hcju/isabelleps huggingface dataset:
ordering3: shows "\<forall>n n' n''. (finite {a, b, c, d} \<longrightarrow> n'' < card {a, b, c, d}) \<and> n < n' \<and> n' < n'' \<longrightarrow> ord (if n = 0 then a else if n = 1 then b else if n = 2 then c else d) (if n' = 0 then a else if n' = 1 then b else if n' = 2 then c else d) (if n'' = 0 then a else if n'' = 1 then b else if n'' = 2 then c else d)"
lower_bound_ex: shows "\<exists>f\<in>BAD P. \<forall>g. (f, g) \<in> gseq \<longrightarrow> g \<notin> BAD P"
chine_simps_2: shows "C.dom src_f.chine = src_f.dom.apex"
LPDi_Times: fixes i :: "nat" shows "LPDi i (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesR (\<Union>j\<le>i. LPDi j r) s \<union> (\<Union>j\<le>i. LPDi j s)"
assms_2: shows "i \<notin> A \<union> branch_nominals ((ps, a) # branch)"
Q1'_a_def: shows "Q1'_a \<equiv> [: \<lambda>(stk, mrk). {(stk', mrk'). root = nil \<and> stk' = [] \<and> mrk' = mrk} :]"
\<sigma>_le: fixes s :: "'a" assumes "s \<in> B" shows "\<sigma> \<le> s \<bullet> (1::'a)"
subresultant_mat_dim_1: fixes j :: "nat"   and p :: "'b poly"   and q :: "'b poly" shows "dim_row (subresultant_mat j p q) = degree p - j + (degree q - j)"
set_mergesort: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "set (mergesort f xs) = set xs"
is_span: shows "C.span HH\<mu>H\<nu>\<pi>\<rho>.dom.leg0 HH\<mu>H\<nu>\<pi>\<rho>.dom.leg1"
collinear_alt: fixes S :: "'a set" shows "collinear S = (\<exists>u v. \<forall>x\<in>S. \<exists>c. x = u + c *\<^sub>R v)"
bij_imp_permutes: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "bij_betw p S S"   and "\<And>x. x \<notin> S \<Longrightarrow> p x = x" shows "p permutes S"
"case": fixes f :: "int list \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a"   and x1 :: "int list"   and x2 :: "int"   and x3 :: "int" shows "(case presb.Eq x1 x2 x3 of presb.Eq x xa xb \<Rightarrow> f x xa xb) = f x1 x2 x3"
beta_preserves_typ_of: fixes r :: "Core.term"   and T :: "typ"   and s :: "Core.term" assumes "typ_of r = Some T"   and "r \<rightarrow>\<^sub>\<beta> s" shows "typ_of s = Some T"
bi_total: fixes T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "bi_total T" shows "bi_total (pcr_ltl_prop_equiv_quotient T)"
partially_ordered_set_axioms: shows "partially_ordered_set A (\<sqsubseteq>)"
ctr_1: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and T :: "('b, 'a) process_term" assumes "isACT sys T" shows "solution sys T = Action (PREF sys T) (solution sys (CONT sys T))"
normalize_uminus: fixes x :: "'a" shows "normalize (- x) = normalize x"
select: fixes x :: "'a"   and y :: "'a" shows "x + y = x \<or> x + y = y"
minimal_sequence_to_failure_extending_det_state_cover_ob: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and V :: "'d list set" assumes "well_formed M1"   and "well_formed M2"   and "observable M2"   and "is_det_state_cover M2 V"   and "\<not> M1 \<preceq> M2" obtains vs :: "('d \<times> 'e) list" and xs :: "('d \<times> 'e) list" where "minimal_sequence_to_failure_extending V M1 M2 vs xs"
healthy_parts: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> bounded_by b (t P)"   and "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> nneg (t P)"   and "\<And>P Q. \<lbrakk>sound P; sound Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q"   and "\<And>P c x. \<lbrakk>sound P; 0 \<le> c\<rbrakk> \<Longrightarrow> c * t P x = t (\<lambda>x. c * P x) x" shows "healthy t"
smult_0_right: fixes a :: "'b" shows "Polynomial.smult a 0 = 0"
has_contour_integral_split: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex"   and c :: "complex"   and j :: "complex"   and b :: "complex"   and k :: "real" assumes "(f has_contour_integral i) (linepath a c)"   and "(f has_contour_integral j) (linepath c b)"   and "0 \<le> k"   and "k \<le> 1"   and "c - a = k *\<^sub>R (b - a)" shows "(f has_contour_integral i + j) (linepath a b)"
power_inductr: fixes z :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and i :: "nat" assumes "z \<union> y ; x \<subseteq> y" shows "z ; rel_d.power x i \<subseteq> y"
assms_4: fixes i :: "'b" assumes "i \<in> I" shows "A i \<in> sets (M' i)"
simps_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm" shows "\<Gamma> \<turnstile> s is t : TUnit = True"
dist_norm: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and x :: "'e"   and y :: "'e" assumes "class.dist_norm minus dist norm" shows "dist x y = norm (minus x y)"
mult_le_prts: fixes a1 :: "'a"   and a :: "'a"   and a2 :: "'a"   and b1 :: "'a"   and b :: "'a"   and b2 :: "'a" assumes "a1 \<le> a"   and "a \<le> a2"   and "b1 \<le> b"   and "b \<le> b2" shows "a * b \<le> pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
final_thread_def: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r2.final_thread s t \<equiv> case thr s t of None \<Rightarrow> False | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> final2 x \<and> ln = no_wait_locks \<and> wset s t = None"
Ass: fixes G :: "prog"   and s0 :: "st"   and va :: "Term.var"   and w :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "state"   and e :: "expr"   and v :: "val"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>va=\<succ>(w, f)\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>v\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>va:=e-\<succ>v\<rightarrow> assign f v s2"
induct: fixes P :: "nat \<Rightarrow> (real \<times> real pdevs) list \<Rightarrow> (real \<times> real pdevs) list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "(real \<times> real pdevs) list"   and a2 :: "(real \<times> real pdevs) list" assumes "\<And>d x xs y ys. P d xs ys \<Longrightarrow> P d (x # xs) (y # ys)"   and "\<And>d uv_. P d [] uv_"   and "\<And>d uu_. P d uu_ []" shows "P a0 a1 a2"
e2frame_frame: fixes s :: "recf \<times> nat list \<times> nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2frame (encode_config (s # ss, rv)) = encode_frame s"
coeff_mult_closed: fixes A :: "'a set"   and p :: "'a mpoly"   and q :: "'a mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" assumes "ring_closed A"   and "\<And>x. MPoly_Type.coeff p x \<in> A"   and "\<And>x. MPoly_Type.coeff q x \<in> A" shows "MPoly_Type.coeff (p * q) x \<in> A"
simps_1: shows "bst_of_list [] = \<langle>\<rangle>"
subgroup_indicator_modify_carrier: fixes H' :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "subgroup_indicator (G\<lparr>carrier := H'\<rparr>) H a = subgroup_indicator G H a"
iprev_image: fixes n :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> nat" assumes "n \<in> I"   and "SetInterval2.strict_mono_on f I" shows "iprev (f n) (f ` I) = f (iprev n I)"
add_closed': shows "\<forall>x\<in>U\<^sub>V\<^sub>S. \<forall>y\<in>U\<^sub>V\<^sub>S. x +\<^sub>V\<^sub>S y \<in> U\<^sub>V\<^sub>S"
nextl_init_state: fixes xs :: "'a list" shows "M.nextl (init M) xs \<in> states M"
Diff_insert0: fixes x :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "x \<notin> A" shows "A - insert x B = A - B"
pos: fixes rs :: "'rule stream"   and r :: "'rule" assumes "fair rs"   and "r \<in> R" shows "shd (sdrop (pos rs r) rs) = r"
r_call_eq: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and c :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and t :: "state" assumes "\<Gamma> c = Some rs"   and "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> [Rule m (Call c)] \<Rightarrow>\<^sub>r t = \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t"
deg_shifts_mono: fixes fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list"   and gs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list"   and d :: "nat" assumes "set fs \<subseteq> set gs" shows "set (deg_shifts d fs) \<subseteq> set (deg_shifts d gs)"
mono: shows "mono (\<lambda>T. T \<union> {\<langle>\<rangle>} \<union> (\<Union>l\<in>T. \<Union>v\<in>S. \<Union>r\<in>T. {\<langle>l, v, r\<rangle>}))"
atLeastAtMost_singleton_iff: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" shows "(signed.atLeastAtMost a b = {c}) = (a = b \<and> b = c)"
dom_inv: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.dom (CC.inv f) = CC.cod f"
tSplice_tLast: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "tLength t = tLength t'"   and "envObs a (tLast t) = envObs a (tLast t')" shows "tLast (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t') = tLast t"
Un_singleton_iff: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" shows "(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
imonad2: fixes M :: "'c nres" shows "M \<bind> RETURN = M"
left_commute: fixes b :: "('a \<times> 'a set) set"   and a :: "('a \<times> 'a set) set"   and c :: "('a \<times> 'a set) set" shows "b \<parallel> (a \<parallel> c) = a \<parallel> (b \<parallel> c)"
get_mode_def: shows "get_mode = (\<lambda>shadow_root_ptr. get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode)"
reflexion_is_sound: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and D :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "finite (cl_ecl C)"   and "basic_superposition.reflexion trm_ord sel filter_trms C D \<sigma> k C'" shows "clause_entails_clause (cl_ecl C) (cl_ecl D)"
preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D'.trg (F \<mu>) = F'.map\<^sub>0 (C'.trg \<mu>)"
intro: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'b set" assumes "GroupHom G T"   and "GroupIso_axioms G T H" shows "GroupIso G T H"
surjective: fixes r :: "'a StoreImpl_scheme" shows "r = \<lparr>newOID = newOID r, newAID = newAID r, vals = vals r, \<dots> = more r\<rparr>"
lexord_def: fixes r :: "('a \<times> 'a) set" shows "lexord r = {(x, y). \<exists>a v. y = x @ a # v \<or> (\<exists>u a b v w. (a, b) \<in> r \<and> x = u @ a # v \<and> y = u @ b # w)}"
simple_matches_extended_packet: fixes m :: "'a simple_match"   and iifce :: "char list"   and oifce :: "char list"   and s :: "'a word"   and d :: "'a word"   and prot :: "8 word"   and sport :: "16 word"   and dport :: "16 word"   and tcp_flags :: "tcp_flag set"   and payload1 :: "char list"   and tcp_flags2 :: "tcp_flag set"   and payload2 :: "char list"   and aux :: "'b" shows "simple_matches m \<lparr>p_iiface = iifce, p_oiface = oifce, p_src = s, p_dst = d, p_proto = prot, p_sport = sport, p_dport = dport, p_tcp_flags = tcp_flags, p_payload = payload1\<rparr> = simple_matches m \<lparr>p_iiface = iifce, p_oiface = oifce, p_src = s, p_dst = d, p_proto = prot, p_sport = sport, p_dport = dport, p_tcp_flags = tcp_flags2, p_payload = payload2, \<dots> = aux\<rparr>"
lift_sum: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "lift (sum f A) = (\<Sum>x\<in>A. lift (f x))"
hyps_1: shows "P \<turnstile>\<^sub>1 \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"
Un_Int_eq_2: fixes S :: "'e set"   and T :: "'e set" shows "(S \<union> T) \<inter> T = T"
that: fixes es\<^sub>1 :: "('a \<times> bool) list"   and es\<^sub>2 :: "('a \<times> bool) list" assumes "e = es\<^sub>1 @ es\<^sub>2"   and "es\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)"   and "es\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)" shows "thesis"
init_segment_empty: fixes S :: "nat set" shows "init_segment {} S"
cor_cmod_real: fixes a :: "complex" assumes "is_real a" shows "cor (cmod a) = a \<or> cor (cmod a) = - a"
9: shows "components g \<le> forest_components (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T)"
Int: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<inter> b \<in> sets M"
continuous_on_cases_local: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "closedin (top_of_set (s \<union> t)) s"   and "closedin (top_of_set (s \<union> t)) t"   and "continuous_on s f"   and "continuous_on t g"   and "\<And>x. x \<in> s \<and> \<not> P x \<or> x \<in> t \<and> P x \<Longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
create_name_gba_def: fixes \<phi> :: "'b ltlr" shows "create_name_gba \<phi> \<equiv> create_gba \<phi> \<bind> (\<lambda>G. ASSERT (nds_invars (g_V G)) \<bind> (\<lambda>_. RETURN (gba_rename name G)))"
dvdm_def: fixes f :: "int poly"   and g :: "int poly" shows "(f dvdm g) = (\<exists>h. g =m f * h)"
leadsTo_wf_induct_lemma: fixes r :: "('c \<times> 'c) set"   and F :: "'d program"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and B :: "'d set"   and m :: "'c" assumes "wf r"   and "\<forall>m. F \<in> A \<inter> f -` {m} \<longmapsto> A \<inter> f -` r\<inverse> `` {m} \<union> B" shows "F \<in> A \<inter> f -` {m} \<longmapsto> B"
callT: fixes \<xi> :: "'e"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and pn :: "'g"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and p' :: "('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, \<Gamma> pn), a, \<xi>', p') \<in> seqp_sos \<Gamma>" shows "((\<xi>, call(pn)), a, \<xi>', p') \<in> seqp_sos \<Gamma>"
enum_v_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and n :: "nat" assumes "flow_network \<Delta> g" shows "flow_network.enum_v \<Delta> n = from_nat_into (\<^bold>V\<^bsub>\<Delta>\<^esub> - {source \<Delta>, sink \<Delta>}) (fst (prod_decode n))"
elims_2: fixes x :: "('fsym, 'psym) lit" assumes "wtL x" obtains a :: "('fsym, 'psym) atm" where "x = Pos a__" and "wtA a__"   | a :: "('fsym, 'psym) atm" where "x = Neg a" and "wtA a"
Sup_aform_def: fixes X :: "'a \<times> 'a pdevs" shows "Sup_aform X = fst X + Radius X"
l4_5: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "Cong A C A' C'" shows "\<exists>B'. Bet A' B' C' \<and> A B C Cong3 A' B' C'"
simps_4: fixes f1 :: "'l"   and f2 :: "'m \<Rightarrow> 'l" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
elements_mat_h_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "elements_mat_h A = range (\<lambda>(i, j). A $h i $h j)"
vector_component_6: fixes x :: "('a, 'b) vec"   and i :: "'b" shows "(- x) $ i = - x $ i"
exec_xs_ys: shows "\<Gamma>\<turnstile> \<langle>sequence Seq (xs @ ys),Normal s\<rangle> \<Rightarrow> t"
simps_1: shows "sngl \<langle>\<rangle> = False"
prod_1_def: fixes p1 :: "'e \<Rightarrow> 'g decision option"   and p2 :: "'f \<Rightarrow> 'h decision option" shows "p1 \<Otimes>\<^sub>1 p2 \<equiv> \<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>"
qfree_lift_eq_qe: fixes qe :: "'a list \<Rightarrow> 'a fm"   and as :: "'a list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)"   and "\<forall>a\<in>set as. depends\<^sub>0 a" shows "qfree (lift_eq_qe qe as)"
fair_state_seq_def: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" shows "fair_state_seq Sts = (N_of_state (Liminf_state Sts) = {} \<and> P_of_state (Liminf_state Sts) = {})"
f0_b_1: shows "inverse (lead_coeff (F (i - 1))) \<noteq> 0"
list_decode_inverse: fixes n :: "nat" shows "list_encode (list_decode n) = n"
assms: shows "locally_compatible_r R1 R2 (A + B) (C + D) (E + F)"
SUP_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
subcls_induct: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and Q :: "char list \<Rightarrow> bool"   and C :: "char list" assumes "wf_prog wf_md P"   and "\<And>C. \<forall>D. (C, D) \<in> (subcls1 P)\<^sup>+ \<longrightarrow> Q D \<Longrightarrow> Q C" shows "Q C"
list_of_oalist_of_list_id: fixes xs :: "('b \<times> 'c) list \<times> 'b nat_term_order" assumes "ko_ntm.oalist_inv xs" shows "list_of_oalist_ntm (OAlist_ntm xs) = xs"
program_inv_def: fixes prog :: "'a program_scheme" shows "program_inv prog = (0 < IArray.length (states prog) \<and> IArray.length (states prog) = IArray.length (processes prog) \<and> (\<forall>s\<in>set (IArray.list_of (states prog)). 0 < IArray.length s) \<and> lm.ball (proc_data prog) (\<lambda>(uu_, sidx). sidx < IArray.length (processes prog) \<and> fst (processes prog !! sidx) = sidx) \<and> (\<forall>(sidx, start, procArgs, args)\<in>set (IArray.list_of (processes prog)). \<exists>s. start = Index s \<and> s < IArray.length (states prog !! sidx)))"
float_defs_10: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "flt a b = (fcompare a b = Lt)"
Scope: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and \<alpha> :: "'d action"   and P' :: "('d, 'e, 'f) psi"   and x :: "name" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<alpha> \<prec> P')"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>" shows "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> (\<lparr>\<nu>x\<rparr>P) (\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P')"
simps_4: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<not> (\<forall>r\<in>R. Resumption.resumption.is_Done r)" shows "Resumption.resumption.output (resumption_lub R) = (THE out. out \<in> Resumption.resumption.output ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r}))"
recv_marker_other_channels_not_shrinking: fixes ev :: "('a, 'b, 'c) event"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and c :: "('a, 'b, 'c) configuration"   and c' :: "('a, 'b, 'c) configuration"   and i' :: "nat" assumes "ev = RecvMarker i p q"   and "c \<turnstile> ev \<mapsto> c'" shows "(length (msgs c i') \<le> length (msgs c' i')) = (i \<noteq> i')"
hyps: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "I < n" shows "det (swap_col_to_front A I) = (- (1::'a)) ^ I * det A"
SUP_diff_ennreal: fixes c :: "ennreal"   and f :: "'k \<Rightarrow> ennreal"   and I :: "'k set" assumes "c < \<top>" shows "(\<Squnion>i\<in>I. f i - c) = \<Squnion> (f ` I) - c"
nop_onto_n: shows "n \<in>\<^sub>\<circ> \<omega>"
GE_\<Phi>_def: shows "GE_\<Phi> \<equiv> ProblemIkTpartG.gPB wtFsym arOf resOf \<Phi> tpD tpFD tpCD"
RS2Set: fixes p :: "('b, 'c) Combinators list" shows "set (removeShadowRules2 p) \<subseteq> set p"
pos_oriented_opposite_ocircline_cmat: fixes H :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H \<and> H \<noteq> mat_zero" shows "pos_oriented_cmat (opposite_ocircline_cmat H) = (\<not> pos_oriented_cmat H)"
comb_sys_h: shows "snd (snd (reduce_system p (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) = snd (snd (reduce_system p (qs1 @ qs2, M_mat (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))), get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))), get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))))"
strict_trans2: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term"   and c :: "('d, 'e) Term.term" assumes "a \<rhd> b"   and "b \<unrhd> c" shows "a \<rhd> c"
finite_fresh_pvar: fixes P :: "pvar set" assumes "finite P" obtains p :: "pvar" where "p \<notin> P"
tab\<^sub>1_simps_1: fixes r :: "'a" assumes "ide r" shows "ide (tab\<^sub>1 r)"
FDOM_eff_subset_prob_dom_pair: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "a \<in> PROB" shows "fmdom' (snd a) \<subseteq> prob_dom PROB"
midpoint_sym: fixes a :: "'b"   and b :: "'b" shows "midpoint a b = midpoint b a"
vint_mult_def: shows "vint_mult = (\<lambda>x\<in>\<^sub>\<circ>\<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (int_of_vint (x\<lparr>0\<^sub>\<nat>\<rparr>) * int_of_vint (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<int>)"
to_fun_const: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "monom (UP Zp) a 0 \<bullet> b = a"
reversepath_def: fixes g :: "real \<Rightarrow> 'c" shows "reversepath g = (\<lambda>x. g (1 - x))"
upd_rev_cs_Cons_Call_Cons_Empty: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "CFG.upd_rev_cs kind cs as = []" shows "CFG.upd_rev_cs kind cs (a # as) = []"