hcju/isabelleps huggingface dataset:
Smith_mxn_HMA_def: fixes Smith_1x2 :: "('b, 2) Finite_Cartesian_Product.vec \<Rightarrow> ('b, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and Smith_2x2 :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<Rightarrow> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and div_op :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Smith_mxn_HMA Smith_1x2 Smith_2x2 div_op A = (let Smith_1x2_JNF = \<lambda>A'. let (S', Q') = Smith_1x2 (Mod_Type_Connect.to_hma\<^sub>v (Matrix.row A' 0)) in (mat_of_row (Mod_Type_Connect.from_hma\<^sub>v S'), Mod_Type_Connect.from_hma\<^sub>m Q'); Smith_2x2_JNF = \<lambda>A'. let (P', S', Q') = Smith_2x2 (Mod_Type_Connect.to_hma\<^sub>m A') in (Mod_Type_Connect.from_hma\<^sub>m P', Mod_Type_Connect.from_hma\<^sub>m S', Mod_Type_Connect.from_hma\<^sub>m Q'); (P, S, Q) = Smith_Impl.Smith_mxn Smith_1x2_JNF Smith_2x2_JNF div_op (Mod_Type_Connect.from_hma\<^sub>m A) in (Mod_Type_Connect.to_hma\<^sub>m P, Mod_Type_Connect.to_hma\<^sub>m S, Mod_Type_Connect.to_hma\<^sub>m Q))"
Skip: fixes P :: "'d set"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> Q" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Skip Q,A"
recv_marker_means_snapshotted_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and i :: "nat"   and q :: "'d"   and p :: "'d" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "Marker \<in> set (msgs c i)"   and "Marker \<notin> set (msgs c' i)"   and "channel i = Some (q, p)" shows "ps c' p \<noteq> None"
final_step_of_path': fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u"   and "p \<notin> \<P> k (u - 1)" shows "\<exists>q ts. 0 < u \<and> p = q @ ts \<and> is_continuation k (u - 1) q ts"
Assign_tE: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and p :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(x ::= a, s) \<Rightarrow> p \<Down> t" obtains "p = Suc 0" and "t = s(x := aval a s)"
const_OclIncluding: fixes x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and S :: "'d state \<times> 'd state \<Rightarrow> Bag('e)" assumes "const x"   and "const S" shows "const S->including\<^sub>B\<^sub>a\<^sub>g(x)"
rep_eq: fixes x :: "'a['c, 'b]"   and xa :: "'a['c, 'b]" shows "fromQ (x - xa) = \<lparr>mag = mag (fromQ x) - mag (fromQ xa), dim = Quantity.dim (fromQ x), unit_sys = unit\<rparr>"
inject_3: fixes x31 :: "'a"   and x32 :: "'b"   and y31 :: "'a"   and y32 :: "'b" shows "(<x31, x32> = <y31, y32>) = (x31 = y31 \<and> x32 = y32)"
Domainp_HMA_I: shows "Domainp Mod_Type_Connect.HMA_I = (\<lambda>i. i < CARD('b))"
consistent_def3: shows "B.consistent = (\<forall>\<phi>\<in>fmla. \<not> (bprv \<phi> \<and> bprv (neg \<phi>)))"
slength_bottom_iff: fixes xs :: "[:'r:]" shows "(slength\<cdot>xs = \<bottom>) = (xs = \<bottom>)"
seq_CSP: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is CSP1 healthy"   and "P is R healthy"   and "is_CSP_process Q" shows "is_CSP_process (P ;; Q)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_tag_name type_wf set_tag_name set_tag_name_locs"   and "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_tag_name_get_disconnected_nodes_axioms set_tag_name_locs get_disconnected_nodes_locs" shows "l_set_tag_name_get_disconnected_nodes type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"
ipassmt_sanity_nowildcards_def: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option" shows "ipassmt_sanity_nowildcards ipassmt \<equiv> \<forall>iface\<in>dom ipassmt. \<not> iface_is_wildcard iface"
simps_1: fixes uu :: "'b" shows "is_exn (Some uu) = False"
nsg_lr_cst_eq: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G" shows "a \<diamondsuit> N = N \<bullet> a"
Fvars_ground_aux: fixes t :: "trm"   and B :: "name set" assumes "Fvars t \<subseteq> B" shows "ground_aux t (atom ` B)"
restrict_relp_True: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "R \<upharpoonleft> (\<lambda>_. True) \<otimes> (\<lambda>_. True) = R"
mat_to_list_def: fixes A :: "'a mat" shows "mat_to_list A = map (\<lambda>i. map (\<lambda>j. A $$ (i, j)) [0..<dim_col A]) [0..<dim_row A]"
diff_frac_eq_2: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and b :: "real"   and a :: "real" assumes "PL_Indis cval indis"   and "b \<noteq> 1" shows "1 - (a - b) / (1 - b) = (1 - a) / (1 - b)"
filter_nths_diff: fixes i :: "nat"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and A :: "nat set" assumes "i < length xs"   and "\<not> P (xs ! i)" shows "filter P (nths xs (A - {i})) = filter P (nths xs A)"
constant_on_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
assms_2: shows "(state0, state) \<in> transitionRelation decisionVars"
subspace_left_null_space: fixes A :: "(('a, 'c) vec, 'b) vec" shows "vec.subspace (left_null_space A)"
exec: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
gen_dfs_fe_inv_imp_pre: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a"   and it :: "'a set"   and V' :: "'a set"   and t :: "'a" assumes "gen_dfs_pre E U S V u0"   and "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 it V' False"   and "t \<in> it"   and "it \<subseteq> E `` {u0}"   and "t \<notin> V'" shows "gen_dfs_pre E U (insert u0 S) V' t"
map_fst: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "0 < n" shows "map f [0..<n] = f 0 # map f [1..<n]"
Confidentiality_Tgs_bis: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Says Kas A \<lbrace>Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "\<not> expiredAK Ta evs"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "Key servK \<notin> analz (knows Spy evs)"
evalRecvArgs_cl_inv: fixes g :: "gState\<^sub>I"   and p :: "pState"   and rargs :: "recvArg list"   and xs :: "integer list" assumes "cl_inv (g, p)" shows "cl_inv (evalRecvArgs rargs xs g p)"
ib: shows "i + 1 < j \<and> j \<le> length t \<and> ps (S t (i + 1)) p \<noteq> None \<and> snd (cs (S t j) cid) = Done"
IH_1: fixes x :: "'a list list"   and xa :: "'a list"   and xb :: "nat"   and xc :: "'a"   and xd :: "'a list \<times> 'a list \<times> 'a list"   and xe :: "'a list"   and y :: "'a list \<times> 'a list"   and xda :: "'a list"   and ya :: "'a list"   and xdb :: "nat"   and xdc :: "nat" assumes "\<not> length xs \<le> 20"   and "x = chop 5 xs"   and "xa = map slow_median x"   and "xb = ((length xs + 4) div 5 - 1) div 2"   and "xc = mom_select xb xa"   and "xd = partition3 xc xs"   and "(xe, y) = xd"   and "(xda, ya) = y"   and "xdb = length xe"   and "xdc = length xda"   and "k < xdb" shows "T_mom_select k xe \<le> T'_mom_select (length xe)"
concat_filter_neq_Nil: fixes xs :: "'a list list" shows "concat (filter (\<lambda>ys. ys \<noteq> []) xs) = concat xs"
le_of_nat_term_order_plus_monotone: fixes to :: "'b nat_term_order"   and s :: "'b"   and t :: "'b"   and u :: "'b" assumes "le_of_nat_term_order to s t" shows "le_of_nat_term_order to (u + s) (u + t)"
at_set_avoiding2: fixes xs :: "atom set"   and c :: "'a"   and x :: "'b" assumes "finite xs"   and "finite (supp c)"   and "finite (supp x)"   and "xs \<sharp>* x" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
code: fixes x :: "complex"   and y :: "complex" shows "x - y = Complex (Re x - Re y) (Im x - Im y)"
less_imp_not_eq2: fixes x :: "nat list"   and y :: "nat list" assumes "x <\<^sub>v y" shows "(y = x) = False"
set_Pi_pmf_subset': fixes A :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A" shows "set_pmf (Pi_pmf A dflt p) \<subseteq> PiE_dflt A dflt (set_pmf \<circ> p)"
trie_lookup_empty: shows "trie_lookup trie_empty = Map.empty"
is_Write\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_Write\<^sub>s\<^sub>b r = (case r of Write\<^sub>s\<^sub>b volatile a x v xa xb xc xd \<Rightarrow> True | _ \<Rightarrow> False)"
nop_axioms: shows "nop A (1\<^sub>\<nat>) f"
list_length_hint1: fixes l :: "'a list" assumes "l \<noteq> []" shows "0 < length l"
that: fixes a :: "V"   and c :: "V"   and b :: "V" assumes "ac = \<langle>a, c\<rangle>"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s"   and "\<langle>b, c\<rangle> \<in>\<^sub>\<circ> r" shows "thesis"
red_rtrancl_uminus: 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 "(punit.red F)\<^sup>*\<^sup>* (- p) (- q)"
index_row_2: fixes A :: "'a mat"   and i :: "nat" shows "dim_vec (row A i) = dim_col A"
length_rotater: fixes n :: "nat"   and xs :: "'b list" shows "length (rotater n xs) = length xs"
l2_inv1_def: shows "l2_inv1 \<equiv> {s. \<forall>A B. can_signal s A B \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s}"
simps: fixes I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE" shows "semBV I ODE = Inl ` ODE_vars I ODE \<union> Inr ` ODE_vars I ODE"
T1T: shows "poly_mod.Mp p T1 = poly_mod.Mp p T"
f_Exec_Stream_expand_aggregate_map_nth: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and f :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f"   and ag :: "'e list \<Rightarrow> 'e" assumes "0 < k"   and "n < length xs" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c)) k ag ! n = ag (map f (f_Exec_Comp_Stream trans_fun (xs ! n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (xs \<down> n \<odot> k) c)))"
elementary_cartesian_closed_category_axioms: shows "elementary_cartesian_closed_category C pr0 pr1 \<one> trm exp eval \<Lambda>"
polytope_imp_closed: fixes S :: "'b set" assumes "polytope S" shows "closed S"
e'_eq: fixes i :: "nat" assumes "i < min d da" shows "e' i = e i"
subresultant_prs_locale2_def: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly" shows "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2 \<equiv> subresultant_prs_locale F n \<delta> f k \<beta> G1 G2 \<and> subresultant_prs_locale2_axioms \<delta> f k \<beta>"
isuccs_def: fixes i :: "instr"   and n :: "int" shows "isuccs i n = (case i of JMP j \<Rightarrow> {n + 1 + j} | JMPLESS j \<Rightarrow> {n + 1 + j, n + 1} | JMPGE j \<Rightarrow> {n + 1 + j, n + 1} | _ \<Rightarrow> {n + 1})"
cases: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "accessible_on P A a" obtains x :: "'b" where "a = x" and "x \<in> A" and "\<And>y. \<lbrakk>y \<in> A; P y x\<rbrakk> \<Longrightarrow> accessible_on P A y"
pathsConverge'_def: fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and y :: "'node"   and ys :: "'node list"   and z :: "'node" shows "old.pathsConverge' g x xs y ys z \<equiv> g \<turnstile> x-xs\<rightarrow>z \<and> g \<turnstile> y-ys\<rightarrow>z \<and> 1 < length xs \<and> 1 < length ys \<and> x \<noteq> y \<and> set (butlast xs) \<inter> set (butlast ys) = {}"
union: fixes A :: "'a set"   and B :: "'a set" shows "local.MaxR.MaxR_opt (A \<union> B) = (case local.MaxR.MaxR_opt A of None \<Rightarrow> local.MaxR.MaxR_opt B | Some mA \<Rightarrow> Some (case local.MaxR.MaxR_opt B of None \<Rightarrow> mA | Some mB \<Rightarrow> local.MaxR.maxR mA mB))"
quorum_for_def: fixes Q :: "process set"   and v :: "val"   and v_f :: "process \<Rightarrow> val option" shows "quorum_for Q v v_f = (Q \<in> Quorum \<and> (\<forall>p\<in>Q. v_f p = Some v))"
frange_vsingleton: fixes a :: "V"   and b :: "V" shows "\<R>\<^sub>\<bullet> (set {[a, b]\<^sub>\<circ>}) = set {b}"
assms_4: shows "h \<turnstile> get_elements_by_tag_name ptr idd \<rightarrow>\<^sub>r results"
approx_un_argE: fixes p :: "nat"   and F :: "float interval \<Rightarrow> float interval option"   and X :: "((real \<times> real pdevs) \<times> real) option"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "approx_un p F X = Some Y" obtains X' :: "(real \<times> real pdevs) \<times> real" where "X = Some X'"
\<epsilon>_uh_nonneg: fixes seed_gen :: "'a spmf"   and hash :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "0 \<le> PRF_UHF.hash.\<epsilon>_uh seed_gen hash"
conth: shows "continuous_on (T - \<xi> ` K) h"
induct: fixes P :: "'a tree23s \<Rightarrow> bool"   and a0 :: "'a tree23s" assumes "\<And>t1 a t2. P (TTs t1 a (T t2))"   and "\<And>t1 a t2 b t3. P (TTs t1 a (TTs t2 b (T t3)))"   and "\<And>t1 a t2 b v va vb. P (TTs v va vb) \<Longrightarrow> P (TTs t1 a (TTs t2 b (TTs v va vb)))"   and "\<And>v. P (T v)" shows "P a0"
Nmlize_Trg_1: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Trg t\<^bold>\<rfloor> = Trg \<^bold>\<lfloor>t\<^bold>\<rfloor>"
the_var_AVar_def: fixes G :: "prog"   and s :: "state"   and a :: "expr"   and i :: "expr" shows "the_var G s (a.[i]) = avar G (the_val i) (the_val a) s"
UnI2: fixes c :: "'g"   and B :: "'g set"   and A :: "'g set" assumes "c \<in> B" shows "c \<in> A \<union> B"
Seq_decomp_Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p\<^sub>2 :: "('e, 'd, 'f) com"   and z :: "'e"   and p' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Seq Throw p\<^sub>2, Normal z) \<rightarrow> (p', s')" shows "s' = Normal z \<and> p' = Throw"
fch_equivalence: fixes h :: "'a" assumes "forest h" shows "equivalence (forest_components h)"
bp_bins_finite: fixes P :: "'a set set" assumes "bp P" shows "\<forall>B\<in>P. finite B"
ae_eq: shows "\<forall>w\<in>space M. (cls_val_process Mkt pf1 0 w = 0) = (cls_val_process Mkt pf2 0 w = 0)"
homeomorphism_on_compose: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and u :: "'e \<Rightarrow> 'c"   and F :: "'e filter" assumes "homeomorphism_on S f"   and "x \<in> S"   and "\<forall>\<^sub>F n in F. u n \<in> S" shows "(u \<longlongrightarrow> x) F = ((\<lambda>n. f (u n)) \<longlongrightarrow> f x) F"
orthogonal_cone_1: fixes X :: "'a vec set"   and W :: "'a vec set"   and Ls :: "'a vec list"   and Ws :: "'a vec list"   and Bs :: "'a vec list" assumes "X \<subseteq> carrier_vec n"   and "W \<subseteq> carrier_vec n"   and "finite X"   and "local.span (set Ls \<union> W) = carrier_vec n"   and "\<And>w x. \<lbrakk>w \<in> W; x \<in> set Ls\<rbrakk> \<Longrightarrow> w \<bullet> x = (0::'a)"   and "W = set Ws"   and "local.span (set Ls) = local.span X"   and "set Ls \<subseteq> X"   and "lin_indpt_list (Ls @ Bs)"   and "length (Ls @ Bs) = n" shows "cone (X \<union> set Bs) \<inter> {x \<in> carrier_vec n. \<forall>w\<in>W. w \<bullet> x = (0::'a)} = cone X"
SUP_simple_integral_sequences: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "incseq f"   and "\<And>i. simple_function M (f i)"   and "incseq g"   and "\<And>i. simple_function M (g i)"   and "AE x in M. (SUP i. f i x) = (SUP i. g i x)" shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
qSwap_involutive: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "X #[[x \<and> y]]_zs #[[x \<and> y]]_zs = X"
D: shows "ipurge_tr_rev_aux I D U (xs @ ys @ zs) = ipurge_tr_rev_aux I D U (xs @ ipurge_tr_aux I D (D ` set ys) zs)"
sumCases': fixes P :: "pi"   and Q :: "pi"   and Rs :: "residual" assumes "P \<oplus> Q \<longmapsto> Rs" obtains P :: "pi" and Qa :: "pi" where "Pa__ = Pb__ \<and> Q = Qa__" and "Pb__ \<longmapsto> Rs"   | Qa :: "pi" and P :: "pi" where "Pa__ = P \<and> Q = Qa" and "Qa \<longmapsto> Rs"
e: fixes x :: "'b"   and n :: "nat" shows "\<exists>i. dist x (e i) < 1 / real (Suc n)"
select_convs_7: fixes inpt :: "Proc \<Rightarrow> InputsOrNi"   and outpt :: "Proc \<Rightarrow> InputsOrNi"   and disk :: "Disk \<Rightarrow> Proc \<Rightarrow> DiskBlock"   and dblock :: "Proc \<Rightarrow> DiskBlock"   and phase :: "Proc \<Rightarrow> nat"   and disksWritten :: "Proc \<Rightarrow> Disk set"   and blocksRead :: "Proc \<Rightarrow> Disk \<Rightarrow> BlockProc set"   and allInput :: "InputsOrNi set"   and chosen :: "InputsOrNi"   and more :: "'a" shows "blocksRead \<lparr>inpt = inpt, outpt = outpt, disk = disk, dblock = dblock, phase = phase, disksWritten = disksWritten, blocksRead = blocksRead, allInput = allInput, chosen = chosen, \<dots> = more\<rparr> = blocksRead"
setdist_unique: fixes a :: "'b"   and S :: "'b set"   and b :: "'b"   and T :: "'b set" assumes "a \<in> S"   and "b \<in> T"   and "\<And>x y. x \<in> S \<and> y \<in> T \<Longrightarrow> dist a b \<le> dist x y" shows "setdist S T = dist a b"
f_def: fixes i :: "nat" shows "f i = \<omega> (P $ i)"
Sr_tOp_closed: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "sr R S"   and "x \<in> S"   and "y \<in> S" shows "x \<cdot>\<^sub>r y \<in> S"
index_le_size: fixes xs :: "'c list"   and x :: "'c" shows "index xs x \<le> length xs"
Pi'_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set" shows "Pi' I A = {P. domain P = I \<and> (\<forall>i. i \<in> I \<longrightarrow> (P)\<^sub>F i \<in> A i)}"
bottom_right_bottom: fixes a :: "'b" shows "lcm a (0::'b) = (0::'b)"
degree_aform_err_affine_unop: fixes X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat"   and p :: "nat"   and a :: "real"   and b :: "real"   and d :: "real" assumes "degree_aform_err X \<le> n" shows "degree_aform_err (affine_unop p a b d X) \<le> n"
fmdrop_fset_fmdom: fixes A :: "('c, 'd) fmap" shows "fmdrop_fset (fmdom A) A = fmempty"
SD_iff_expected_utilities_le: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> lotteries_on carrier"   and "q \<in> lotteries_on carrier" shows "SD le p q = (\<forall>u. is_vnm_utility u \<longrightarrow> measure_pmf.expectation p u \<le> measure_pmf.expectation q u)"
submoduleE_1: fixes H :: "'c set" assumes "Module.submodule H K V" shows "H \<subseteq> carrier V"
r_neg: fixes x :: "'a" shows "x + - x = (0::'a)"
Pi'_I: fixes f :: "'d \<Rightarrow>\<^sub>F 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "domain f = A"   and "\<And>x. x \<in> A \<Longrightarrow> (f)\<^sub>F x \<in> B x" shows "f \<in> Pi' A B"
Pluss_None: fixes r :: "'c rexp" shows "(Full \<in> toplevel_summands r) = (Pluss r = None)"
ZFTThd: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF" shows "ZFTThd (ZFTriple a b c) = c"
P_mod_I_J: fixes p :: "'a"   and I :: "'a set"   and J :: "'a set" assumes "p \<in> carrier R"   and "ideal S I"   and "ideal S J"   and "I \<subseteq> J"   and "P_mod R S X I p" shows "P_mod R S X J p"
lin_dep_def: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and S :: "'c set" assumes "Module.module R M" shows "module.lin_dep R M S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R \<and> module.lincomb M a A = \<zero>\<^bsub>M\<^esub> \<and> v \<in> A \<and> a v \<noteq> \<zero>\<^bsub>R\<^esub>)"
mintree_exists: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "(bq \<noteq> []) = (BinomialHeapStruc.getMinTree bq \<in> set bq)"
sub_num_simps_8: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num"   and l :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit1 k) (num.Bit0 l) = neg_numeral.dbl_inc plus one (neg_numeral.sub minus plus one k l)"
ax: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and seq :: "('a, 'b) sequent"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "r \<in> R"   and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> Ax" shows "extendRule seq r \<in> ext R R' M N"
timpl_apply_term_def: fixes a :: "'c set"   and b :: "'c set"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" shows "\<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle> \<equiv> term_variants ((\<lambda>_. [])(Abs a := [Abs b])) t"
list_combE_2: fixes f :: "'a"   and xs :: "'a list"   and S :: "name fset"   and x :: "'a" assumes "closed_except (list_comb f xs) S"   and "x \<in> set xs" shows "closed_except x S"
of_bool_eq_1: shows "of_bool False = (0::'a)"
assms_1: shows "f \<triangleright> n \<noteq> g \<triangleright> n"
reflexive_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "reflexive R \<equiv> \<forall>x. R x x"
prod_unit_r: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes> b \<in> Units G"   and "b \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<in> Units G"
eliminate_entries_i: fixes B' :: "'a mat"   and vs :: "integer \<Rightarrow> 'i"   and vs' :: "nat \<Rightarrow> 'a"   and i :: "nat"   and B :: "'i mat"   and j :: "nat" assumes "\<And>j. j < dim_row B' \<Longrightarrow> R (vs (integer_of_nat j)) (vs' j)"   and "i < dim_row B'"   and "mat_rel R B B'" shows "mat_rel R (eliminate_entries_i ops vs B i j) (eliminate_entries vs' B' i j)"
subdivision_with_projD: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "subdivision_pair G H" shows "pair_sd G H"
span_fst_QR_decomposition: fixes A :: "((real, 'b) vec, 'a) vec" shows "vec.span (columns A) = vec.span (columns (fst (QR_decomposition A)))"
Zp_residue_mult_zero_1: fixes a :: "nat \<Rightarrow> int"   and k :: "nat"   and b :: "nat \<Rightarrow> int" assumes "a k = 0" shows "(a \<otimes> b) k = 0"
simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (x # ys) = (Ball (set ys) (less x) \<and> linorder.strict_sorted less ys)"
z: shows "z \<in> Set.filter (\<lambda>(x, y). x\<^sup>2 + y\<^sup>2 = p) (SIGMA x:{0..p}. {x..p})"