hcju/isabelleps huggingface dataset:
A_1_2_2: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and \<Gamma>1 :: "nat \<Rightarrow> nat option"   and \<Gamma>2 :: "nat \<Rightarrow> nat option"   and xs :: "nat list"   and Bs :: "nat list"   and ds :: "exp list"   and e :: "exp"   and D :: "nat" assumes "CT OK"   and "\<Gamma> = \<Gamma>1 ++ \<Gamma>2"   and "\<Gamma>2 = [xs [\<mapsto>] Bs]"   and "length xs = length ds"   and "length Bs = length ds"   and "\<exists>As. CT;\<Gamma>1 \<turnstile>+ ds : As \<and> CT \<turnstile>+ As <: Bs"   and "CT;\<Gamma> \<turnstile> e : D" shows "\<exists>C. CT;\<Gamma>1 \<turnstile> (ds/xs)e : C \<and> CT \<turnstile> C <: D"
rb_aux_inv1_lt_inj_on: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "rb_aux_inv1 bs" shows "inj_on lt (set bs)"
distinct_1: fixes name1 :: "name"   and name2 :: "name" shows "name1[name2] \<noteq> \<tau>"
radj_Inf_pres: fixes g :: "'c \<Rightarrow> 'b" assumes "\<exists>f. f \<stileturn> g" shows "Inf_pres g"
Acute_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Acute Bet Cong A B C \<equiv> \<exists>A' B' C'. Tarski_neutral_dimensionless.Per Bet Cong A' B' C' \<and> Tarski_neutral_dimensionless.LtA Bet Cong A B C A' B' C'"
subtuple_mono: fixes xm :: "'e \<rightharpoonup>\<^sub>f 'c"   and ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> fmran' xm; y \<in> fmran' ym\<rbrakk> \<Longrightarrow> f x y \<longrightarrow> g x y" shows "subtuple f xm ym \<longrightarrow> subtuple g xm ym"
Fail_HInv3_t: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "Fail s s' t"   and "HInv3_inner s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_inner s' p q d"
Throwable_def: shows "Throwable \<equiv> STR ''java/lang/Throwable''"
asIMOD_igAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom"   and xs :: "'varSort"   and x :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> 'a" shows "igAbs (asIMOD SEM) xs x X = (\<lambda>val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then X (val (x := sX)_xs) else sDummy SEM (SOME s. sWls SEM s (X val))) else undefined)"
dsr5: fixes x :: "'a"   and y :: "'a" shows "ad (ad (x + y)) = ad (ad x) + ad (ad y)"
prems_1: shows "\<rho> \<in> space (state_measure V' \<Gamma>)"
simps_2: fixes A :: "(('a, 'b) vec, 'c) vec"   and n :: "nat"   and i :: "'c"   and j :: "'b" shows "row_add_iterate A (Suc n) i j = (if Suc n = mod_type_class.to_nat i then row_add_iterate A n i j else row_add_iterate (row_add A (mod_type_class.from_nat (Suc n)) i (- A $ mod_type_class.from_nat (Suc n) $ j)) n i j)"
elims: fixes x :: "'fsym trm"   and y :: "('fsym, 'tp) efsym trm" assumes "tT x = y" obtains xa :: "Preliminaries.var" where "x = Var xa" and "y = (if unprot (TE.tpOfV xa) then Var xa else Fn (Tag (TE.tpOfV xa)) [Var xa])"   | v :: "'fsym" and va :: "'fsym trm list" where "x = Fn v va" and "y = tNN (Fn v va)"
prime_1: shows "prime (of_nat x + \<i>\<^sub>\<int> * of_nat y)"
elims_2: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and x :: "'a ast_effect" assumes "wf_effect ty_ent x" obtains a :: "'a atom formula list" and d :: "'a atom formula list" where "x = Effect a d" and "Ball (set a) (wf_fmla_atom ty_ent) \<and> Ball (set d) (wf_fmla_atom ty_ent)"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "n_nodes (Br a t1 t2) = Suc (n_nodes t1 + n_nodes t2)"
divide_sgn: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "divide a (sgn b) = times a (sgn b)"
elims_3: fixes x :: "nat"   and xa :: "(bool list \<times> bool list) insec_query" assumes "\<not> valid_mac_query x xa" obtains \<eta> :: "nat" and a :: "bool list" and m :: "bool list" where "x = \<eta>" and "xa = Edit (a, m)" and "\<not> (a \<in> vld \<eta> \<and> m \<in> vld \<eta>)"
HIGH_COMP: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and c2 :: "IMP" assumes "G \<rhd> c1 : twiddle"   and "G \<rhd> c2 : twiddle" shows "G \<rhd> Comp c1 c2 : twiddle"
carrier_FactGroup: fixes G :: "('b, 'c) monoid_scheme"   and N :: "'b set" shows "carrier (G Mod N) = (#>\<^bsub>G\<^esub>) N ` carrier G"
flatten_nonEmpty: fixes c :: "('d, 'e, 'f) com" shows "flatten c \<noteq> []"
get_map_def: fixes M :: "('c \<times> 'd) set"   and k :: "'c" shows "get_map M k = the_elem (Set.filter (\<lambda>(k', uu_). k = k') M)"
inf_sup_aci_5: fixes inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and x :: "'k"   and y :: "'k" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
wpl_opt_bst: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" shows "wpl.wpl w i j (Optimal_BST.opt_bst w i j) = Optimal_BST.min_wpl w i j"
finite_bounded_log2: fixes a :: "complex"   and b :: "real"   and w :: "complex" assumes "a \<noteq> 0" shows "finite {z. cmod z \<le> b \<and> exp (a * z) = w}"
inflation: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d" assumes "galois_connection G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x)"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "Ik.SAT \<Phi> \<equiv> \<forall>\<xi>. GE.wtE \<xi> \<longrightarrow> Ik.satPB \<xi> \<Phi>"
wt_subst_db: fixes i :: "nat"   and x :: "'v"   and s :: "('s, 'v) tm"   and j :: "nat" shows "wt (subst_db i x s) = wt (subst (raw_db_subst j x) s)"
lookup_map_of: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c" assumes "sorted1 (inorder t)" shows "lookup t x = AList_Upd_Del.map_of (inorder t) x"
geodesic_segment_subsegment: fixes G :: "'b set"   and H :: "'b set" assumes "geodesic_segment G"   and "H \<subseteq> G"   and "compact H"   and "connected H"   and "H \<noteq> {}" shows "geodesic_segment H"
l_sub_smult: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K " shows "H \<subseteq> H \<diamondop>\<^bsub>G\<^esub> K"
ydeg: shows "\<forall>i\<le>degree p. degree (coeff p i) \<le> d"
sum_closed: fixes f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int" assumes "f \<in> carrier Zp"   and "g \<in> carrier Zp" shows "f \<oplus> g \<in> carrier Zp"
map\<^sub>1_simps_1: fixes f :: "'c" assumes "C.ide f" shows "D.arr (\<rho>\<^sub>1 f)"
predecessors_def: fixes inEdges' :: "'b \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'c) list"   and g :: "'b"   and n :: "'a" shows "graph_path_base.predecessors inEdges' g n \<equiv> map getFrom (graph_path_base.inEdges inEdges' g n)"
map_cong0: fixes x :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>z. z \<in> range x \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> x"
convex_Times: fixes s :: "'c set"   and t :: "'d set" assumes "convex s"   and "convex t" shows "convex (s \<times> t)"
alpha: shows "X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs"
process_prop_3_def: fixes P :: "('e list \<times> 'e set) set \<times> 'e list set" shows "process_prop_3 P \<equiv> \<forall>xs X Y. (xs, Y) \<in> fst P \<and> X \<subseteq> Y \<longrightarrow> (xs, X) \<in> fst P"
isTrivialPhi_det: fixes g :: "'g"   and v :: "'val" assumes "uninst_code.ssa.trivial uses phis g v" shows "\<exists>!v'. uninst_code.ssa.isTrivialPhi phis g v v'"
orbits_2: shows "pmf (sds R29) d = pmf (sds R29) a"
ereal_truncation_bottom: fixes x :: "ereal" shows "(\<lambda>n. max x (ereal (- real n))) \<longlonglongrightarrow> x"
lexn_conv: fixes r :: "('a \<times> 'a) set"   and n :: "nat" shows "lexn r n = {(xs, ys). length xs = n \<and> length ys = n \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y) \<in> r)}"
LeftDerivationLadder_ladder_n_bound: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and b :: "symbol list"   and index :: "nat" assumes "LeftDerivationLadder a D L b"   and "index < length L" shows "ladder_n L index \<le> length D"
satisfies_T0: shows "is_left_adjoint t\<^sub>0u\<^sub>1.p\<^sub>0"
le_iff_sup: fixes x :: "'d"   and y :: "'d" shows "(x \<le> y) = (x \<sqinter> y = y)"
vsv_ex1_app1: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(b = r\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
map_bind_pmf: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'f pmf"   and g :: "'f \<Rightarrow> 'e pmf" shows "map_pmf f (M \<bind> g) = M \<bind> (\<lambda>x. map_pmf f (g x))"
domains_char: fixes f :: "'a" shows "domains f = (if arr f then {local.dom f} else {})"
inject: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and x23 :: "('d, 'e) ctxt"   and x24 :: "('d, 'e) Term.term list"   and y21 :: "'d"   and y22 :: "('d, 'e) Term.term list"   and y23 :: "('d, 'e) ctxt"   and y24 :: "('d, 'e) Term.term list" shows "(More x21 x22 x23 x24 = More y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
fml_equiv_subst: fixes \<phi> :: "fml"   and \<psi> :: "fml"   and P :: "(variable \<Rightarrow> real) set \<Rightarrow> bool"   and I :: "interp" assumes "fml_equiv \<phi> \<psi>"   and "P (fml_sem I \<phi>)" shows "P (fml_sem I \<psi>)"
vdisjnt_nin_left: fixes B :: "V"   and A :: "V"   and a :: "V" assumes "B \<inter>\<^sub>\<circ> A = []\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> A" shows "a \<notin>\<^sub>\<circ> B"
intro_spec_refine_iff: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'b nres"   and R :: "('b \<times> 'd) set"   and M :: "'d nres" shows "(RES X \<bind> f \<le> \<Down> R M) = (\<forall>x\<in>X. f x \<le> \<Down> R M)"
parts_crypt_pwd: fixes s :: "(agent \<times> msg) set"   and K :: "key"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt K (Pwd n) \<in> parts (used s)" shows "(\<exists>SK. K = SesK SK \<and> (Owner n, Crypt (SesK SK) (Pwd n)) \<in> s) \<or> {Pwd n, Key K} \<subseteq> spied s"
block_closed: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> Partition" shows "a \<in> G"
DisjConj: fixes C :: "fm"   and A :: "fm"   and B :: "fm" shows "C OR A IMP (C IMP B) IMP A OR B \<in> boolean_axioms"
dim_def: fixes V :: "('a \<Rightarrow> real) set" shows "diff_fun_space.m1.dim V = (if \<exists>b\<subseteq>diff_fun_space. \<not> diff_fun_space.m1.dependent b \<and> diff_fun_space.m1.span b = diff_fun_space.m1.span V then card (SOME b. b \<subseteq> diff_fun_space \<and> \<not> diff_fun_space.m1.dependent b \<and> diff_fun_space.m1.span b = diff_fun_space.m1.span V) else 0)"
simps_2: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "map_idx f (x # xs) n = f x n # map_idx f xs (Suc n)"
verit_less_mono_div_int2: fixes A :: "int"   and B :: "int"   and n :: "int" assumes "A \<le> B"   and "0 < - n" shows "B div n \<le> A div n"
cCons_Cons_eq: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "cCons x (y # ys) = x # y # ys"
one_closed: shows "\<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub> \<in> carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"
le_fininf_le_inf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
P2: shows "A B Perp P A \<and> Col A B T \<and> Bet C T P"
P_2: shows "B = P * (mat_of_rows n G)\<^sup>T"
norm_power: fixes x :: "'a"   and n :: "nat" shows "norm (x ^ n) = norm x ^ n"
rtrancl_while_invariant: fixes p :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b list"   and x :: "'b"   and st :: "'b list \<times> 'b set" assumes "While_Combinator.rtrancl_while_invariant p f x st"   and "While_Combinator.rtrancl_while_test p st" shows "While_Combinator.rtrancl_while_invariant p f x (While_Combinator.rtrancl_while_step f st)"
MInfty_neq_PInfty_1: shows "\<infinity> \<noteq> - \<infinity>"
word_rotl_eq: fixes n :: "nat"   and w :: "'b word" shows "word_rotl n w = of_bl (rotate n (to_bl w))"
fps_cos_deriv: fixes c :: "'a" shows "fps_deriv (fps_cos c) = fps_const (- c) * fps_sin c"
setc_def_to_dest: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "A \<equiv> {x. P x}"   and "x \<in> A" shows "P x"
trans_not_less: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b \<noteq> Less"   and "cmp b c \<noteq> Less" shows "cmp a c \<noteq> Less"
prob_space_imp_subprob_space: fixes M :: "'d measure" assumes "prob_space M" shows "subprob_space M"
\<I>_eqI: fixes \<I> :: "('d, 'e) \<I>"   and \<I>' :: "('d, 'e) \<I>" assumes "outs_\<I> \<I> = outs_\<I> \<I>'"   and "\<And>x. x \<in> outs_\<I> \<I>' \<Longrightarrow> responses_\<I> \<I> x = responses_\<I> \<I>' x" shows "\<I> = \<I>'"
InterpEqEq: fixes A :: "'t"   and e1 :: "('t, 'f) Expression"   and B :: "'t"   and g :: "'m"   and e2 :: "('t, 'f) Expression" assumes "L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk> \<rightarrow> IMor g"   and "L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk> \<rightarrow> IMor g" shows "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk> \<rightarrow> IBool True"
T_spaceM_stable_1: fixes x :: "'a" assumes "x \<in> space M" shows "Tinv x \<in> space M"
pwd_spy_iv_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_iv n)"
nth_mem: fixes n :: "nat"   and xs :: "'c list" assumes "n < |xs|" shows "xs ! n \<in> set xs"
exhaust: fixes y :: "('b, 'c, 'a) writerT" obtains x :: "'a" where "y = WriterT x"
peak_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and p :: "('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> ('b \<times> 'a) list" shows "peak ars p = (let (\<tau>, \<sigma>) = p in {\<tau>, \<sigma>} \<subseteq> seq ars \<and> fst \<tau> = fst \<sigma>)"
asBinp_def: fixes qbinp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs option" shows "asBinp qbinp \<equiv> lift asAbs qbinp"
to_metric_completion_isometry: shows "isometry_on UNIV to_metric_completion"
cont_set_spmf: shows "cont lub_spmf (ord_spmf (=)) \<Union> (\<subseteq>) set_spmf"
dres_cases: fixes x :: "'b dres" obtains "x = dSUCCEED"   | r :: "'b" where "x = dRETURN r"   | "x = dFAIL"
ideD_2: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.ide a" shows "S.dom a = a"
map\<^sub>1_simps_5: fixes f :: "'c" assumes "C.ide f" shows "D.cod (\<tau>\<^sub>1 f) = \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f"
take_0: shows "\<psi>\<^sup>E\<^bsub>0\<^esub> = \<bottom>"
slsd_def: fixes x :: "'a" shows "slsd x = (\<top> \<cdot> x = x)"
Amax_mem: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" assumes "f \<in> {i. i \<le> n} \<rightarrow> Z\<^sub>-\<^sub>\<infinity>" shows "Amax n f \<in> Z\<^sub>-\<^sub>\<infinity>"
path_from_to_def: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and xs :: "'a list"   and w :: "'a" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w \<equiv> (Digraph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
unit_imp_dvd: fixes b :: "'c"   and a :: "'c" assumes "is_unit b" shows "b dvd a"
omega_absorb2: fixes x :: "'a"   and y :: "'a" shows "\<Omega> x y \<cdot> ad (x \<cdot> y) = \<Omega> x y"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>i. (local.has_left i \<Longrightarrow> P (local.the_left i)) \<Longrightarrow> P i" shows "P a0"
freshResidual3_1: fixes x :: "name"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" assumes "x \<sharp> bn \<alpha>"   and "x \<sharp> \<alpha> \<prec> P" shows "x \<sharp> \<alpha>"
subclseq: shows "G\<turnstile>dynC\<preceq>\<^sub>C statC"
derf: fixes x :: "real" assumes "a < x"   and "x < b" shows "(f has_derivative f' x) (at x)"
that: fixes v\<^sub>2 :: "'item list" assumes "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2" shows "thesis"
continuous_map_subtopology_fst: fixes X :: "'a topology"   and Y :: "'b topology"   and Z :: "('a \<times> 'b) set" shows "continuous_map (subtopology (prod_topology X Y) Z) X fst"
hpsafe_Test_simps: fixes p :: "('d, 'e, 'f) formula" shows "hpsafe (? p) = fsafe p"
update_def: fixes get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'f"   and m :: "'e" shows "monad_state_base.update get put f m = get (\<lambda>s. put (f s) m)"
ub_cos_def: fixes prec :: "nat"   and x :: "float" shows "ub_cos prec x = (let horner = \<lambda>x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x); half = \<lambda>x. float_plus_up prec (Float 1 1 * x * x) (- 1) in if x < Float 1 (- 1) then horner x else if x < 1 then half (horner (x * Float 1 (- 1))) else half (half (horner (x * Float 1 (- 2)))))"
antimonoI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f y \<le> f x" shows "order.antimono less_eq f"
l11_44_1_b: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C"   and "B A C CongA B C A" shows "Cong B A B C"
bin_wc_eq: fixes xs :: "cell list" shows "bl_bin xs = bl2wc xs"
cases: fixes x :: "int list" obtains "x = []"   | i :: "int" and "is" :: "int list" where "x = i # is"
pre_rule': fixes P :: "assn"   and R :: "assn"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and P' :: "assn" assumes "\<not> <P * R> f <Q>"   and "P \<Longrightarrow>\<^sub>A P'" shows "\<not> <P' * R> f <Q>"
perfect_choose_dist: fixes r :: "real"   and x :: "'a" assumes "0 < r" shows "\<exists>a. a \<noteq> x \<and> dist a x < r"
group_byE: fixes c :: "'a"   and cs :: "'a set"   and b :: "'b"   and f :: "'c \<Rightarrow> 'b \<times> 'a"   and as :: "'c set" assumes "c \<in> cs"   and "(b, cs) \<in> group_by f as" obtains a :: "'c" where "a \<in> as" and "f a = (b, c)"
T_def: shows "T \<equiv> \<lambda>n. pi\<^sup>2 / 6 - 1 / n + 1 / (2 * n\<^sup>2) - (\<Sum>i = 1..N. bernoulli (2 * i) / n ^ (2 * i + 1))"
dg_GRPH_Cod_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_GRPH \<alpha>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> dg_GRPH \<alpha>\<lparr>Obj\<rparr>"
log_pow_cancel: fixes a :: "real"   and b :: "nat" assumes "0 < a"   and "a \<noteq> 1" shows "log a (a ^ b) = real b"
finite_dimensional_real_vector_space_on_axioms: shows "finite_dimensional_real_vector_space_on S1 Basis1"
that: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "valid_decomp X qs"   and "standard_decomp 0 qs"   and "monomial_decomp qs"   and "cone_decomp (normal_form F ` P[X]) qs"   and "exact_decomp 0 qs"   and "\<And>g. \<lbrakk>\<And>f. f \<in> F \<Longrightarrow> homogeneous f; g \<in> punit.reduced_GB F\<rbrakk> \<Longrightarrow> poly_deg g \<le> \<b> qs 0" shows "thesis"
map_gpv_conv_bind: fixes f :: "'d \<Rightarrow> 'a"   and x :: "('d, 'b, 'c) gpv" shows "map_gpv f (\<lambda>x. x) x = x \<bind> (\<lambda>x. Done (f x))"
order_iff_strict: fixes a :: "'c list"   and b :: "'c list" shows "prefix a b = (strict_prefix a b \<or> a = b)"
DmdRec: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> (\<diamond>P) = (Init P \<or> \<diamond>P$)"
PiE_UNIV_domain: fixes T :: "'c \<Rightarrow> 'd set" shows "Pi\<^sub>E UNIV T = Pi UNIV T"
cmif: shows "m\<inverse> O f \<subseteq> m\<inverse>"
sel_1: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "\<not> lnull xs"   and "\<not> lnull ys" shows "lhd (lzip xs ys) = (lhd xs, lhd ys)"
mn: shows "is_left_adjoint m \<and> is_left_adjoint n \<and> w \<cong> n \<star> m\<^sup>*"
Cauchy: fixes f :: "'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'b" assumes "bounded_clinear f"   and "Cauchy X" shows "Cauchy (\<lambda>n. f (X n))"
is_projection_on_insert: fixes S :: "'b set"   and a :: "'b"   and \<pi> :: "'b \<Rightarrow> 'b"   and \<pi>a :: "'b \<Rightarrow> 'b" assumes "\<And>s. s \<in> S \<Longrightarrow> is_orthogonal a s"   and "is_projection_on \<pi> (closure (cspan S))"   and "is_projection_on \<pi>a (cspan {a})" shows "is_projection_on (\<lambda>x. \<pi>a x + \<pi> x) (closure (cspan (insert a S)))"
projs_1: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "\<pi>\<^sub>2 (dsn, dsk, flag, hops, nhip, pre) = dsn"
Rep_singleton_matrix: fixes j :: "nat"   and i :: "nat"   and e :: "'a"   and m :: "nat"   and n :: "nat" shows "Rep_matrix (singleton_matrix j i e) m n = (if j = m \<and> i = n then e else (0::'a))"
poly_mult_Nil2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list"   and x :: "'b" assumes "class.semiring_0 plus zero times" shows "semiring_0.poly plus zero times (semiring_0.pmult plus zero times p []) x = zero"
Sem_def: fixes x :: "'a" shows "I\<lbrakk>x\<rbrakk> \<equiv> x"
simps: fixes cs :: "'b list"   and fs :: "'b list"   and n :: "nat" shows "lhr cs fs n = (if cs = [] \<or> last cs = (0::'b) \<or> length fs < length cs - 1 then undefined else if n < length fs then fs ! n else (\<Sum>k<length cs - 1. cs ! k * lhr cs fs (n + 1 - length cs + k)) / - last cs)"
infinite_Pow: fixes A :: "'c set" assumes "infinite A" shows "infinite (Pow A)"
atoms_monotone: fixes G :: "chmsg set"   and H :: "chmsg set" assumes "G \<subseteq> H" shows "atoms G \<subseteq> atoms H"