hcju/isabelleps huggingface dataset:
bisim_simple_\<R>\<^sub>1: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and c :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "(((c, mds), mem), (c', mds'), mem') \<in> sifum_types.\<R>\<^sub>1 ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P" shows "c = c'"
retractionE: fixes g :: "'a \<times> 'a" assumes "VV.retraction g" obtains f :: "'a \<times> 'a" where "VV.ide (VV.comp g f)"
relcomp_distrib2: fixes S :: "('a \<times> 'c) set"   and T :: "('a \<times> 'c) set"   and R :: "('c \<times> 'b) set" shows "(S \<union> T) O R = S O R \<union> T O R"
a1: shows "low_equal s1 s2 \<and> Some t1 = store_word_mem s1 addr data bm 10 \<and> Some t2 = store_word_mem s2 addr data bm 10"
p_residues: fixes m :: "nat" assumes "0 < m" shows "residues (p ^ m)"
dsjAssoc2: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> (A OR B) OR C" shows "H \<turnstile> A OR B OR C"
iT_Minus_cut_ge2: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" shows "(k \<ominus> I) \<down>\<ge> (k - t) = k \<ominus> I \<down>\<le> t"
assms_2: shows "\<Psi>'' \<simeq> \<Psi>'''"
vec_carrier_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "(vec n f \<in> carrier_vec m) = (n = m)"
pmf_eq_0_set_pmf: fixes M :: "'b pmf"   and p :: "'b" shows "(pmf M p = 0) = (p \<notin> set_pmf M)"
rep_eq: shows "semilattice_set_apply inf_sls = inf"
label_incr_simp: fixes n :: "w_node"   and i :: "nat"   and m :: "w_node"   and j :: "nat" assumes "n \<oplus> i = m \<oplus> i + j" shows "n = m \<oplus> j"
fact_eq: fixes m :: "nat" assumes "Suc m < p" shows "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
pause_option_def: fixes pause :: "'k \<Rightarrow> ('l \<Rightarrow> 'j) \<Rightarrow> 'j"   and out :: "'k"   and c :: "'l \<Rightarrow> ('i, 'j) optionT" shows "pause_option pause out c = OptionT (pause out (\<lambda>i. run_option (c i)))"
RHT_Ch_card: fixes h :: "'c" shows "card (Ch h (fp_cop_F ds)) \<le> card (Ch h X)"
natUnion_shift: fixes C :: "nat \<Rightarrow> 'b set"   and m :: "nat" assumes "chain C" shows "natUnion (\<lambda>n. C (n + m)) = natUnion C"
least_mem: fixes L :: "('c, 'd) gorder_scheme"   and l :: "'c"   and A :: "'c set" assumes "least L l A" shows "l \<in> A"
and_not_mask_twice: fixes w :: "'a word"   and n :: "nat"   and m :: "nat" shows "(w && ~~ mask n) && ~~ mask m = w && ~~ mask (max m n)"
neg_numeral_neq_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "uminus (numeral.numeral one plus m) \<noteq> numeral.numeral one plus n"
obs_consistent_med11: shows "obs_consistent R11 med11 l1 skl1"
inv_naturality: fixes f :: "'a" assumes "arr f" shows "(\<I> \<otimes> f) \<cdot> local.inv (\<ll> (local.dom f)) = local.inv (\<ll> (cod f)) \<cdot> local.map f"
underS_E: fixes i :: "'a"   and R :: "('a \<times> 'a) set"   and j :: "'a" assumes "i \<in> underS R j" shows "i \<noteq> j \<and> (i, j) \<in> R"
ospec: fixes A :: "'a option"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "! x:A: P x"   and "A = Some x" shows "P x"
Alist_impl_of: fixes xs :: "('a, 'b) alist" shows "Alist (alist.impl_of xs) = xs"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) 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"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent_child_rel"   and "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf"   and "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs type_wf"   and "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"   and "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name heap_is_wellformed" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed 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"
radical_nonzero: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "radical x = \<Prod>(prime_factors x)"
assms_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "LINT x|M. indicat_real A x * g x = LINT x|M. indicat_real A (T x) * f x"
DynPDG_path_ddep: fixes n :: "'node"   and V :: "'var"   and as :: "'edge list"   and n' :: "'node" assumes "n -{V}as\<rightarrow>\<^sub>d\<^sub>d n'" shows "n -as\<rightarrow>\<^sub>d* n'"
1: shows "u \<^bold>\<Down> v \<in> VHom (u \<^bold>\<star> v) \<^bold>\<lfloor>u \<^bold>\<star> v\<^bold>\<rfloor>"
assms_4: fixes l1 :: "'a list"   and l2 :: "'a list"   and l3 :: "'a list" assumes "x \<noteq> y"   and "l = l1 @ x # l2 @ y # l3" shows "P"
pivot_vars': fixes s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar_code (\<T> s) x\<^sub>i)" shows "let s' = pivot_code x\<^sub>i x\<^sub>j s in rvars (\<T> s') = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i} \<and> lvars (\<T> s') = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"
ide_char: fixes a :: "'a \<times> 'a \<times> 'a" shows "VVV.ide a = (VVV.arr a \<and> VxVxV.ide a)"
emeasure_eq_ennreal_measure: fixes M :: "'a measure"   and A :: "'a set" assumes "emeasure M A \<noteq> \<top>" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
surjective: fixes r :: "('c, 'd) strips_operator_scheme" shows "r = \<lparr>strips_operator.precondition_of = strips_operator.precondition_of r, add_effects_of = add_effects_of r, delete_effects_of = delete_effects_of r, \<dots> = strips_operator.more r\<rparr>"
pred_of_rel_def: fixes s :: "('c \<times> 'd) set" shows "pred_of_rel s \<equiv> \<lambda>a b. (a, b) \<in> s"
assms: shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
lres_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> z) = (x \<le> z / y)"
mod_add: fixes a :: "nat"   and k :: "nat"   and m :: "nat"   and b :: "nat" shows "((a + k) mod m = (b + k) mod m) = (a mod m = b mod m)"
permutes_convert_perm: fixes \<sigma> :: "nat \<Rightarrow> nat" assumes "\<sigma> permutes {..<n}" shows "convert_perm \<sigma> permutes Roots"
assoc_2: fixes shows_prec :: "nat \<Rightarrow> 'c \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'c list \<Rightarrow> char list \<Rightarrow> char list"   and xs :: "'c list"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_list xs r @ s = shows_list xs (r @ s)"
assms: shows "u \<in> \<langle>G\<rangle>"
affine: fixes V :: "'b set" shows "affine V = (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
Cauchy_convergent_iff: fixes X :: "nat \<Rightarrow> 'd" shows "Cauchy X = convergent X"
bex_Edges: fixes P :: "'s \<times> ('c, real) cconstraint \<times> ('c set \<times> 's) pmf \<times> 'c set \<times> 's \<Rightarrow> bool" shows "(\<exists>x\<in>Edges A. P x) = (\<exists>l g \<mu> X l'. A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l' \<and> P (l, g, \<mu>, X, l'))"
simps_2: fixes i :: "'i"   and x :: "nat"   and c :: "'a"   and s :: "('i, 'a) state" shows "assert_bound (i, Geq x c) s = assert_bound' Negative i x c s"
simps_2: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "lift f (Inner x, Inner y) = f (x, y)"
card_of_leq: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set" assumes "A \<subseteq> S.Univ" shows "|A| <o cardSuc (cmax |UNIV| natLeq)"
v_e_nz_1: shows "0 < real (card (uverts H))"
low_mds_eq_def: fixes mds :: "Mode \<Rightarrow> Eg1.var set"   and mem\<^sub>1 :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>2 :: "Eg1.var \<Rightarrow> nat" shows "A.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. dma mem\<^sub>1 x = Low \<and> (x \<in> \<C> \<or> x \<notin> mds AsmNoReadOrWrite) \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
scaleR_diff_right: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
simps_15: fixes e :: "expr" shows "fv (throw e) = fv e"
group_l_invI: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.monoid G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
cop_F_RH_mono: fixes x :: "'a" assumes "Xd x \<notin> Xd ` Ch (Xh x) fp"   and "x \<in> Rf CH fp" shows "x \<in> Rf CH (cop_F ds fp)"
expY: shows "distributed M lborel Y (\<lambda>x. ennreal (exponential_density u x))"
disj_zero_left: fixes x :: "'a" shows "\<zero> \<^bold>\<squnion> x = x"
WRP_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> WRP t u IFF WRP t' u'"
sset_induct: fixes y :: "'d"   and s :: "'d stream"   and P :: "'d \<Rightarrow> 'd stream \<Rightarrow> bool" assumes "y \<in> sset s"   and "\<And>s. P (shd s) s"   and "\<And>s y. \<lbrakk>y \<in> sset (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s" shows "P y s"
semIntAll_prCons: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prCons (semInt SEM) (semIntAbs SEM) SEM"
order_iff_strict: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "(a \<preceq> b) = (a \<prec> b \<or> a = b)"
i': fixes k :: "nat" shows "integral (cbox a b) (f k) \<le> i"
wlp_is_weakest_liberal_precondition: fixes S :: "com"   and P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "is_weakest_liberal_precondition (wlp S P) S P"
base: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "rtrancl_tab r xs x x"
neg_diag_empty_spec: fixes i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
closedin_compact_space: fixes X :: "'c topology"   and S :: "'c set" assumes "compact_space X"   and "closedin X S" shows "compactin X S"
Range_snd: fixes r :: "('d \<times> 'c) set" shows "Range r = snd ` r"
ss: shows "(a, [(\<alpha>, b)]) \<in> seq ars"
Digamma_of_nat: fixes n :: "nat" shows "Digamma (of_nat (Suc n)) = harm n - euler_mascheroni"
le_cases3: fixes x :: "'c word"   and y :: "'c word"   and z :: "'c word" obtains "x \<le>s y" and "y \<le>s z"   | "y \<le>s x" and "x \<le>s z"   | "x \<le>s z" and "z \<le>s y"   | "z \<le>s y" and "y \<le>s x"   | "y \<le>s z" and "z \<le>s x"   | "z \<le>s x" and "x \<le>s y"
properfactor_mult_imp_properfactor: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "properfactor (Ring_Divisibility.mult_of (K [X])) b a" shows "properfactor (K [X]) b a"
monotonic: fixes P :: "pi"   and A :: "(pi \<times> pi) set"   and P' :: "pi"   and B :: "(pi \<times> pi) set" assumes "P \<leadsto>[A] P'"   and "A \<subseteq> B" shows "P \<leadsto>[B] P'"
maxsimpchain_def: fixes xs :: "'a set list" shows "maxsimpchain xs \<equiv> Ball (set xs) maxsimp \<and> adjacentchain xs"
split_scaleC_pos_le: fixes a :: "complex"   and b :: "'b" assumes "0 \<le> a \<and> (0::'b) \<le> b \<or> a \<le> 0 \<and> b \<le> (0::'b)" shows "(0::'b) \<le> a *\<^sub>C b"
K_imply_member: fixes p :: "'b fm"   and ps :: "'b fm list" assumes "p \<in> set ps" shows "\<turnstile> imply ps p"
simps_2: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "quadratic_sub var a b c d (Less p) = (let p1 = quadratic_part_1 var a b d (Less p); p2 = quadratic_part_2 var c p1; A = isolate_variable_sparse p2 var 0; B = isolate_variable_sparse p2 var 1 in Or (And (fm.Atom (Less A)) (fm.Atom (Less (B\<^sup>2 * c - A\<^sup>2)))) (And (fm.Atom (Leq B)) (Or (fm.Atom (Less A)) (fm.Atom (Less (A\<^sup>2 - B\<^sup>2 * c))))))"
homeomorphic_rel_interiors_same_dimension: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "aff_dim S = aff_dim T" shows "rel_interior S homeomorphic rel_interior T"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) L"
atLeast_iff: fixes i :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c" shows "(i \<in> ord.atLeast less_eq k) = less_eq k i"
fermat_theorem_power_poly_R: fixes a :: "'a mod_ring"   and n :: "nat" shows "[:a:] [^]\<^bsub>R\<^esub> CARD('a) ^ n = [:a:]"
primitive_primitive_extension: assumes "\<Phi> \<noteq> principal_dchar c" shows "primitive_dchar \<Phi>.conductor \<Phi>.primitive_extension"
assms: shows "run_one_step d i (s, vs, ves, $Relop_i x291 x292) = (s', vs', res)"
domain_subset_carrier: fixes c :: "('c, 'd) chart"   and charts :: "('c, 'd) chart set" assumes "c \<in> charts" shows "domain c \<subseteq> manifold.carrier charts"
path_entry_append: fixes E :: "(nat \<times> nat) set"   and l :: "nat list"   and v :: "nat"   and w :: "nat" assumes "path_entry E l v"   and "(v, w) \<in> E" shows "path_entry E (v # l) w"
assign_Some: fixes f :: "'a \<Rightarrow> state \<Rightarrow> state"   and v :: "'a"   and x :: "abrupt"   and s :: "st" shows "assign f v (Some x, s) = (Some x, s)"
fact_double: fixes n :: "nat" shows "fact (2 * n) = (2::'b) ^ (2 * n) * pochhammer ((1::'b) / (2::'b)) n * fact n"
simps_3: fixes xs :: "('c, 'd) bintree option list"   and y :: "('c, 'd) bintree option"   and ys :: "('c, 'd) bintree option list" shows "meld (None # xs) (y # ys) = y # meld xs ys"
lsc_sequentially_gen: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (\<forall>x c c0. x \<longlonglongrightarrow> x0 \<and> c \<longlonglongrightarrow> c0 \<and> (\<forall>n. f (x n) \<le> c n) \<longrightarrow> f x0 \<le> c0)"
cINF_lower: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_below less_eq (f ` A)"   and "x \<in> A" shows "less_eq (Inf (f ` A)) (f x)"
V_Cons: fixes \<omega> :: "('e \<times> 'f) stream"   and a :: "'e"   and as :: "'e list"   and o :: "'f"   and os :: "'f list" shows "(\<exists>\<omega>'. \<omega> = zip (a # as) (o # os) @- \<omega>') = (fst (shd \<omega>) = a \<and> snd (shd \<omega>) = o \<and> (\<exists>\<omega>'. stl \<omega> = zip as os @- \<omega>'))"
simps_1: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "tMap f (tInit x) = tInit (f x)"
induct: fixes xa :: "'a"   and K :: "'a set"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "xa \<in> simple_extension K x"   and "P \<zero>"   and "\<And>k1 k2. \<lbrakk>k1 \<in> simple_extension K x; P k1; k2 \<in> K\<rbrakk> \<Longrightarrow> P (k1 \<otimes> x \<oplus> k2)" shows "P xa"
is_Inl_def: shows "is_Inl \<equiv> \<lambda>uu. case uu of Inl uu_ \<Rightarrow> True | Inr uv_ \<Rightarrow> False"
bad: fixes s' :: "'s"   and x :: "'a" assumes "I s'"   and "\<not> bad s'"   and "count s' < n + count s"   and "consider x"   and "x \<in> outs_\<I> \<I>" shows "spmf (map_spmf (bad \<circ> snd) (callee s' x)) True \<le> k"
star_n_folk: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "!z \<cdot> x = x \<cdot> !z"   and "!z \<cdot> y = y \<cdot> !z" shows "(!z \<cdot> x + t z \<cdot> y)\<^sup>\<star> \<cdot> !z = !z \<cdot> (!z \<cdot> x)\<^sup>\<star>"
lookup_map_of: fixes t :: "('c \<times> 'd) tree"   and x :: "'c" assumes "sorted1 (inorder t)" shows "lookup t x = AList_Upd_Del.map_of (inorder t) x"
call_only_one_intra_edge: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and a :: "'g"   and Q :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and r :: "'k"   and p :: "'l"   and fs :: "(('i \<Rightarrow> 'j option) \<Rightarrow> 'j option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<exists>!a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<and> intra_kind (kind a')"
other_member: fixes I :: "'e set"   and i :: "'e" assumes "I \<noteq> {}"   and "i \<in> I"   and "card I \<noteq> 1" obtains j :: "'e" where "j \<in> I" and "j \<noteq> i"
rel_S_def: fixes L1 :: "'z \<Rightarrow> 'ag \<Rightarrow> bool"   and L2 :: "'aa \<Rightarrow> 'ah \<Rightarrow> bool"   and Co1 :: "'ab \<Rightarrow> 'ai \<Rightarrow> bool"   and Co2 :: "'ac \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra1 :: "'ad \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra2 :: "'ae \<Rightarrow> 'al \<Rightarrow> bool" shows "rel_S L1 L2 Co1 Co2 Contra1 Contra2 = vimage2p Rep_S Rep_S (rel_G L1 L2 Co1 Co2 Contra1 Contra2)"
derivation_ge_cons: fixes d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" shows "derivation_ge (d # D) i = (i \<le> fst d \<and> derivation_ge D i)"
Hset2_alt: shows "Hset2 = \<Union> \<circ> BNF_Def.collect {(`) F1set2 \<circ> Gset1, (`) F2set2 \<circ> Gset2, (`) F3set2 \<circ> Gset3}"
base: fixes x :: "variable" assumes "x \<notin> BVG \<alpha>" shows "{x} \<in> Bignorabimus \<alpha>"
pathVertices_fwd_simps_5: fixes s :: "nat"   and p :: "(nat \<times> nat) list" shows "s \<in> set (pathVertices_fwd s p)"
integrable_real_indicator: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "integrable M (indicat_real A)"
Sigma_mono: fixes A :: "'c set"   and C :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and D :: "'c \<Rightarrow> 'd set" assumes "A \<subseteq> C"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x" shows "Sigma A B \<subseteq> Sigma C D"
Unifier_support: fixes t :: "('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and u :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "Unifier \<delta> t u"   and "\<delta> supports \<theta>" shows "Unifier \<theta> t u"
count_roots_interval_sf_rat: fixes p :: "int poly" assumes "p \<noteq> 0" shows "root_info_cond (Sturm_Rat.count_roots_interval_sf_rat p) p"
Abs_fmap_inverse: fixes y :: "'d \<Rightarrow> 'e option" assumes "y \<in> {m. finite (dom m)}" shows "fmlookup (Abs_fmap y) = y"
g_pos: fixes t :: "real" assumes "t \<in> {a..b}" shows "0 < g t"
zero_in_verts: shows "0 \<in> set (g_V G)"
le: shows "\<forall>\<^sub>F z in at_left 1. p x y n * p y x m * z ^ (n + m) \<le> (1 - gf_U y y z) / (1 - gf_U x x z)"
presSubstAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presSubstAbs h hA MOD \<equiv> \<forall>ys Y y us s A. wls (asSort ys) Y \<and> wlsAbs (us, s) A \<longrightarrow> hA (A $[Y / y]_ys) = gSubstAbs MOD ys Y (h Y) y A (hA A)"
WCR: shows "WCR_on r {x. SN_on r {x}}"
subcomplex_gallery: fixes X :: "'b set set"   and Y :: "'b set set"   and Cs :: "'b set list" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y"   and "SimplicialComplex.maxsimpchain Y Cs" shows "SimplicialComplex.maxsimpchain X Cs"
that: fixes f' :: "'a \<Rightarrow> 'e" assumes "diff_fun k charts f'"   and "\<And>x. x \<in> B \<Longrightarrow> f' x = f x"   and "csupport_on carrier f' \<inter> carrier \<subseteq> sub.carrier" shows "thesis"
Hilbert_fun_alt: fixes X :: "'x set"   and A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and z :: "nat" assumes "finite X"   and "A \<subseteq> P[X]"   and "phull.subspace A" shows "Hilbert_fun A z = card (lpp ` (hom_deg_set z A - {0}))"
f1'_def: shows "f1' = min (f1 * c) (f2 / e)"
set_of_neg_inc: fixes A :: "'a interval"   and A' :: "'a interval" assumes "set_of A \<subseteq> set_of A'" shows "set_of (- A) \<subseteq> set_of (- A')"
order_equiv_trans: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "attractive A less_eq"   and "less_eq x y"   and "equivpartp less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"