hcju/isabelleps huggingface dataset:
deduction_Diff: fixes H :: "fmla set"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> B" shows "H - {C} \<turnstile> C IMP B"
Cont: fixes b' :: "'a"   and conv' :: "('b, 'a, 'c, 'd) converter"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('b, 'a, 'c, 'd) converter"   and a :: "'b"   and b :: "'a"   and \<I> :: "('b, 'a) \<I>" assumes "(b', conv') \<in> results_gpv \<I>' (run_converter conv a)"   and "b \<in> results_converter \<I> \<I>' conv'"   and "a \<in> outs_\<I> \<I>" shows "b \<in> results_converter \<I> \<I>' conv"
span_eq: fixes S :: "'b set"   and T :: "'b set" shows "(local.span S = local.span T) = (S \<subseteq> local.span T \<and> T \<subseteq> local.span S)"
prv_imp_trans1: 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 num :: "'b set"   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 prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<chi> \<psi>) (imp (imp \<phi> \<chi>) (imp \<phi> \<psi>)))"
star_n_abs: fixes X :: "nat \<Rightarrow> 'c" shows "\<bar>star_n X\<bar> = star_n (\<lambda>n. \<bar>X n\<bar>)"
agiop_giop: shows "iop (b_ag M) = mop M"
softSubst_implies_prv_S: fixes \<phi> :: "'fmla" shows "let SS = \<lambda>t1 t2. psubst S [(t1, xx), (t2, yy)] in \<phi> \<in> fmla \<longrightarrow> Fvars \<phi> = {xx} \<longrightarrow> bprv (SS \<langle>\<phi>\<rangle> \<langle>softSubst \<phi> \<langle>\<phi>\<rangle> xx\<rangle>)"
cfs: shows "factorize_rat_poly_monic (ps ! i) = (c, fs)"
assms_1: shows "(\<box>\<^sub>c\<gamma>) t n"
IS_LEFT_UNIQUED: fixes r :: "('d \<times> 'e) set"   and y :: "'d"   and x :: "'e"   and z :: "'d" assumes "IS_LEFT_UNIQUE r"   and "(y, x) \<in> r"   and "(z, x) \<in> r" shows "y = z"
bb: fixes n :: "nat"   and k :: "nat" shows "Suc n choose k = n choose k + (if k = 0 then 0 else n choose (k - 1))"
tj_stack_discovered: shows "set (tj_stack s) \<subseteq> dom (discovered s)"
Inf_lower2: fixes u :: "'a"   and A :: "'a set"   and v :: "'a" assumes "u \<in> A"   and "u \<le> v" shows "\<Sqinter> A \<le> v"
aType_lemma: fixes a :: "nat \<Rightarrow> aType list"   and t :: "nat" assumes "msg (Suc 0) a" shows "a t = [] \<or> a t = [sc_ack]"
sublist_append: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "sublist xs (ys @ zs) = (sublist xs ys \<or> sublist xs zs \<or> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> suffix xs1 ys \<and> prefix xs2 zs))"
discrete_topology_def: fixes U :: "'a set" shows "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"
equivalence_is_left_adjoint: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.equivalence_map f" shows "Span.is_left_adjoint f"
subgroup_imp_normal: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'c set" assumes "comm_group G"   and "subgroup A G" shows "A \<lhd> G"
mask_out_add_aligned: fixes p :: "'b word"   and n :: "nat"   and q :: "'b word" assumes "is_aligned p n" shows "p + (q AND NOT (mask n)) = p + q AND NOT (mask n)"
box_empty_imp: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'b set \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b" assumes "operative f z g"   and "box a b = {}" shows "g (cbox a b) = z"
get_roots_UN_get_work_locs_def: shows "get_roots_UN_get_work_locs = hs_get_roots_locs \<union> hs_get_work_locs"
sample_const: fixes p :: "int pmf"   and m :: "'m" shows "sample p (\<lambda>_. m) = m"
simps_1: fixes f :: "('c \<Rightarrow> 'b) tree"   and x :: "'c tree" shows "root (f \<diamondop> x) = root f (root x)"
decr_grading_idI: fixes d :: "'c \<Rightarrow> nat"   and t :: "'c"   and n :: "nat" assumes "hom_grading d"   and "d t \<le> n" shows "decr_grading d n t = t"
distinct_2: fixes nat :: "nat" shows "\<infinity> \<noteq> enat nat"
square_free_factorizationD_1: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs)" shows "p = Polynomial.smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
path_split_Cons: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n as n'"   and "as \<noteq> []" obtains a' :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" and as' :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" where "as = a' # as'" and "n = src a'" and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a'" and "CFGExit.path (trg a') as' n'"
borel_measurable_iff_halfspace_less: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)"
prv_eqv_sym: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (eqv \<phi> \<chi>)" shows "bprv (eqv \<chi> \<phi>)"
standard_decomp_Nil: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "ps\<^sub>+ = []" shows "standard_decomp k ps"
decl_eval\<^sub>w: fixes mu :: "'Var ModeUpd"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>Skip@[mu], mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>Stop, update_modes mu mds, mem\<rangle>\<^sub>w"
coprime_iff_prime_factors_disjoint: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)" shows "coprime x y = (prime_factors x \<inter> prime_factors y = {})"
eventually_frequently_const_simps_3: fixes P :: "'a \<Rightarrow> bool"   and C :: "bool"   and F :: "'a filter" shows "(\<forall>\<^sub>F x in F. P x \<or> C) = (eventually P F \<or> C)"
assms_1: shows "list_all2 P (nths xs A) (nths ys A)"
th0: shows "inverse (of_nat (Suc k) * fps_radical r (Suc k) a ^ k) * (of_nat (Suc k) * fps_radical r (Suc k) a ^ k) = 1"
pred_iffI: fixes A :: "'a Predicate.pred"   and B :: "'a Predicate.pred" assumes "\<And>x. pred.eval A x \<Longrightarrow> pred.eval B x"   and "\<And>x. pred.eval B x \<Longrightarrow> pred.eval A x" shows "A = B"
distinct_9: fixes x1 :: "'e" shows "prot_atom.Atom x1 \<noteq> OccursSecType"
set_fold_insert: fixes xs1 :: "'b list"   and xs2 :: "'b list" shows "set (fold List.insert xs1 xs2) = set xs1 \<union> set xs2"
vsv_vimageE: fixes b :: "V"   and A :: "V" assumes "b \<in>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr> `\<^sub>\<circ> A" obtains x :: "V" where "\<FF>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = b" and "x \<in>\<^sub>\<circ> A"
subseqs_set_subset: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<in> set (subseqs xs)" shows "set ys \<subseteq> set xs"
i_Exec_Stream_Pre_Post2_iAll: fixes result :: "nat \<Rightarrow> 'a"   and trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "nat set" assumes "result = i_Exec_Comp_Stream trans_fun input c"   and "\<forall>x_n c_n. P c_n \<longrightarrow> Q x_n (trans_fun x_n c_n)" shows "\<box> t I. P (result\<^bsup>\<leftarrow> c\<^esup> t) \<longrightarrow> Q (input t) (result t)"
safe_cosafe: fixes r :: "'a regex"   and m :: "modality" assumes "\<And>x. \<lbrakk>x \<in> atms r; safe Strict x\<rbrakk> \<Longrightarrow> safe Lax x"   and "local.safe_regex m Strict r" shows "local.safe_regex m Lax r"
ring_simprules_14: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
init_ProcessingE: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "sh C = \<lfloor>(sfs, Processing)\<rfloor>"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "e' = unit \<and> s' = (h, l, sh)"
maximal_independent_subset: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "vs1.independent B" and "V \<subseteq> vs1.span B"
A4_def: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "A4 x a Q = ((\<lambda>(ps, t). x \<in> dom ps \<and> vars a \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval a ps), t)) \<and>* $ 1)"
IH_1: shows "fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S) \<subseteq> vars\<^sub>s\<^sub>s\<^sub>t S"
pair_code: fixes d :: "('fun, 'a) Term.term \<times> ('fun, 'a) Term.term" shows "pair d = pair' Pair d"
distinct_3: fixes a1 :: "'d"   and a2 :: "'d" shows "a1\<lparr>a2\<rparr> \<noteq> \<tau>"
lprod_2_3: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> R" shows "([a, c], [b, c]) \<in> lprod R"
Landau_Picard: obtains R :: "complex \<Rightarrow> real" where "\<And>z. 0 < R z" and "\<And>f. \<lbrakk>f holomorphic_on cball 0 (R (f 0)); \<And>z. cmod z \<le> R (f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1\<rbrakk> \<Longrightarrow> cmod (deriv f 0) < 1"
funionM_def: fixes m1 :: "'m"   and m2 :: "'m" shows "funionM m1 m2 \<equiv> bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A |\<union>| B)))"
connected_sing: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.topological_space open" shows "topological_space.connected open {x}"
subtuple_mono: fixes xm :: "'f \<rightharpoonup>\<^sub>f 'd"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> fmran' xm; y \<in> fmran' ym\<rbrakk> \<Longrightarrow> f x y \<longrightarrow> g x y" shows "subtuple f xm ym \<longrightarrow> subtuple g xm ym"
no_img_tick: shows "\<forall>k. (\<nexists>k\<^sub>0. f k\<^sub>0 = k) \<longrightarrow> (\<forall>c. \<not> hamlet (Rep_run r k c))"
raise2: fixes ck :: "bool"   and s1 :: "v sem_env"   and env :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and err :: "v error_result" assumes "evaluate ck s1 env e (s2, Rerr err)" shows "evaluate ck s1 env (Raise e) (s2, Rerr err)"
cases: fixes x :: "('b \<times> nat) tree" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> nat) tree" and a :: "'b" and h :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (a, h), r\<rangle>"
that_1: assumes "x \<in> A"   and "f x \<in> B x" shows "thesis"
typ_of_imp_has_typ: fixes t :: "Core.term"   and ty :: "typ" assumes "typ_of t = Some ty" shows "\<turnstile>\<^sub>\<tau> t : ty"
arctan_ms_aux_neg_inf: fixes xs :: "('a \<times> real) msllist"   and basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real" assumes "\<exists>e>0. ms_aux_hd_exp xs = Some e"   and "fst (dominant_term_ms_aux xs) < 0"   and "trimmed_ms_aux xs"   and "basis_wf basis"   and "is_expansion_aux xs f basis" shows "is_expansion_aux (minus_ms_aux (const_ms_aux (- pi / 2)) (arctan_ms_aux (inverse_ms_aux xs))) (\<lambda>x. arctan (f x)) basis"
that: fixes l :: "'b" assumes "(f \<longlongrightarrow> l) (at x within X)" shows "thesis"
absolute_liveness_eventually: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "absolute_liveness P = ((\<exists>\<sigma>. P \<sigma>) \<and> P = \<diamond>P)"
set_mset_union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "set_mset (M + N) = set_mset M \<union> set_mset N"
borel_measurable_has_bochner_integral': fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and g :: "'e \<Rightarrow> 'c"   and N :: "'e measure" assumes "has_bochner_integral M f x"   and "g \<in> N \<rightarrow>\<^sub>M M" shows "(\<lambda>x. f (g x)) \<in> borel_measurable N"
def_gfp_unfold: fixes A :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "A \<equiv> \<mu> f"   and "mono f" shows "A = f A"
Q2_def: shows "Q2 = (\<lambda>F X. \<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X))"
prop_in_f_3_d: fixes v :: "i" shows "[\<^bold>\<not>Indiscriminate A! in v]"
word_add_less_dest: fixes i :: "'b word"   and k :: "'b word"   and j :: "'b word" assumes "i + k < j + k"   and "unat i + unat k < 2 ^ LENGTH('b)"   and "unat j + unat k < 2 ^ LENGTH('b)" shows "i < j"
Diff_insert0: fixes x :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "x \<notin> A" shows "A - insert x B = A - B"
Pbij_compose_def: fixes \<beta> :: "(Location \<times> Location) set"   and \<gamma> :: "(Location \<times> Location) set" shows "Pbij_compose \<beta> \<gamma> = {(l, ll). \<exists>l1. (l, l1) \<in> \<beta> \<and> (l1, ll) \<in> \<gamma>}"
inclusion: fixes f :: "'a" assumes "Arr f" shows "B.arr f"
Rep_lifted_distrib_lattice_top_cases: fixes y :: "('c, 'd) phi \<Rightarrow> 'd" assumes "y \<in> {xf. True}" obtains x :: "('c, 'd) lifted_distrib_lattice_top" where "y = Rep_lifted_distrib_lattice_top x"
assm_rule: assumes "expand_inv \<phi> (n', ns')"   and "((n', ns'), n, ns) \<in> expand_ord \<phi>" shows "f (n', ns') \<le> SPEC P"
integrable_subinterval: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c"   and c :: "real"   and d :: "real" assumes "integrable (lebesgue_on {a..b}) f"   and "{c..d} \<subseteq> {a..b}" shows "integrable (lebesgue_on {c..d}) f"
leg1_composite: shows "HH\<nu>\<pi>\<rho>.leg1 = \<nu>\<pi>.leg1 \<cdot> \<nu>\<pi>_\<rho>.prj\<^sub>1"
hd_inv: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Inverse $$ e \<Rightarrow> (\<lambda>\<sigma> x. f \<sigma> (op_sem Inverse x) * ennreal ((inverse (extract_real x))\<^sup>2))"
Skip_normal_steps_end: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "'d"   and c :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate" assumes "\<Gamma> \<turnstile> (Skip, Normal s) \<rightarrow>\<^sup>* (c, s')" shows "c = Skip \<and> s' = Normal s"
simps_5: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a"   and I :: "'a"   and b :: "bexp"   and P :: "'a"   and C :: "'a acom"   and Q :: "'a" shows "Step f g S ({I} WHILE b DO {P} C {Q}) = {S \<squnion> post C} WHILE b DO {g b I} Step f g P C {g (bexp.Not b) I}"
neg: shows "dim_vec v = dim_col A \<and> inner_prod v (A *\<^sub>v v) < 0"
r_step: fixes e :: "nat" shows "eval r_step [e] \<down>= estep e"
cef: assumes "f \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "e O f = f"
fixed_point_imp_\<alpha>_limit_set: fixes x :: "'a" assumes "x \<in> X"   and "f x = (0::'a)" shows "\<alpha>_limit_set x = {x}"
th0: shows "row i A + (c *s z + y) = row i A + y + c *s z"
orderEq: shows "p ^ a * m = card (subgroups_of_size (p ^ a)) * card (conj.stabilizer P)"
assms_4: fixes x :: "nat \<Rightarrow> 'a" shows "dim_col (matricize I (T x)) = nc"
elims_3: fixes x :: "('b \<times> nat) tree" assumes "\<not> ltree x" obtains l :: "('b \<times> nat) tree" and a :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (a, n), r\<rangle>" and "\<not> (min_height r \<le> min_height l \<and> n = min_height r + 1 \<and> ltree l \<and> ltree r)"
loose_bvar1_decr''': fixes t :: "Core.term"   and lev :: "nat"   and lev' :: "nat" assumes "loose_bvar1 t (Suc lev)"   and "lev' \<le> lev" shows "loose_bvar1 (decr lev' t) lev"
ptd_lookup_unchanged_4: 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) 4 = ptd_lookup va ptp ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) 4"
path_cons: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "\<pi> m = \<pi>' 0" shows "IFC_def.is_path prob (IFC_def.path_append \<pi> m \<pi>')"
mult_cancel_right1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times b c) = (c = zero \<or> b = one)"
open_segment_commute: fixes a :: "'a"   and b :: "'a" shows "open_segment a b = open_segment b a"
fall2_elim: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "fall2 F \<pi>l" obtains "\<And>\<pi> \<pi>'. \<lbrakk>wfp AP' \<pi>; wfp AP' \<pi>'; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0; stateOf \<pi>' = stateOf \<pi>\<rbrakk> \<Longrightarrow> F \<pi> \<pi>' \<pi>l"
put_env_def: fixes put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f"   and m :: "('d, 'e) envT" shows "put_env put s m = EnvT (\<lambda>r. put s (run_env m r))"
in_degree_eq_card_arc_ends: fixes v :: "'a" shows "in_degree G v = card (arc_to_ends G ` in_arcs G v)"
intro: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and create_document :: "((_) heap, exception, (_) document_ptr) prog" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document" shows "l_get_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name create_document"
simpnum_ci: fixes bs :: "real list"   and t :: "Ferrack.num" shows "Inum bs (simpnum t) = Inum bs t"
not_real_square_gt_zero: fixes x :: "real" shows "(\<not> 0 < x \<cdot> x) = (x = 0)"
fst_scaleC: fixes r :: "complex"   and A :: "'c \<times> 'd" shows "fst (r *\<^sub>C A) = r *\<^sub>C fst A"
sig_red_singleD4: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "ord_term_lin.is_le_rel sing_reg"
RSpan_single: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b" assumes "R_scalar_mult R" shows "R_scalar_mult.RSpan R smult [m] = {smult r m |r. r \<in> R}"
assms_3: shows "A \<inter> space M \<noteq> {}"
s2max: shows "\<forall>x. x \<in># {#t1, t1, s1, s1#} \<longrightarrow> (x, s2) \<in> trm_ord"
lesspoll_trans: fixes X :: "'a set"   and Y :: "'b set"   and Z :: "'c set" assumes "X \<prec> Y"   and "Y \<prec> Z" shows "X \<prec> Z"
permutation_split_ran: fixes j :: "'b"   and S :: "'b set" assumes "j \<in> S" shows "{p. p permutes S} = (\<Union>i\<in>S. {p. p permutes S \<and> p i = j})"
strict_gR_rel_def: fixes x :: "'a"   and y :: "'a" shows "(x \<prec>\<^sub>R y) = (x \<preceq>\<^sub>R y \<and> \<not> y \<preceq>\<^sub>R x)"
CallUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and s :: "'d"   and n :: "nat" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> =Suc n\<Rightarrow> Stuck"
notin_set_remove1: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "x \<notin> set xs" shows "x \<notin> set (remove1 y xs)"
u_1: shows "u \<in> {x--z} \<union> Gzt"
DD': shows "D \<cdot>\<^sub>l\<^sub>s \<mu> = D'"
config'_n_bv: fixes init :: "('c list \<times> bool list \<times> 'c list) pmf"   and s0 :: "'c list"   and qs :: "'c list" assumes "map_pmf (snd \<circ> snd) init = return_pmf s0"   and "map_pmf (fst \<circ> snd) init = bv (length s0)" shows "map_pmf (snd \<circ> snd) (config'_rand BIT init qs) = return_pmf s0 \<and> map_pmf (fst \<circ> snd) (config'_rand BIT init qs) = bv (length s0)"
rel_implies_denot: fixes \<rho> :: "(nat \<times> val) list"   and e :: "exp"   and v :: "val" assumes "\<rho> \<turnstile> e \<Rightarrow> v" shows "v \<in> E e \<rho>"
obj_simps_1: fixes a :: "'b" assumes "B.obj a" shows "B.arr a"
case2: assumes "x \<le> 0"   and "0 \<le> y" shows "- 4611686018427387904 \<le> x * y"
winding: shows "\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0"
mult_pCons_0: shows "(*) (pCons (0::'a) 1) = pCons (0::'a)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x alg_dvd g x"