hcju/isabelleps huggingface dataset:
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_set_disconnected_nodes_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root get_shadow_root_locs"
linear_axioms: shows "Vector_Spaces.linear (*a) local.scale (\<lambda>x. (x, 0::'c))"
i_set_defs_5: shows "iMODb_set \<equiv> {[ r, mod m, c ] |r m c. True}"
under_underS_trans: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "trans r"   and "antisym r"   and "a \<in> under r b"   and "b \<in> underS r c" shows "a \<in> underS r c"
simps_3: fixes uv :: "nat"   and var :: "nat \<times> char list"   and \<beta> :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "\<A> (R uv var) \<beta> ve = (case \<beta> (binder var) of None \<Rightarrow> DI 0 | Some l \<Rightarrow> case ve (var, l) of None \<Rightarrow> DI 0 | Some d \<Rightarrow> d)"
of_int_mult: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w * z) = times (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
dichotomous_Lazard_main: fixes n :: "nat"   and x :: "'b"   and y :: "'b" assumes "\<And>i. i \<le> n \<Longrightarrow> to_fract x ^ i / to_fract y ^ (i - 1) \<in> range to_fract" shows "to_fract (dichotomous_Lazard x y n) = to_fract x ^ n / to_fract y ^ (n - 1)"
empty_not_insert: fixes a :: "'b"   and A :: "'b set" shows "{} \<noteq> a \<triangleright> A"
IH: fixes s :: "'a list"   and v :: "'a val" assumes "s \<in> deriv c r \<rightarrow> v" shows "(c # s) \<in> r \<rightarrow> injval r c v"
ffb_fbd_galois_var: fixes f :: "'d \<Rightarrow> 'c set"   and X :: "'d set"   and Y :: "'c set" shows "(bd\<^sub>\<F> f X \<subseteq> Y) = (X \<subseteq> fb\<^sub>\<F> f Y)"
flat_pm_of_poly_in_Polys: fixes p :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) poly"   and Y :: "'a set"   and x :: "'a" assumes "range (poly.coeff p) \<subseteq> P[Y]" shows "flat_pm_of_poly x p \<in> P[insert x Y]"
ord_not_coprime: fixes n :: "'a"   and a :: "'a" assumes "\<not> coprime n a" shows "ord n a = 0"
resChainFresh: fixes x :: "name"   and xvec :: "name list"   and P :: "('d, 'e, 'f) psi" shows "x \<sharp> \<lparr>\<nu>*xvec\<rparr>P = (x mem xvec \<or> x \<sharp> P)"
a: shows "\<forall>i. f i \<in> A1 \<inter> A2"
finsum_sum: fixes B :: "'a mod_ring poly set"   and a :: "'a mod_ring poly \<Rightarrow> 'a mod_ring"   and R :: "('a mod_ring, 'b) partial_object_scheme" assumes "finite B"   and "a \<in> B \<rightarrow> carrier R"   and "B \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})" shows "(\<Oplus>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub>v\<in>B. a v \<odot>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub> v) = (\<Sum>v\<in>B. Polynomial.smult (a v) v)"
inf_aci_1: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<inter># y = y \<inter># x"
CR3_implies_CR4: fixes \<tau> :: "ty" assumes "CR3 \<tau>" shows "CR4 \<tau>"
aux_comp_with_sqrt2_bis: fixes n :: "nat" shows "2 ^ n / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) = 1 / sqrt 2"
runit_commutes_with_R: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "monoidal_category.runit C T \<alpha> \<iota> (T (a, monoidal_category.unity C \<iota>)) = T (monoidal_category.runit C T \<alpha> \<iota> a, monoidal_category.unity C \<iota>)"
f: shows "f \<in> borel_measurable (distr M M' T)"
stable_on_blocking_onD: fixes ds :: "'d set"   and X :: "'x set"   and X'' :: "'x set"   and h :: "'h" assumes "stable_on ds X"   and "X'' = Ch h (X \<union> X'')"   and "X'' \<subseteq> CD_on ds (X \<union> X'')" shows "X'' = Ch h X"
infer_sound: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> M = Some \<tau>" shows "\<Gamma> \<turnstile> M : \<tau>"
pr1_ext: fixes a :: "'a"   and b :: "'a" assumes "\<not> (ide a \<and> ide b)" shows "\<pp>\<^sub>1[a, b] = null"
lift_transient: fixes i :: "nat"   and F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" shows "(lift i F \<in> transient (lift_set i A)) = (F \<in> transient A)"
wsumT': shows "(\<Sum>t\<in>T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
homs_pass1': fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "homs (pass\<^sub>1' hs) = Pairing_Heap_Tree.pass\<^sub>1 (homs hs)"
1: shows "set adjs \<subseteq> {(v, d). (u, v) \<in> edges g \<and> w {u, v} = d}"
norm_blinfun_def: shows "norm \<equiv> map_fun blinfun_apply id onorm"
le_oInv: fixes z :: "ordinal"   and a :: "ordinal" assumes "F z \<le> a" shows "z \<le> oInv F a"
gInduct_elim: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme"   and phi :: "'sort \<Rightarrow> 'c \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'sort"   and X :: "'c"   and us :: "'varSort"   and s' :: "'sort"   and A :: "'d" assumes "gInduct MOD"   and "\<And>xs x. phi (asSort xs) (gVar MOD xs x)"   and "\<And>delta inp' inp binp' binp. \<lbrakk>wlsInp delta inp'; wlsBinp delta binp'; gWlsInp MOD delta inp; gWlsBinp MOD delta binp; liftAll2 phi (arOf delta) inp; liftAll2 phiAbs (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (gOp MOD delta inp' inp binp' binp)"   and "\<And>xs s x X' X. \<lbrakk>isInBar (xs, s); wls s X'; gWls MOD s X; phi s X\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (gAbs MOD xs x X' X)" shows "(gWls MOD s X \<longrightarrow> phi s X) \<and> (gWlsAbs MOD (us, s') A \<longrightarrow> phiAbs (us, s') A)"
take_derivation_shift: fixes n :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" shows "take n (derivation_shift D left right) = derivation_shift (take n D) left right"
dil_tick_count: fixes sub :: "'b run"   and r :: "'b run"   and a :: "clock"   and b :: "clock"   and n :: "nat" assumes "sub \<lless> r"   and "\<forall>n. #\<^sub>\<le> sub a n \<le> #\<^sub>\<le> sub b n" shows "#\<^sub>\<le> r a n \<le> #\<^sub>\<le> r b n"
Henstock_lemma_part1: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and e :: "real"   and d :: "'c \<Rightarrow> 'c set"   and p :: "('c \<times> 'c set) set" assumes "f integrable_on cbox a b"   and "0 < e"   and "gauge d"   and "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) - integral (cbox a b) f) < e"   and "p tagged_partial_division_of cbox a b"   and "d fine p" shows "norm (\<Sum>(x, K)\<in>p. content K *\<^sub>R f x - integral K f) \<le> e"
simps: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "'a"   and f :: "'c \<Rightarrow> 'b"   and a2 :: "'a"   and x :: "'c" shows "plus.ap_monoid_add plus (Monoid_ap a1 f) (Monoid_ap a2 x) = Monoid_ap (plus a1 a2) (f x)"
countable_image: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "countable A" shows "countable (f ` A)"
s0_\<gamma>_s': shows "s0\<^bsub>SES\<^esub> \<gamma>\<Longrightarrow>\<^bsub>SES\<^esub> s'"
pos_2: fixes a' :: "'a" assumes "a' \<in> set al" shows "0 < f a'"
yoneda_functor_fixed_object_def: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and \<phi> :: "'ag \<times> 'ag \<Rightarrow> 'ag \<Rightarrow> 'ah"   and a :: "'ag" shows "yoneda_functor_fixed_object C S \<phi> a \<equiv> yoneda_functor C S \<phi> \<and> yoneda_functor_fixed_object_axioms C a"
prems: shows "KD_Tree.size_kdt kdt = 2 ^ height kdt"
induced_arrowI_2: fixes \<chi>' :: "'j \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "D.cones_map (\<chi>.induced_arrow a' \<chi>') \<chi> = \<chi>'"
in_chain_finite: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "finite A"   and "A \<noteq> {}" shows "Sup A \<in> A"
elims_2: fixes x :: "'b simple_match"   and xa :: "('b, 'c) simple_packet_scheme" assumes "simple_matches x xa" obtains m :: "'b simple_match" and p :: "('b, 'c) simple_packet_scheme" where "x = m" and "xa = p" and "match_iface (iiface m) (p_iiface p) \<and> match_iface (oiface m) (p_oiface p) \<and> simple_match_ip (src m) (p_src p) \<and> simple_match_ip (dst m) (p_dst p) \<and> match_proto (proto m) (p_proto p) \<and> simple_match_port (sports m) (p_sport p) \<and> simple_match_port (dports m) (p_dport p)"
rep_eq: fixes x :: "'a" shows "Rep_ttree (single x) = {[], [x]}"
Theap_thunk: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and p :: "Vars.var list"   and e :: "exp"   and a :: "Arity" assumes "x \<in> thunks \<Gamma>"   and "p \<in> paths (Theap \<Gamma> e\<cdot>a)"   and "\<not> one_call_in_path x p" shows "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>0"
assms_2: shows "hermitean H2 \<and> H2 \<noteq> mat_zero"
typing_cond_prot_impl_typing_cond: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" assumes "typing_cond_prot \<P>"   and "\<A> \<in> \<P>" shows "typing_cond (unlabel \<A>)"
indexed_const_def: fixes k :: "('a list \<times> nat) multiset \<Rightarrow> 'a" shows "M.indexed_const k = (\<lambda>m. if m = {#} then k else \<zero>\<^bsub>M\<^esub>)"
term_to_exp_def: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(Term_Class.term \<times> Term_Class.term) fset"   and t :: "Term_Class.term" shows "term_to_exp C_info rs t = cakeml.mk_con C_info (heads_of rs |\<union>| constructors.C C_info) (pterm_to_sterm (nterm_to_pterm (fresh_frun (Term_to_Nterm.term_to_nterm [] t) (heads_of rs |\<union>| constructors.C C_info))))"
lt_le: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>u. v \<prec>\<^sub>t u \<Longrightarrow> lookup p u = (0::'c)" shows "lt p \<preceq>\<^sub>t v"
is_sublist_distinct_prefix: fixes v :: "'a"   and as :: "'a list"   and vs :: "'a list" assumes "is_sublist (v # as) (v # vs)"   and "distinct (v # vs)" shows "is_prefix as vs"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1 @ rs\<^sub>2, s\<rangle> \<Rightarrow> t"
SUP_eq_top_iff: 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 f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Sup (f ` A) = top) = (\<forall>x. less x top \<longrightarrow> (\<exists>i\<in>A. less x (f i)))"
inj_on_id: fixes A :: "'a set" shows "inj_on id A"
ins_list_snoc: fixes xs :: "'b list"   and x :: "'b" assumes "sorted (xs @ [x])" shows "ins_list x xs = xs @ [x]"
mono_char: fixes f :: "'s" shows "local.mono f = (arr f \<and> inj_on (Fun f) (Dom f))"
rref_upt_condition2: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'d)) = (1::'d)"
F_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "comm_monoid_fset f z" shows "comm_monoid_fset.F f z \<equiv> (id ---> fset ---> id) (comm_monoid_set.F f z)"
simps_4: fixes j :: "nat"   and i :: "nat" shows "prev j (Suc i) = Some (j, i)"
index_of_aux_lookup_2: fixes p :: "'a"   and ls :: "'a list"   and n :: "nat" assumes "p \<in> set ls" shows "ls ! (index_of_aux ls p n - n) = p"
simps_2: fixes uv :: "'a" shows "(extC.Val uv \<le> extC.Bot) = False"
wpQ_is_pre: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2\<^sub>' {wp\<^sub>Q c Q} c {Q}"
lesubs'_1: shows "raw_match_term u u' lefsubs = Some lesubs'"
exec_ta_satisfiable: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and t :: "'b"   and s :: "'a option \<times> 'c \<times> 'a frame list"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and s' :: "'a option \<times> 'c \<times> 'a frame list" assumes "JVM_heap addr2thread_id thread_id2addr allocate typeof_addr heap_write P"   and "JVM_heap_base.exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t s ta s'" shows "\<exists>s. final_thread.actions_ok JVM_final s t ta"
PReqS: fixes evsPReqS :: "event list"   and C :: "agent"   and k :: "nat"   and KC2 :: "nat"   and Transaction :: "msg"   and M :: "agent"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and HOD :: "msg"   and OIData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and Chall_C :: "nat"   and Chall_M :: "nat"   and PIHead :: "msg"   and PANData :: "msg"   and PIData :: "msg"   and PIDualSigned :: "msg"   and EKj :: "nat"   and OIDualSigned :: "msg"   and P :: "agent" assumes "evsPReqS \<in> set_pur"   and "C = Cardholder k"   and "CardSecret k \<noteq> 0"   and "Key KC2 \<notin> used evsPReqS"   and "KC2 \<in> symKeys"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "HOD = Hash \<lbrace>Number OrderDesc, Number PurchAmt\<rbrace>"   and "OIData = \<lbrace>Number LID_M, Number XID, Nonce Chall_C, HOD, Nonce Chall_M\<rbrace>"   and "PIHead = \<lbrace>Number LID_M, Number XID, HOD, Number PurchAmt, Agent M, Hash \<lbrace>Number XID, Nonce (CardSecret k)\<rbrace>\<rbrace>"   and "PANData = \<lbrace>Pan (pan C), Nonce (PANSecret k)\<rbrace>"   and "PIData = \<lbrace>PIHead, PANData\<rbrace>"   and "PIDualSigned = \<lbrace>sign (priSK C) \<lbrace>Hash PIData, Hash OIData\<rbrace>, EXcrypt KC2 EKj \<lbrace>PIHead, Hash OIData\<rbrace> PANData\<rbrace>"   and "OIDualSigned = \<lbrace>OIData, Hash PIData\<rbrace>"   and "Gets C (sign (priSK M) \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Nonce Chall_M, cert P EKj onlyEnc (priSK RCA)\<rbrace>) \<in> set evsPReqS"   and "Says C M \<lbrace>Number LID_M, Nonce Chall_C\<rbrace> \<in> set evsPReqS"   and "Notes C \<lbrace>Number LID_M, Transaction\<rbrace> \<in> set evsPReqS" shows "Says C M \<lbrace>PIDualSigned, OIDualSigned\<rbrace> # Notes C \<lbrace>Key KC2, Agent M\<rbrace> # evsPReqS \<in> set_pur"
finite_lin_spanTr3_0: fixes g :: "nat \<Rightarrow> 'a"   and l :: "nat"   and A :: "'b set"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "'e \<Rightarrow> 'b"   and m :: "'e"   and f :: "'e \<Rightarrow> 'a"   and sa :: "nat \<Rightarrow> 'b" assumes "bij_to g {j. j \<le> l} (g ` {j. j \<le> l})"   and "ideal R A"   and "\<forall>naa. \<forall>s\<in>{j. j \<le> naa} \<rightarrow> A. \<forall>f\<in>{j. j \<le> naa} \<rightarrow> h ` {j. j \<le> n}. \<exists>t\<in>{j. j \<le> n} \<rightarrow> A. l_comb R M naa s f = l_comb R M n t h"   and "h \<in> {j. j \<le> Suc n} \<rightarrow> carrier M"   and "s \<in> {j. j \<le> m} \<rightarrow> A"   and "f \<in> {j. j \<le> m} \<rightarrow> h ` {j. j \<le> Suc n}"   and "l \<le> Suc n"   and "sa \<in> {j. j \<le> l} \<rightarrow> A"   and "g \<in> {j. j \<le> l} \<rightarrow> h ` {j. j \<le> Suc n}"   and "0 < l"   and "f ` {j. j \<le> m} = g ` {j. j \<le> l}"   and "h (Suc n) = g l" shows "\<exists>t\<in>{j. j \<le> Suc n} \<rightarrow> A. l_comb R M l sa g = l_comb R M (Suc n) t h"
e: shows "card UNIV < card ((\<tau> ^^ i) {})"
size_list_simp_2: fixes xs :: "'a list" shows "|xs| = (if xs = [] then 0 else Suc |tl xs|)"
succ_it_is_iterator: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) graph"   and invar :: "'a \<Rightarrow> bool"   and succ_list_it :: "'a \<Rightarrow> 'b \<Rightarrow> (('c \<times> 'b) list \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'b \<Rightarrow> ('c \<times> 'b) list \<Rightarrow> ('c \<times> 'b) list) \<Rightarrow> ('c \<times> 'b) list \<Rightarrow> ('c \<times> 'b) list"   and g :: "'a"   and v :: "'b" assumes "graph_succ_it \<alpha> invar succ_list_it"   and "invar g" shows "set_iterator (graph_succ_it_defs.succ_it succ_list_it g v) (succ (\<alpha> g) v)"
ac_Zp_def: fixes p :: "int"   and x :: "nat \<Rightarrow> int" assumes "padic_integers p" shows "padic_integers.ac_Zp p x = padic_integers.normalizer p (nat (padic_integers.ord_Zp p x)) x"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and f5 :: "char list \<Rightarrow> 'c"   and f6 :: "'c"   and f7 :: "char list \<Rightarrow> 'c"   and f8 :: "'c"   and f9 :: "'c"   and action :: "action" shows "P (case action of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = ((action = Accept \<longrightarrow> P f1) \<and> (action = Drop \<longrightarrow> P f2) \<and> (action = Log \<longrightarrow> P f3) \<and> (action = Reject \<longrightarrow> P f4) \<and> (\<forall>x5. action = Call x5 \<longrightarrow> P (f5 x5)) \<and> (action = Return \<longrightarrow> P f6) \<and> (\<forall>x7. action = Goto x7 \<longrightarrow> P (f7 x7)) \<and> (action = Empty \<longrightarrow> P f8) \<and> (action = Unknown \<longrightarrow> P f9))"
r_neg1: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> (x \<oplus>\<^bsub>G\<^esub> y) = y"
strip_while_dropWhile_commute: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "strip_while P (dropWhile Q xs) = dropWhile Q (strip_while P xs)"
inj_on_g: shows "inj_on g (tangent_space p)"
read_actions_not_write_actions: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> read_actions E"   and "a \<in> write_actions E" shows "False"
A_FactGroup_def': fixes G :: "('e, 'f) ring_scheme"   and H :: "'e set" shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = (<+>\<^bsub>G\<^esub>), one = H\<rparr>"
borel_measurable_iff_halfspace_ge: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)"
geom_rand_walk_pseudo_proj_False: fixes n :: "nat" shows "geom_proc n = geom_proc n \<circ> pseudo_proj_False n"
prv_prv_imp_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> \<chi>)"   and "prv (imp \<chi> \<psi>)" shows "prv (imp \<phi> \<psi>)"
hom_def: fixes G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
beukers_nn_integral3_pos: fixes n :: "nat" shows "0 < beukers_nn_integral3 n"
update_keys: fixes k :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" shows "map fst (AList.update k v al) = (if k \<in> set (map fst al) then map fst al else map fst al @ [k])"
polypoly_def: fixes bs :: "'a list"   and p :: "poly" shows "polypoly bs p = map (Ipoly bs) (coefficients p)"
outSetCorelatedEmpty2: fixes x :: "chanID" assumes "outSetCorelated x = {}"   and "OUTfromV_VARto" shows "OUTfromV x = {}"
a: shows "{x \<in> local.degenerate_lotteries. \<forall>y\<in>local.degenerate_lotteries. x \<succeq>[\<R>] y} = {}"
gen_gbg_impl_rel_eext_def_internal: fixes Rm :: "('f \<times> 'h) set"   and Rf :: "('e \<times> 'g set set) set" shows "gen_gbg_impl_rel_eext Rm Rf \<equiv> {(\<lparr>gbgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>gbg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
simps_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(TargetTerm x2 = TargetTerm y2) = (x2 = y2)"
mru_of_set_def: fixes vs :: "'d \<Rightarrow> 'c \<Rightarrow> 'e option" shows "mru_of_set vs \<equiv> \<lambda>Q. option_Max_by fst (vote_set vs Q)"
ChiStar_Image: fixes S :: "'d"   and M :: "'d set"   and A :: "('d, 'e, 'f) hierauto" assumes "S \<in> M" shows "S \<in> ChiStar A `` M"
assms_1: shows "ipPartition (set (map wordinterval_to_set (extract_IPSets_generic0 src rs))) A"
gen_dfs_def: shows "a.a.gen_dfs.gen_dfs \<equiv> gds_init a.a.gen_dfs.gds \<bind> WHILE a.a.gen_dfs.gen_cond a.a.gen_dfs.gen_step"
\<sigma>\<^sub>2'_def: shows "\<sigma>\<^sub>2' = final \<tau>\<^sub>2"
runit_hcomp_1: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<r>\<^sub>D[f \<star>\<^sub>D g] = (f \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D \<a>\<^sub>D[f, g, src\<^sub>D g]"
tendsto_vec_nth: fixes f :: "'c \<Rightarrow> ('d, 'e) vec"   and a :: "('d, 'e) vec"   and net :: "'c filter"   and i :: "'e" assumes "(f \<longlongrightarrow> a) net" shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
singleton_sub: fixes a :: "'h"   and A :: "'h set" assumes "a \<in> A" shows "{a} \<subseteq> A"