hcju/isabelleps huggingface dataset:
prems_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
hyps_6: fixes b :: "trm" assumes "e\<^sub>1 \<Down> b" shows "Lam [x].e\<^sub>1' = b"
WHILE_def: shows "aWHILE \<equiv> aWHILEI (\<lambda>_. True)"
carrier_reduced_homology_group_subset: fixes p :: "int"   and X :: "'a topology" shows "carrier (reduced_homology_group p X) \<subseteq> carrier (homology_group p X)"
ptd_lookup_unchanged_3: fixes va :: "word32"   and ptp :: "word32"   and s1 :: "('b, 'c) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" shows "ptd_lookup va ptp (mem s1) 3 = ptd_lookup va ptp ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) 3"
distrib: fixes g :: "'d \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "aggregation.sum_0 (\<lambda>x. g x + h x) A = aggregation.sum_0 g A + aggregation.sum_0 h A"
ldv': fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<cdot> (z \<sqinter> \<top> \<cdot> x) = y \<cdot> z \<sqinter> \<top> \<cdot> x"
pre_back_edge_def: fixes u :: "'v"   and v :: "'v"   and s0 :: "'s"   and s :: "'s" shows "pre_back_edge u v s0 s \<equiv> post_get_pending u (Some v) s0 s \<and> gds_is_discovered gds v s \<and> \<not> gds_is_finished gds v s"
min_ground_head_Sym: fixes f :: "'s" shows "min_ground_head (Sym f) = f"
Sync_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat"   and e2 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (sync\<^bsub>V\<^esub> (e) e2, xs) (sync\<^bsub>V\<^esub> (e') e2, xs')"
lt_dbm_le: fixes a :: "'c"   and b :: "'c" assumes "Lt a \<preceq> Le b" shows "a \<le> b"
span_empty: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module scale" shows "module.span scale {} = {0::'e}"
hyps_2: assumes "\<nexists>ev. ev \<in> set t \<and> regular_event ev \<and> occurs_on ev = p"   and "states c p = st" shows "states c' p = st"
valid_typeI_div: shows "valid_typeI_division s two_chain_typeI"
take_bit_mult: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "take_bit n (take_bit n k * take_bit n l) = take_bit n (k * l)"
\<beta>_def: shows "\<beta> = radiant_of (B - center)"
take_f_join_eq2: fixes I :: "nat set"   and xs :: "'b list"   and n :: "nat" assumes "card (I \<down>< length xs) \<le> n" shows "xs \<Join>\<^sub> I \<down> n = xs \<Join>\<^sub> I"
scaleQ_add_left: fixes a :: "'e"   and b :: "'e"   and x :: "'e['f, 'g]" shows "a + b *\<^sub>Q x = (a *\<^sub>Q x) + (b *\<^sub>Q x)"
that: fixes ct :: "'s \<Rightarrow> 's pmf" assumes "ct \<in> Pi\<^sub>E S K"   and "v \<circ> simple ct = F_sup (v \<circ> simple ct)" shows "thesis"
subspace_image: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "module_hom s1 s2 f"   and "module.subspace s1 S" shows "module.subspace s2 (f ` S)"
map_add_assoc: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and m3 :: "'a \<Rightarrow> 'b option" shows "m3 \<Oplus> m2 \<Oplus> m1 = m3 \<Oplus> (m2 \<Oplus> m1)"
cross3_simps_50: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "(a - b) * x = a * x - b * x"
get_exp_self_fulfills: fixes a :: "'a" assumes "a \<in> carrier G" shows "a = a [^] get_exp a a"
cblinfun_assoc_right_3: fixes a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and c :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "a + b + c = a + (b + c)"
sorted_Nil: fixes cmp :: "'b comparator" shows "Sorting_Algorithms.sorted cmp [] = True"
pref_pow_ext: fixes x :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "x \<le>p r \<^sup>@ k" shows "x \<le>p r \<^sup>@ Suc k"
groupF_code: fixes f :: "'c \<Rightarrow> 'd"   and as :: "'c list" shows "groupF f as = groupF_code f as"
poly_lift_hom_X_plus: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and a :: "'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "a \<in> carrier R" shows "poly_lift_hom R S \<phi> (X_poly_plus R a) = X_poly_plus S (\<phi> a)"
decl_eval\<^sub>w': fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and mem :: "'a \<Rightarrow> 'b"   and mds' :: "Mode \<Rightarrow> 'a set"   and upd :: "'a ModeUpd"   and mds :: "Mode \<Rightarrow> 'a set" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "mem' = mem"   and "mds' = sifum_lang_no_dma.update_modes upd mds" shows "(((Skip@[upd], mds), mem), (Stop, mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
cInf_mono: fixes B :: "'b set"   and A :: "'b set"   and c :: "'b" assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b"   and "\<And>a. a \<in> A \<Longrightarrow> c \<le> a"   and "B \<noteq> {}" shows "Inf A \<le> Inf B"
zero_in_radical_ideal: fixes F :: "'c set" shows "(0::'c) \<in> \<surd>ideal F"
intros_4: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and p :: "'f"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None (Some (p, True)) True n'" shows "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ [n'']) n'"
distinct_mset_mset_distinct: fixes xs :: "'a list" shows "distinct_mset (mset xs) = distinct xs"
simps: fixes cs :: "('i \<times> constraint) list"   and J :: "'i set"   and ds :: "'a ns_constraint list"   and s :: "'s" shows "checked_cs cs J (ds, s) = (ds = map snd (to_ns cs) \<and> checked_nsc (to_ns cs) J s)"
ConstP_Eats: fixes t1 :: "tm"   and t2 :: "tm" shows "{ConstP t1, ConstP t2} \<turnstile> ConstP (Q_Eats t1 t2)"
simps: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" shows "a1 \<rhd> a2 \<sim> a3 = (\<exists>P \<Psi> Q. a1 = \<Psi> \<and> a2 = P \<and> a3 = Q \<and> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi> \<and> \<Psi> \<rhd> P \<leadsto>[bisim] Q \<and> (\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q) \<and> \<Psi> \<rhd> Q \<sim> P)"
cmp_simps'_3: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "trg\<^sub>D (\<Phi> (\<mu>, \<nu>)) = map\<^sub>0 (trg\<^sub>C \<mu>)"
mds\<^sub>A'_def: shows "mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C'"
zcount_add_zmset: fixes b :: "'b"   and A :: "'b zmultiset"   and a :: "'b" shows "zcount (add_zmset b A) a = (if b = a then zcount A a + 1 else zcount A a)"
NSCauchy_Cauchy: fixes X :: "nat \<Rightarrow> 'a" assumes "NSCauchy X" shows "Cauchy X"
closed_map_const: fixes X :: "'c topology"   and Y :: "'d topology"   and c :: "'d" shows "closed_map X Y (\<lambda>x. c) = (topspace X = {} \<or> closedin Y {c})"
determ_maximalD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "determ t" shows "maximal t"
F2set1_IF2set: fixes x :: "('d, 'd IF1, 'd IF2) F2" shows "F2set1 x \<subseteq> IF2set (ctor2 x)"
fmrel_on_fset_refl_strong: fixes S :: "'c fset"   and m :: "('c, 'd) fmap"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x |\<in>| S; fmlookup m x = Some y\<rbrakk> \<Longrightarrow> P y y" shows "fmrel_on_fset S P m m"
typing_alt_simps_3: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "real"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal (RealLiteral c) : \<tau>) = (\<tau> = Real[1])"
sin_cos_eq: fixes a :: "real"   and b :: "real" assumes "cos a = cos b"   and "sin a = sin b" shows "\<exists>x. a - b = 2 * real_of_int x * pi"
span_minimal: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T"   and "vs2.subspace T" shows "vs2.span S \<subseteq> T"
sel_1: fixes x1 :: "object atom formula"   and x2 :: "object ast_effect" shows "ground_action.precondition (Ground_Action x1 x2) = x1"
ncoplanar_perm_15: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet C B D A"
solved: fixes v\<^sub>1 :: "'vertex"   and p\<^sub>1 :: "('form, 'var) out_port"   and v\<^sub>2 :: "'vertex"   and p\<^sub>2 :: "('form, 'var) antecedent" assumes "((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> edges" shows "labelAtOut v\<^sub>1 p\<^sub>1 = labelAtIn v\<^sub>2 p\<^sub>2"
iNext_def: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNext t0 I P \<equiv> P (inext t0 I)"
1: shows "\<S>\<bullet>\<C> \<turnstile> $* es : t1s _> t2s"
assms_2: fixes x :: "nat" assumes "x < Suc n" shows "f x \<down>= vs ! x"
insert_def: fixes x :: "'a \<times> 'b"   and xs :: "'x" shows "local.insert x xs = oalist_of_list (update_by_raw x (list_of_oalist xs))"
all_tran_formulas_validated: fixes F :: "'a formula"   and \<A> :: "'a \<Rightarrow> bool" shows "\<forall>J\<in>set (subformulae F). tseytin_update \<A> F \<Turnstile> J = tseytin_update \<A> F \<Turnstile> tseytin_toatom F J"
dual_neg: fixes x :: "'a" shows "x * \<top> \<sqinter> (x ^ o * \<bottom>) = \<bottom>"
trd_def: fixes cmp_term :: "(('a, 'b) pp \<times> 'c) nat_term_order"   and fs :: "(('a, 'b) pp \<times> 'c \<Rightarrow>\<^sub>0 'd) list"   and p :: "('a, 'b) pp \<times> 'c \<Rightarrow>\<^sub>0 'd" shows "trd_pprod cmp_term fs p = trd_aux_pprod cmp_term fs p 0"
RBT_Mapping_def: fixes t :: "('b, 'c) mapping_rbt" shows "RBT_Mapping t = Mapping (RBT_Mapping2.lookup t)"
join_related2: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> x" shows "x \<squnion> y = x"
simps_2: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" shows "timpls_transformable_to TI (Fun f T) (Fun g S) = ((f = g \<or> is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> set TI) \<and> list_all2 (timpls_transformable_to TI) T S)"
yivp_1: shows " D Y\<^sub>2 = (\<lambda>t. f t (Y\<^sub>2 t)) on U"
ltake_eSuc_LCons: fixes n :: "enat"   and x :: "'c"   and xs :: "'c llist" shows "ltake (eSuc n) (LCons x xs) = LCons x (ltake n xs)"
bind_def: fixes f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog" shows "f \<bind> g = Prog (\<lambda>h. case the_prog f h of Inl exception \<Rightarrow> Inl exception | Inr (x, h') \<Rightarrow> the_prog (g x) h')"
induced_automorphism_on_chamber_vertices_f: fixes C :: "'a set"   and v :: "'a" assumes "folding_g.chamber C"   and "v \<in> C" shows "\
 v = (if C \<in> f \<turnstile> folding_g.\<C> then g v else f v)"
qGood_qTerm_rawInduct: fixes phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>|{i. inp i \<noteq> None}| <o |UNIV|; |{i. binp i \<noteq> None}| <o |UNIV|; liftAll (\<lambda>X. qGood X \<and> phi X) inp; liftAll (\<lambda>A. qGoodAbs A \<and> phiAbs A) binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>qGood X; phi X\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "(qGood X \<longrightarrow> phi X) \<and> (qGoodAbs A \<longrightarrow> phiAbs A)"
finsert_fminus_if: fixes x :: "'b"   and A :: "'b fset"   and B :: "'b fset" shows "finsert x A |-| B = (if x |\<in>| B then A |-| B else finsert x (A |-| B))"
a_inv_closed: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> x \<in> carrier R"
Collect_case_prod_Sigma: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
ts_process_failures: fixes T :: "'a list set" assumes "trace_set T" shows "failures (ts_process T) = {(xs, X). xs \<in> T \<and> (\<forall>x\<in>X. xs @ [x] \<notin> T)}"
sing_pow: fixes i :: "nat"   and m :: "nat"   and a :: "'b" assumes "i < m" shows "([a] \<^sup>@ m) ! i = a"
Union: fixes S :: "'d set set"   and B :: "'d set"   and CC :: "'d set set"   and F :: "'d program" assumes "\<forall>A\<in>S. (A, B) \<in> elt CC F" shows "(\<Union> S, B) \<in> elt CC F"
RecvMarker: shows "ev' = RecvMarker i'' (occurs_on ev') s"
a1: shows "a * (1 - a\<^sup>2 - b\<^sup>2) = b"
d_approx: fixes ii :: "nat" assumes "ii < m" shows "rat_of_int (d fs ii) \<le> rat_of_nat (N ^ ii)"
HMA_elements_mat: shows "rel_fun HMA_M (=) elements_mat elements_mat_h"
rp_defs_6: fixes b :: "bool"   and b' :: "bool"   and P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "spec b b' P \<equiv> \<lambda>(A, A'). P (A\<lparr>wait := b'\<rparr>, A'\<lparr>ok := b\<rparr>)"
distinct_71: fixes x4 :: "'c set"   and x71 :: "'b"   and x72 :: "nat" shows "Abs x4 \<noteq> PubConstAtom x71 x72"
Z1_cyclic_group: shows "Finite_And_Cyclic_Groups.cyclic_group (Z 1) 0"
SIGT_plus2: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<sigma> :: "('b, 'c, 'a, 'd) subst_scheme"   and t1 :: "('b, 'e) trm"   and t2 :: "('b, 'e) trm" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (Plus t1 t2). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT t2. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
upd_incoming_def: fixes n :: "('b, 'd) node_scheme"   and ns :: "('b, 'c) node_scheme set" shows "upd_incoming n ns \<equiv> upd_incoming_f n ` ns"
simps_2: fixes callee :: "'i \<Rightarrow> 'j \<Rightarrow> ('h \<times> 'i) spmf"   and p :: "'i spmf"   and a :: "'j"   and b :: "'h"   and xs :: "('j \<times> 'h) list"   and x :: "'j" shows "trace_callee callee p ((a, b) # xs) x = trace_callee callee (cond_spmf_fst (p \<bind> (\<lambda>s. callee s a)) b) xs x"
N1_iff_N2: fixes A :: "('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step list" shows "set (N1 A) = N2 A"
replicate_prop: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and n :: "nat" assumes "P x" shows "\<forall>y\<in>set (replicate n x). P y"
zero_aezfun_transfer: shows "(0::'a) \<delta>\<delta> (0::'b) = 0"
inf_sup_absorb: fixes x :: "'a"   and y :: "'a" shows "inf_class.inf x (sup_class.sup x y) = x"
oreach_def: fixes S :: "('f, 'e) spec" shows "oreach S \<equiv> obs S ` reach S"
lookup_mult_vec: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.mult_vec pair_of_term term_of_pair p q) v = lookup (term_powerprod.proj_poly term_of_pair (term_powerprod.component_of_term pair_of_term v) p * term_powerprod.proj_poly term_of_pair (term_powerprod.component_of_term pair_of_term v) q) (term_powerprod.pp_of_term pair_of_term v)"
ih_12: shows "InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
I0: shows "I (dom (\<alpha> m)) \<sigma>0"
method_simp: fixes sig :: "sig"   and C :: "qtname"   and m :: "methd" shows "method sig (C, m) = (C, mdecl (sig, m))"
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A" shows "prod g (insert x A) = g x * prod g (A - {x})"
f3: shows "\<pi>\<^sub>* (\<pi>\<^sup>* (\<pi>\<^sub>* y)) \<in> carrier \<X>"
degree_Q: fixes x :: "nat \<Rightarrow> complex"   and n :: "nat" shows "Polynomial.degree (power_sum_puzzle.Q x n) = n"
vec_hom_inj: fixes v :: "'a vec"   and w :: "'a vec" assumes "vec\<^sub>h v = vec\<^sub>h w" shows "v = w"
alw_destR: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.alw \<phi> \<pi>l" shows "Shallow.alw \<phi> (map stl \<pi>l)"
fv_snd_rcv_strand_subset_2: fixes S :: "('b, 'a) strand_step list" shows "\<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<subseteq> fv\<^sub>s\<^sub>t S"
lookup_row_to_poly: fixes ts :: "'d list"   and r :: "'e vec"   and i :: "nat" assumes "distinct ts"   and "dim_vec r = length ts"   and "i < length ts" shows "lookup (row_to_poly ts r) (ts ! i) = r $ i"
fpxs_X_power_add: fixes a :: "rat"   and b :: "rat" shows "fpxs_X_power (a + b) = fpxs_X_power a * fpxs_X_power b"
Int_UNIV_left: fixes B :: "'c set" shows "UNIV \<inter> B = B"
assms_1: shows "e \<in> UNIV \<rightarrow> {- 1..1}"
weight_bind_spmf: fixes x :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "weight_spmf (x \<bind> f) = integral\<^sup>L (measure_spmf x) (weight_spmf \<circ> f)"
is_aligned_over_length: fixes p :: "'b word"   and n :: "nat" assumes "is_aligned p n"   and "LENGTH('b) \<le> n" shows "p = 0"
dist_fls_sym: fixes a :: "'a fls"   and b :: "'a fls" shows "dist a b = dist b a"
rem_inclusion: fixes B :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "B \<in> A .\<bottom>. \<phi>" shows "B \<subseteq> A"
assms_2: shows "comm_monoid_mult_class.coprime (prod_list ps) q"
C: shows "\<exists>t1 u1. t = t1 @ [tick] \<and> u = u1 @ [tick] \<and> s setinterleaves ((t1, u1), insert tick (ev ` A))"
fmdom_add: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmdom (m ++\<^sub>f n) = fmdom m |\<union>| fmdom n"
bot_less: fixes a :: "'a list" shows "(a \<noteq> []) = strict_prefix [] a"
funcs_agree: shows "list_all2 (cl_typing \<S>) (s.funcs s) (s_funcs \<S>)"
sum_rel_sv: fixes Rl :: "('a \<times> 'b) set"   and Rr :: "('c \<times> 'd) set" assumes "single_valued Rl"   and "single_valued Rr" shows "single_valued (\<langle>Rl, Rr\<rangle>sum_rel)"
eqd_pref1: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "x \<cdot> x\<inverse>\<^sup>>u = u"
prj_emb: fixes x :: "'b" shows "PRJ('a)\<cdot>(EMB('b)\<cdot>x) = COERCE('b, 'a)\<cdot>x"
sF': fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and ps :: "char list \<Rightarrow> int option"   and n :: "nat" shows "wp\<^sub>3\<^sub>' (WHILE b DO c) Q (ps, n) = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) (ps, n)"
simps_6: fixes \<phi> :: "Formula.formula" shows "Formula.future_bounded (formula.Neg \<phi>) = Formula.future_bounded \<phi>"
Equals_def: fixes \<theta> :: "trm"   and \<theta>' :: "trm" shows "Equals \<theta> \<theta>' = (Geq \<theta> \<theta>' && Geq \<theta>' \<theta>)"
combine_options_cases: fixes x :: "'b option"   and P :: "'b option \<Rightarrow> 'c option \<Rightarrow> bool"   and y :: "'c option" assumes "x = None \<Longrightarrow> P x y"   and "y = None \<Longrightarrow> P x y"   and "\<And>a b. \<lbrakk>x = \<lfloor>a\<rfloor>; y = \<lfloor>b\<rfloor>\<rbrakk> \<Longrightarrow> P x y" shows "P x y"
PPs_closed_lpp: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'x set" assumes "p \<in> P[X]" shows "lpp p \<in> .[X]"
power_0_left: fixes n :: "nat" shows "(0::'a) ^ n = (if n = 0 then 1::'a else (0::'a))"
that: fixes f :: "'a \<Rightarrow> nat" assumes "\<forall>x\<in>S. \<forall>y\<in>S. (f x < f y) = (x \<prec> y)"   and "range f \<subseteq> {0..card S - 1}" shows "thesis"
intros_7: fixes T1 :: "ty"   and T2 :: "ty" shows "T1 \<rightarrow> T2 <:: \<top>"
cong__le: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "A B Le C D"
sim: shows "strong_reduction_simulation (Rel\<inverse>) (STCal Source Target)"
min_gallery_betw_chamber_distance: fixes X :: "'b set set"   and C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.min_maxsimpchain X (C # Cs @ [D])" shows "ChamberComplex.chamber_distance X C D = Suc (length Cs)"
f_dvd_hc: shows "f dvd (\<Prod>c\<in>UNIV. h - [:c:])"
A: fixes y :: "'b" assumes "y \<in> Y" shows "dist y (f (g y)) \<le> 3 * C * lambda"
simps: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "('a \<times> 'd) list"   and ox :: "'o"   and ys :: "('a \<times> 'e) list \<times> 'o" shows "prod_ord_raw f (xs, ox) ys = prod_ord_pair ox f xs (sort_oalist_aux ox ys)"
inv_dghm_ArrMap_app: fixes a' :: "V"   and a :: "V" assumes "a' = \<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "inv_dghm \<FF>\<lparr>ArrMap\<rparr>\<lparr>a'\<rparr> = a"
simps: fixes g :: "game" shows "neg_game g = Game (zimage neg_game (right_options g)) (zimage neg_game (left_options g))"
Inverse_relE: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "Category C"   and "cinv\<^bsub>C\<^esub> f g" obtains "f \<approx>>\<^bsub>C\<^esub> g" and "f ;;\<^bsub>C\<^esub> g = id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f)" and "g ;;\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f)"
bitnot_zero: shows "(\<not>\<^sub>b \<zero>) = \<one>"
N_def: shows "N = (\<lambda>f. \<Sqinter> {eNorm N1 f1 + eNorm N2 f2 |f1 f2. f = f1 + f2})"
lemma_2_10_26: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a l\<rightarrow> b) r\<rightarrow> b"
i_Intersect: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set set" assumes "ring R"   and "\<And>I. I \<in> S \<Longrightarrow> ideal I R"   and "S \<noteq> {}" shows "ideal (\<Inter> S) R"
fib2: fixes n :: "nat" shows "fib (Suc (Suc n)) = fib (Suc n) + fib n"
nn_integral_fst: fixes f :: "'c \<times> 'b \<Rightarrow> ennreal"   and M1 :: "'c measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M2 \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
assms_4: shows "\<Gamma>',\<gamma>,p\<turnstile> \<langle>rs', Undecided\<rangle> \<Rightarrow> Undecided"
in_set_of_decomp: fixes p :: "ref"   and t :: "dag" shows "(p \<in> set_of t) = (\<exists>l r. t = Node l p r \<or> subdag t (Node l p r))"
same_fw_behaviour_def: fixes a :: "'b word"   and b :: "'b word"   and rs :: "'b simple_rule list" shows "same_fw_behaviour a b rs \<equiv> \<forall>p. simple_fw rs (p\<lparr>p_src := a\<rparr>) = simple_fw rs (p\<lparr>p_src := b\<rparr>) \<and> simple_fw rs (p\<lparr>p_dst := a\<rparr>) = simple_fw rs (p\<lparr>p_dst := b\<rparr>)"