
minus_closed: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<ominus>\<^bsub>P\<^esub> y \<in> carrier P"
eq1: shows "(\<Sum>v\<in>keys r \<union> keys s. lookup r v *s v) = (\<Sum>v\<in>keys r. lookup r v *s v)"
assms: shows "\<forall>c1 c2. c1 \<in> A \<and> c2 \<in> A \<and> c1 \<noteq> c2 \<longrightarrow> comm_monoid_mult_class.coprime (h - [:c1:]) (h - [:c2:])"
frac_gtz: shows "0 < 1 / (2 * a)"
mset_eq_setD: fixes xs :: "'a list"   and ys :: "'a list" assumes "mset xs = mset ys" shows "set xs = set ys"
f_nonneg: fixes x :: "real" shows "0 \<le> f x"
r: fixes u :: "'a"   and v :: "'a" shows "content (g ` cbox u v) = r * content (cbox u v)"
are_operator_effects_consistent_def: fixes op\<^sub>1 :: "('a, 'b) strips_operator_scheme"   and op\<^sub>2 :: "('a, 'c) strips_operator_scheme" shows "are_operator_effects_consistent op\<^sub>1 op\<^sub>2 \<equiv> let add\<^sub>1 = add_effects_of op\<^sub>1; add\<^sub>2 = add_effects_of op\<^sub>2; del\<^sub>1 = delete_effects_of op\<^sub>1; del\<^sub>2 = delete_effects_of op\<^sub>2 in \<not> list_ex (\<lambda>v. list_ex ((=) v) del\<^sub>2) add\<^sub>1 \<and> \<not> list_ex (\<lambda>v. list_ex ((=) v) add\<^sub>2) del\<^sub>1"
xs_def: shows "xs = \<Oplus>Gs\<leftarrow> x"
InitSucState_Conf_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "S \<notin> Target TS"   and "A \<in> the (CompFun (HA ST) S)"   and "S \<in> Conf ST"   and "S \<in> ChiStar (HA ST) `` Source TS" shows "InitState A \<in> StepConf (HA ST) (Conf ST) TS"
TAbs_type: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and s :: "trm"   and S :: "ty"   and T :: "ty"   and U' :: "ty"   and U :: "ty" assumes "\<Gamma> \<turnstile> (\<lambda>X<:S. s) : T"   and "\<Gamma>\<turnstile>T<:(\<forall>X<:U. U')"   and "X \<sharp> \<Gamma>"   and "X \<sharp> S"   and "X \<sharp> U" obtains S' :: "ty" where "\<Gamma>\<turnstile>U<:S" and "TVarB X U # \<Gamma> \<turnstile> s : S'" and "(TVarB X U # \<Gamma>)\<turnstile>S'<:U'"
proj_finite2: fixes t :: "cnf llist"   and n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "\<forall>n'. enat n' \<le> llength t \<longrightarrow> n \<le> n' \<longrightarrow> \<not> active c (lnth t n')" shows "lfinite (dynamic_component.proj tCMP active c t)"
simps_92: fixes x3 :: "'a formula" shows "atoms (\<^bold>\<not> x3) = atoms x3"
aadd_neg_negg: fixes a :: "ant"   and b :: "ant" assumes "a \<le> 0"   and "b < 0" shows "a + b < 0"
table_of_Some_in_set: fixes l :: "('b \<times> 'a) list"   and k :: "'b"   and x :: "'a" assumes "table_of l k = Some x" shows "(k, x) \<in> set l"
exec_move_ThrowI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (throw e) h s ta h' s'"
HARootStates_notmem_Chi: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and T :: "'d" assumes "S \<in> HAStates A"   and "T \<in> States (HARoot A)" shows "T \<notin> Chi A S"
\<omega>_limit_set_in_compact_subset: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and K :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "auto_ll_on_open.trapped_forward f X x K" shows "auto_ll_on_open.\<omega>_limit_set f X x \<subseteq> K"
pow_nat_eq_1_imp_unit: fixes x :: "nat \<times> nat"   and n :: "nat" assumes "x [^]\<^bsub>S\<^esub> n = \<one>\<^bsub>S\<^esub>"   and "0 < n"   and "x \<in> carrier S" shows "x \<in> Units S"
assms: fixes x :: "'b" assumes "x \<in># A" shows "f x \<le> g x"
heap_wellformed_induct_rev_si: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "\<And>child. \<lbrakk>\<And>parent child_node. \<lbrakk>child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node; h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent\<rbrakk> \<Longrightarrow> P parent; \<And>host shadow_root. \<lbrakk>child = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root; h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host\<rbrakk> \<Longrightarrow> P (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)\<rbrakk> \<Longrightarrow> P child" shows "P ptr"
strict_implies_not_eq: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<prec> b" shows "a \<noteq> b"
Fun_in_terms_of_comp: fixes f :: "'s" assumes "S.arr f" shows "S.Fun f = restrict (S.img \<circ> (\<cdot>\<^sub>S) f \<circ> S.mkPoint (S.dom f)) (S.Dom f)"
Seq_decomp_star_Fault: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d"   and f :: "'f" assumes "\<Gamma> \<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>* (Skip, Fault f)" shows "\<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Fault f) \<or> (\<exists>s'. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Normal s') \<and> \<Gamma> \<turnstile> (p\<^sub>2, Normal s') \<rightarrow>\<^sup>* (Skip, Fault f))"
assms_1: shows "constructive_security_obsf real1 ideal1 sim1 (\<I>_real1_core \<oplus>\<^sub>\<I> \<I>_real1_rest) (\<I>_ideal1_core \<oplus>\<^sub>\<I> \<I>_ideal1_rest) (\<I>_common1_core \<oplus>\<^sub>\<I> \<I>_common1_rest) (absorb \<A> (obsf_converter (fused_wiring \<odot> (1\<^sub>C |\<^sub>\<propto> converter_of_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2))))) adv1"
red_rtranclp_diff_in_pmdl: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "(punit.red F)\<^sup>*\<^sup>* p q" shows "p - q \<in> local.punit.pmdl F"
prob_space_exponential_escape_rate: fixes x :: "'a" shows "prob_space (exponential (escape_rate x))"
wf_tr': shows "wf_action_translation E (J (Suc n))"
square_free_m_smultI: fixes f :: "int poly"   and ci :: "int"   and c :: "int" assumes "square_free_m f"   and "M (ci * c) = 1" shows "square_free_m (Polynomial.smult c f)"
distinct_17: fixes x6 :: "ty" shows "Boolean \<noteq> x6\<lfloor>\<rceil>"
test_import_2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot" shows "sup x (inf y (il_semiring.neg inf sup bot top x)) = sup x y"
simps: fixes g :: "'a"   and n :: "'node"   and v :: "'var" shows "i.lookupDef g n v = (if n \<notin> set (\<alpha>n2 g) then undefined else if v \<in> defs2 g n then (v, n, SimpleDef) else case i.predecessors g n of [] \<Rightarrow> (v, n, PhiDef) | [m] \<Rightarrow> i.lookupDef g m v | m # aa # lista \<Rightarrow> (v, n, PhiDef))"
exhaust: fixes y :: "'b sort_key" obtains x11 :: "'b \<times> sort_dir" and x12 :: "'b sort_key" where "y = NonFinal x11 x12"   | x2 :: "'b" where "y = Final x2"
backward_reduction_P: fixes D :: "'a literal multiset"   and L' :: "'a literal"   and N :: "'a literal multiset set"   and L :: "'a literal"   and \<sigma> :: "'s"   and C :: "'a literal multiset"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" assumes "D + {#L'#} \<in> N"   and "- L = L' \<cdot>l \<sigma>"   and "D \<cdot> \<sigma> \<subseteq># C" shows "(N, P \<union> {C + {#L#}}, Q) \<leadsto> (N, P \<union> {C}, Q)"
safe2_inv1_imp_safe: fixes s :: "('c, 'd) configuration stream" assumes "SafeStickyNrecVacantUpto s"   and "holds InvGlobVacantUptoImpliesNrec s" shows "SafeGlobVacantUptoImpliesStickyNrec s"
atomic_step_ev_wait_all_def: fixes tid :: "thread_id_t"   and s :: "state_t" shows "atomic_step_ev_wait_all tid s = s\<lparr>thread := (thread s)(tid := thread s tid\<lparr>ev_counter := 0\<rparr>)\<rparr>"
asprod_1: fixes x :: "ant" shows "1 *\<^sub>a x = x"
is_imgu_imp_is_mgu: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "is_imgu \<sigma> E" shows "is_mgu \<sigma> E"
inner_minus_right: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'e"   and y :: "'e" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (uminus y) = - inner x y"
AE_notin_null_part: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "AE x in M. x \<notin> null_part M S"
that: fixes p :: "int poly" assumes "p \<noteq> 0"   and "poly.coeff p 0 \<noteq> 0"   and "poly (of_int_poly p) x = (0::'a)" shows "thesis"
simps_2: fixes n :: "nat" shows "nat2Nat (Suc n) = SucNat (nat2Nat n)"
simps: fixes x :: "'c"   and xs :: "'c list" shows "last (x # xs) = (if xs = [] then x else last xs)"
degree_pCons_0: fixes a :: "'a" shows "degree [:a:] = 0"
disj_matrices_add_x_le: fixes A :: "'a matrix"   and B :: "'a matrix"   and C :: "'a matrix" assumes "disj_matrices A B"   and "disj_matrices B C" shows "(A \<le> B + C) = (A \<le> C \<and> 0 \<le> B)"
unif_UNIF1_conv: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" shows "((E, E') \<in> unif) = (\<exists>\<sigma>. UNIF1 \<sigma> E E')"
part_eq: fixes x :: "i"   and y :: "i" assumes "MM3" shows "P x y = (PP x y \<or> x = y)"
tendsto_at_iff_tendsto_nhds: fixes g :: "'d \<Rightarrow> 'e"   and l :: "'d" shows "g \<midarrow>l\<rightarrow> g l = (g \<longlongrightarrow> g l) (nhds l)"
steps_z_sound: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and Z :: "('e \<Rightarrow> 'f) set"   and l' :: "'d"   and Z' :: "('e \<Rightarrow> 'f) set"   and u' :: "'e \<Rightarrow> 'f" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "u' \<in> Z'" shows "\<exists>u\<in>Z. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l', u'\<rangle>"
simplex_map_oriented_simplex: fixes p :: "nat"   and q :: "nat"   and l :: "nat \<Rightarrow> nat \<Rightarrow> real"   and r :: "nat"   and S :: "('c \<Rightarrow> real) set"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "simplicial_simplex p (standard_simplex q) (oriented_simplex p l)"   and "simplicial_simplex r S g"   and "q \<le> r" shows "simplex_map p g (oriented_simplex p l) = oriented_simplex p (g \<circ> l)"
bfs_correct: fixes succ :: "'a \<Rightarrow> 'a set"   and src :: "'a"   and dst :: "'a" assumes "Graph succ" shows "Graph.bfs succ src dst \<le> Graph.bfs_spec succ src dst"
newabc: shows "\<alpha> ^ 4 + \<beta> ^ 4 = \<gamma>\<^sup>2"
MP_rule_def: shows "MP_rule \<equiv> ([Pred pid1 nothing, Pred pid1 nothing \<rightarrow> Pred pid2 nothing], Pred pid2 nothing)"
r_constn_def: fixes n :: "nat"   and c :: "nat" shows "r_constn n c \<equiv> if n = 0 then r_const c else r_dummy n (r_const c)"
effect_heapE: fixes f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect (Heap_Monad.heap f) h h' r" obtains "h' = snd (f h)" and "r = fst (f h)"
completely_multiplicativeI: assumes "\<And>p k. \<lbrakk>prime p; 0 < k\<rbrakk> \<Longrightarrow> f (p ^ k) = f p ^ k" shows "completely_multiplicative_function f"
a0i_defs_2: shows "a0i_init \<equiv> {\<lparr>signals = \<lambda>s. 0\<rparr>}"
ordinal_div_1: fixes x :: "ordinal" shows "x div oSuc 0 = x"
c: shows "\<forall>ja\<in>{k - j..i - j}. (\<pi> \<guillemotleft> j) ja \<noteq> ipd ((\<pi> \<guillemotleft> j) (k - j))"
select_convs_2: fixes Obj :: "'b set"   and Mor :: "'a set"   and Dom :: "'a \<Rightarrow> 'b"   and Cod :: "'a \<Rightarrow> 'b"   and Id :: "'b \<Rightarrow> 'a"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'c" shows "mor\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Mor"
expand_case_formula: fixes Q :: "'a \<Rightarrow> bool"   and fatom :: "signs \<Rightarrow> predicate \<Rightarrow> vbl list \<Rightarrow> 'a"   and fconj :: "signs \<Rightarrow> formula \<Rightarrow> formula \<Rightarrow> 'a"   and fall :: "signs \<Rightarrow> formula \<Rightarrow> 'a"   and F :: "formula" shows "Q (case F of FAtom x xa xb \<Rightarrow> fatom x xa xb | FConj x xa xb \<Rightarrow> fconj x xa xb | FAll x xa \<Rightarrow> fall x xa) = ((\<forall>z P vs. F = FAtom z P vs \<longrightarrow> Q (fatom z P vs)) \<and> (\<forall>z A0 A1. F = FConj z A0 A1 \<longrightarrow> Q (fconj z A0 A1)) \<and> (\<forall>z A. F = FAll z A \<longrightarrow> Q (fall z A)))"
mult_rel_def: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "mult_rel X Y \<equiv> let x = SOME x. x \<in> X; y = SOME y. y \<in> Y in mult_rel_aux (fst x) (snd x) (fst y) (snd y)"
delay_bisimulation_diverge_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2" shows "delay_bisimulation_diverge trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
tllist_all2_tfiniteD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys" shows "tfinite xs = tfinite ys"
rep_eq: fixes x :: "('c, 'd) uexpr"   and xa :: "'b \<Longrightarrow> 'd" shows "\<lbrakk>x \<restriction>\<^sub>e xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x\<rbrakk>\<^sub>e (create\<^bsub>xa\<^esub> b))"
IntegralE: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and k :: "real"   and e :: "real" assumes "Integral (a, b) f k"   and "0 < e" obtains \<delta> :: "real \<Rightarrow> real" where "gauge {a..b} \<delta>" and "\<forall>D. fine \<delta> (a, b) D \<longrightarrow> \<bar>rsum D f - k\<bar> < e"
ClosureIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E ClosureIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "to_single_type \<rho> \<le> \<sigma>" and "to_unique_collection \<tau>' \<tau>"
Px_nn: fixes x :: "'c" assumes "x \<in> space S" shows "0 \<le> Px x"
S21: shows "g_comp (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair n x) = c_cons (c_pair (c_pair n x) (c_assoc_value (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair (c_fst (c_snd n)) y))) (pr_gr (c_pair (c_pair n x) (upb1 + upb2)))"
wp1_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp\<^sub>1 c Q = (\<lambda>l s. \<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> Q l t)"
simps_4: fixes uv :: "pdf_operator"   and e :: "expr" shows "free_vars (uv $$ e) = free_vars e"
discr_ParT: fixes cl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl"   and "discrL cl" shows "discr (ParT cl)"
List_islist: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" assumes "List p h ps" shows "islist p h"
down_cast_same: assumes "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) = SCAST('b \<rightarrow> 'c)"
size_4: fixes x21 :: "'b"   and x22 :: "'b list" shows "\<^bold>|x21 # x22\<^bold>| = \<^bold>|x22\<^bold>| + Suc 0"
set_intros: fixes x2 :: "'b" shows "x2 \<in> set_option (Some x2)"
simps_4: fixes l1 :: "value"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (L l1) (Minus v va) = True"
S_def: fixes insert_message :: "('a, 'b) insert_message" shows "S insert_message = (case insert_message of InsertMessage x1 x2 x3 x4 \<Rightarrow> x3)"
in_Span_obtain_same_length_coeffs: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and n :: "'b"   and ms :: "'b list" assumes "VectorSpace smult V"   and "n \<in> R_scalar_mult.RSpan UNIV smult ms" shows "\<exists>rs. set rs \<subseteq> UNIV \<and> length rs = length ms \<and> n = scalar_mult.lincomb smult rs ms"
in_path_event: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and a :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "Q \<in> \<P>"   and "a \<in> Q" shows "a \<in> \<E>"
assms: shows "T = get_trms D (dom_trms C E) k"
5: fixes i :: "int" assumes "i \<noteq> 0"   and "x [^] i = \<one>" shows "finite (carrier (subgroup_generated G {x}))"
that: fixes S :: "('fun, 'var) Term.term list" assumes "Fun f S \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>s\<^sub>t A \<or> Fun f S \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t assignment_rhs\<^sub>s\<^sub>t A"   and "Fun f T = Fun f S \<cdot> \<I>" shows "thesis"
s: shows "vec.span (columns (fst (QR_decomposition A))) = UNIV"
simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and name :: "name"   and u :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.match unapp unconst (Const name) u = (case unconst u of None \<Rightarrow> None | Some name' \<Rightarrow> if name = name' then Some fmempty else None)"
red_diff_rtrancl: fixes F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "(punit.red F)\<^sup>*\<^sup>* (p - q) 0" obtains s :: "'a \<Rightarrow>\<^sub>0 'd" where "(punit.red F)\<^sup>*\<^sup>* p s" and "(punit.red F)\<^sup>*\<^sup>* q s"
prems_1: shows "CT \<turnstile> Da \<not><: D"
pmult_Cons: fixes h :: "'a"   and t :: "'a list"   and l2 :: "'a list" shows "(h # t) *** l2 = (if t = [] then h %* l2 else h %* l2 +++ ((0::'a) # t *** l2))"
queue_invar_simps_3: fixes q :: "('c, 'd) SkewBinomialTree list"   and q' :: "('c, 'd) SkewBinomialTree list" shows "queue_invar (q @ q') = (queue_invar q \<and> queue_invar q')"
Subst_all_preserves_finite_rhs: fixes ES :: "('b list set \<times> 'b trm set) set"   and yrhs :: "'b trm set"   and Y :: "'b list set" assumes "finite_rhs ES"   and "finite yrhs" shows "finite_rhs (Subst_all ES Y yrhs)"
pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor_axioms: shows "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
vwalk_to_vpath_cons: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "vwalk_to_vpath (x # xs) = x # vwalk_to_vpath xs"
ipassmt_inj: shows "\<forall>k. ipassmt k = Some i_ips \<longrightarrow> k = ifce"
nth_less_prefix_length: fixes n :: "nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "n < prefix_length P xs" shows "P (xs ! n)"
bsok_simps_15: fixes e :: "'a expr1"   and M :: "String.literal"   and ps :: "'a expr1 list"   and n :: "nat" shows "bsok (e\<bullet>M(ps)) n = (bsok e n \<and> bsoks ps n)"
sup_level: fixes l1 :: "level"   and l2 :: "level" shows "sup l1 l2 \<equiv> if Hi \<in> {l1, l2} then Hi else Lo"
show_real_def: fixes x :: "real" shows "show_real x \<equiv> if \<exists>y. x = Ratreal y then show (THE y. x = Ratreal y) else ''Irrational''"
ts''_def_1: shows "\<C> \<turnstile> [C ConstInt32 n] : ts _> ts''"
inorder_lookup: fixes t :: "'t"   and a :: "'a" assumes "inv t \<and> sorted1 (inorder t)" shows "lookup t a = map_of (inorder t) a"
Rep_quot_inverse: fixes x :: "'b quot" shows "Abs_quot (Rep_quot x) = x"
Max_wrt_among_total_preorder: fixes A :: "'alt set" shows "Max_wrt_among (R i) A = {x \<in> alts \<inter> A. \<forall>y\<in>alts \<inter> A. R i y x}"