hcju/isabelleps huggingface dataset:
bottom_lower: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a" assumes "weak_partial_order_bottom L"   and "x \<in> carrier L" shows "\<bottom>\<^bsub>L\<^esub> \<sqsubseteq>\<^bsub>L\<^esub> x"
sx: shows "p\<inverse>\<^sup>>x \<cdot> s = x"
g1: shows "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>CI T\<^esub> \<rightarrow> IMor g"
ofilter_subset_embedS_iso: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and B :: "'c set" assumes "Well_order r"   and "ofilter r A"   and "ofilter r B" shows "(A \<subset> B) = embedS (Restr r A) (Restr r B) id \<and> (A = B) = iso (Restr r A) (Restr r B) id"
ffilter_singleton: fixes f :: "'b \<Rightarrow> bool"   and e :: "'b" assumes "f e" shows "ffilter f {|e|} = {|e|}"
simps_2: fixes x :: "char list"   and a :: "aexp"   and X :: "char list set" shows "bury (x ::= a) X = (if x \<in> X then x ::= a else SKIP)"
fineD: fixes d :: "'c \<Rightarrow> 'd set"   and s :: "('c \<times> 'd set) set"   and x :: "'c"   and k :: "'d set" assumes "d fine s"   and "(x, k) \<in> s" shows "k \<subseteq> d x"
ex1E: fixes P :: "'d \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'d" where "P x" and "\<forall>y. P y \<longrightarrow> y = x"
field_accessible_from_permits_acc_inheritance: fixes G :: "prog"   and accC :: "qtname"   and membr :: "qtname \<times> memberdecl"   and statC :: "qtname"   and dynC :: "qtname" assumes "G \<turnstile> membr of statC accessible_from accC"   and "is_field membr"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC" shows "G \<turnstile> membr in dynC permits_acc_from accC"
lbrules_2: shows "\<Union> (A ` {1..m}) = {1..j}"
prj_simps_11: shows "C.seq \<pi>\<rho>.prj\<^sub>1 \<p>\<^sub>0[\<mu>\<nu>.leg0, \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1]"
filter_remove1: fixes Q :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
rupture_surj_hom_2: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (A [X])"   and "p \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)" shows "ring_hom_ring (K [X]\<^bsub>A [X]\<^esub>) (Rupt\<^bsub>A [X]\<^esub> K p) (UP_A.rupture_surj K p)"
X: shows "X \<in> M \<rightarrow>\<^sub>M M1"
bisim1_length_xs: 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" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "length xs = length loc"
diff: fixes D :: "'a set"   and E :: "'a set" assumes "D \<in> M"   and "E \<in> M"   and "D \<subseteq> E" shows "E - D \<in> M"
poly_eval_focus_in_Polys: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and a :: "'c \<Rightarrow> 'd"   and Y :: "'c set" assumes "p \<in> P[X]" shows "poly_eval (\<lambda>x. monomial (a x) 0) (focus Y p) \<in> P[X - Y]"
has_derivative_subst: fixes f :: "'a \<Rightarrow> 'b"   and df :: "'a \<Rightarrow> 'b"   and x :: "'a"   and d :: "'a \<Rightarrow> 'b" assumes "(f has_derivative df) (at x)"   and "df = d" shows "(f has_derivative d) (at x)"
subcomplex_chamber: fixes Y :: "'a set set"   and C :: "'a set" assumes "folding_g.ChamberSubcomplex Y"   and "SimplicialComplex.maxsimp Y C" shows "folding_g.chamber C"
Range_insert: fixes a :: "'c"   and b :: "'b"   and r :: "('c \<times> 'b) set" shows "Range (insert (a, b) r) = insert b (Range r)"
simps: fixes l :: "('b \<times> bal) tree"   and a :: "'b"   and b :: "bal"   and r :: "('b \<times> bal) tree" shows "is_bal \<langle>l, (a, b), r\<rangle> = (b = Bal)"
distinct_1: fixes a1 :: "'d"   and a2 :: "'d"   and psi :: "('d, 'e, 'f) psi"   and a' :: "'d"   and boundOutput' :: "('d, 'e, 'f) boundOutput" shows "RIn a1 a2 psi \<noteq> ROut a' boundOutput'"
phull_subset_module: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "phull B \<subseteq> pmdl B"
pascal_pappus1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points"   and R :: "Points" assumes "pascal_prop" shows "is_pappus1 A B C A' B' C' P Q R"
im_sheaf_is_presheaf: shows "presheaf_of_rings S is_open idim.im_sheaf idim.im_sheaf_morphisms b idim.add_im_sheaf idim.mult_im_sheaf idim.zero_im_sheaf idim.one_im_sheaf"
subcategory_axioms: shows "subcategory VxVxV.comp (\<lambda>\<tau>\<mu>\<nu>. arr (fst \<tau>\<mu>\<nu>) \<and> VV.arr (snd \<tau>\<mu>\<nu>) \<and> src (fst \<tau>\<mu>\<nu>) = trg (fst (snd \<tau>\<mu>\<nu>)))"
closed_sum_assoc: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A +\<^sub>M (B +\<^sub>M C) = A +\<^sub>M B +\<^sub>M C"
nil_par_nil: shows "nil \<parallel> nil \<sqsubseteq> nil"
subsetI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "order.greater_eq B A"
simps_2: fixes I :: "('b, 'c, 'a) interp"   and P :: "'a"   and terms :: "'a \<Rightarrow> ('b, 'a) trm" shows "fml_sem I ($\<phi> P terms) = {\<nu>. Predicates I P (\<chi>i. dterm_sem I (terms i) \<nu>)}"
QR_lemma_13: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.a p q + QR.d p q + QR.n p q = QR.r p q"
C: shows "(\<not> check_eq (fst M ! q) (fst M ! p) T) = (\<exists>n<m. dist_nodes M (Suc n) v p q)"
Gateway_L7: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "ts lose"   and "msg (Suc 0) a"   and "msg (Suc 0) stop"   and "msg (Suc 0) req"   and "req (Suc t) = [init]"   and "\<forall>m<k + 3. req (t + m) \<noteq> [send]"   and "req (t + 3 + k) = [send]"   and "ack t = [init_state]"   and "\<forall>j\<le>k + d + 3. lose (t + j) = [False]"   and "\<forall>t1\<le>t. req t1 = []" shows "\<forall>t2<t + 3 + k + d. i t2 = []"
assms_2: shows "(r|\<tau> @ \<sigma>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r"
left_uvar_def: fixes x :: "'d \<Longrightarrow> 'f" shows "0-x = x ;\<^sub>L mrg_left"
preserve_min_or_delete_completely: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and C :: "'a literal multiset"   and i :: "nat" assumes "St \<leadsto>\<^sub>w St'"   and "(C, i) \<in># wP_of_wstate St"   and "\<forall>k. (C, k) \<in># wP_of_wstate St \<longrightarrow> i \<le> k" shows "(C, i) \<in># wP_of_wstate St' \<or> (\<forall>j. (C, j) \<notin># wP_of_wstate St')"
assms: shows "R' = Ax \<union> g3up \<union> g3mod1 \<union> g3mod2"
hhom_is_subcategory: fixes a :: "'a"   and b :: "'a" shows "subcategory (\<cdot>\<^sub>C) (\<lambda>\<mu>. \<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright>)"
map_topology_def: fixes p :: "'d \<Rightarrow> 'c"   and X :: "'d topology" shows "map_topology p X = final_topology (p ` topspace X) (\<lambda>_. X) (\<lambda>_. p)"
zero_pratt1_def: shows "0 \<equiv> \<bottom>\<^sub>1"
F1: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "F 1 = map_poly to_fract G1"
totalize_Constrains_iff: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" shows "(totalize F \<in> A Co B) = (F \<in> A Co B)"
WHILE_refine': fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R :: "('d \<times> 'c) set"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres" assumes "I' x' \<Longrightarrow> (x, x') \<in> R"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "WHILE b f x \<le> \<Down> R (WHILE\<^bsup>I'\<^esup> b' f' x')"
cycle_free_diag_dest: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "linordered_ab_monoid_add.cycle_free plus zero less_eq m n" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"
set_val_get_shadow_root: fixes ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
order_less_irrefl: fixes x :: "'a" shows "\<not> x < x"
bounded_def: fixes S :: "'b set" shows "bounded S = (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
t_1: shows "t \<in> V\<^bsub>T\<^esub>"
lfilter_const_true: fixes xs :: "'a llist" shows "lfilter\<cdot>(\<Lambda> x. TT)\<cdot>xs = xs"
da_elim_cases_20: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (c1 Finally c2)\<guillemotright> A" obtains C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<turnstile> B \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C1 \<union> nrm C2" and "brk A = (brk C1 \<Rightarrow>\<union>\<^sub>\<forall> nrm C2) \<Rightarrow>\<inter>  brk C2"
rel2p_def: fixes R :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c" shows "rel2p R x y \<equiv> (x, y) \<in> R"
PeqQ: shows "\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q"
stable_match: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw"   and "StableMarriage.stable Pm Pw \<mu>" shows "StableMarriage.match Pm Pw \<mu>"
compl_open_closed: fixes m :: "'a set" assumes "m open" shows "carrier - m closed"
p_geI: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and p' :: "real" assumes "akra_bazzi_params_nonzero k as bs"   and "1 \<le> (\<Sum>i<k. as ! i * bs ! i powr p')" shows "p' \<le> akra_bazzi_exponent as bs"
nsubst_dterm: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<theta> :: "('sf + 'b, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "TadmitFO \<sigma> \<theta>"   and "dsafe \<theta>"   and "\<And>i. dsafe (\<sigma> i)" shows "dterm_sem I (TsubstFO \<theta> \<sigma>) \<nu> = dterm_sem (adjointFO I \<sigma> \<nu>) \<theta> \<nu>"
IH_2: assumes "i < nr"   and "j < nc"   and "A $$ (i, j) \<noteq> (0::'a)" shows "pivot_positions_main (mat\<^sub>h A) nr nc (Suc i) (Suc j) = pivot_positions_main A nr nc (Suc i) (Suc j)"
da_elim_cases_10: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (If(e) c1 Else c2)\<guillemotright> A" obtains E :: "assigned" and C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> B \<union> assigns_if True e \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<turnstile> B \<union> assigns_if False e \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C1 \<inter> nrm C2" and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2"
oalist_inv_list_of_oalist_tc: fixes xs :: "('b, 'c) oalist_tc" shows "tc.oalist_inv_raw (list_of_oalist_tc xs)"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H G" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
BIT: shows "T\<^sub>p_on_rand' BIT s qs \<le> 175 / 10\<^sup>2 * real (T\<^sub>p [x, y] qs (OPT2 qs [x, y]))"
TensorDiag_preserves_Diag_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t"   and "monoidal_language.Diag C u" shows "monoidal_language.Diag C (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
adversary2_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf"   and h :: "'a" shows "adversary2 \<A> h = \<A> h \<bind> (\<lambda>(w1, w2). return_spmf (w1 + x * w2))"
advantage_def: fixes key_gen :: "'a spmf"   and "prf" :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and rand :: "'c spmf"   and \<A> :: "(bool, 'b, 'c) gpv" shows "prf.advantage key_gen prf rand \<A> = \<bar>spmf (prf.game_0 key_gen prf \<A>) True - spmf (prf.game_1 rand \<A>) True\<bar>"
comparator_def: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp" shows "comparator cmp \<equiv> ((\<forall>a. cmp a a = Equiv) \<and> (\<forall>a b c. cmp a b = Equiv \<longrightarrow> cmp b c = Equiv \<longrightarrow> cmp a c = Equiv)) \<and> (\<forall>a b c. cmp a b = Less \<longrightarrow> cmp b c = Less \<longrightarrow> cmp a c = Less) \<and> (\<forall>b a. (cmp b a = Greater) = (cmp a b = Less))"
cong: fixes unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and unconst :: "'b \<Rightarrow> name option" shows "term.match unapp unconst \<equiv> term.match unapp unconst"
S10: shows "hd (nat_to_list u) = c_fst (c_snd (u - 1))"
edges_undef_simps_4: fixes l :: "('c \<times> 'd \<times> nat) list"   and w :: "'c \<times> 'd \<Rightarrow> enat"   and u :: "'c"   and v :: "'d"   and d :: "nat" shows "edges_undef l (w((u, v) := enat d)) = (edges_undef l w \<and> (\<forall>d'. (u, v, d') \<notin> set l))"
insert_remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "Max (insert x A) = (if A - {x} = {} then x else max x (Max (A - {x})))"
measure_Diff: fixes M :: "'c measure"   and A :: "'c set"   and B :: "'c set" assumes "emeasure M A \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
dom_cod: fixes f :: "'s" assumes "arr f" shows "local.dom (cod f) = cod f"
irreflp_transp_imp_antisymP: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "irreflp p"   and "transp p" shows "antisymp p"
assms_3: shows "\<Gamma> (Fun f T) = Fun f (map \<Gamma> T)"
diff_union_swap: fixes a :: "'b"   and b :: "'b"   and M :: "'b multiset" assumes "a \<noteq> b" shows "add_mset b (remove1_mset a M) = remove1_mset a (add_mset b M)"
fresh_conv_MOST: fixes a :: "atom"   and x :: "'c" shows "a \<sharp> x = (\<forall>\<^sub>\<infinity>b. (a \<rightleftharpoons> b) \<bullet> x = x)"
exhaust: fixes y :: "ed_ou_witness" obtains "y = ed_ou_witness.a"   | "y = ed_ou_witness.b"   | "y = ed_ou_witness.c"   | "y = d"   | "y = e"
elims_3: fixes x :: "'b dres"   and xa :: "'b dres" assumes "\<not> x \<le> xa" obtains a :: "'b" and b :: "'b" where "x = dRETURN a" and "xa = dRETURN b" and "a \<noteq> b"   | "x = dFAILi" and "xa = dSUCCEEDi"   | v :: "'b" where "x = dFAILi" and "xa = dRETURN v__"   | v :: "'b" where "x = dRETURN v" and "xa = dSUCCEEDi"
bij_betw_empty1: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set" assumes "bij_betw f {} A" shows "A = {}"
a_complement_inf_double_complement: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> - - y) = - (x \<sqinter> y)"
interaction_any_bounded_converter_parallel_converter2: fixes conv1 :: "('a, 'b, 'c, 'd) converter"   and n :: "enat"   and conv2 :: "('e, 'f, 'g, 'h) converter" assumes "interaction_any_bounded_converter conv1 n"   and "interaction_any_bounded_converter conv2 n" shows "interaction_any_bounded_converter (parallel_converter2 conv1 conv2) n"
mrconc: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'b"   and p :: "'a list"   and a :: "'a"   and b :: "'a" assumes "applied_rule_rev C x p = \<lfloor>a\<rfloor>" shows "applied_rule_rev C x (b # p) = \<lfloor>a\<rfloor>"
l_scc: fixes G :: "('a, 'b) graph_rec_scheme"   and l :: "'a set list"   and D :: "'a set" assumes "cscc_invar_ext G l D" shows "set l \<subseteq> Collect (is_scc (g_E G))"
four_block_mat_adjoint: fixes A :: "'c mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'c mat"   and nc2 :: "nat"   and C :: "'c mat"   and nr2 :: "nat"   and D :: "'c mat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "adjoint (four_block_mat A B C D) = four_block_mat (adjoint A) (adjoint C) (adjoint B) (adjoint D)"
borel_closed: fixes A :: "'c set" assumes "closed A" shows "A \<in> sets borel"
append_same_eq: fixes ys :: "'e list"   and xs :: "'e list"   and zs :: "'e list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
put_inj: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e" assumes "weak_lens x" shows "inj (put\<^bsub>x\<^esub> \<sigma>)"
d_restrict_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<le> d x \<cdot> y)"
add_eq_0_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(plus a b = zero) = (b = uminus a)"
Procedure_defs_5: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'b" shows "Return ch p v \<equiv> $Calling ch p \<and> rbit<ch!p>$ = $cbit<ch!p> \<and> res<ch!p>$ = $v"
independent_bound_general: fixes S :: "'a set" assumes "independent S" shows "finite S \<and> card S \<le> dim S"
2: fixes theta :: "('a \<times> 'a) set" assumes "sym theta" shows "sym (Retr theta)"
extract_merge_id: fixes A :: "'e cset"   and f :: "'e \<Rightarrow> 'd cset Monomorphic_Monad.id" shows "extract (merge_id A f) = cUnion (cimage (extract \<circ> f) A)"
fundamental_theorem_of_algebra: fixes p :: "complex poly" assumes "\<not> constant (poly p)" shows "\<exists>z. poly p z = 0"
ldrop_eq_LConsD: fixes n :: "enat"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" assumes "ldrop n xs = LCons y ys" shows "n < llength xs"
Rep_OrdSum_inverse: fixes x :: "('b, 'c) OrdSum" shows "Abs_OrdSum (Rep_OrdSum x) = x"
norm_diff2: fixes y :: "'a"   and y1 :: "'a"   and y2 :: "'a"   and x :: "'a"   and x1 :: "'a"   and x2 :: "'a"   and e :: "real"   and e1 :: "real"   and e2 :: "real" assumes "y = y1 + y2"   and "x = x1 + x2"   and "e = e1 + e2"   and "norm (y1 - x1) \<le> e1"   and "norm (y2 - x2) \<le> e2" shows "norm (y - x) \<le> e"
ask_state_def: fixes ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> ('a, 'b) stateT" shows "ask_state ask f = StateT (\<lambda>s. ask (\<lambda>r. run_state (f r) s))"
IS_TO_SORTED_LIST_def: fixes \<Omega> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rs :: "'c set \<Rightarrow> 'd \<Rightarrow> assn"   and Rk :: "'c \<Rightarrow> 'e \<Rightarrow> assn"   and tsl :: "'d \<Rightarrow> 'e list Heap" shows "IS_TO_SORTED_LIST \<Omega> Rs Rk tsl \<equiv> (tsl, it_to_sorted_list \<Omega>) \<in> Rs\<^sup>k \<rightarrow>\<^sub>a list_assn Rk"
sgn_is_unit: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "sgn x dvd (1::'a)"
prems_9: shows "fmpred (\<lambda>_. vwelldefined') \<Gamma>"
in_cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> O[F](h)) = (g \<in> O[F](h))"
irreducible\<^sub>d_smultI: fixes c :: "'c"   and p :: "'c poly" assumes "c \<noteq> (0::'c)"   and "irreducible\<^sub>d (smult c p)" shows "irreducible\<^sub>d p"
distinct_nns: fixes kdt :: "'b kdt"   and ps :: "(real, 'b) vec list"   and n :: "nat"   and p :: "(real, 'b) vec" assumes "invar kdt"   and "distinct ps"   and "set ps \<inter> set_kdt kdt = {}" shows "distinct (nearest_nbors n ps p kdt)"
convex_real_interval_4: fixes b :: "real" shows "convex {..<b}"
UN_extend_simps_8: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set set" shows "(\<Union>y\<in>A. \<Union> (B ` y)) = \<Union> (B ` \<Union> A)"
overwrite: shows "\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"
tanh_0: shows "tanh (0::'a) = (0::'a)"
ennreal_suminf_cmult: fixes r :: "ennreal"   and f :: "nat \<Rightarrow> ennreal" shows "(\<Sum>i. r * f i) = r * (\<Sum>i. f i)"
succs_def: fixes v :: "nat" shows "succs v = g_E G `` {v}"
wcode_lemma_pre': fixes args :: "nat list"   and m :: "nat" assumes "args \<noteq> []" shows "\<exists>stp rn. steps0 (Suc 0, [], <m # args>) (t_wcode_prepare |+| t_wcode_main |+| t_wcode_adjust) stp = (0, [Bk], Oc \<up> Suc m @ Bk # Oc \<up> Suc (bl_bin (<args>)) @ Bk \<up> rn)"
sink_sum_stimcomm: fixes A :: "'a"   and B :: "'a"   and C :: "'a" assumes "\<exists>s t. s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, A) \<and> \<not> (t \<circ> B \<le>\<^sub>\<K> B + C \<and> t \<circ> C \<le>\<^sub>\<K> B + C)" shows "A \<rightarrow>\<^sub>\<S> B + C"
leg_simps_1: shows "C.arr H\<mu>H\<nu>\<pi>.leg0"
strategy_conforming_path_exists: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v0 \<in> V\<^bsub>G\<^esub>"   and "ParityGame.strategy G p \<sigma>"   and "ParityGame.strategy G p** \<sigma>'" obtains P :: "'c llist" where "vmc2_path G P v0 p \<sigma> \<sigma>'"
h_def: shows "h \<equiv> \<lambda>x. g x /\<^sub>R norm (g x)"
lq_reassoc: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v" shows "u\<inverse>\<^sup>>v \<cdot> w = u\<inverse>\<^sup>>(v \<cdot> w)"
path_component_of_sym_iff: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "path_component_of X x y = path_component_of X y x"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "max (max a b) c = max a (max b c)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "semilattice_set_apply f b (semilattice_set_apply f a c) = semilattice_set_apply f a (semilattice_set_apply f b c)"
x_facts_7: shows "x \<notin> Fvars \<tau>"
pcompose_pCons: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" shows "pCons a p \<circ>\<^sub>p q = [:a:] + q * p \<circ>\<^sub>p q"
pres_transf: fixes p :: "nat" shows "\<forall>ss\<in>A. p < n \<longrightarrow> (\<forall>x<length ss. ss ! x < p) \<longrightarrow> transf p ss \<in> A"
transp_on_converse: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A" shows "transp_on P\<inverse>\<inverse> A"
nonneg: fixes n :: "nat" shows "0 \<le> f n"
in_prime_factors_iff: 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 p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "(p \<in># factorial_semiring.prime_factorization zero times one normalize x) = (x \<noteq> zero \<and> dvd.dvd times p x \<and> normalization_semidom.prime zero times one normalize p)"