hcju/isabelleps huggingface dataset:
facedist_chdist_mingal_btwmap: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and D :: "'b set"   and z :: "'b set"   and Cs :: "'b set list" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D"   and "order.greater_eq C z"   and "ChamberComplex.face_distance X (f ` z) (f ` D) = ChamberComplex.face_distance X z D"   and "ChamberComplex.face_distance X z D = ChamberComplex.chamber_distance X C D"   and "SimplicialComplex.min_maxsimpchain X (C # Cs @ [D])" shows "SimplicialComplex.min_maxsimpchain X (f \<Turnstile> (C # Cs @ [D]))"
intros: fixes n :: "'node"   and h :: "'outPort"   and p :: "'inPort" assumes "hyps n h = Some p" shows "hyps_for'p n p h"
flip_at_simps_1: fixes a :: "'e"   and b :: "'e" shows "(a \<leftrightarrow> b) \<bullet> a = b"
chine_hcomp_props_6: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"   and "span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu>" shows "C (prj1 (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>))) (span_bicategory.chine_hcomp C prj0 prj1 \<nu> \<mu>) = C (Chn \<nu>) (prj1 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>)))"
MTF_competitive2: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list" assumes "MTF_Off as rs s0"   and "s0 \<noteq> []"   and "\<forall>i<length rs. rs ! i \<in> set s0" shows "real (T_on MTF s0 rs) \<le> (2 - 1 / real (length s0)) * real (T s0 rs as)"
DotCallNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and call :: "'a call_expr"   and b :: "'a expr" assumes "\<Gamma> \<turnstile> Call src DotCall call \<Rrightarrow> b" obtains src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = Call src\<^sub>2__ DotCall call\<^sub>2__" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2__" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2__ : \<tau>__" and "\<tau>__ \<le> OclAny[?] \<or> \<tau>__ \<le> Tuple fmempty" and "(\<Gamma>, \<tau>__) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2__"   | src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = CollectIteratorCall src\<^sub>2 ArrowCall [new_vname \<Gamma>] (Some \<sigma>) (Call (Var (new_vname \<Gamma>)) DotCall call\<^sub>2)" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2 : \<tau>" and "element_type \<tau> \<sigma>" and "(\<Gamma>, \<sigma>) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2"
WWeakControlDependence_aux: fixes prog :: "cmd" shows "WeakControlDependencePDG sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (_Exit_)"
bisimT: shows "weak_reduction_bisimulation (indRelRTPO TRel) (STCal Source Target)"
assms_3: shows "u \<in> [M]\<^bsub>v,n\<^esub>"
proc\<^sub>2_correct: fixes p :: "prog"   and s\<^sub>L :: "stmt"   and s\<^sub>H :: "stmt" assumes "body_split p s\<^sub>L s\<^sub>H"   and "wfs \<Gamma>\<^sub>0 s\<^sub>L"   and "wfs \<Gamma>\<^sub>0 s\<^sub>H"   and "low_proc_non0\<^sub>2 s\<^sub>L"   and "low_proc_0\<^sub>2 s\<^sub>L"   and "low_proc_no_input_change s\<^sub>L"   and "high_proc\<^sub>2 s\<^sub>H"   and "high_proc_no_low_output_change s\<^sub>H" shows "low_proc_non0\<^sub>1 p \<and> low_proc_0\<^sub>1 p \<and> high_proc\<^sub>1 p"
not_choice_simps_4: fixes l :: "'d"   and fip :: "'a \<Rightarrow> nat"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp"   and q :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}unicast(fip, fmsg) . p \<triangleright> q)"
multi_member_this: fixes x :: "'b"   and XS :: "'b multiset" shows "x \<in># {#x#} + XS"
l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs ShadowRootClass.known_ptrs Shadow_DOM.get_parent"
All2_Eats_E1: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" assumes "atom i \<sharp> (t, u)"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (All2 i (Eats t u) A) H \<turnstile> All2 i t A"
locally_open_subset: fixes P :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and t :: "'a set" assumes "locally P S"   and "openin (top_of_set S) t" shows "locally P t"
set_interval_defs_4: fixes l :: "'a" shows "{l..} \<equiv> {x. l \<le> x}"
polyn_expr_tOpTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "pol_coeff S (n, f)" shows "\<forall>g. pol_coeff S (m, g) \<longrightarrow> (\<exists>h. pol_coeff S (n + m, h) \<and> h (n + m) = f n \<cdot>\<^sub>r\<^bsub>S\<^esub> g m \<and> polyn_expr R X (n + m) (n + m, h) = polyn_expr R X n (n, f) \<cdot>\<^sub>r polyn_expr R X m (m, g))"
permute_pure: fixes permute :: "perm \<Rightarrow> 'd \<Rightarrow> 'd"   and p :: "perm"   and x :: "'d" assumes "class.pure permute" shows "permute p x = x"
collapse_empty_empty: fixes s :: "nat \<Rightarrow> 'b" assumes "emptyseq (\<natural> s)" shows "emptyseq s"
inj_on_eq_iff: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and x :: "'f"   and y :: "'f" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
min_kp_def: fixes a :: "'d \<times> 'e"   and l :: "('f \<times> 'g \<times> 'd \<times> 'e) tree"   and r :: "('h \<times> 'i \<times> 'd \<times> 'e) tree" shows "min_kp a l r \<equiv> case (l, r) of (\<langle>\<rangle>, \<langle>\<rangle>) \<Rightarrow> a | (\<langle>\<rangle>, \<langle>x, (xa, xb, kpr), xc\<rangle>) \<Rightarrow> min2 a kpr | (\<langle>x, (xa, xb, kpl), xc\<rangle>, \<langle>\<rangle>) \<Rightarrow> min2 a kpl | (\<langle>x, (xa, xb, kpl), xc\<rangle>, \<langle>xd, (xe, xf, kpr), xg\<rangle>) \<Rightarrow> min2 a (min2 kpl kpr)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>gt ys xs x. ext gt ys xs \<Longrightarrow> ext gt (x # ys) (x # xs)" shows "ext_compat_cons_axioms ext"
div_power: 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 b :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b a" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
ccloses_subset: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and \<Delta> :: "(coname \<times> ty) list"   and \<Delta>' :: "(coname \<times> ty) list" assumes "\<theta>_c ccloses \<Delta>"   and "set \<Delta>' \<subseteq> set \<Delta>" shows "\<theta>_c ccloses \<Delta>'"
atomsInSequentI: fixes z :: "signs"   and P :: "predicate"   and vs :: "vbl list"   and ps :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "(z, P, vs) \<in> set (fst ps)" shows "FAtom z P vs \<in> set (sequent ps)"
fBall_ffilter: fixes X :: "'a fset"   and f :: "'a \<Rightarrow> bool" assumes "fBall X f" shows "ffilter f X = X"
holds_mono: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and Q :: "'b \<Rightarrow> bool" assumes "holds P xs"   and "\<And>x. P x \<Longrightarrow> Q x" shows "holds Q xs"
conf_RefTD: fixes G :: "prog"   and s :: "st"   and a' :: "val"   and T :: "ref_ty" assumes "G,s\<turnstile>a'\<Colon>\<preceq>RefT T" shows "a' = Null \<or> (\<exists>a obj T'. a' = Addr a \<and> heap s a = Some obj \<and> obj_ty obj = T' \<and> G\<turnstile>T'\<preceq>RefT T)"
whilefalse: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and \<iota> :: "nat"   and c :: "('b, 'c) MWLsCom" assumes "BMap (E b m) = False" shows "((while\<^bsub>\<iota>\<^esub> b do c od, m), [], None, m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
exec_gpv_oracle_bisim_bad_plossless: fixes bad2 :: "'b \<Rightarrow> bool"   and s2 :: "'b"   and X_bad :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'d"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and bad1 :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and oracle1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and oracle2 :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and gpv :: "('g, 'e, 'f) gpv" assumes "if bad2 s2 then X_bad s1 s2 else X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. \<lbrakk>X s1 s2; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>s2. bad2 s2 \<Longrightarrow> callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"   and "\<And>s1. bad1 s1 \<Longrightarrow> callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"   and "\<And>s1 x. \<lbrakk>bad1 s1; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. \<lbrakk>bad2 s2; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "plossless_gpv \<I> gpv"   and "\<And>s1. \<I> \<turnstile>c oracle1 s1 \<surd>"   and "\<And>s2. \<I> \<turnstile>c oracle2 s2 \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
\<nu>\<kappa>_proper: fixes x :: "\<nu>" shows "proper (x\<^sup>P)"
seq_maps_to_n: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "index_to_residue s k \<in> UNIV \<rightarrow> carrier (residue_ring (p ^ k))"
ltrm_coeff: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "coeff P (monom P (f (deg R f)) (deg R f)) n = (if n = deg R f then f (deg R f) else \<zero>)"
swap_cancel_1: fixes a :: "atom"   and b :: "atom" shows "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
polytope_imp_bounded: fixes S :: "'a set" assumes "polytope S" shows "bounded S"
ev: shows "\<forall>\<^sub>F z in nhds s. z \<in> {z. 1 < Re z}"
comp_diagram_functor: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and J' :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and F :: "'ai \<Rightarrow> 'ag" assumes "diagram J C D"   and "functor J' J F" shows "diagram J' C (D \<circ> F)"
rel_sum_sel: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'e + 'g"   and b :: "'f + 'h" shows "rel_sum R1 R2 a b = (isl a = isl b \<and> (isl a \<longrightarrow> isl b \<longrightarrow> R1 (projl a) (projl b)) \<and> (\<not> isl a \<longrightarrow> \<not> isl b \<longrightarrow> R2 (projr a) (projr b)))"
cball_sing: fixes e :: "real"   and x :: "'a" assumes "e = 0" shows "cball x e = {x}"
step_defNode: fixes v :: "'val" assumes "v \<in> allVars g"   and "v \<noteq> chooseNext g" shows "step.defNode g v = defNode g v"
constructive_security_sim_obsf_axioms: shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv"
in_rel_Collect_case_prod_eq: fixes X :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "in_rel {(x, y). X x y} = X"
mult_neg_numeral_simps_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and m :: "num"   and n :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus (numeral.numeral one plus m)) (uminus (numeral.numeral one plus n)) = numeral.numeral one plus (m * n)"
monomials_lowerPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'b mpoly" shows "monomials (lowerPoly b i mp) = higherPowers b i ` (monomials mp \<inter> {x. \<forall>j\<in>{b..<b + i}. lookup x j = 0})"
dec_word_stream_enc: fixes w :: "'a list"   and I :: "(nat + nat set) list" shows "dec_word (stream_enc (w, I)) = cut_same any (w @- sconst any)"
meet_comm: fixes l :: "Lines"   and m :: "Lines"   and P :: "Points" assumes "meet_in l m P" shows "meet_in m l P"
abs_summable_on_cmult_left: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "c \<noteq> (0::'a) \<Longrightarrow> f abs_summable_on A" shows "(\<lambda>x. f x * c) abs_summable_on A"
pdevs_val_coll: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list"   and f :: "nat \<Rightarrow> real" assumes "list_all (coll 0 x) xs"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"   and "x \<noteq> 0"   and "f \<in> UNIV \<rightarrow> {- 1..1}" obtains e :: "real" where "e \<in> {- 1..1}" and "pdevs_val f (pdevs_of_list xs) = e *\<^sub>R sum_list xs"
MP_Rel_def: fixes f :: "int poly"   and f' :: "'a mod_ring poly" shows "MP_Rel f f' \<equiv> Mp f = to_int_poly f'"
assms_2: shows "(\<box>(\<lambda>s. taken s \<longrightarrow> taken' s)) \<sigma>"
\<theta>_le_\<psi>: fixes x :: "real" shows "\<theta> x \<le> \<psi> x"
new_keys_not_used: fixes evs :: "event list"   and K :: "nat" assumes "evs \<in> set_mr"   and "Key K \<notin> used evs"   and "K \<in> symKeys" shows "K \<notin> keysFor (parts (knows Spy evs))"
seq_if_composable: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "S.hcomp \<nu> \<mu> \<noteq> S.null" shows "S.src \<nu> = S.trg \<mu>"
order_iff: fixes b :: "'t"   and a :: "'t" shows "(b \<preceq>\<^sub>t a) = (a = ord_term_lin.max a b)"
compact_attains_inf: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.linorder_topology less_eq less open"   and "topological_space.compact open S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. \<forall>t\<in>S. less_eq s t"
maps_to_n_infinite_seq: fixes k :: "nat"   and f :: "nat \<Rightarrow> int" assumes "maps_to_n (int k) f" shows "\<exists>l. \<forall>m. \<exists>n>m. f n = l"
bindE: fixes R :: "'b Predicate.pred"   and Q :: "'b \<Rightarrow> 'a Predicate.pred"   and y :: "'a" assumes "pred.eval (Predicate.bind R Q) y" obtains x :: "'b" where "pred.eval R x" and "pred.eval (Q x) y"
zero: assumes "vectorspace V" shows "f (0::'a) = 0"
hcomp_in_hhomE: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and c :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hhom (\<nu> \<circ> \<mu>) a c" obtains "Span.arr \<mu>" and "Span.arr \<nu>" and "Span.src \<nu> = Span.trg \<mu>" and "Span.src \<mu> = a" and "Span.trg \<nu> = c"
simps_2: fixes X :: "char list set" shows "m\<^sub>o None X = h * card X + 1"
if_split_fmem1: fixes Q :: "bool"   and x :: "'x"   and y :: "'x"   and b :: "'x fset" shows "((if Q then x else y) |\<in>| b) = ((Q \<longrightarrow> x |\<in>| b) \<and> (\<not> Q \<longrightarrow> y |\<in>| b))"
Cons_mem_set_subseqs_Cons: fixes y :: "'c"   and ys :: "'c list"   and x :: "'c"   and xs :: "'c list" shows "(y # ys \<in> set (subseqs (x # xs))) = (y = x \<and> ys \<in> set (subseqs xs) \<or> y # ys \<in> set (subseqs xs))"
bin_last_bl_to_bin: fixes bs :: "bool list" shows "odd (bl_to_bin bs) = (bs \<noteq> [] \<and> last bs)"
approx_bin_def: fixes p :: "nat"   and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and rd :: "(real \<times> real pdevs) \<times> real"   and sd :: "(real \<times> real pdevs) \<times> real" shows "approx_bin p f rd sd = f (ivl_of_aform_err p rd) (ivl_of_aform_err p sd) \<bind> (\<lambda>ivl. Some (ivl_err (real_interval ivl)))"
node_in_scc_of_node: fixes v :: "'b"   and E :: "('b \<times> 'b) set" shows "v \<in> scc_of E v"
close_fresh': shows "\<forall>(x, y)\<in>set pi. x \<sharp> close \<Gamma> T\<^sub>1 \<and> y \<sharp> close \<Gamma> T\<^sub>1"
swap_def: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" shows "swap xs i j = xs[i := xs ! j, j := xs ! i]"
not_irreducible_one: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.irreducible times one zero one"
all: fixes pats :: "Term_Class.term list"   and rhs :: "'a" assumes "(pats, rhs) |\<in>| rs" shows "pats = []"
map_iterator_abs_genord_rule_insert_P: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and m :: "'h \<Rightarrow> 'f option"   and R :: "'h \<times> 'f \<Rightarrow> 'h \<times> 'f \<Rightarrow> bool"   and I :: "'h set \<Rightarrow> 'g \<Rightarrow> bool"   and \<sigma>0 :: "'g"   and c :: "'g \<Rightarrow> bool"   and f :: "'h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'g \<Rightarrow> bool" assumes "map_iterator_abs_genord \<alpha> invar iti m R"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; invar v; m k = Some (\<alpha> v); it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> dom m - it - {k} \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k, \<alpha> v) (k', \<alpha> v'); \<forall>k' v'. k' \<in> it \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k', \<alpha> v') (k, \<alpha> v)\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<in> it \<and> invar v \<and> m k = Some (\<alpha> v) \<and> k' \<notin> it \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k, \<alpha> v) (k', \<alpha> v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
strongly_preferred_def: fixes a :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a" shows "a \<prec>[R] b \<equiv> R a b \<and> \<not> R b a"
int_simps_14: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> #False) = (\<not> P)"
in_set_merge_coeffsD: fixes a :: "char list list"   and b :: "int"   and p :: "(char list list \<times> int) list" assumes "(a, b) \<in> set (merge_coeffs p)" shows "\<exists>b. (a, b) \<in> set p"
2: shows "T\<^bsub>a\<^esub> f = lin_part (T\<^bsub>a\<^esub> f) \<oplus>\<^bsub>P\<^esub> shift 2 (T\<^bsub>a\<^esub> f) \<otimes>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> 2"
lfp_const: fixes t :: "'c" shows "lfp (\<lambda>x. t) = t"
lt_Integer_bottom_iff: fixes x :: "Integer"   and y :: "Integer" shows "(lt\<cdot>x\<cdot>y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
sup_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "sup x y = (\<chi>i. sup (x $ i) (y $ i))"
tiny_dg_Dom_vrange: shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
bij_image_INT: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'c set"   and A :: "'e set" assumes "bij f" shows "f ` \<Inter> (B ` A) = (\<Inter>x\<in>A. f ` B x)"
rvars_of_lvar_rvars: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "EqForLVar eq_idx_for_lvar"   and "x \<in> lvars t" shows "rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar t x) \<subseteq> rvars t"
sorted_map: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f xs) = sorted_wrt (\<lambda>x y. less_eq (f x) (f y)) xs"
mapping_of: fixes x :: "'q mpoly" shows "mapping_of x \<in> UNIV"
dagger_unfoldr_distr: fixes y :: "'a"   and x :: "'a" shows "y + x\<^sup>\<star> \<cdot> x \<cdot> y = x\<^sup>\<star> \<cdot> y"
intros: fixes x :: "'c"   and r :: "('c \<times> 'c) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
mergeable_envs_Ex: fixes n :: "nat"   and S :: "'c list set"   and \<alpha> :: "'d MFOTL.formula"   and \<beta> :: "'e MFOTL.formula"   and v :: "'c list" assumes "mergeable_envs n S"   and "MFOTL.nfv \<alpha> \<le> n"   and "MFOTL.nfv \<beta> \<le> n"   and "\<exists>v'\<in>S. \<forall>x\<in>fv \<alpha>. v' ! x = v ! x"   and "\<exists>v'\<in>S. \<forall>x\<in>fv \<beta>. v' ! x = v ! x" shows "\<exists>v'\<in>S. \<forall>x\<in>fv \<alpha> \<union> fv \<beta>. v' ! x = v ! x"
rcos_self: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and H :: "'a set" assumes "Group.group G"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> H #>\<^bsub>G\<^esub> x"
R: shows "range w \<subseteq> set (map set (subseqs (atoms_list (ltlc_to_ltln \<phi>\<^sub>c))))"
acyclic_5d_def: fixes x :: "'a" shows "acyclic_5d x \<equiv> \<forall>y\<le>x. coreflexive (y\<^sup>+ \<sqinter> (x \<sqinter> - y)\<^sup>+)"
u_simps_4: shows "local.dom s\<^sub>0 = s\<^sub>0"
MN_Rel_Derivs: fixes x :: "'b list"   and y :: "'b list"   and A :: "'b list set" shows "x \<approx>A y = (Derivs x A = Derivs y A)"
nn_integral_lookup_default: fixes m :: "('b, real) mapping" assumes "finite (Mapping.keys m)"   and "All_mapping m (\<lambda>_. (\<le>) 0)" shows "\<integral>\<^sup>+ k. ennreal (Mapping.lookup_default 0 m k) \<partial>count_space UNIV = ennreal (sum (Mapping.lookup_default 0 m) (Mapping.keys m))"
prts: fixes a :: "'a" shows "a = pprt a + nprt a"
cring_simprules_16: fixes x :: "int" shows "x + 0 = x"
Cons_prefix_Cons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
bounded: shows "- pi < arg x / real n \<and> arg x / real n \<le> pi"