hcju/isabelleps huggingface dataset:
map_spmf_parametric: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_fun A B) (rel_fun (rel_spmf A) (rel_spmf B)) map_spmf map_spmf"
equiv_utrans_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "equiv_utrans t u = (le_utrans t u \<and> le_utrans u t)"
equiintegrable_on_def: fixes F :: "('c \<Rightarrow> 'd) set"   and I :: "'c set" shows "F equiintegrable_on I \<equiv> (\<forall>f\<in>F. f integrable_on I) \<and> (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>f \<D>. f \<in> F \<and> \<D> tagged_division_of I \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R f x) - integral I f) < e))"
dterms_psimps_2: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and u :: "'b \<Rightarrow> 'b"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "dterms \<Gamma> ({l}\<lbrakk>u\<rbrakk> p) = sterms \<Gamma> p"
carrier_eq: shows "f -` carrier = {x. map_relation f le x x}"
\<gamma>_fun_def: fixes \<gamma> :: "'d \<Rightarrow> 'c set"   and F :: "'b \<Rightarrow> 'd" shows "\<gamma>_fun \<gamma> F = {f. \<forall>x. f x \<in> \<gamma> (F x)}"
tranclp_induct: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r a y \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r\<^sup>+\<^sup>+ a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
in_space_lim_stream: fixes \<omega> :: "(real \<times> 'a) stream"   and x :: "real \<times> 'a" shows "\<omega> \<in> space (K.lim_stream x)"
simps: fixes s1 :: "'e"   and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" shows "P10 (s1, s2) = (\<not> initialized s2 \<longrightarrow> dstate s2 = \<bottom>)"
assms: fixes b :: "int"   and a :: "int" assumes "0 < b"   and "coprime a b" shows "P (Fract a b)"
imp_E: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> A"   and "insert B H \<turnstile> C" shows "insert (A IMP B) H \<turnstile> C"
that_1: fixes ti :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], Undecided\<rangle> \<Rightarrow> ti"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, ti\<rangle> \<Rightarrow> t"   and "no_matching_Goto \<gamma> p [r]" shows "thesis"
reachable_restrict: fixes r :: "'a" assumes "r \<sqinter> top = r" shows "id r \<sqinter> top = id r \<sqinter> top"
ivp: shows "Y \<in> Sols f U S t\<^sub>0 s"
InterE: fixes A :: "'b"   and C :: "'b set set"   and X :: "'b set" assumes "A \<in> \<Inter> C" obtains "X \<notin> C"   | "A \<in> X"
pt_open_onD: fixes Q :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and A :: "('b \<Rightarrow> 'c) set"   and f :: "'b \<Rightarrow> 'c" assumes "pt_open_on Q A"   and "Q f"   and "f \<in> A" shows "\<exists>n. \<forall>g\<in>A. (\<forall>i<n. g i = f i) \<longrightarrow> Q g"
are_all_operators_non_interfering_tail: fixes op :: "'a strips_operator"   and ops :: "'a strips_operator list" assumes "are_all_operators_non_interfering (op # ops)" shows "are_all_operators_non_interfering ops"
card_Diff1_less: fixes A :: "'e set"   and x :: "'e" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) < card A"
cl9: fixes x :: "('a \<times> 'a set) set" shows "x \<inter> 1\<^sub>\<sigma> \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma> = x \<inter> 1\<^sub>\<sigma>"
one_hcoords_def: shows "1\<^sub>h\<^sub>c \<equiv> Abs_complex_homo_coords 1\<^sub>v"
subgroup_img_is_subgroup: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and I :: "'a set" assumes "group_hom G H h"   and "subgroup I G" shows "subgroup (h ` I) H"
Un_insert_left: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "(a \<triangleright> B) \<union> C = a \<triangleright> B \<union> C"
inverse_equality: fixes u :: "'a"   and v :: "'a" assumes "u + v = \<zero>"   and "v + u = \<zero>"   and "u \<in> R"   and "v \<in> R" shows "- u = v"
t: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
assms_2: assumes "x dvd a"   and "x dvd b"   and "\<And>y. \<lbrakk>y dvd a; y dvd b\<rbrakk> \<Longrightarrow> y dvd x" shows "thesis"
expand: fixes phantom :: "('b, 'a) phantom"   and phantom' :: "('b, 'a) phantom" assumes "of_phantom phantom = of_phantom phantom'" shows "phantom = phantom'"
avl_delete_1: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t" shows "avl (delete x t)"
dvdI: fixes a :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and k :: "'d" assumes "a = times b k" shows "dvd.dvd times b a"
poincare_mapsto_def: fixes P :: "'e set"   and X0 :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set"   and S :: "'e set"   and CX :: "'e set"   and Y :: "('e \<times> 'f \<Rightarrow>\<^sub>L 'e) set" shows "poincare_mapsto P X0 S CX Y = (\<forall>(x, d)\<in>X0. returns_to P x \<and> fst ` X0 \<subseteq> S \<and> return_time P differentiable at x within S \<and> (\<exists>D. (poincare_map P has_derivative blinfun_apply D) (at x within S) \<and> (poincare_map P x, D o\<^sub>L d) \<in> Y) \<and> (\<forall>t\<in>{0<..<return_time P x}. flow0 x t \<in> CX))"
ltl_FG_to_generalized_rabin_wellformed: fixes \<phi> :: "'a ltl" shows "finite (reach \<Sigma> (fst (\<P> \<phi>)) (fst (snd (\<P> \<phi>))))"
vsubsetD: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "a \<subseteq>\<^sub>\<circ> b"   and "c \<in>\<^sub>\<circ> a" shows "c \<in>\<^sub>\<circ> b"
ImageI: fixes a :: "'g"   and b :: "'h"   and r :: "('g \<times> 'h) set"   and A :: "'g set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
bn_eqvt: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and bn :: "'e \<Rightarrow> atom set"   and p :: "perm"   and \<alpha> :: "'e" assumes "class.bn permute bn" shows "p \<bullet> bn \<alpha> = bn (permute p \<alpha>)"
tmbound0_tmbound_iff: fixes t :: "tm" shows "tmbound 0 t = tmbound0 t"
assms_4: shows "call_of_return_node (parent_node n\<^sub>4) (parent_node n\<^sub>1)"
concat_len_one: fixes us :: "'b list list" assumes "\<^bold>|us\<^bold>| = 1" shows "concat us = hd us"
rem_implicit_pres_ops_lookup_op_None: fixes vars :: "(char list \<times> nat option \<times> char list list) list"   and init :: "nat list"   and goal :: "(nat \<times> nat) list"   and ops :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list" shows "(ast_problem.lookup_operator (vars, init, goal, ops) name = None) = (ast_problem.lookup_operator (rem_implicit_pres_ops (vars, init, goal, ops)) name = None)"
effect_run: fixes c :: "'d Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'d" assumes "effect c h h' r" shows "run c (Some h) (Some h') r"
ab: shows "(a, b) \<in> set (half_segments_of_aform X)"
assign_def: fixes f :: "'c control_state_scheme \<Rightarrow> 'c control_state_scheme" shows "assign f = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else Some ((), f \<sigma>))"
NT_ND: fixes c :: "'a event list"   and P :: "'a process" assumes "c \<notin> T P" shows "c \<notin> D P"
Q_min_is_light: fixes u :: "'v"   and d :: "nat" assumes "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" obtains v :: "'v" where "\<pi> u = Some v" and "light_edge (S (A Q \<pi>)) v u"
P1: shows "B Out C' C \<and> Cong B C' B A"
less_eqE: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b" obtains c :: "'b multiset" where "b = a + c"
right_zero: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c" assumes "ring R addition multiplication zero unit"   and "a \<in> R" shows "multiplication a zero = zero"
f: shows "current (\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>) f"
index_transpose_mat_2: fixes A :: "'a mat" shows "dim_row A\<^sup>t = dim_col A"
prod_mset_multiplicity: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and M :: "'a multiset" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one M = comm_monoid_mult.prod times one (\<lambda>x. power.power one times x (count M x)) (set_mset M)"
lookup_order1_prod: fixes Bs :: "'b tensor list"   and "is" :: "nat list" assumes "\<And>B. B \<in> set Bs \<Longrightarrow> Tensor.order B = 1"   and "is \<lhd> dims (prod_list Bs)" shows "lookup (prod_list Bs) is = prod_list (map2 (\<lambda>i B. lookup B [i]) is Bs)"
trd_in_pmdl: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and fs :: "('a \<Rightarrow>\<^sub>0 'c) list" shows "p - punit.trd fs p \<in> local.punit.pmdl (set fs)"
q_a_simp: shows "map h (map ((\<otimes>) a) q @ replicate (length p) \<zero>) = map ((\<otimes>\<^bsub>S\<^esub>) (h a)) (map h q) @ replicate (length (map h p)) \<zero>\<^bsub>S\<^esub>"
fst_splitAt_last: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "distinct xs" shows "fst (splitAt (last xs) xs) = butlast xs"
homeomorphic_map_subtopologies: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set" assumes "homeomorphic_map X Y f"   and "f ` (topspace X \<inter> S) = topspace Y \<inter> T" shows "homeomorphic_map (subtopology X S) (subtopology Y T) f"
insertion_polyneg: fixes e :: "nat \<Rightarrow> 'b"   and b :: "'b poly" shows "insertion e (~\<^sub>p b) = - insertion e b"
coboundedI: fixes A :: "'a Word.word set"   and a :: "'a Word.word" assumes "finite A"   and "a \<in> A" shows "a \<le>s signed.Max A"
bind_def: shows "bind = COERCE(udom\<cdot>'c \<rightarrow> (udom \<rightarrow> udom\<cdot>'c) \<rightarrow> udom\<cdot>'c, 'b\<cdot>'c \<rightarrow> ('b \<rightarrow> 'd\<cdot>'c) \<rightarrow> 'd\<cdot>'c)\<cdot>bindU"
split_sel_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (\<not> (sum = Inl (projl sum) \<and> \<not> P (f1 (projl sum)) \<or> sum = Inr (projr sum) \<and> \<not> P (f2 (projr sum))))"
CondOr: fixes Env :: "env"   and B :: "lname set"   and e1 :: "expr"   and E1 :: "assigned"   and e2 :: "expr"   and E2 :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1"   and "Env\<turnstile> B \<union> assigns_if False e1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2"   and "nrm A = B \<union> assigns_if True (BinOp CondOr e1 e2) \<inter> assigns_if False (BinOp CondOr e1 e2)"   and "brk A = (\<lambda>l. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>BinOp CondOr e1 e2\<rangle>\<guillemotright> A"
intros: fixes s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and x :: "'a"   and s' :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and x' :: "'a"   and ln' :: "'b \<Rightarrow>f nat"   and \<tau>move :: "('b, 'c, 'a, 'd, 'e, 'f) \<tau>moves"   and ta :: "('b, 'c, 'a, 'd, 'e, 'f) thread_action" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "thr s' t = \<lfloor>(x', ln')\<rfloor>"   and "\<tau>move (x, shr s) ta (x', shr s')" shows "\<tau>multithreaded.m\<tau>move \<tau>move s (t, ta) s'"
has_integral_diff: fixes f :: "'b \<Rightarrow> 'c"   and k :: "'c"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "(f has_integral k) S"   and "(g has_integral l) S" shows "((\<lambda>x. f x - g x) has_integral k - l) S"
WHILE_PARALLEL: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and p :: "L"   and G :: "Var \<Rightarrow> L"   and q :: "L"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" assumes "X \<rhd> c : Sec (Q p G) (EQ G q) (EQ G q) \<Phi>"   and "\<forall>s ss. EQ G p s ss \<longrightarrow> evalB b s = evalB b ss"   and "LEQ p q" shows "X \<rhd> While b c : Sec (Q p G) (EQ G q) (EQ G q) (PhiWhileP (EQ G q) b \<Phi>)"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "indRelp a1 a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2))"
less_irrefl: fixes x :: "'b list" shows "\<not> x <p x"
simps_3: fixes b1 :: "bool"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Bc b1) (Gt v va) = True"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c fset" shows "g |`| f |`| v = (g \<circ> f) |`| v"
S_reduced_forI_flip_altsublist_adjacent: fixes w :: "'w"   and ss :: "'w list"   and ts :: "'w list" assumes "S_reduced_for w ss"   and "flip_altsublist_adjacent ss ts" shows "S_reduced_for w ts"
SMP_mono: fixes A :: "('fun, 'var) Term.term set"   and B :: "('fun, 'var) Term.term set" assumes "A \<subseteq> B" shows "SMP A \<subseteq> SMP B"
lnth_rel_chain: fixes xs :: "'b llist"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<not> lnull xs"   and "\<forall>j. enat (j + 1) < llength xs \<longrightarrow> R (lnth xs j) (lnth xs (j + 1))" shows "chain R xs"
unsafe_safe_eq_2: shows "OF_match_linear OF_match_fields_unsafe oft = OF_match_linear OF_match_fields_safe oft"
multisetUnionLessMono2: fixes r :: "('b \<times> 'b) set"   and b1 :: "'b multiset"   and b2 :: "'b multiset"   and a :: "'b multiset" assumes "trans r"   and "(b1, b2) \<in> mult r" shows "(a + b1, a + b2) \<in> mult r"
alphaAbs_preserves_qGoodAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "A $= A'" shows "qGoodAbs A = qGoodAbs A'"
order_iff: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a = min a b)"
tensor_vec_from_lookup_Cons: fixes d :: "nat"   and ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" shows "tensor_vec_from_lookup (d # ds) e = concat (map (\<lambda>i. tensor_vec_from_lookup ds (\<lambda>is. e (i # is))) [0..<d])"
simps_6: fixes v :: "char list"   and va :: "aexp"   and vb :: "'a"   and vc :: "'a" shows "(v ::= va {vb} \<sqsubseteq> SKIP {vc}) = False"
fmlookup_fold_not_mem: fixes x :: "nat"   and k1 :: "nat list"   and P1 :: "(nat, rat) fmap"   and P2 :: "(nat, rat) fmap" assumes "x \<notin> set k1" shows "fmlookup (foldl (\<lambda>lp v. add_monom (get_var_coeff P1 v) v lp) P2 k1) x = fmlookup P2 x"
integral_iff: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "domain R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>) = (a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
tail_def: fixes t :: "('c, 'd) FingerTree" shows "tail t \<equiv> if t = local.empty then local.empty else Abs_FingerTree (FingerTreeStruc.tail (Rep_FingerTree t))"
butlast_inorder_app_id: fixes xs :: "('b btree \<times> 'b) list"   and xs' :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b" assumes "xs = xs' @ [(sub, sep)]" shows "inorder_list xs' @ inorder sub @ [sep] = inorder_list xs"
Hv_def: fixes v :: "(real, 7) vec" shows "Hv v \<equiv> Octo 0 (v $ 1) (v $ 2) (v $ 3) (v $ 4) (v $ 5) (v $ 6) (v $ 7)"
chine_is_induced_map: shows "TTfgh_TfTgh.is_induced_map TTfgh_TfTgh.chine"
assms_6: shows "kKS k \<notin> LocalSecrets Q"
pref_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>p b"   and "b \<le>p c" shows "a \<le>p c"
filter_M_reads: fixes xs :: "'x list"   and P :: "'x \<Rightarrow> ('y, 'z, bool) prog"   and h :: "'y"   and h' :: "'y"   and S :: "('y \<Rightarrow> 'y \<Rightarrow> bool) set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h'"   and "\<And>x. x \<in> set xs \<Longrightarrow> reads S (P x) h h'"   and "\<forall>P\<in>S. reflp P \<and> transp P" shows "reads S (filter_M P xs) h h'"
vc_sound: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "vc C Q" shows "\<turnstile>\<^sub>2\<^sub>' {pre C Q} strip C {Q}"
hyps: assumes "x \<in> set ys" shows "Missing_List.min_list ys \<le> x"
simps_1: fixes ys :: "brick list" shows "[] \<otimes> ys = ys"
cases: fixes x :: "'a tree" obtains "x = \<langle>\<rangle>"   | l :: "'a tree" and xa :: "'a" and r :: "'a tree" where "x = \<langle>l, xa, r\<rangle>"
arccos_cos2: fixes x :: "real" assumes "x \<le> 0"   and "- pi \<le> x" shows "arccos (cos x) = - x"
right: assumes "height l \<le> height r" shows "fib (height \<langle>l, (a, h), r\<rangle> + 2) \<le> size1 \<langle>l, (a, h), r\<rangle>"
L2P_inj: fixes l :: "proj2_line"   and m :: "proj2_line" assumes "L2P l = L2P m" shows "l = m"
us_facts_5: shows "z \<notin> FvarsT t2"
happens_before_into_action_order: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" assumes "P,E \<turnstile> a \<le>hb a'" shows "E \<turnstile> a \<le>a a'"
equivI: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "refl_on A r"   and "sym r"   and "trans r" shows "equiv A r"
cases: fixes a1 :: "'a"   and a2 :: "'a \<Rightarrow> real"   and a3 :: "Regions.intv" assumes "Regions.intv_elem a1 a2 a3" obtains u :: "'a \<Rightarrow> real" and x :: "'a" and d :: "nat" where "a1 = x__" and "a2 = u__" and "a3 = Regions.intv.Const d__" and "u__ x__ = real d__"   | d :: "nat" and u :: "'a \<Rightarrow> real" and x :: "'a" where "a1 = xa__" and "a2 = ua__" and "a3 = Regions.intv.Intv d" and "real d < ua__ xa__" and "ua__ xa__ < real (d + 1)"   | c :: "nat" and u :: "'a \<Rightarrow> real" and x :: "'a" where "a1 = x" and "a2 = u" and "a3 = Regions.intv.Greater c" and "real c < u x"
rank_ins: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t' :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.rank_invar (t # bq)" shows "BinomialHeapStruc.rank t \<le> BinomialHeapStruc.rank (hd (BinomialHeapStruc.ins t' (t # bq))) \<or> BinomialHeapStruc.rank t' \<le> BinomialHeapStruc.rank (hd (BinomialHeapStruc.ins t' (t # bq)))"
WCFG_CondFalse: fixes b :: "expr"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> (_ 0 _) -(\<lambda>s. interpret b s = Some false)\<^sub>\<surd>\<rightarrow> (_ 0 _) \<oplus> #:c\<^sub>1 + 1"
move_left1_def: shows "move_left1 \<equiv> \<lambda>(l, r, buf). if l \<noteq> 0 then ASSERT (r - 1 < length buf \<and> l - 1 < length buf) \<bind> (\<lambda>_. RETURN (l - 1, r - 1, buf[r - 1 := buf ! (l - 1)])) else RETURN (l, r, buf)"
keys_subset_singleton_imp_monomial: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b" assumes "keys p \<subseteq> {t}" shows "monomial (lookup p t) t = p"
enc_preserves_pred_iff_indRelRPO_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_preserves_pred Pred = rel_preserves_pred indRelRPO Pred"
inject_1: fixes x1 :: "('c, 'd) Term.term"   and y1 :: "('c, 'd) Term.term" shows "(send\<langle>x1\<rangle>\<^sub>s\<^sub>t = send\<langle>y1\<rangle>\<^sub>s\<^sub>t) = (x1 = y1)"
pwD2: fixes S :: "'a nres"   and S' :: "'a nres"   and x :: "'a" assumes "S \<le> S'"   and "inres S x" shows "inres S' x"
iterate_eq_map_sum: fixes t :: "'a tree"   and xs :: "(nat \<times> nat) list" assumes "local.valid t"   and "windows as xs"   and "case xs of [] \<Rightarrow> True | x # xs' \<Rightarrow> l t \<le> fst x \<and> r t \<le> snd x" shows "local.iterate t xs = map (\<lambda>w. local.sum (fst w) (snd w)) xs"
arev_dom: fixes a :: "'a \<times> 'a" shows "(a \<in> arcs (with_proj G)) = (swap_in (parcs G) a \<noteq> a)"
llength_LCons: fixes x :: "'a"   and xs :: "'a llist" shows "llength (x #\<^sub>l xs) = eSuc (llength xs)"
trms\<^sub>s\<^sub>s\<^sub>t_setops\<^sub>s\<^sub>s\<^sub>t_wt_instance_ex: fixes B :: "('a strand_label \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step) list"   and A :: "('a strand_label \<times> ('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step) list" assumes "\<forall>b\<in>set B. \<exists>a\<in>set A. \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "\<forall>t\<in>trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t B \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel B). \<exists>s\<in>trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A). \<exists>\<delta>. t = s \<cdot> \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
gcd_mult: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "c \<otimes> somegcd G a b \<sim> somegcd G (c \<otimes> a) (c \<otimes> b)"
red_subset: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "red G p q"   and "G \<subseteq> F" shows "red F p q"
diagonal_to_Smith_row_i_preserves_previous: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and b :: "'b" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "to_nat a \<noteq> to_nat b" shows "Diagonal_to_Smith_row_i A i bezout $ a $ b = A $ a $ b"
ind_sheaf_is_presheaf: shows "presheaf_of_rings U it.ind_is_open local.ind_sheaf ind_ring_morphisms b ind_add_str ind_mult_str ind_zero_str ind_one_str"
ideal_induct_homomorphism: fixes B :: "'a set"   and p :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "B \<subseteq> range h"   and "p \<in> range h"   and "p \<in> ideal B"   and "P (0::'a)"   and "\<And>c b a. \<lbrakk>c \<in> range h; b \<in> B; P a; a \<in> range h\<rbrakk> \<Longrightarrow> P (c * b + a)" shows "P p"
ra_ia_rb_ib_rc_ic_def_1: shows "ra = Re a"
ccBinds_strict: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallAnalysis.ccBinds ccExp \<Gamma>\<cdot>\<bottom> = \<bottom>"
i2: shows "\<forall>D\<in>MajoritySet. \<forall>qa. \<exists>d\<in>D. bal (dblock s' q) < mbal (disk s' d qa) \<longrightarrow> hasRead s' q d qa"
wt: shows "Env\<turnstile>e\<Colon>-T"
Domainp_iff: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b" shows "Domainp T x = (\<exists>y. T x y)"
neg_equal_iff_equal: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(uminus a = uminus b) = (a = b)"
diagonalize_2x2_JNF_def: fixes A :: "'b mat" shows "diagonalize_2x2_JNF A = (let (P, D, Q) = diagonalize_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 D, Mod_Type_Connect.from_hma\<^sub>m Q))"
labelAtIn_def: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent" shows "labelAtIn v p = subst (inst v) (freshen (vidx v) (labelsIn (nodeOf v) p))"
spmf_False_conv_True: fixes p :: "bool spmf" assumes "lossless_spmf p" shows "spmf p False = 1 - spmf p True"
compare_cases: fixes y :: "'b \<Rightarrow> 'b \<Rightarrow> comp" assumes "y \<in> {cmp. comparator cmp}" obtains x :: "'b comparator" where "y = compare x"