hcju/isabelleps huggingface dataset:
subset_Compl_singleton: fixes A :: "'c set"   and b :: "'c" shows "(A \<subseteq> - {b}) = (b \<notin> A)"
iteration_denest3: fixes y :: "'a"   and x :: "'a" shows "(y\<^sup>\<star> \<cdot> x)\<^sup>\<infinity> \<cdot> y\<^sup>\<infinity> = (x + y)\<^sup>\<infinity>"
gdd_index_options: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "\<B> index ps = 0 \<or> int (\<B> index ps) = 1"
merge_id_def: fixes A :: "'d cset"   and f :: "'d \<Rightarrow> 'c cset Monomorphic_Monad.id" shows "merge_id A f = return_id (cUnion (cimage (extract \<circ> f) A))"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b"
rank_skew_cons_up: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list"   and t' :: "('c, 'd) SkewBinomialTree" assumes "rank_invar (t # bq)"   and "rank t' \<le> rank t" shows "rank_skew_invar (t' # t # bq)"
assms_2: shows "Env\<turnstile>t\<Colon>T"
cinner_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "\<langle>x, y\<rangle> = \<langle>fst x, fst y\<rangle> + \<langle>snd x, snd y\<rangle>"
fst_to_pair_unit: fixes x :: "'a" shows "fst (to_pair_unit x) = x"
rpsem_Choice_simps: fixes \<alpha> :: "prog"   and \<beta> :: "prog"   and \<nu> :: "fin_string \<Rightarrow> real"   and b :: "fin_string \<Rightarrow> real" shows "([\<alpha> \<union>\<union> \<beta>]\<nu> \<downharpoonleft> b) = (([\<alpha>]\<nu> \<downharpoonleft> b) \<or> ([\<beta>]\<nu> \<downharpoonleft> b))"
elims_3: fixes x :: "'b literal set" assumes "\<not> ground_clause x" obtains C :: "'b literal set" where "x = C" and "vars_of_cl C \<noteq> {}"
base_paths: fixes p :: "(symbol \<times> character list) list"   and k :: "nat" assumes "p \<in> \<P> k 0"   and "0 < k" shows "\<exists>u. p \<in> \<P> (k - 1) u"
SNF_1x2_works': shows "\<forall>A\<in>carrier_mat 1 2. is_SNF A (1\<^sub>m 1, Smith_1x2_JNF A)"
digraph_mapI_permutes: fixes M :: "'b pre_map" assumes "edge_rev M permutes arcs G"   and "\<And>a. a \<in> arcs G \<Longrightarrow> edge_rev M a \<noteq> a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> edge_rev M (edge_rev M a) = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> tail G (edge_rev M a) = head G a"   and "edge_succ M permutes arcs G"   and "\<And>v. \<lbrakk>v \<in> verts G; out_arcs G v \<noteq> {}\<rbrakk> \<Longrightarrow> cyclic_on (edge_succ M) (out_arcs G v)" shows "digraph_map G M"
fold_invariant: fixes xs :: "'b list"   and Q :: "'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and s :: "'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> Q x"   and "P s"   and "\<And>x s. \<lbrakk>Q x; P s\<rbrakk> \<Longrightarrow> P (f x s)" shows "P (fold f xs s)"
codim_map: fixes C :: "'a set"   and y :: "'a set" assumes "domain.chamber C"   and "order.greater_eq C y" shows "card (f ` C - f ` y) = card (C - y)"
fempty_iff: fixes c :: "'b" shows "(c |\<in>| {||}) = False"
minimal_witness_ex: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat" assumes "P k" shows "\<exists>k0\<le>k. P k0 \<and> (\<forall>k<k0. \<not> P k)"
hyps: assumes "(e, d) \<in> set M'"   and "uniq (elements M')" shows "(e, \<not> d) \<notin> set M'"
elims_3: fixes x :: "'a set \<times> (nat \<Rightarrow> 'b)"   and xa :: "('a \<times> 'b atom) set" assumes "\<not> x \<Turnstile>\<^sub>i\<^sub>a\<^sub>s xa" obtains I :: "'a set" and v :: "nat \<Rightarrow> 'b" and as :: "('a \<times> 'b atom) set" where "x = (I, v)" and "xa = as" and "\<not> v \<Turnstile>\<^sub>a\<^sub>s (snd ` (as \<inter> I \<times> UNIV))"
1: shows "finite (lift (liset visible (llist_of_stream u)))"
pow_four_block_mat: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat"   and m :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat m m" shows "four_block_mat A (0\<^sub>m n m) (0\<^sub>m m n) B ^\<^sub>m k = four_block_mat (A ^\<^sub>m k) (0\<^sub>m n m) (0\<^sub>m m n) (B ^\<^sub>m k)"
submonoid_axioms: shows "submonoid (translation ` G) (G \<rightarrow>\<^sub>E G) (compose G) (identity G)"
distinct_mset_set_mset_ident: fixes M :: "'b multiset" assumes "distinct_mset M" shows "mset_set (set_mset M) = M"
comm_monoid_set_axioms: shows "comm_monoid_set (+) (0::'a)"
countable_Diff_eq: fixes A :: "'b set"   and x :: "'b" shows "countable (A - {x}) = countable A"
SOCom_iff_indRelRTPO_is_strong_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" shows "((\<forall>S S'. S \<longmapsto>Source S' \<longrightarrow> (\<exists>T. \<lbrakk>S\<rbrakk> \<longmapsto>Target T \<and> (\<lbrakk>S'\<rbrakk>, T) \<in> TRel\<^sup>*)) \<and> strong_reduction_simulation (TRel\<^sup>+) Target) = (\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q'))"
simps_29: fixes vb :: "'b"   and vc :: "bexp"   and vd :: "'b acom"   and ve :: "'b"   and v :: "'b acom"   and va :: "'b acom" shows "({vb} WHILE vc DO vd {ve} \<sqsubseteq> v;; va) = False"
IH: fixes A :: "nat set"   and B :: "nat set" assumes "\<forall>i\<in>A. \<forall>a\<in>B. i < a" shows "nths xs (A \<union> B) = nths xs A @ nths xs B"
substInp_def: fixes xs :: "'varSort"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and x :: "'var" shows "substInp xs X x \<equiv> psubstInp (idEnv [x \<leftarrow> X]_xs)"
\<psi>_def: fixes x :: "'c"   and g :: "'d" shows "\<psi> x g = (THE f. \<guillemotleft>f : F (D.dom g) \<rightarrow>\<^sub>C x\<guillemotright> \<and> \<phi> (D.dom g) f = g)"
card_le_dim_span: fixes V :: "'a vec set"   and W :: "'a vec set" assumes "V \<subseteq> carrier_vec n"   and "V \<subseteq> local.span W"   and "lin_indpt V" shows "card V \<le> dim_span W"
ladder_every_stage_3: fixes index :: "nat" assumes "index < length L"   and "0 < index" shows "ladder_ix L' index = ladder_ix L index"
intros_1: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and control_dependence :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n"
unit_simps_1: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.src (Span.unit a) = a"
rt_fresh_asI: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1" shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
IH: fixes w' :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl" shows "(xs \<frown> w') \<Turnstile> \<phi> = w' \<Turnstile> af \<phi> xs"
join_sub: fixes L :: "nat set"   and R :: "nat set"   and n :: "nat"   and t1 :: "'b option list set"   and t2 :: "'b option list set"   and pos :: "bool" assumes "L \<subseteq> R"   and "table n L t1"   and "table n R t2" shows "join t2 pos t1 = {as \<in> t2. proj_tuple_in_join pos (join_mask n L) as t1}"
2: shows "snd ` set (pairs (add_pairs_single_naive data) False hs) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
var_pow_finsum_decomp: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" shows "p = (\<Oplus>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub>t\<in>set (UP.dense_repr p). [fst t] \<otimes>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> X\<^bsub>poly_ring R\<^esub> [^]\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> snd t)"
get_parent_reads_pointers: fixes ptr :: "(_) object_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing) \<in> Shadow_DOM.get_parent_locs"
ib: shows "mod_type_class.to_nat i < mod_type_class.to_nat b"
are_all_operators_applicable_cons_2: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and ops :: "'a strips_operator list" assumes "are_all_operators_applicable s (op # ops)" shows "are_all_operators_applicable s ops"
matrix_to_iarray_det_euclidean_ring: fixes A :: "(('c, 'd) vec, 'd) vec" shows "det A = det_iarrays_rings (matrix_to_iarray A)"
uminus_bit0_def: fixes x :: "'b bit0" shows "- x = Abs_bit0' (- Rep_bit0 x)"
canonical_proj_ext_ring_hom: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R" shows "ring_hom_ring R (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n])) (canonical_proj_ext I n)"
expand: fixes tree :: "'b tree"   and tree' :: "'b tree" assumes "root tree = root tree' \<and> left tree = left tree' \<and> right tree = right tree'" shows "tree = tree'"
zero_smult: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and m :: "'b" assumes "VectorSpace smult V"   and "m \<in> V" shows "smult (0::'a) m = (0::'b)"
powser_sums_if: fixes m :: "nat"   and z :: "'a" shows "(\<lambda>n. (if n = m then 1::'a else (0::'a)) * z ^ n) sums z ^ m"
b: shows "Suc (filter (\<lambda>x. Suc x < length init) (paid_A' ! n) ! (length (paid_A ! n) - Suc m)) < length init"
IH: fixes k :: "nat" shows "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
disjnt_Un2: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set" shows "disjnt C (A \<union> B) = (disjnt C A \<and> disjnt C B)"
ffbd_ffb_dual_var: fixes f :: "'c \<Rightarrow> 'd set" shows "\<partial> \<circ> bd\<^sub>\<F> f = fb\<^sub>\<F> (op\<^sub>K f) \<circ> \<partial>"
neighbors_adm: fixes v :: "nat"   and u :: "nat" assumes "v \<in> adjacent_nodes u - n u" shows "(u, v) \<notin> adm_edges f l"
intros_7: fixes uu :: "'d \<Rightarrow> 'e"   and uv :: "'d option"   and uw :: "'d option" shows "dbm_entry_val uu uv uw \<infinity>"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('j \<Rightarrow> 'j) \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme"   and v :: "'j" assumes "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v" shows "ptr |\<in>| element_ptr_kinds h"
Cons_7: shows "\<forall>c. (c \<in> set newWl \<or> c \<in> set (clause # Wl')) = (Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c)"
fls_divide_times: fixes f :: "'a fls"   and g :: "'a fls"   and h :: "'a fls" shows "f * g / h = f * (g / h)"
path0E: fixes l1 :: "'loc"   and l2 :: "'loc" assumes "path l1 l2 []" obtains "l2 = l1"
eq_divide_imp: 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero"   and "times a c = b" shows "a = divide b c"
factorization': shows "lr_fps_denominator cs = interp_alt_factorization fctrs"
xor_one_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "xor a one = minus (plus a (zero_neq_one.of_bool one zero (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a))) (zero_neq_one.of_bool one zero (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a))"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and c :: "'a"   and trm :: "trm" assumes "\<And>c. P c zer"   and "\<And>x c. P c (Var x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (suc x)"   and "\<And>x1 x2a c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (pls x1 x2a)"   and "\<And>x1 x2a c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (tms x1 x2a)" shows "P c trm"
perm_set_eq: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs <~~> ys" shows "set xs = set ys"
square_free_smult_iff: fixes c :: "'c"   and f :: "'c poly" assumes "c \<noteq> (0::'c)" shows "square_free (smult c f) = square_free f"
weakSimI: fixes Q :: "ccs"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "\<And>\<alpha> Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>\<^sup>^<Rel> Q"
learn_limI: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i. \<psi> i = f \<and> (\<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)" shows "learn_lim \<psi> U s"
mod_diff_right_eq: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b" shows "(b - a) mod m = (b - a mod m) mod m"
assms_3: fixes x :: "real" assumes "x \<in> closed_segment a b" shows "f x \<le> g x"
assms: shows "Keys (set fs) = set ks \<union> set vs"
that: assumes "arr_Rel \<alpha> S"   and "S\<lparr>ArrDom\<rparr> = A"   and "S\<lparr>ArrCod\<rparr> = B" shows "thesis"
assoc\<^sub>F\<^sub>M\<^sub>C_def: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" shows "\<a>[a, b, c] = \<F>C.mkarr \<^bold>\<a>\<^bold>[\<F>C.rep a, \<F>C.rep b, \<F>C.rep c\<^bold>]"
less_int_natI: fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "nat x < nat y" shows "x < y"
intros: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "tyenv_wellformed mds \<Gamma> \<S> P"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "pred P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c, mds, mem\<^sub>2\<rangle>"
homotopic_loops_sym_eq: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_loops s p q = homotopic_loops s q p"
ChamberSubcomplex_def: fixes Y :: "'a set set" shows "folding_g.ChamberSubcomplex Y \<equiv> order.greater_eq X Y \<and> ChamberComplex Y \<and> (\<forall>C. SimplicialComplex.maxsimp Y C \<longrightarrow> folding_g.chamber C)"
inv_genPoly: fixes g :: "graph"   and g' :: "graph"   and i :: "nat"   and v :: "nat"   and f :: "face" assumes "Invariants.inv g"   and "g' \<in> set (generatePolygon i v f g)"   and "f \<in> set (nonFinals g)"   and "2 < i"   and "v \<in> \<V> f" shows "Invariants.inv g'"
Join_stable: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set" shows "(F \<squnion> G \<in> stable A) = (F \<in> stable A \<and> G \<in> stable A)"
bet_conga__bet: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "A B C CongA A' B' C'" shows "Bet A' B' C'"
assms_3: shows "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)"
inf_sup_ord_1: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod (inf_prod x y) x"
less_irrefl: fixes ko :: "'a key_order"   and x :: "'a" shows "\<not> ko.lt ko x x"
\<phi>_biuniversal_prop_2: fixes u :: "'a"   and v :: "'a"   and w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide u"   and "ide v"   and "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : p\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : p\<^sub>1 \<star> w \<Rightarrow> p\<^sub>1 \<star> w'\<guillemotright>"   and "(s\<^sub>1 \<star> \<theta>) \<cdot> \<a>[s\<^sub>1, p\<^sub>0, w] \<cdot> (r\<^sub>0s\<^sub>1.\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[r\<^sub>0, p\<^sub>1, w] = (s\<^sub>1 \<star> \<theta>') \<cdot> \<a>[s\<^sub>1, p\<^sub>0, w'] \<cdot> (r\<^sub>0s\<^sub>1.\<phi> \<star> w') \<cdot> \<a>\<^sup>-\<^sup>1[r\<^sub>0, p\<^sub>1, w'] \<cdot> (r\<^sub>0 \<star> \<beta>)" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<theta> = \<theta>' \<cdot> (p\<^sub>0 \<star> \<gamma>) \<and> p\<^sub>1 \<star> \<gamma> = \<beta>"
prefix_same_cases: fixes xs\<^sub>1 :: "'b list"   and ys :: "'b list"   and xs\<^sub>2 :: "'b list" assumes "prefix xs\<^sub>1 ys"   and "prefix xs\<^sub>2 ys" shows "prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
fw_invariant_aux_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and i :: "nat"   and n :: "nat"   and j :: "nat"   and k :: "nat"   and i'' :: "nat"   and j'' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i' :: "nat"   and j' :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n"   and "i'' \<le> i"   and "j'' \<le> j" shows "less_eq (linordered_ab_monoid_add.fw plus less_eq m n k i j i' j') (linordered_ab_monoid_add.fw plus less_eq m n k i'' j'' i' j')"
Shleg_def: shows "local.Shleg = poly (of_int_poly local.shleg_poly)"
sub: fixes V :: "'a set" shows "{Z \<in> (\<lambda>i. T i \<inter> f i -` U) ` I. Z \<inter> V \<noteq> {}} \<subseteq> (\<lambda>i. T i \<inter> f i -` U) ` {i \<in> I. T i \<inter> V \<noteq> {}}"
graph_map'_def: fixes R :: "('c \<times> 'd) set" shows "graph_map' R = (if functional R then Some (graph_map R) else None)"
assms_2: assumes "poly_cond p"   and "1 < degree p" shows "thesis"
ord_strict_p_monomial_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" shows "(p \<prec>\<^sub>p monomial c v) = (c \<noteq> (0::'c) \<and> (p = 0 \<or> lt p \<prec>\<^sub>t v))"
split_stream_first': fixes x :: "'a"   and xs :: "'a stream" assumes "x \<in> sset xs" obtains ys :: "'a list" and zs :: "'a stream" where "xs = ys @- x ## zs" and "x \<notin> set ys"
phis_wf: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and args :: "'val list" assumes "phis g (n, v) = Some args" shows "length (old.predecessors g n) = length args"
assms_2: shows "t \<turnstile> s2 -2-ta2\<rightarrow> s2'"
elims_2: fixes x :: "formula" assumes "future_bounded x" obtains uu_ :: "String.literal" and uv_ :: "trm list" where "x = Pred uu_ uv_"   | p :: "String.literal" and \<phi> :: "formula" and \<psi> :: "formula" where "x = formula.Let p \<phi>'__ \<psi>__" and "future_bounded \<phi>'__ \<and> future_bounded \<psi>__"   | uw_ :: "trm" and ux_ :: "trm" where "x = formula.Eq uw_ ux_"   | uy_ :: "trm" and uz_ :: "trm" where "x = Less uy_ uz_"   | va_ :: "trm" and vb_ :: "trm" where "x = LessEq va_ vb_"   | \<phi> :: "formula" where "x = Neg \<phi>''__" and "future_bounded \<phi>''__"   | \<phi> :: "formula" and \<psi> :: "formula" where "x = Or \<phi>'''__ \<psi>'__" and "future_bounded \<phi>'''__ \<and> future_bounded \<psi>'__"   | \<phi> :: "formula" and \<psi> :: "formula" where "x = And \<phi>''''__ \<psi>''__" and "future_bounded \<phi>''''__ \<and> future_bounded \<psi>''__"   | l :: "formula list" where "x = Ands l" and "list_all future_bounded l"   | \<phi> :: "formula" where "x = Exists \<phi>'''''__" and "future_bounded \<phi>'''''__"   | y :: "nat" and \<omega> :: "agg_type \<times> event_data" and b :: "nat" and f :: "trm" and \<phi> :: "formula" where "x = Agg y \<omega> b f \<phi>''''''__" and "future_bounded \<phi>''''''__"   | I :: "\<I>" and \<phi> :: "formula" where "x = Prev I__ \<phi>'''''''__" and "future_bounded \<phi>'''''''__"   | I :: "\<I>" and \<phi> :: "formula" where "x = Next Ia__ \<phi>''''''''__" and "future_bounded \<phi>''''''''__"   | \<phi> :: "formula" and I :: "\<I>" and \<psi> :: "formula" where "x = Since \<phi>'''''''''__ Ib__ \<psi>'''__" and "future_bounded \<phi>'''''''''__ \<and> future_bounded \<psi>'''__"   | \<phi> :: "formula" and I :: "\<I>" and \<psi> :: "formula" where "x = Until \<phi> Ic__ \<psi>" and "future_bounded \<phi> \<and> future_bounded \<psi> \<and> right Ic__ \<noteq> \<infinity>"   | I :: "\<I>" and r :: "formula Regex.regex" where "x = MatchP Id__ r__" and "regex.pred_regex future_bounded r__"   | I :: "\<I>" and r :: "formula Regex.regex" where "x = MatchF I r" and "regex.pred_regex future_bounded r \<and> right I \<noteq> \<infinity>"
append_self_conv2: fixes xs :: "'e list"   and ys :: "'e list" shows "(xs @ ys = ys) = (xs = [])"
linT: fixes F :: "behavior \<Rightarrow> bool"   and G :: "behavior \<Rightarrow> bool" shows "\<turnstile> \<diamond>F \<and> \<diamond>G \<longrightarrow> \<diamond>(F \<and> \<diamond>G) \<or> \<diamond>(G \<and> \<diamond>F)"
length_nth_simps_2: fixes x :: "'b"   and xs :: "'b list" shows "length (x # xs) = Suc (length xs)"
fmran'I: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "fmlookup m x = Some y" shows "y \<in> fmran' m"
intros_8: fixes X :: "freemsg"   and Y :: "freemsg"   and Z :: "freemsg" assumes "X \<sim> Y"   and "Y \<sim> Z" shows "X \<sim> Z"
path_not_Nil: fixes g :: "'g"   and ns :: "'node list" assumes "local.path g ns" shows "ns \<noteq> []"
gcd_condition_monoid_axioms: shows "gcd_condition_monoid G"
mwc_sarpI: fixes f :: "'a set \<Rightarrow> 'a set" assumes "\<And>Xs. \<lbrakk>1 < length Xs; distinct (map f Xs); \<forall>i. f (Xs !% i) \<subset> Xs !% i \<inter> Xs !% (i + 1)\<rbrakk> \<Longrightarrow> False" shows "mwc_sarp f"
qSwap_preserves_qGood: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm" shows "qGood (X #[[x \<and> y]]_zs) = qGood X"
inv\<^sub>2E_1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"
chine_in_hom_1: shows "\<guillemotleft>TTfgh_TfTgh.chine : src (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<rightarrow> src ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)\<guillemotright>"
less_irrefl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.preorder less_eq less" shows "\<not> less x x"