hcju/isabelleps huggingface dataset:
substitutable_preferences_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "substitutable_preferences_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>B. \<forall>b\<in>B. a \<noteq> b \<and> a \<in> f B \<longrightarrow> a \<in> f (B - {b}))"
join: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a" assumes "merge_on A h bo m"   and "h a = h b"   and "a \<in> A" shows "\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u)"
assms: shows "(X, add_mset y Y) \<in> multpw ns"
strategy_attracts_via_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and A :: "'a set"   and W :: "'a set" shows "strategy_attracts_via p \<sigma> v0 A W \<equiv> \<forall>P. vmc_path G P v0 p \<sigma> \<longrightarrow> visits_via P A W"
count_filter_zmset: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c zmultiset"   and a :: "'c" shows "zcount (filter_zmset P M) a = (if P a then zcount M a else 0)"
dllf: shows "matrix_seq.lowner_lub f \<in> carrier_mat d d"
list_def: shows "list = filter (\<lambda>na. snd na = l) n_as"
axioms_2: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "ring R addition multiplication zero unit"
assms_8: fixes exp :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and c :: "'a"   and var :: "Vars.var" assumes "\<And>c. P1 c exp"   and "\<And>c. P2 c \<Gamma>" shows "P2 c ((var, exp) # \<Gamma>)"
spr_simRels_def: shows "spr_simRels \<equiv> \<lambda>a. {(s, s') |s s'. envObs a (sprFst s) = envObs a (sprFst s') \<and> envObs a (sprLst s) = envObs a (sprLst s') \<and> sprCRel s = sprCRel s'}"
smods_0_1: fixes q :: "real poly" shows "smods 0 q = []"
wfp_on_def: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" shows "wfp_on P A = (\<nexists>f. \<forall>i. f i \<in> A \<and> P (f (Suc i)) (f i))"
nonprincipal: shows "\<chi> \<noteq> principal_dchar n"
countable: fixes x :: "'a" assumes "x \<in> A" shows "countable (B' x)"
Loss_L5Suc: fixes d :: "nat"   and a :: "nat \<Rightarrow> aType list"   and t :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and y :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list"   and j :: "nat" assumes "\<forall>j\<le>d. a (t + Suc j) = []"   and "Loss lose a i2 y i"   and "Suc j \<le> d"   and "ts lose" shows "y (t + Suc j) = []"
lea_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F" shows "C B A LeA F E D"
sim_cpoke: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and e :: "'event" assumes "S p q"   and "e \<in> E" shows "trace_eq_simcl S (mk_lossless (p \<bind> (\<lambda>s. cpoke core1 s e))) (mk_lossless (q \<bind> (\<lambda>s. cpoke core2 s e)))"
ipresCons_imp_ipresFreshAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.ipresCons Delta h hA MOD"   and "FixSyn.igFreshCls Delta MOD"   and "FixSyn.igConsIPresIGWls Delta MOD" shows "FixSyn.ipresFreshAll Delta h hA MOD"
fold_congs_1: fixes r :: "'a thread_state_scheme"   and r' :: "'a thread_state_scheme"   and v' :: "V + bool \<Rightarrow> nat"   and f :: "(V + bool \<Rightarrow> nat) \<Rightarrow> V + bool \<Rightarrow> nat"   and f' :: "(V + bool \<Rightarrow> nat) \<Rightarrow> V + bool \<Rightarrow> nat" assumes "r = r'"   and "l r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "l_update f r = l_update f' r'"
mustT_If_R: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and tst :: "'a"   and s :: "'b"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com" assumes "\<not> tval tst s"   and "PL.mustT tval aval (if tst then c1 else c2) s" shows "PL.mustT tval aval c2 s"
intro: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d" assumes "SecurityInvariant_preliminaries sinvar"   and "SecurityInvariant_ACS_axioms sinvar default_node_properties" shows "SecurityInvariant_ACS sinvar default_node_properties"
wf_pddl_ground_act_is_sound_opr: fixes g_opr :: "ground_action" assumes "wf_ground_action g_opr" shows "sound_opr g_opr (pddl_opr_to_act g_opr)"
single_comp_support: fixes asset :: "'c"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" shows "support_set (qty_single asset qty) \<subseteq> {asset}"
characteristic_iso_1: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>local.inv \<a>[b, b, f] : b \<star> b \<star> f \<Rightarrow>\<^sub>S (b \<star> b) \<star> f\<guillemotright>"
fresh_4: fixes a :: "name"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2 \<rightarrow> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
a_less_lub: fixes S :: "'a set"   and a :: "'a"   and L :: "'a" assumes "S \<subseteq> A"   and "S \<noteq> {}"   and "\<forall>x\<in>S. (a, x) \<in> r"   and "\<forall>y\<in>S. (y, L) \<in> r" shows "(a, L) \<in> r"
a_zpz: fixes m :: "int"   and n :: "int" shows "ant m + ant n = ant (m + n)"
bisim1FAcc: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and F :: "String.literal"   and D :: "String.literal" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e\<bullet>F{D},h \<turnstile> (e'\<bullet>F{D}, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
le_sqrtI: fixes x :: "nat"   and y :: "nat" assumes "x\<^sup>2 \<le> y" shows "x \<le> Discrete.sqrt y"
simps_1: fixes G :: "('b, 'a) labeled_graph"   and a :: "'b allegorical_term"   and b :: "'b allegorical_term" shows ":G:\<lbrakk>A_Int a b\<rbrakk> = (:G:\<lbrakk>a\<rbrakk>) \<inter> (:G:\<lbrakk>b\<rbrakk>)"
insert_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "(C::'b set) \<subseteq> (D::'b set)" shows "(a::'b) \<triangleright> (C::'b set) \<subseteq> a \<triangleright> (D::'b set)"
has_derivative_unique: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and x :: "'a"   and F' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative F) (at x)"   and "(f has_derivative F') (at x)" shows "F = F'"
fls_subdegree_pow: fixes f :: "'a fls"   and n :: "nat" shows "fls_subdegree (f ^ n) = int n * fls_subdegree f"
vec_to_iarray_def: fixes A :: "('d, 'e) vec" shows "vec_to_iarray A = IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat i) CARD('e)"
pumping_lemma_aux: fixes z :: "'a list"   and A :: "'a list set" assumes "z \<in> A"   and "finite (range (\<lambda>w. Derivs w A))"   and "card (range (\<lambda>w. Derivs w A)) \<le> length z" shows "\<exists>u v w. z = u @ v @ w \<and> length (u @ v) \<le> card (range (\<lambda>w. Derivs w A)) \<and> v \<noteq> [] \<and> (\<forall>i. u @ repeat i v @ w \<in> A)"
cancel1: fixes xs :: "'d list"   and gs :: "'d list"   and gs' :: "'d list" assumes "xs \<noteq> []" shows "cancel1 (gs @ xs @ xs @ gs') (gs @ xs @ gs')"
typeof_NullPointer: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt NullPointer)) = \<lfloor>Class NullPointer\<rfloor>"
assms_2: shows "Q \<longmapsto>\<alpha> \<prec> Q'"
tr_par_wf_trms: fixes A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list" assumes "A' \<in> set (tr\<^sub>p\<^sub>c A [])"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>l\<^sub>s\<^sub>t A')"
cyclic_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "Elementary_Groups.cyclic_group G \<equiv> \<exists>x\<in>carrier G. subgroup_generated G {x} = G"
interleaves_swap: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "xs \<simeq> {ys, zs, P} = xs \<simeq> {zs, ys, \<lambda>w ws. \<not> P w ws}"
irreducible_imp_prime_elem_gcd: fixes x :: "'a" assumes "irreducible x" shows "prime_elem x"
empty_neutral_2: fixes x :: "'a multiset" shows "x + {#} = x"
mod_relH: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap"   and P :: "assn" assumes "relH as h h'"   and "pHeap h as \<Turnstile> P" shows "pHeap h' as \<Turnstile> P"
log_remove: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and m :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ [Rule m Log] @ rs\<^sub>2, s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, s\<rangle> \<Rightarrow> t"
assms_7: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e"   and "\<not> isVal e"   and "I (\<Gamma>, Var x, S)" shows "P (\<Gamma>, Var x, S) (delete x \<Gamma>, e, Upd x # S)"
assms_9: shows "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"
E'': shows "E' = lconcat (lmap (\<lambda>(t, ta). llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) E'')"
rep_eq: fixes x :: "'a cfg" shows "action x = (case rep_cfg x of (s, x) \<Rightarrow> action_sch x)"
Wax_def: 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 infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "ProblemIkTpart.Wax wtFsym resOf protFw \<equiv> {[Pos (Eq (ProblemIkTpart.lOfWax \<sigma>) (ProblemIkTpart.rOfWax \<sigma>))] |\<sigma>. \<not> ProblemIkTpart.isRes wtFsym resOf \<sigma> \<and> protFw \<sigma>}"
abs_eq: shows "list_generator = Generator (list_has_next, list_next)"
beta_rho: fixes q :: "nat"   and i :: "nat"   and p :: "nat"   and l :: "nat" assumes "te q (i + 1) \<le> te p (i + 1)"   and "correct p (te p (i + 1))"   and "correct q (te p (i + 1))"   and "correct l (te p (i + 1))" shows "\<bar>PC l (te p (i + 1)) - PC l (te q (i + 1)) - (te p (i + 1) - te q (i + 1))\<bar> \<le> \<beta> * \<rho>"
flushing_def: fixes x :: "'d" shows "flushing x = id"
gt0: shows "0 < Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z)"
le_Suc: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "a + (1::'a) \<le> b"
inner_Pair_0_1: fixes x :: "'c \<times> 'd"   and b :: "'d" shows "inner x (0::'c, b) = inner (snd x) b"
I_minWeightBasisD_3: fixes T :: "'a set"   and E :: "'a set" assumes "I_minWeightBasis (T, E)" shows "E \<subseteq> carrier"
prems: shows "\<exists>a. inf_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) a"
Un_Int_crazy: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> B \<union> B \<inter> C \<union> C \<inter> A = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
dom_char: fixes f :: "'c \<times> 'c" shows "CopxC.dom f = (if Cop.arr (fst f) \<and> C.arr (snd f) then (Cop.dom (fst f), C.dom (snd f)) else (Cop.null, C.null))"
subset_iff: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq B A = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
extensional_restrict: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" assumes "f \<in> extensional A" shows "restrict f A = f"
all_formulas_entailed: fixes N2 :: "'g set"   and N1 :: "'g set" assumes "\<forall>C\<in>N2. N1 \<Turnstile>G {C}" shows "N1 \<Turnstile>G N2"
less_extended_simps_7: fixes l :: "'a extended" shows "(l < -\<infinity>) = False"
simps_9: fixes i :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (AllN i \<phi>) \<Gamma> = (\<forall>l. length l = i \<longrightarrow> eval \<phi> (l @ \<Gamma>))"
no_Private_override: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old" shows "accmodi old \<noteq> Private"
same_postrec_prefix: shows "\<forall>k<i. postrecording_event t k = postrecording_event (swap_events i j t) k"
set1_FGl_def: fixes x :: "(('a, 'b, 'c, 'd, 'h, 'i, 'j) G, 'a, 'k, 'c, 'l, 'm, 'h, 'n, 'o, 'p) F" shows "set1_FGl x = \<Union> (set1_G ` set1_F x) \<union> set2_F x"
bilateral_substitutes_lemma: fixes x :: "'a"   and h :: "'c"   and X :: "'a set"   and d :: "'b" assumes "Xd x \<notin> Xd ` Ch h X"   and "d \<notin> Xd ` Ch h X"   and "d \<noteq> Xd x" shows "d \<notin> Xd ` Ch h (insert x X)"
t: shows "Can t \<and> Arr t \<and> Can (Inv t) \<and> Arr (Inv t) \<and> Ide (Dom t) \<and> Ide (Cod t)"
elims_3: fixes x :: "('c, 'd) SkewBinomialTree list" assumes "\<not> rank_skew_invar x" obtains t :: "('c, 'd) SkewBinomialTree" and t' :: "('c, 'd) SkewBinomialTree" and bq :: "('c, 'd) SkewBinomialTree list" where "x = t # t' # bq" and "\<not> (rank t \<le> rank t' \<and> rank_invar (t' # bq))"
add_comm: fixes x :: "'c"   and y :: "'c" assumes "D x y" shows "D y x \<and> x \<oplus> y = y \<oplus> x"
infinite_chain_eSuc_esize: fixes C :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain (\<le>) C"   and "infinite C"   and "x \<in> C" obtains y :: "'a" where "y \<in> C" and "eSuc (esize x) \<le> esize y"
finite_combination_1: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and \<tau> :: "'e \<Rightarrow> ('d, 'e, 'f) val option" assumes "substitution subst"   and "finite (RID\<^sub>S \<sigma>)"   and "finite (RID\<^sub>S \<tau>)" shows "finite (RID\<^sub>S (\<sigma>;;\<tau>))"
uniY: shows "S \<union> X \<union> insert e Y = {0..<m}"
integrable_Un': fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "f integrable_on A"   and "f integrable_on B"   and "negligible (A \<inter> B)"   and "C = A \<union> B" shows "f integrable_on C"
bounded_interior: fixes S :: "'b set" assumes "bounded S" shows "bounded (interior S)"
embed_axioms: shows "Pi_Derivatives.embed \<Sigma> project embed"
basic_subtype_x_OclVoid: fixes \<tau> :: "'a basic_type"   and P :: "bool" assumes "\<tau> \<sqsubset>\<^sub>B OclVoid" shows "P"
exi: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a"   and \<phi> :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "x \<in> var"   and "\<phi> \<in> fmla" shows "exi x \<phi> \<in> fmla"
clearBit_no: fixes bin :: "num"   and n :: "nat" shows "clearBit (numeral bin) n = word_of_int (bin_sc n False (numeral bin))"
Float_le_zero_iff: fixes a :: "int"   and b :: "int" shows "(Float a b \<le> 0) = (a \<le> 0)"
open_prod_elim: fixes S :: "('c \<times> 'd) set"   and x :: "'c \<times> 'd" assumes "open S"   and "x \<in> S" obtains A :: "'c set" and B :: "'d set" where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
assms: fixes n :: "int" assumes "\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i" shows "P n"
wilsons_construction_index_1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set"   and I :: "'c set"   and f :: "'b set \<Rightarrow> ('b \<times> 'c) set multiset"   and \<K>' :: "int set"   and G1 :: "('b \<times> 'c) set"   and G2 :: "('b \<times> 'c) set"   and x :: "'b"   and ix :: "'c"   and y :: "'b"   and iy :: "'c" assumes "K_GDD point_set block_collection sizes groups"   and "\<And>B. B \<in># block_collection \<Longrightarrow> K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}"   and "G1 \<in> {G \<times> I |G. G \<in> groups}"   and "G2 \<in> {G \<times> I |G. G \<in> groups}"   and "G1 \<noteq> G2"   and "(x, ix) \<in> G1"   and "(y, iy) \<in> G2" shows "int (\<Sum>\<^sub># (image_mset f block_collection) index {(x, ix), (y, iy)}) = 1"
FINAL: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "modifies modset s s\<^sub>0"   and "I s"   and "\<not> bval b s" shows "Q s"
proj_def: fixes c :: "'id" shows "proj c = lmap (tCMP c) \<circ> lfilter (active c)"
lemma1: fixes r :: "dB"   and r' :: "dB"   and s :: "dB"   and s' :: "dB" assumes "r \<rightarrow>\<^sub>s r'"   and "s \<rightarrow>\<^sub>s s'" shows "r \<degree> s \<rightarrow>\<^sub>s r' \<degree> s'"
always_eventually: fixes P :: "'e \<Rightarrow> bool"   and F :: "'e filter" assumes "\<forall>x. P x" shows "eventually P F"
lambda_system_additive: fixes f :: "'a set \<Rightarrow> ennreal" shows "Measure_Space.additive (lambda_system \<Omega> M f) f"
prime_factorization_exists: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> comm_semiring_1.prime_elem times one zero x) \<and> normalize (comm_monoid_mult.prod_mset times one A) = normalize x"
that: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r x y"   and "r\<^sup>*\<^sup>* x z" shows "\<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u"
cod_simp: fixes f :: "'a \<times> 'a" assumes "B.VxV.arr f" shows "B.VxV.cod f = (B.cod (fst f), B.cod (snd f))"
SimpleDef: assumes "v \<in> defs g n"   and "lookupDef g n v = (v, n, SimpleDef)" shows "thesis"
distinct_finite_set: fixes x :: "'a set" shows "finite {ys. set ys = x \<and> distinct ys}"
less_induct1: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "class.well_founded less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
assms_1: fixes xset :: "('a, 'b, 'c) formula set['a]" assumes "finite (supp xset)"   and "\<And>x. x \<in> set_bset xset \<Longrightarrow> P x" shows "P (Conj xset)"
lunit_coherence: fixes f :: "'a" assumes "ide f" shows "S.lunit (S.UP f) = S.UP \<l>[f] \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit (trg f) \<star>\<^sub>S S.UP f)"
fdia_export_1: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "d y \<cdot>  |x\<rangle> z =  |d y \<cdot> x\<rangle> z"
equiv_norm_Arr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.equiv \<^bold>\<parallel>t\<^bold>\<parallel> t"
less_tag: fixes x :: "'b tag_const"   and y :: "'b tag_const" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
circline_A0_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "circline_A0_cmat H = (let (A, B, C, D) = H in A = 0)"
length_drop_mask: fixes w :: "'a word"   and n :: "nat" shows "length (dropWhile Not (to_bl (w && mask n))) \<le> n"
propagate_all_imp_InvGlobPointstampsEq: fixes c1 :: "('loc, 't) Propagate.configuration"   and c0 :: "('loc, 't) Propagate.configuration" assumes "Some c1 = propagate_all c0" shows "c_pts c0 = c_pts c1"
dist_concat_def: fixes ls1 :: "'b list set"   and ls2 :: "'b list set" shows "ls1 \<^sup>\<frown> ls2 = uncurry (@) ` (ls1 \<times> ls2)"
Rep_lazy_inverse: fixes x :: "'a lazy" shows "Abs_lazy (Rep_lazy x) = x"
vimage_singleton_eq: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and b :: "'c" shows "(a \<in> f -` {b}) = (f a = b)"
of_nat_le_0_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
ord_strict_canc: fixes s :: "'a"   and u :: "'a"   and t :: "'a" assumes "s + u \<prec> t + u" shows "s \<prec> t"
borel_measurable_eq: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w = g w} \<in> sets M"
abs_conv_abscissa_diff_le: fixes f :: "'a fds"   and g :: "'a fds" shows "abs_conv_abscissa (f - g) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa g)"
coprime_self: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a a = dvd.dvd times a one"
additive: fixes g :: "'v"   and g' :: "'v" assumes "g \<in> V"   and "g' \<in> V" shows "T (g + g') = T g + T g'"
1: shows "\<forall>k. i \<noteq> k \<longrightarrow> f (i, k) = bot"
complement_same_b: shows "size \<B>\<^sup>C = size \<B>"
thetaParWT_WretrT: shows "thetaParWT \<subseteq> WretrT (thetaParWT \<union> WbisT)"
function_ring_is_abelian_group: shows "abelian_group F"
ext_irrefl_trans_strong_axioms: shows "ext_irrefl_trans_strong lexext"
fun_cong: fixes f :: "'p \<Rightarrow> 'q"   and g :: "'p \<Rightarrow> 'q"   and x :: "'p" assumes "f = g" shows "f x = g x"
holo: fixes m :: "nat" shows "g holomorphic_on A m - S m"
dist_s_mtf: fixes n :: "nat" shows "distinct (s_mtf n)"
matrix_vector_mul_linear: fixes A :: "(('a, 'b) vec, 'c) vec" shows "linear ((*v) A)"
vcard_vfinite_omega: fixes A :: "V" assumes "vcard A \<in>\<^sub>\<circ> \<omega>" shows "vfinite A"
ia2: shows "ia = mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1"
elims_1: fixes x :: "'a ext0"   and xa :: "'a ext0"   and y :: "bool" assumes "(x \<le> xa) = y" obtains uu_ :: "'a ext0" where "xa__ = ext0.Bot" and "xa = uu_" and "ya__"   | uv_ :: "'a" where "xa__ = ext0.Val uv_" and "xa = ext0.Bot" and "\<not> ya__"   | x :: "'a" and y :: "'a" where "xa__ = ext0.Val x" and "xa = ext0.Val y" and "ya__ = (x \<le> y)"   | uw_ :: "'a" where "xa__ = ext0.Val uw_" and "xa = ext0.Top" and "ya__"   | "xa__ = ext0.Top" and "xa = ext0.Bot" and "\<not> ya__"   | ux_ :: "'a" where "xa__ = ext0.Top" and "xa = ext0.Val ux_" and "\<not> ya__"   | "xa__ = ext0.Top" and "xa = ext0.Top" and "ya__"
simps_20: fixes v :: "'a rexp"   and vc :: "'a rexp"   and vb :: "'a" shows "local.le_rexp (Plus v vc) (Atom vb) = False"