hcju/isabelleps huggingface dataset:
cnt_lt: shows "count (image_mset f M) (f a) + count N a < count (image_mset f N) (f a) + count M a"
alist_add_Cons: fixes x :: "('d, 'c) bintree option"   and xs :: "('d, 'c) bintree option list" assumes "distinct (vals (x # xs))" shows "set (elements (add x xs)) = set (elements (x # xs))"
eq_invI: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bijection f"   and "inv f a = inv f b" shows "a = b"
igSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igSwapCls MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbs MOD \<and> igSwapIGOp MOD"
accessible_def: shows "accessible \<equiv> {q. left_lang q \<noteq> {}}"
continuous_at_eps_delta: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f x = (\<forall>e>0. \<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
foldl_append: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'b list"   and ys :: "'b list" shows "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
vconn_implies_hpar_4: fixes \<mu> :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "trg g = trg f"
equalityCE: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A = B" obtains "s2p A c" and "s2p B c"   | "c \<notin> A" and "c \<notin> B"
f_aggregate_drop_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'d list"   and ag :: "'d list \<Rightarrow> 'd" shows "f_aggregate (xs \<up> (n * k)) k ag = f_aggregate xs k ag \<up> n"
mult_cancel_right2: fixes a :: "'a"   and c :: "'a" shows "(a * c = c) = (c = (0::'a) \<or> a = (1::'a))"
assms_5: fixes x :: "nat" assumes "x\<^sub>1 \<le> x" shows "b * real x + h x = real (t x)"
conj_superexpansion1: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> .\<not> (p .\<and>. q) .\<and>. .\<not> p) = (A \<turnstile> .\<not> p)"
hom: fixes i :: "'c" assumes "i \<in> I" shows "f i \<in> hom (A i) G"
subset_interior_of_eq: fixes S :: "'d set"   and X :: "'d topology" shows "(S \<subseteq> X interior_of S) = openin X S"
"1_3": fixes disc_node :: "(_) node_ptr"   and disc_doc :: "(_) document_ptr"   and ancestors :: "(_) object_ptr list" assumes "child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_node"   and "h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc"   and "h \<turnstile> get_ancestors_si (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_doc) \<rightarrow>\<^sub>r ancestors"   and "ancestor \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_doc"   and "ancestor \<in> set ancestors" shows "((\<forall>children ancestor_child. h \<turnstile> get_child_nodes ancestor \<rightarrow>\<^sub>r children \<longrightarrow> ancestor_child \<in> set children \<longrightarrow> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_child \<in> set ancestors \<longrightarrow> thesis) \<longrightarrow> thesis) \<or> ((\<forall>ancestor_element shadow_root. ancestor = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_element \<longrightarrow> h \<turnstile> get_shadow_root ancestor_element \<rightarrow>\<^sub>r Some shadow_root \<longrightarrow> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root \<in> set ancestors \<longrightarrow> thesis) \<longrightarrow> thesis)"
perp_up_degenerate_in_S_incident_2: fixes a :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "proj2_incident a l" shows "proj2_incident (perp_up a l) (drop_perp a l)"
map_mem_spmf_of_set: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "B \<noteq> {}" shows "map_spmf (\<lambda>x. x \<in> A) (spmf_of_set B) = spmf_of_pmf (bernoulli_pmf (real (card (A \<inter> B)) / real (card B)))"
vector_smult_lzero: fixes x :: "('b, 'c) vec" shows "(0::'b) *s x = 0"
hyps: assumes "k < min (nrows A) (ncols A)"   and "isDiagonal A" shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"
shows_prec_mat_def: fixes p :: "nat"   and A :: "'a mat" shows "shows_prec p A \<equiv> shows_mat A"
instInpP_fmla: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<tau> :: "'c"   and n :: "nat" assumes "Deduct_with_False_Disj_Rename var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "\<phi> \<in> fmla"   and "\<tau> \<in> fmla" shows "Deduct_with_False_Disj_Rename.instInpP var Var subst cnj exi \<phi> n \<tau> \<in> fmla"
valid_Cn_not: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(not_PL p \<in> Cn {}) = (Cn {p} = UNIV)"
path_P_k: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" assumes "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size" shows "Digraph.walk G (ProofStepInduct_NonTrivial.P_k G v0 v1 paths P_new sep_size) \<and> distinct (ProofStepInduct_NonTrivial.P_k G v0 v1 paths P_new sep_size)"
rec_zero: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" shows "rec e r zero = e"
orbit_nonempty: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "orbit f s \<noteq> {}"
return_time_exivl: fixes P :: "'d set"   and x :: "'d" assumes "returns_to P x"   and "closed P" shows "return_time P x \<in> existence_ivl0 x"
absn_subst: fixes n :: "nat"   and s :: "dB"   and t :: "dB"   and k :: "nat" shows "(Abs ^^ n) s[t/k] = (Abs ^^ n) (s[liftn n t 0/k + n])"
invariant_restrict_inD: fixes A :: "('e, 'f) automaton"   and P :: "'e \<Rightarrow> bool"   and QI :: "'f \<Rightarrow> bool" assumes "A \<TTurnstile> P" shows "A \<TTurnstile> (QI \<rightarrow>) P"
app_subst_Cons: fixes S :: "nat \<Rightarrow> typ"   and x :: "'a"   and l :: "'a list" shows "$ S (x # l) = $ S x # $ S l"
len_gt_false: assumes "length (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))) < dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))" shows "False"
qSkelBinp_def: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qAbs option" shows "qSkelBinp qbinp = lift qSkelAbs qbinp"
fds_converges_norm_iff: fixes f :: "'a fds"   and s :: "'a" shows "fds_converges (fds_norm f) (s \<bullet> (1::'a)) = fds_abs_converges f s"
simps_4: fixes tst :: "'a"   and c :: "('a, 'b, 'c) cmd" shows "proper (While tst c) = proper c"
universal_property_for_stalk: fixes A :: "'b set"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and \<psi> :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> 'b" assumes "ring A add mult zero one"   and "\<And>U. U \<in> top.neighborhoods \<pp> \<Longrightarrow> ring_homomorphism (\<psi> U) (\<O> U) (add_sheaf_spec U) (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U) A add mult zero one"   and "\<And>U V s. \<lbrakk>U \<in> top.neighborhoods \<pp>; V \<in> top.neighborhoods \<pp>; V \<subseteq> U; s \<in> \<O> U\<rbrakk> \<Longrightarrow> (\<psi> V \<circ> sheaf_spec_morphisms U V) s = \<psi> U s" shows "\<forall>V\<in>top.neighborhoods \<pp>. \<exists>!u. ring_homomorphism u st.carrier_stalk st.add_stalk st.mult_stalk (st.zero_stalk V) (st.one_stalk V) A add mult zero one \<and> (\<forall>U\<in>top.neighborhoods \<pp>. \<forall>s\<in>\<O> U. (u \<circ> st.canonical_fun U) s = \<psi> U s)"
simps_3: fixes p :: "real \<times> real \<times> real"   and L :: "atomUni list" shows "separateAtoms (LessUni p # L) = (let (a, b, c, d) = separateAtoms L in (a, p # b, c, d))"
map_closed: fixes a :: "'a \<times> 'a" assumes "a \<in> R \<times> S" shows "rel.Class a \<in> rel.Partition"
axioms_4: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "yoneda_lemma C S \<phi> F a" shows "set_valued_functor (dual_category.comp C) S F"
restrict0_in_fun_space: fixes f :: "'a \<Rightarrow> real" assumes "diff_fun k charts f" shows "restrict0 carrier f \<in> diff_fun_space"
bij_betwE: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "\<forall>a\<in>A. f a \<in> B"
coreflexive_pseudo_complement: fixes x :: "'a"   and y :: "'a" assumes "x \<le> top" shows "(x \<sqinter> y = bot) = (x \<le> - y \<sqinter> top)"
Hermite_of_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "Hermite_of A ass res bezout = (let A' = echelon_form_of A bezout in Hermite_of_upt_row_i A' (nrows A) ass res)"
h_1: shows "nonstop (code tp) (bl2wc (<lm>)) stpa = 0"
\<sigma>_\<rho>_nb: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and t :: "MIR.num"   and k :: "int" assumes "iszlfm p (a # bs)"   and "numbound0 t" shows "bound0 (\<sigma>_\<rho> p (t, k))"
monoid_axioms: shows "monoid M' (\<cdot>') \<one>'"
R2_imp_total2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> \<R>\<^sup>2" shows "total2 f"
assms_2: fixes x :: "('a, 'b) pre_digraph" assumes "P x" shows "Q x"
simps: fixes "consider" :: "'b \<Rightarrow> bool"   and gpv :: "('c, 'b, 'd) gpv"   and n :: "enat" shows "interaction_bounded_by consider gpv n = (interaction_bound consider gpv \<le> n)"
arr_mkArr: fixes A :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and B :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and F :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr \<Rightarrow> (('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" shows "S.arr (S.mkArr A B F) = (A \<subseteq> S.Univ \<and> B \<subseteq> S.Univ \<and> F \<in> A \<rightarrow> B)"
init_index_valid: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and as :: "('a \<times> 'b atom) set"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "index_valid as (init t)"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "('d, 'b) dds"   and b :: "('d, 'c) dds" assumes "rel_dds' R a b" obtains xa :: "'d \<Rightarrow> 'b \<times> ('d, 'b) dds" and ya :: "'d \<Rightarrow> 'c \<times> ('d, 'c) dds" where "a = DDS xa" and "b = DDS ya" and "((=) ===> rel_prod R (rel_dds' R)) xa ya"
fg_2: shows "g(x := y) \<in> FINFUNC"
step_leaves_empty_stack_empty: fixes t :: "nat"   and f :: "recf"   and x :: "nat"   and v :: "nat"   and t' :: "nat" assumes "iterate t step ([(f, Partial_Recursive.list_decode x, [])], None) = ([], Some v)" shows "iterate (t + t') step ([(f, Partial_Recursive.list_decode x, [])], None) = ([], Some v)"
dtor2_ctor2: fixes x :: "('d, 'd JF1, 'd JF2) F2" shows "dtor2 (ctor2 x) = x"
modaliseEmpty: fixes a :: "'b" shows "a \<cdot> (\<Empt>) = \<Empt>"
conc_abs_swap: fixes R :: "('b \<times> 'c) set"   and m' :: "'b nres"   and m :: "'c nres" assumes "single_valued R" shows "(m' \<le> \<Down> R m) = (\<Up> R m' \<le> m)"
norm_triangle_ineq4: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) \<le> norm a + norm b"
order_less_imp_not_eq: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "(x = y) = False"
gr_implies_not_zero: fixes m :: "'d multiset"   and n :: "'d multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
connect: assumes "x \<in> closure S"   and "x \<in> closure T" shows "f x = g x"
diff_divide_eq_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "minus x (divide y z) = divide (minus (times x z) y) z"
exec_move_SyncI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and V :: "nat"   and o' :: "'addr expr1" assumes "exec_move ci P t e h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (sync\<^bsub>V\<^esub> (o') e) h (stk, loc, Suc (Suc (Suc (length (compE2 o') + pc))), xcp) ta h' (stk', loc', Suc (Suc (Suc (length (compE2 o') + pc'))), xcp')"
individually_rational_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "individually_rational_on ds X = (CD_on ds X = X \<and> CH X = X)"
path_connected_imp_connected: fixes S :: "'c set" assumes "path_connected S" shows "connected S"
restrict_def: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
ENR_retract_of_ENR: fixes T :: "'b set"   and S :: "'b set" assumes "ENR T"   and "S retract_of T" shows "ENR S"
errMOD_igSubstIGVar2STR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igVarIPresIGWls MOD"   and "igSubstIGVar2 MOD" shows "igSubstIGVar2STR (errMOD MOD)"
ContractsWithBilateralSubstitutesAndIRC_axioms: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch"
Q1_a_def: shows "Q1_a \<equiv> [: X, mrk \<leadsto> X', mrk' . root = nil \<and> X' = {} \<and> mrk' = mrk :]"
perm_self_inverseI: fixes p :: "perm"   and q :: "perm"   and a :: "atom"   and a' :: "atom" assumes "- p = q"   and "a \<sharp> p"   and "a' \<sharp> p" shows "- ((a \<rightleftharpoons> a') + p) = (a \<rightleftharpoons> a') + q"
empty_multisets: fixes F :: "'b set" shows "{#} \<in> multisets F"
simps_4: fixes \<A> :: "'b ltl set"   and \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" shows "\<A> \<Turnstile>\<^sub>P \<phi> or \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<or> \<A> \<Turnstile>\<^sub>P \<psi>)"
subprob_space_def: fixes M :: "'a measure" shows "subprob_space M \<equiv> finite_measure M \<and> subprob_space_axioms M"
ghom_inv_inv: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "x \<in> carrier F" shows "f (\<rho>\<^bsub>F\<^esub> x) = \<rho>\<^bsub>G\<^esub> f x"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" assumes "a1 \<rightarrow>c a2" obtains c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1a__ ;; c2a__, sa__)" and "a2 = (c1'__ ;; c2a__, s'a__)" and "(c1a__, sa__) \<rightarrow>c (c1'__, s'a__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1b__ ;; c2b__, sb__)" and "a2 = (c2b__, s'b__)" and "(c1b__, sb__) \<rightarrow>t s'b__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tst__ then c1c__ else c2c__, sc__)" and "a2 = (c1c__, sc__)" and "tval tst__ sc__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tsta__ then c1d__ else c2d__, sd__)" and "a2 = (c2d__, sd__)" and "\<not> tval tsta__ sd__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, se__)" and "a2 = (c ;; while tst do c, se__)" and "tval tst se__"   | c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1e__ | c2e__, sf__)" and "a2 = (c1' | c2e__, s'c__)" and "(c1e__, sf__) \<rightarrow>c (c1', s'c__)"   | c2 :: "('test, 'atom) com" and s :: "'state" and c2' :: "('test, 'atom) com" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1f__ | c2f__, sg__)" and "a2 = (c1f__ | c2', s'd__)" and "(c2f__, sg__) \<rightarrow>c (c2', s'd__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1g__ | c2g__, sh__)" and "a2 = (c2g__, s'e__)" and "(c1g__, sh__) \<rightarrow>t s'e__"   | c2 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1 | c2, s)" and "a2 = (c1, s')" and "(c2, s) \<rightarrow>t s'"
semiring_normalization_rules_18: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lx :: "'a"   and rx :: "'a"   and ry :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times lx (times rx ry) = times (times lx rx) ry"
simps_7: fixes f1 :: "'h"   and f2 :: "nat \<Rightarrow> 'h \<Rightarrow> 'h"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
InitFinal: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and b :: "bool" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C ([],b) \<leftarrow> e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
case_1: fixes f1 :: "'c"   and f2 :: "enat \<Rightarrow> 'c" shows "(case 0 of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = f1"
Hilbert_fun_direct_decomp: fixes X :: "'c set"   and A :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and ps :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set list"   and z :: "nat" assumes "finite X"   and "A \<subseteq> P[X]"   and "direct_decomp A ps"   and "\<And>s. s \<in> set ps \<Longrightarrow> homogeneous_set s"   and "\<And>s. s \<in> set ps \<Longrightarrow> phull.subspace s" shows "Hilbert_fun A z = (\<Sum>p\<in>set ps. Hilbert_fun p z)"
isomorphic_reflexive: fixes f :: "'a" assumes "S.ide f" shows "S.isomorphic f f"
gauss_i_neq_0_1: shows "\<i>\<^sub>\<int> \<noteq> 0"
induct: fixes x1 :: "exp"   and x2 :: "exp"   and P :: "exp \<Rightarrow> exp \<Rightarrow> bool" assumes "x1 \<longrightarrow>* x2"   and "\<And>e. P e e"   and "\<And>e1 e2 e3. \<lbrakk>e1 \<longrightarrow> e2; e2 \<longrightarrow>* e3; P e2 e3\<rbrakk> \<Longrightarrow> P e1 e3" shows "P x1 x2"
ipresIGWls_def: fixes h :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'w, 'ae, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'ai, 'ab, 'aj, 'ak, 'x, 'al, 'am) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
IH_2: fixes ds :: "bool list" assumes "internal ds T\<^sub>2" shows "path ds T\<^sub>2"
st_mat_cases: fixes y :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "y \<in> {a. non_neg_mat a \<and> right_stoch_mat a}" obtains x :: "'a st_mat" where "y = st_mat x"
assms_2: shows "\<forall>\<^sub>F x in at_top. f x = g x"
array_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>array_rel \<equiv> {(Array xs, Array ys) |xs ys. (xs, ys) \<in> \<langle>R\<rangle>list_rel}"
sorted_poly_list_rel_wrt_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "('a \<times> char list multiset) set" shows "sorted_poly_list_rel_wrt S R = {(xs, ys). (xs, ys) \<in> \<langle>R \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel \<and> sorted_wrt S (map fst xs) \<and> distinct (map fst xs) \<and> 0 \<notin># snd `# ys}"
index_mat_multcol_4: fixes k :: "nat"   and a :: "'a"   and A :: "'a mat" shows "dim_row (multcol k a A) = dim_row A"
ts_tuple_rel_ext_Cons': fixes tas :: "nat \<times> 'a option list"   and tass :: "(nat \<times> 'a option list set) list"   and nt :: "nat"   and X :: "'a option list set" assumes "tas \<in> ts_tuple_rel (set tass)" shows "tas \<in> ts_tuple_rel (set ((nt, X) # tass))"
partition_onD3: fixes A :: "'a set"   and P :: "'a set set" assumes "partition_on A P" shows "{} \<notin> P"
cases: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'a literal multiset" assumes "gr.ord_resolve a1 a2 a3 a4 a5" obtains CAs :: "'a literal multiset list" and n :: "nat" and Cs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and D :: "'a literal multiset" where "a1 = CAs" and "a2 = D + negs (mset As)" and "a3 = AAs" and "a4 = As" and "a5 = \<Sum>\<^sub># (mset Cs) + D" and "length CAs = n" and "length Cs = n" and "length AAs = n" and "length As = n" and "n \<noteq> 0" and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)" and "\<forall>i<n. AAs ! i \<noteq> {#}" and "\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i" and "gr.eligible As (D + negs (mset As))" and "\<forall>i<n. gr.strictly_maximal_wrt (As ! i) (Cs ! i)" and "\<forall>i<n. S_Q (CAs ! i) = {#}"
Inf_vec_def: fixes X :: "('a, 'b) vec set" shows "Inf X = (\<chi>i. INF x\<in>X. x $ i)"
mrg_def: fixes cert :: "'a list" shows "merge cert \<equiv> merge cert f r \<top>"