hcju/isabelleps huggingface dataset:
kth_obs_le_iff: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and l :: "nat"   and j :: "nat" assumes "IFC prob"   and "IFC_def.is_kth_obs prob \<pi> k i"   and "IFC_def.is_kth_obs prob \<pi> l j" shows "(k < l) = (i < j)"
label_incr_ge: fixes l :: "nat"   and n :: "label"   and i :: "nat" assumes "Label l = n \<oplus> i" shows "i \<le> l"
disc_iff_1: fixes abort :: "bool"   and x :: "real"   and n :: "real"   and acc :: "real" shows "(gbinomial_series_aux abort x n acc = MSLNil) = (abort \<and> acc = 0)"
scaleC_add_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "complex"   and x :: "'a"   and y :: "'a" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (plus x y) = plus (scaleC a x) (scaleC a y)"
wb_lens_weak: fixes x :: "'e \<Longrightarrow> 'f" assumes "wb_lens x" shows "weak_lens x"
cont2monofunE: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "cont f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
s1: shows "ccw'.sortedP 0 ((u + v) # w # y # ys)"
hseq_char': fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "hseq \<nu> \<mu> = (\<nu> \<star> \<mu> \<noteq> null)"
onl_def: fixes \<Gamma> :: "'i \<Rightarrow> ('g, 'h, 'i, 'j) seqp"   and P :: "'f \<times> 'j \<Rightarrow> bool" shows "onl \<Gamma> P \<equiv> \<lambda>(\<xi>, p). \<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"
chaos_par_chaos: shows "chaos \<sqsubseteq> chaos \<parallel> chaos"
Suc: fixes w :: "'a" shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
zero_iff_1: shows "(x = 0) = (x' = 0)"
assms_2: fixes x :: "hf" assumes "Ord x"   and "\<And>y. y \<^bold>\<in> x \<Longrightarrow> P y" shows "P x"
domain_invol: fixes x :: "('a \<times> 'a) set" shows "rdom (rdom x) = rdom x"
all_lt_Max_imp_lt_mset: fixes N :: "'b multiset"   and M :: "'b multiset" assumes "N \<noteq> {#}"   and "\<forall>a\<in>#M. a < Max_mset N" shows "M < N"
nsum_eq: fixes A :: "('c, 'd) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>n. g j \<in> carrier A"   and "\<forall>j\<le>n. f j = g j" shows "\<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A g n"
embed_measure_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "embed_measure M f = measure_of (f ` space M) {f ` A |A. A \<in> sets M} (\<lambda>A. emeasure M (f -` A \<inter> space M))"
fresh': shows "(supp p \<union> supp p') \<sharp>* \<Gamma>"
call_2\<^sub>C_def: fixes M :: "'c \<Rightarrow> 'd \<Rightarrow> 'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and A\<^sub>1 :: "'a control_state_scheme \<Rightarrow> 'c"   and A\<^sub>2 :: "'a control_state_scheme \<Rightarrow> 'd" shows "call_2\<^sub>C M A\<^sub>1 A\<^sub>2 = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else M (A\<^sub>1 \<sigma>) (A\<^sub>2 \<sigma>) \<sigma>)"
is_bound_typ_instance: fixes t :: "typ"   and sch :: "type_scheme" shows "t <| sch = (\<exists>S. t = bound_typ_inst S sch)"
cases: fixes a :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "a \<in> cterms \<Gamma>" obtains p :: "('a, 'b, 'c, 'd) seqp" and pn :: "'c" where "a = p__" and "p__ \<in> sterms \<Gamma> (\<Gamma> pn)"   | pp :: "('a, 'b, 'c, 'd) seqp" and p :: "('a, 'b, 'c, 'd) seqp" where "a = p" and "pp \<in> cterms \<Gamma>" and "p \<in> dterms \<Gamma> pp"
knows_Gets: fixes A :: "agent"   and A' :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows A (Gets A' X # evs) = knows A evs"
branchStops: fixes subs :: "'b \<Rightarrow> 'b set"   and Gamma :: "'b"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "branch subs Gamma f"   and "terminal subs (f n)" shows "f (Suc n) = f n"
lift_subst: fixes j :: "nat"   and i :: "nat"   and t :: "db"   and s :: "db" assumes "j < i + 1" shows "lift (t<s/j>) i = lift t (i + 1)<lift s i/j>"
PORI_2: shows "(\<lambda>i. ValFF) \<in> unlr POR.delta"
intros: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "\<And>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<Longrightarrow> d_OUT f x = 0" shows "wave \<Gamma> f"
gb_schema_dummy_domI2: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and args :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<times> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl" shows "gb_schema_dummy_dom (data, D, args)"
simps_53: fixes f1 :: "('b, 'c) trm \<Rightarrow> ('b, 'c) trm \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c \<Rightarrow> ('b, 'c) trm) \<Rightarrow> 'a"   and f3 :: "('b, 'd, 'c) formula \<Rightarrow> 'a"   and f4 :: "('b, 'd, 'c) formula \<Rightarrow> ('b, 'd, 'c) formula \<Rightarrow> 'a"   and f5 :: "'c \<Rightarrow> ('b, 'd, 'c) formula \<Rightarrow> 'a"   and f6 :: "('b, 'd, 'c) hp \<Rightarrow> ('b, 'd, 'c) formula \<Rightarrow> 'a"   and f7 :: "'d \<Rightarrow> ('b, 'd, 'c) formula \<Rightarrow> 'a"   and x41 :: "('b, 'd, 'c) formula"   and x42 :: "('b, 'd, 'c) formula" shows "(case x41 && x42 of Geq x xa \<Rightarrow> f1 x xa | $\<phi> x xa \<Rightarrow> f2 x xa | ! x \<Rightarrow> f3 x | x && xa \<Rightarrow> f4 x xa | Exists x xa \<Rightarrow> f5 x xa | \<langle> x \<rangle> xa \<Rightarrow> f6 x xa | InContext x xa \<Rightarrow> f7 x xa) = f4 x41 x42"
lenR': shows "length (map of_int_hom.vec_hom fs') = m"
m_inv_closed: fixes x :: "'a" assumes "x \<in> A" shows "inv x \<in> A"
nodes_target: fixes p :: "'state"   and r :: "'action list" assumes "p \<in> R.nodes"   and "R.path r p" shows "target r p \<in> R.nodes"
multi_nonempty_split: fixes M :: "'b multiset" assumes "M \<noteq> \<Empt>" shows "\<exists>A a. M = add_mset a A"
every_position_is_determined: fixes A :: "'b llist set"   and N :: "enat"   and p :: "'b list" assumes "GaleStewartDeterminedGames.closed_GSgame A N"   and "GSgame.position N p" shows "GSgame.position N p \<and> (\<exists>\<sigma>. GSgame.strategy_winning_by_Even A N \<sigma> p) \<or> GSgame.position N p \<and> (\<exists>\<sigma>. GSgame.strategy_winning_by_Odd A N \<sigma> p)"
neg_le_iff_le: fixes b :: "'a"   and a :: "'a" shows "(- b \<le> - a) = (a \<le> b)"
num_params_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" shows "num_params (map_poly f p) = num_params p"
e2ennreal_mult: fixes a :: "ereal"   and b :: "ereal" assumes "0 \<le> a" shows "e2ennreal (a * b) = e2ennreal a * e2ennreal b"
geom_spick_lt: fixes m :: "nat"   and n :: "nat"   and w :: "bool stream"   and x :: "bool" assumes "m < n" shows "geom_proc m (spick w n x) = geom_proc m w"
inf_sup_ord_2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) y"
rel_fun_def: fixes A :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'k \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
asTerm_def: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm" shows "asTerm qX \<equiv> proj alphaGood qX"
asymp_equiv_norm: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g" shows "(\<lambda>x. norm (f x)) \<sim>[F] (\<lambda>x. norm (g x))"
asIMOD_igSubstIGVar1: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstIGVar1 (asIMOD SEM)"
dim_row_take_cols: fixes A :: "'a mat"   and ls :: "nat list" shows "dim_row (take_cols A ls) = dim_row A"
bind_UNION: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'g set" shows "A \<bind> f = \<Union> (f ` A)"
d: shows "prime (3 * 2 ^ (7 - 1) - 1)"
elements_with_ord_restrict_totatives: fixes n :: "nat"   and d :: "nat" assumes "1 < n"   and "0 < d" shows "{x \<in> {..<n}. ord n x = d} = {x \<in> totatives n. ord n x = d}"
span_sum: fixes f :: "'c \<Rightarrow> 'a"   and Sa :: "'a set"   and A :: "'c set" assumes "\<forall>x. f x \<in> S"   and "Sa \<subseteq> S"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> span Sa" shows "sum f A \<in> span Sa"
pmdl_Macaulay_list: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pmdl (set (Macaulay_list ps)) = pmdl (set ps)"
ideal_prod_assocTr0: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and y :: "'a"   and z :: "'a" assumes "ideal R A"   and "ideal R B"   and "ideal R C"   and "y \<in> C"   and "z \<in> set_mult R A B" shows "z \<cdot>\<^sub>r y \<in> sum_mult R A (B \<diamondsuit>\<^sub>r C)"
set_mult_one_imp_inc: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'c set"   and B :: "'c set" assumes "Group.group G"   and "\<one>\<^bsub>G\<^esub> \<in> A"   and "A \<subseteq> carrier G"   and "B \<subseteq> carrier G" shows "B \<subseteq> B <#>\<^bsub>G\<^esub> A"
has_contour_integral_subpath: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and u :: "real"   and v :: "real" assumes "f contour_integrable_on g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<le> v" shows "(f has_contour_integral integral {u..v} (\<lambda>x. f (g x) * vector_derivative g (at x))) (subpath u v g)"
space_prob_algebra: fixes M :: "'c measure" shows "space (prob_algebra M) = {N. sets N = sets M \<and> prob_space N}"
simps_1: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set" shows "\<lbrakk>M; D; []\<rbrakk>\<^sub>s = (\<lambda>\<I>. True)"
lcl_trans_l: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset"   and D :: "'b multiset"   and E :: "'b multiset"   and F :: "'b multiset" assumes "locally_compatible_l R1 R2 (A + B) (C + D) (E + F)" shows "locally_compatible_l R1 R2 A C E"
image2pE: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and fx :: "'d"   and gy :: "'f" assumes "BNF_Greatest_Fixpoint.image2p f g R fx gy" obtains x :: "'c" and y :: "'e" where "fx = f x" and "gy = g y" and "R x y"
fold_def: fixes f :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c"   and al :: "('d, nat) alist" shows "DAList_Multiset.fold f e al = fold_impl f e (alist.impl_of al)"
set_le_in: fixes n :: "'b"   and I :: "'b set" shows "{a. a < n \<and> a \<in> I} = {a \<in> I. a < n}"
smallstep_ideal_deterministic: fixes t :: "Syntax.term"   and u :: "Syntax.term"   and u' :: "Syntax.term" assumes "\<lless>[], t\<ggreater> I\<rightarrow> \<lless>[], u\<ggreater>"   and "\<lless>[], t\<ggreater> I\<rightarrow> \<lless>[], u'\<ggreater>" shows "u = u'"
lapp_fin_fin_iff: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" shows "(r @@ s \<in> A\<^sup>\<star>) = (r \<in> A\<^sup>\<star> \<and> s \<in> A\<^sup>\<star>)"
vrat_plus_closed: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x +\<^sub>\<rat> y \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
sigma_array: fixes mm :: "nat"   and j :: "nat"   and l :: "nat" assumes "mm \<le> length fs"   and "j < mm"   and "l \<le> j" shows "sigma_array (IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (if i = mm then Suc j else Suc i)) (Suc mm)) (IArray.of_fun (\<mu>' mm) (Suc j)) (IArray.of_fun (\<mu>' (Suc j)) (if Suc j = mm then Suc j else Suc (Suc j))) (\<mu>' l l) l = \<sigma>s l mm (Suc j)"
orthogonal_to_span: fixes a :: "'c"   and S :: "'c set"   and x :: "'c" assumes "a \<in> span S"   and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" shows "orthogonal x a"
redundant_set_def: fixes g :: "'g"   and P :: "'val set" shows "redundant_set g P \<equiv> P \<noteq> {} \<and> P \<subseteq> dom (phi g) \<and> (\<exists>v'\<in>allVars g. \<forall>\<phi>\<in>P. \<forall>\<phi>'. phiArg g \<phi> \<phi>' \<longrightarrow> \<phi>' \<in> P \<union> {v'})"
select_convs_2: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c" shows "flows_fix \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_fix"
in_carrier_atlasI: fixes c :: "('a, 'b) chart"   and x :: "'a" assumes "c \<in> sub.atlas"   and "x \<in> domain c" shows "x \<in> sub.carrier"
valid_node_def: fixes prog :: "cmd"   and n :: "w_node" shows "While_CFG.valid_node prog n \<equiv> \<exists>a. valid_edge prog a \<and> (n = sourcenode a \<or> n = targetnode a)"
optim_ys: shows "optim 0 ys (replicate ar None) = Ok (zs, \<Sigma>2)"
init_inv_mut: fixes m :: "'e" shows "\<forall>s. initial_state \<lparr>PGMs = gc_coms, INIT = gc_system_init, FAIR = \<lambda>s. True\<rparr> s \<longrightarrow> mut_m.invsL m \<lparr>GST = s, HST = []\<rparr>"
assms_4: fixes y :: "'al" assumes "y \<in> U"   and "\<And>z. z \<in> A \<Longrightarrow> z \<le>\<^sub>o\<^sub>w y" shows "x \<le>\<^sub>o\<^sub>w y"
infer_from_def: fixes CC :: "'b literal multiset set"   and \<gamma> :: "'b inference" shows "infer_from CC \<gamma> = (set_mset (prems_of \<gamma>) \<subseteq> CC)"
adopt_node_child_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr" assumes "h \<turnstile> ok adopt_node document_ptr child" shows "child |\<in>| node_ptr_kinds h"
older_seniors_card: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n) = card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n)) = (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n)"
that: fixes y :: "'c" assumes "p = Inr y"   and "f y = Inr x" shows "thesis"
apd_d_def: fixes x :: "('b \<times> 'b) set" shows "rel_antirange_kleene_algebra.ardual.apd_d x = rel_ar (rel_ar x)"
mono: fixes F :: "('d \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "is_filter F"   and "\<forall>x. P x \<longrightarrow> Q x"   and "F P" shows "F Q"
add_mset_disjoint_2: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "({#} = (a, A) \<inter># B) = (a \<notin># B \<and> {#} = A \<inter># B)"
isFun_Lambda: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "isFun (Lambda A f)"
leg_simps_1: shows "C.arr assoc'_fgh.cod.leg0"
Ball_cfuncset_continuous_on: fixes a :: "'d"   and b :: "'d"   and X :: "'e set" shows "\<forall>f\<in>Abstract_Rigorous_Numerics.cfuncset a b X. continuous_on {a..b} f"
id_right: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "f \<bullet>\<^bsub>CC\<^esub> Id\<^bsub>CC\<^esub> (Dom\<^bsub>CC\<^esub> f) = f"
unit_simps_4: fixes a :: "'c" assumes "C.obj a" shows "D.dom (H.unit a) = H.map\<^sub>0 a"
intersect_ts_iff: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and k :: "'b" shows "(c \<in> (A \<Rightarrow>\<inter>  B) k) = (c \<in> A k \<and> c \<in> B k)"
seq_step_invariant_sterms: fixes A :: "('e \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and I :: "'f seq_action \<Rightarrow> bool"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and P :: "('e \<times> 'h) \<times> 'f seq_action \<times> 'e \<times> 'h \<Rightarrow> bool"   and l' :: "'h"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<xi> :: "'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and p' :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"   and "wellformed \<Gamma>"   and "automaton.trans A = seqp_sos \<Gamma>"   and "l' \<in> labels \<Gamma> q"   and "(\<xi>, p) \<in> reachable A I"   and "((\<xi>, p'), a, \<xi>', q) \<in> automaton.trans A"   and "I a"   and "p' \<in> sterms \<Gamma> p" shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
RestrpI: fixes a :: "'c"   and A :: "'c set"   and b :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "a \<in> A"   and "b \<in> A"   and "r a b" shows "(r \<restriction> A) a b"
elims: fixes x :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and y :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "translate_osig x = y" obtains cs :: "(String.literal \<times> String.literal) list" and arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" where "x = (cs, arss)" and "y = (if exe_osig_conds (cs, arss) then (set cs, translate_ars arss) else ({}, [STR ''A'' \<mapsto> [STR ''A'' \<mapsto> [insert STR ''A'' full_sort]]]))"
out_arcs_app_iso_eq: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'c" assumes "iG.digraph_isomorphism hom"   and "u \<in> verts (app_iso hom G)" shows "out_arcs (app_iso hom (app_iso hom G)) (iso_verts hom u) = iso_arcs hom ` out_arcs (app_iso hom G) u"
assms: shows "nofail (m0 \<bind> WHILE cond step)"
surjective: fixes r :: "('c, 'd) Graph_scheme" shows "r = \<lparr>verts = V\<^bsub>r\<^esub>, arcs = E\<^bsub>r\<^esub>, \<dots> = Graph.more r\<rparr>"
ball_SReal_eq: fixes P :: "real star \<Rightarrow> bool" shows "(\<forall>x\<in>\<real>. P x) = (\<forall>x. P (hypreal_of_real x))"
downset_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "downset_on A r = (\<forall>x y. (x, y) \<in> r \<and> y \<in> A \<longrightarrow> x \<in> A)"
linear_continuous_compose: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "continuous F f"   and "linear g" shows "continuous F (\<lambda>x. g (f x))"
Deduction1: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" shows "(F \<triangleright> AX1 \<union> \<Gamma> \<turnstile>\<^sub>H \<bottom>) = (AX1 \<union> \<Gamma> \<turnstile>\<^sub>H F \<^bold>\<rightarrow> \<bottom>)"
inv_closed: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (inv\<^bsub>G\<^esub> x) \<in> carrier H"
cball_subset_ball_iff: fixes a :: "'b"   and r :: "real"   and a' :: "'b"   and r' :: "real" shows "(cball a r \<subseteq> ball a' r') = (dist a a' + r < r' \<or> r < 0)"
order_of_orders_def: shows "order_of_orders = {(r, s). r \<subseteq> s \<and> strict_ext base_r r \<and> strict_ext base_r s}"
impossible_Cons: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "length xs \<le> length ys" shows "(xs = x # ys) = False"
assms_2: shows "s \<in> oreachable A (\<lambda>s s'. I) U"
curve_elt_opp: fixes p\<^sub>1 :: "'a point"   and x\<^sub>1 :: "'a"   and y\<^sub>1 :: "'a"   and p\<^sub>2 :: "'a point"   and x\<^sub>2 :: "'a"   and y\<^sub>2 :: "'a"   and a :: "'a"   and b :: "'a" assumes "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"   and "p\<^sub>2 = Point x\<^sub>2 y\<^sub>2"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "x\<^sub>1 = x\<^sub>2" shows "p\<^sub>1 = p\<^sub>2 \<or> p\<^sub>1 = opp p\<^sub>2"
jkbpC_represents: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.represents jkbp envInit envAction envTrans envVal jview (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview)"
ulim_g: fixes x :: "complex" assumes "x \<in> S" shows "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
hyps_2: fixes Z :: "'b" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} c\<^sub>2 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
Standard_hRe: fixes x :: "complex star" assumes "x \<in> SComplex" shows "hRe x \<in> Standard"
var_inj_term_simp: fixes v :: "Term.var" shows "\<langle>v\<rangle> = In2 v"
collision_react_def: fixes time_set :: "real set" shows "collision_react time_set \<equiv> \<exists>t\<in>time_set. u t = other.s t"
S_NS_compat: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and u :: "('f, 'b) Term.term" assumes "S s t"   and "NS t u" shows "S s u"
kE_int: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'a trm" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"   and "Struct.wtE intT (InfModel.kE intT \<xi>)"   and "Signature.wt wtFsym arOf resOf T" shows "Struct.int intF (InfModel.kE intT \<xi>) T = InfModel.fromFull intT (Signature.tpOf resOf T) (Struct.int (InfModel.intFF arOf resOf intT intF) \<xi> T)"
cc_restr_ccProd: fixes S :: "Vars.var set"   and S\<^sub>1 :: "Vars.var set"   and S\<^sub>2 :: "Vars.var set" shows "(S\<^sub>1 G\<times> S\<^sub>2) G|` S = (S\<^sub>1 \<inter> S) G\<times> (S\<^sub>2 \<inter> S)"
a: shows "a \<in> {0..<n} \<rightarrow> S \<rightarrow> carrier_vec n"
no_state_change_if_nonregular_event: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and ev :: "('a, 'b, 'c) event"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "distributed_system channel"   and "\<not> regular_event ev"   and "distributed_system.next channel trans send recv c ev c'" shows "states c p = states c' p"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := conj.stabilizer P\<rparr>)" shows "Group.group (G\<lparr>carrier := conj.stabilizer P, carrier := H\<rparr>)"
not_self_separated_by_disjoint: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<inter> B = {}" shows "\<not> separated_by {A, B} x x"
unitary_simps_2: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "A * adjoint A = 1\<^sub>m n"
braun_heap_of_A: fixes xs :: "'b list" shows "braun (heap_of_A xs)"