hcju/isabelleps huggingface dataset:
IH: assumes "is_red F (tail p)" shows "\<exists>f\<in>F. \<exists>v\<in>keys (tail p). f \<noteq> 0 \<and> lt f adds\<^sub>t v"
cases: fixes a1 :: "'a cfg"   and a2 :: "'a cfg stream" assumes "MC.enabled a1 a2" obtains \<omega> :: "'a cfg stream" and s :: "'a cfg" where "a1 = s" and "a2 = \<omega>" and "MC.enabled (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> set_pmf (K_cfg s)"
B: fixes ys :: "'a list"   and zs :: "'a list" shows "xs @ [x] \<cong> {ys @ [x], zs, P} \<longrightarrow> xs \<cong> {ys, zs, \<lambda>w ws. P w (ws @ [x])}"
Semilat_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "Semilat A r f \<equiv> semilat (A, r, f)"
equivalent_objects_transitive: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.equivalent_objects a b"   and "D.equivalent_objects b c" shows "D.equivalent_objects a c"
P: shows "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q"
bind_returns_result_E2: fixes h :: "'e"   and f :: "('e, 'f, 'h) prog"   and g :: "'h \<Rightarrow> ('e, 'f, 'g) prog"   and y :: "'g" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'h" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
bv_to_nat1: fixes bs :: "bit list" shows "bv_to_nat (\<one> # bs) = 2 ^ length bs + bv_to_nat bs"
uY: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and P :: "'s \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "unitary P" shows "unitary (wlp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P)"
lookup_proj_poly_syz: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and x :: "'t" shows "lookup (proj_poly_syz n p) x = lookup p (map_component (\<lambda>k. k + n) x)"
inv_all_diff_evD: fixes n :: "nat"   and A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.diff_ev A"   and "i < j"   and "j < n"   and "A $$ (i, i) \<noteq> A $$ (j, j)" shows "A $$ (i, j) = (0::'b)"
hyps_1: assumes "consistent t\<^sub>1" shows "cost t\<^sub>1 = (\<Sum>a\<in>alphabet t\<^sub>1. freq t\<^sub>1 a * depth t\<^sub>1 a)"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'b"   and order :: "WS1S_Prelim.order" shows "h (case order of FO \<Rightarrow> f1 | SO \<Rightarrow> f2) = (case order of FO \<Rightarrow> h f1 | SO \<Rightarrow> h f2)"
assms: shows "x \<subseteq>\<^sub>f fmrestrict_set vs s"
less_half_sum: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b" shows "less a (divide (plus a b) (plus one one))"
lexord_def: fixes r :: "('b \<times> 'b) set" shows "lexord r = {(x, y). \<exists>a v. y = x @ a # v \<or> (\<exists>u a b v w. (a, b) \<in> r \<and> x = u @ a # v \<and> y = u @ b # w)}"
lemma_2_5_13_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b l\<rightarrow> c \<le> a l\<rightarrow> c"
degree_mult_not_eq: fixes f :: "'a poly"   and g :: "'a poly" assumes "degree (f * g) \<noteq> degree f + degree g" shows "lead_coeff f * lead_coeff g = (0::'a)"
4: shows "(dom a \<cong> cod (dom a) \<^bold>\<leftarrow> E (a \<cdot> (dom a))) \<^bold>\<and> (E (a \<cdot> (dom a)) \<^bold>\<leftarrow> dom a \<cong> cod (dom a))"
Chi_HAStates_Self: fixes s :: "'a"   and a :: "('a, 'b, 'c) hierauto" assumes "s \<in> HAStates a" shows "s \<notin> Chi a s"
convex_hull_insert_alt: fixes a :: "'a"   and S :: "'a set" shows "convex hull insert a S = (if S = {} then {a} else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
abs_le_D2: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq (uminus a) b"
lsum_def: fixes f :: "'b \<Rightarrow> nat"   and al :: "'b list" shows "lsum f al \<equiv> sum_list (map f al)"
isGuard_simps_2: 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 pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and x :: "Preliminaries.var"   and p :: "'b"   and Tl :: "'a trm list" assumes "ProblemIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf" shows "ProblemIkPol.isGuard pol x (Neg (Pr p Tl)) = (x \<in> \<Union> (set (map ProblemIkPol.nv2T Tl)) \<and> ProblemIkPolMcalc2C.polC pol p = Fext)"
obj_trg: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "horizontal_homs.obj V src (trg \<mu>)"
\<psi>e: shows "\<guillemotleft>\<psi> (a, a') e : a \<rightarrow> a'\<guillemotright>"
liftAll_imp_liftAll2: fixes phi :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and inp :: "'b \<Rightarrow> 'a option"   and inp' :: "'b \<Rightarrow> 'c option" assumes "liftAll (\<lambda>v. \<forall>v'. phi v v') inp" shows "liftAll2 phi inp inp'"
monad_state_altc_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_state_altc return bind get put altc \<equiv> monad_state return bind get put \<and> monad_altc return bind altc \<and> monad_state_altc_axioms get put altc"
mult_cong: fixes x :: "int"   and y :: "int" shows "x mod m \<otimes> (y mod m) = x * y mod m"
trm_subst_ident: fixes t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv t \<inter> subst_domain \<theta> = {}" shows "t \<cdot> \<theta> = t"
of_int_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 < z" shows "less zero (ring_1.of_int one plus zero minus z)"
pnsubst1_correct: fixes l :: "'a list"   and M :: "mon"   and P2 :: "pol"   and P1 :: "pol"   and n :: "nat" assumes "in_carrier l"   and "Imon l M = Ipol l P2" shows "Ipol l (pnsubst1 P1 M P2 n) = Ipol l P1"
fail\<^sub>S\<^sub>E_def: shows "fail\<^sub>S\<^sub>E = Map.empty"
eq_num_simps_4: fixes m :: "num" shows "(num.Bit0 m = num.One) = False"
X: shows "distributed M MX X (\<lambda>x. ennreal (f x))"
CallThrowObj: fixes e :: "char list exp"   and M :: "char list"   and es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw e\<bullet>M(es),s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
msort_idle: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "distinct xs"   and "linorder.sorted less_eq xs" shows "linorder.msort less xs = xs"
subset_\<P>: fixes k' :: "nat"   and k :: "nat"   and u' :: "nat"   and u :: "nat" assumes "k' < k \<or> k' = k \<and> u' \<le> u" shows "\<P> k' u' \<subseteq> \<P> k u"
closedin_Int: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "closedin U S"   and "closedin U T" shows "closedin U (S \<inter> T)"
infinite_insert: fixes a :: "'a"   and A :: "'a set" shows "infinite (insert a A) = infinite A"
Ic: fixes t :: "real" shows "continuous_on {0..t} I"
cond_precise: shows "\<forall>s. s \<in> wa_invar WB \<and> \<alpha>2 s \<in> wa_cond WA \<longrightarrow> s \<in> wa_cond WB"
heap_is_wellformed_children_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "child |\<in>| node_ptr_kinds h"
isomorphism_expand: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" shows "(f \<circ> g = id \<and> g \<circ> f = id) = ((\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x))"
equal_pair_count_mirror_pairwise: fixes ps1 :: "('b \<times> 'b) list"   and ps2 :: "('b \<times> 'b) list" assumes "length ps1 = length ps2"   and "\<And>i. i < length ps2 \<Longrightarrow> ps1 ! i = mirror_pair (ps2 ! i)" shows "equal_pair_count ps1 = equal_pair_count ps2"
ipurge_tr_rev_aux_append_nil: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list"   and ys :: "'a list" shows "(ipurge_tr_rev_aux I D U (xs @ ys) = ipurge_tr_rev_aux I D U xs) = (ipurge_tr_rev_aux I D U ys = [])"
xor_one_left: fixes x :: "'c" shows "- (1::'c) XOR x = NOT x"
ars_r_def: fixes x :: "('a \<times> 'a) set" shows "rdom x = rel_ad (rel_ad x)"
prems: shows "\<forall>x\<in>fv (Var x) - X. \<exists>a. \<Gamma> (Var x) = Var a"
sel_2: fixes x2 :: "'a" shows "atm_of (Neg x2) = x2"
\<X>_def: shows "\<X> \<equiv> (\<times>) (topspace X) ` Collect (openin Y)"
wf_subcls_nCls: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and D :: "char list"   and D' :: "char list" assumes "wf_prog wf_md P"   and "\<not> is_class P C"   and "P \<turnstile> D \<preceq>\<^sup>* D'"   and "D \<noteq> C" shows "D' \<noteq> C"
mset_factorsD_3: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and F :: "'a multiset"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F p" shows "comm_monoid_mult.prod_mset times one F = p"
case_1: fixes f1 :: "b_e \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "cl \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> v list \<Rightarrow> e list \<Rightarrow> 'a"   and x1 :: "b_e" shows "(case $x1 of $x \<Rightarrow> f1 x | Trap \<Rightarrow> f2 | Callcl x \<Rightarrow> f3 x | Label x xa xb \<Rightarrow> f4 x xa xb | Local x xa xb xc \<Rightarrow> f5 x xa xb xc) = f1 x1"
1: shows "(\<lambda>n. fst (X n)) \<longlonglongrightarrow> lim (\<lambda>n. fst (X n))"
to_partial_def: fixes f :: "'a \<Rightarrow> 'b" shows "to_partial f = (\<lambda>x. Some (f x))"
simps_1: shows "Tree.inorder \<langle>\<rangle> = []"
succ_def: fixes G :: "('b, 'a) graph"   and v :: "'b" shows "succ G v \<equiv> {(w, v'). (v, w, v') \<in> edges G}"
assms_15: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
nat_pow_consistent: fixes x :: "'a"   and n :: "nat"   and H :: "'a set" shows "x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> n = x [^]\<^bsub>G\<lparr>carrier := adjoin G H a, carrier := H\<rparr>\<^esub> n"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_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_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs" shows "l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
subset_Un1: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> B" shows "A \<subseteq> B \<union> C"
conj_imp: fixes A :: "'a set"   and p :: "'a"   and q :: "'a"   and r :: "'a" shows "(A \<turnstile> (p .\<and>. q) .\<longrightarrow>. r) = (A \<turnstile> p .\<longrightarrow>. (q .\<longrightarrow>. r))"
silent_moves_same_level_path: fixes S :: "'node SDG_node set"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,kind \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "m -as\<rightarrow>\<^bsub>sl\<^esub>* m'" shows "ms = ms'"
mult_ac_2: fixes a :: "'e"   and b :: "'e" shows "a * b = b * a"
Rel'_def: shows "Rel' = Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q)"
tBB: shows "trace (P * B * B * adjoint P) = trace (adjoint P * P * B * B)"
maps_nth: fixes E :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and x :: "'b"   and y :: "'a" assumes "(E(xs [\<mapsto>] ys)) x = \<lfloor>y\<rfloor>"   and "length xs = length ys"   and "distinct xs" shows "\<forall>i. x = xs ! i \<and> i < length xs \<longrightarrow> y = ys ! i"
simps_10: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Dom \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = Dom t \<^bold>\<star> Dom u \<^bold>\<star> Dom v"
poincare_distance_x_axis_x_axis: fixes x :: "complex_homo"   and y :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "x \<in> circline_set x_axis"   and "y \<in> circline_set x_axis" shows "poincare_distance x y = (let x' = to_complex x; y' = to_complex y in \<bar>ln (Re ((1 + x') * (1 - y') / ((1 - x') * (1 + y'))))\<bar>)"
extremally_disconnected_not_part_not_ed_ou_card: fixes X :: "'b set" assumes "extremally_disconnected X"   and "\<not> part X"   and "\<not> ed_ou X" shows "card {f. CK_nf_pos f} = 5"
intros: fixes state :: "State"   and Vbl :: "nat set" assumes "getSATFlag state = UNDEF \<Longrightarrow> solve_loop_dom (solve_loop_body state Vbl) Vbl" shows "solve_loop_dom state Vbl"
finite_measure_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<in> sets M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
prod_mset_prod_list: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and xs :: "'b list" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one (mset xs) = monoid_mult.prod_list one times xs"
notame_def: fixes g :: "graph" shows "notame g \<equiv> \<not> (tame10ub g \<and> tame11b g)"
Some_image_these_eq: fixes A :: "'e option set" shows "Some ` Option.these A = {x \<in> A. x \<noteq> None}"
InitBlockThrow: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and T :: "ty"   and v :: "val"   and v' :: "val"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> T casts v to v' " shows "P,E \<turnstile> \<langle>{V:T; V:=Val v;; Throw r},s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
distinct_getVars: fixes \<sigma>l :: "'tp list" shows "distinct (getVars \<sigma>l)"
sum_of_squares: fixes n :: "nat" shows "real (sum power2 {..n}) = real (2 * n ^ 3 + 3 * n\<^sup>2 + n) / 6"
assms: fixes x :: "'a" assumes "x \<in> supp_fun f" shows "f x = (0::'b)"
simps_1: fixes c :: "qtname" shows "assignsE (NewC c) = {}"
units_eq: fixes e1 :: "'a"   and e2 :: "'a" assumes "e1 \<in> {1::'a}"   and "e2 \<in> {1::'a}"   and "D e1 e2" shows "e1 = e2"
span_add: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> ideal S"   and "y \<in> ideal S" shows "x + y \<in> ideal S"
composition_closed: fixes M :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d"   and a :: "'d"   and b :: "'d" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M"   and "b \<in> M" shows "composition a b \<in> M"
prems: shows "\<forall>a\<in>set (f # fs). is_cnf a"
image_domain_eq: fixes c :: "('b, 'a) chart" shows "apply_chart c ` domain c = codomain c"
alt_get: fixes f :: "'s \<Rightarrow> 'm"   and g :: "'s \<Rightarrow> 'm" shows "alt (get f) (get g) = get (\<lambda>x. alt (f x) (g x))"
supp_fun_app_eqvt: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "eqvt f" shows "supp (f x) \<subseteq> supp x"
compute_float_plus: fixes m1 :: "int"   and e1 :: "int"   and m2 :: "int"   and e2 :: "int" shows "Float m1 e1 + Float m2 e2 = (if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else if e1 \<le> e2 then Float (m1 + m2 * 2 ^ nat (e2 - e1)) e1 else Float (m2 + m1 * 2 ^ nat (e1 - e2)) e2)"
bound_effect: fixes I :: "('sf, 'sc, 'sz) interp"   and \<alpha> :: "('sf, 'sc, 'sz) hp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "hpsafe \<alpha>"   and "(\<nu>, \<omega>) \<in> prog_sem I \<alpha>" shows "Vagree \<nu> \<omega> (- BVP \<alpha>)"
alpha_refl_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and bs :: "atom set"   and f :: "'a \<Rightarrow> atom set" assumes "R x x" shows "(bs, x) \<approx>set R f 0 (bs, x)"
le_refl: fixes n :: "nat" shows "order.greater_eq n n"
indep_vars_def: fixes M' :: "'c \<Rightarrow> 'd measure"   and X :: "'c \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'd"   and I :: "'c set" shows "P.P.indep_vars M' X I = ((\<forall>i\<in>I. P.P.random_variable (M' i) (X i)) \<and> P.P.indep_sets (\<lambda>i. sigma_sets (space (Pi\<^sub>M UNIV (\<lambda>_. M))) {X i -` A \<inter> space (Pi\<^sub>M UNIV (\<lambda>_. M)) |A. A \<in> sets (M' i)}) I)"
isCont_s: fixes x :: "real" shows "isCont ego.s x"
h2: fixes j :: "nat" assumes "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))" shows "j < length signs"
nucleus_alt_def2: fixes f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "nucleus f" shows "f y \<leftarrow> f x = f y \<leftarrow> x"
lift_eq_qe_def: fixes qe :: "'a list \<Rightarrow> 'a fm"   and as :: "'a list" shows "lift_eq_qe qe as = (let as = filter (\<lambda>a. \<not> trivial a) as in case filter solvable\<^sub>0 as of [] \<Rightarrow> qe as | eq # eqs \<Rightarrow> let ineqs = filter (\<lambda>a. \<not> solvable\<^sub>0 a) as in list_conj (map (Atom \<circ> subst\<^sub>0 eq) (eqs @ ineqs)))"
comp_geq: fixes x :: "'a::refinement_lattice" shows "((H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)) (x::'a::refinement_lattice) \<sqsubseteq> (F \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)) x"
pochhammer_fact: fixes n :: "nat" shows "fact n = pochhammer (1::'b) n"
continuous_closed_vimage: fixes S :: "'c set"   and f :: "'d \<Rightarrow> 'c" assumes "closed S"   and "\<And>x. isCont f x" shows "closed (f -` S)"
d_lb2: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y \<le> d y"
that: fixes k :: "nat"   and r :: "'b stream"   and p :: "'b" assumes "s = fromN (Suc k) ||| r"   and "v = (k, p)" shows "thesis"
pos_imaginary_parts_3_def: fixes p :: "int poly" shows "pos_imaginary_parts_3 p = Complex_Algebraic_Numbers.remdups_gen equal_3 (filter (\<lambda>x. sgn_3 x = 1) (concat (map roots_of_3 (root_poly_Im p))))"
show7: fixes q :: "int"   and p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and n :: "nat" assumes "1 \<le> q"   and "1 \<le> p"   and "\<forall>n. 1 \<le> a n"   and "\<forall>n. 1 \<le> b n"   and "(\<lambda>n. real_of_int (b (n + 1)) / real_of_int (a (n + 1))) sums (real_of_int p / real_of_int q)" shows "1 \<le> real_of_int (q * (\<Prod>j = 1..n. of_int (a j))) * (\<Sum>j. real_of_int (b (j + n + 1)) / real_of_int (a (j + n + 1)))"
mult_cancel_right2: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times a c = c) = (c = zero \<or> a = one)"
MapsToE: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "f maps\<^bsub>C\<^esub> X to Y" obtains "f \<in> mor\<^bsub>C\<^esub>" and "dom\<^bsub>C\<^esub> f = X" and "cod\<^bsub>C\<^esub> f = Y"
lnths_LNil: fixes A :: "nat set" shows "lnths LNil A = LNil"
prems: shows "w \<Turnstile>\<^sub>n Unf (\<phi>1 M\<^sub>n \<phi>2)[X]\<^sub>\<nu>"