hcju/isabelleps huggingface dataset:
get_curr_win_traps_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and some_trap :: "Trap" assumes "low_equal s1 s2" shows "low_equal (snd (fst (get_curr_win () s1))\<lparr>traps := insert some_trap (traps (snd (fst (get_curr_win () s1))))\<rparr>) (snd (fst (get_curr_win () s2))\<lparr>traps := insert some_trap (traps (snd (fst (get_curr_win () s2))))\<rparr>)"
Subring_def: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'f) Ring_scheme" shows "Subring R S \<equiv> Ring S \<and> carrier S \<subseteq> carrier R \<and> ridmap S \<in> rHom S R"
decomp_tfr\<^sub>s\<^sub>t\<^sub>p: fixes t :: "('fun, 'var) Term.term" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (decomp t)"
connected_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "path g" shows "connected (path_image g)"
cod_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "partial_magma.cod (free_monoidal_category.comp C) (free_monoidal_category.mkarr C t) = free_monoidal_category.mkarr C (monoidal_language.Cod C t)"
o_objects_exist_3: fixes v :: "i" shows "[\<^bold>\<box>\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>A!,x\<^sup>P\<rparr>) in v]"
top_def: fixes L :: "('a, 'b) gorder_scheme" shows "\<top>\<^bsub>L\<^esub> = (SOME x. greatest L x (carrier L))"
knows_composition4: fixes Q :: "specID"   and e :: "Expression list"   and PQ :: "specID"   and P :: "specID" assumes "knows Q e"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ"   and "\<forall>m. m mem e \<longrightarrow> (\<exists>z. m = kE z) \<or> (\<exists>z. m = sE z)"   and "notSpecKeysSecretsExpr P e"   and "notSpecKeysSecretsExpr Q e" shows "knows PQ e"
nth_fls_subdegree_nonzero: fixes f :: "'a fls" assumes "f \<noteq> 0" shows "fls_nth f (fls_subdegree f) \<noteq> (0::'a)"
stutter_free_suffix: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "stutter_free \<sigma>" shows "stutter_free \<sigma> [k..]"
alpha_preserves_qGood: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "X #= X'" shows "qGood X = qGood X'"
set_of_eq: fixes x :: "'b interval" shows "set_of x = {lower x..upper x}"
matrix_mul_sum_alt: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" shows "A ** B = (\<chi>i. \<Sum>k\<in>UNIV. A $ i $ k *s B $ k)"
insert_commute: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "(x::'b) \<triangleright> (y::'b) \<triangleright> (A::'b set) = y \<triangleright> x \<triangleright> A"
filter_M_not_more_elements: fixes h :: "'a"   and P :: "'c \<Rightarrow> ('a, 'b, bool) prog"   and xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"   and "x \<in> set ys" shows "x \<in> set xs"
Block_\<tau>execI_None: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and V :: "nat"   and T :: "ty" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t {V:T=None; e} h s s'"
ode_def: fixes ode_ops :: "ode_ops"   and x :: "'e" shows "ode_operations.ode ode_ops x = einterpret (ode_operations.ode_e ode_ops) (list_of_eucl x)"
cStatImp: fixes R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>, R, S) \<in> Rel" shows "\<Psi> \<rhd> R \<lessapprox>\<^sub>w<Rel> S"
subset_Collect_conv: fixes S :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "(S \<subseteq> Collect P) = (\<forall>x\<in>S. P x)"
univ_preserves: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "equiv A r"   and "f respects r"   and "\<forall>x\<in>A. f x \<in> B" shows "\<forall>X\<in>A // r. univ f X \<in> B"
elims_2: fixes x :: "'a list" assumes "sorted x" obtains "xa__ = []"   | x :: "'a" and ys :: "'a list" where "xa__ = x # ys" and "Ball (set ys) ((\<le>) x) \<and> sorted ys"
whisker_left: fixes f :: "'a"   and \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.ide f"   and "B.seq \<nu> \<mu>" shows "f \<star>\<^sub>B \<nu> \<cdot>\<^sub>B \<mu> = (f \<star>\<^sub>B \<nu>) \<cdot>\<^sub>B f \<star>\<^sub>B \<mu>"
inext_nth_lessThan: fixes a :: "nat"   and n :: "nat" assumes "a < n" shows "{..<n} \<rightarrow> a = a"
firsth: fixes x :: "real" assumes "\<forall>y<x. a * y\<^sup>2 + b * y + c < 0" shows "a \<le> 0"
of_list_repr_def: fixes ba :: "'a list \<times> 'a list" shows "of_list_repr ba = rev (fst ba) @ snd ba"
Inf_eq_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf A = Sup {b. \<forall>a\<in>A. less_eq b a}"
image_gen: fixes S :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "finite S" shows "sum h S = (\<Sum>y\<in>g ` S. sum h {x \<in> S. g x = y})"
oalist_inv_raw_foldr_update_by_pair: fixes ys :: "('a \<times> 'c) list"   and xs :: "('a \<times> 'c) list" assumes "cmp2.oalist_inv_raw ys" shows "cmp2.oalist_inv_raw (foldr cmp2.update_by_pair xs ys)"
diff_face: shows "G.face_cycle_set a' \<noteq> G.face_cycle_set a"
vbrelation_vid_on: fixes A :: "V" shows "vbrelation (vid_on A)"
assms_3: fixes \<sigma> :: "'a"   and p :: "'b"   and \<sigma>' :: "'a" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "P \<sigma> p"   and "U \<sigma> \<sigma>'" shows "P \<sigma>' p"
sound_sum: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (\<lambda>s. P s + Q s)"
rotate_Some: fixes y :: "'a option"   and x :: "'a" shows "(y = Some x) = (Some x = y)"
perm_sing_eq2: fixes y :: "'a"   and ys :: "'a list" shows "([y] <~~> ys) = (ys = [y])"
assms: shows "range_vars \<delta> \<inter> set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x) = {}"
weakenSimMonotonic: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and A :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and B :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto>\<^sub>w<A> Q"   and "A \<subseteq> B" shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<B> Q"
elims_2: fixes x :: "'b tree" assumes "wbt x" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "wbt l \<and> wbt r \<and> (let ratio = real (size1 l) / real (size1 l + size1 r) in \<alpha> \<le> ratio \<and> ratio \<le> 1 - \<alpha>)"
OUT_j_finite: fixes x :: "'v option" shows "d_OUT j x \<noteq> \<top>"
path_Append: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n''"   and "n'' -as'\<rightarrow>* n'" shows "n -as @ as'\<rightarrow>* n'"
fmap_rel_alt_def: fixes K :: "('e \<times> 'g) set"   and V :: "('f \<times> 'h) set" shows "\<langle>K, V\<rangle>fmap_rel \<equiv> {(m1, m2). (\<forall>i j. i \<in># dom_m m2 \<longrightarrow> (j, i) \<in> K \<longrightarrow> (the (fmlookup m1 j), the (fmlookup m2 i)) \<in> V) \<and> fset (fmdom m1) \<subseteq> Domain K \<and> fset (fmdom m2) \<subseteq> Range K \<and> (\<forall>i j. (i, j) \<in> K \<longrightarrow> (j \<in># dom_m m2) = (i \<in># dom_m m1))}"
aka_whilei_def: fixes t :: "'a"   and x :: "'a"   and i :: "'a" shows "while t do x inv i = (d t \<cdot> x)\<^sup>\<star> \<cdot> ad t"
nextState_def: fixes alg :: "('d, 'e, 'f, 'g) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'d"   and st :: "'e"   and msgs :: "'d \<Rightarrow> 'f option"   and st' :: "'e" shows "nextState alg r p st msgs st' \<equiv> CnextState alg r p st msgs undefined st'"
Step1: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> IPImp p \<and> HistP rmhist p \<longrightarrow> UPSpec memCh mm (resbar rmhist) p"
poly_mapping_plus_induct: fixes P :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "P 0"   and "\<And>p c t. \<lbrakk>c \<noteq> (0::'b); t \<notin> keys p; P p\<rbrakk> \<Longrightarrow> P (monomial c t + p)" shows "P p"
rep_ivl_def: shows "rep_ivl \<equiv> quot_type.rep Rep_ivl"
outs'_gpv_induct: fixes x :: "'d"   and gpv :: "('e, 'd, 'f) gpv"   and P :: "('e, 'd, 'f) gpv \<Rightarrow> bool" assumes "x \<in> outs'_gpv gpv"   and "\<And>generat gpv. \<lbrakk>generat \<in> set_spmf (the_gpv gpv); x \<in> generat_outs generat\<rbrakk> \<Longrightarrow> P gpv"   and "\<And>generat gpv c input. \<lbrakk>generat \<in> set_spmf (the_gpv gpv); c \<in> generat_conts generat; x \<in> outs'_gpv (c input); P (c input)\<rbrakk> \<Longrightarrow> P gpv" shows "P gpv"
add_strict_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b" shows "less (plus c a) (plus c b)"
hyps: assumes "is_tangle_diagram (zs \<circ> y)" shows "domain_wall y = codomain_wall zs"
ord_iff_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "less_eq x y = (rep_nat x \<le> rep_nat y)"
sup_ge2: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
simps_36: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y4 :: "('b, 'd) fm" shows "\<not> rel_fm R1 R2 \<^bold>\<bottom> (\<^bold>\<forall> y4)"
triangular_columnD: fixes j :: "nat"   and A :: "'a mat"   and i :: "nat" assumes "local.triangular_column j A"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'a)"
poset_is_SimplicialComplex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ordering less_eq less"   and "\<forall>x\<in>P. ordering.simplex_like less_eq less {y \<in> P. less_eq y x}" shows "SimplicialComplex (ordering.PosetComplex less_eq less P)"
monom_eq_1_iff: fixes c :: "'a"   and n :: "nat" shows "(monom c n = 1) = (c = (1::'a) \<and> n = 0)"
resr_eq: fixes x :: "'a" shows "residual ((\<cdot>) x) = (\<rightarrow>) x"
add_mset_subseteq_single_iff: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b" shows "(add_mset a M \<subseteq># \<LM> b  \<RM>) = (M = \<Empt> \<and> a = b)"
transpose_aux_max: fixes xs :: "'g list"   and xss :: "'h list list" shows "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) = Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0))"
square_free_multD_3: fixes f :: "'c poly"   and g :: "'c poly" assumes "square_free (f * g)" shows "square_free g"
state_of_def: fixes s :: "'e Heap" shows "state_of s \<equiv> State (\<lambda>heap. the (execute s heap))"
bounded_pos_less: fixes S :: "'c set" shows "bounded S = (\<exists>b>0. \<forall>x\<in>S. norm x < b)"
substitutes_prm: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and A' :: "'a trm"   and \<pi> :: "'a prm" assumes "substitutes A x M A'" shows "substitutes (\<pi> \<cdot> A) (\<pi> $ x) (\<pi> \<cdot> M) (\<pi> \<cdot> A')"
tyenv_wellformed_def: fixes mds :: "Mode \<Rightarrow> Eg1.var set"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" shows "A.tyenv_wellformed mds \<Gamma> \<S> P \<equiv> A.mds_consistent mds \<Gamma> \<S> P \<and> A.types_wellformed \<Gamma> \<and> A.types_stable \<Gamma> \<S>"
primitive_extractor_correct_2: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and as :: "'c negation_type list"   and ms :: "'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "normalized_nnf_match ms"
tensor_comm: fixes A :: "'b tensor"   and B :: "'b tensor" shows "A + B = B + A"
adjoint_of_deutsch_transform: shows "U\<^sub>f\<^sup>\<dagger> = U\<^sub>f"
simps_3: fixes x :: "'a"   and a :: "('a, 'c) rbt"   and y :: "'a"   and s :: "'c"   and va :: "('a, 'c) rbt"   and vb :: "'a"   and vc :: "'c"   and vd :: "('a, 'c) rbt" shows "rbt_del_from_right x a y s (Branch R va vb vc vd) = Branch R a y s (rbt_del x (Branch R va vb vc vd))"
l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf \<equiv> type_wf = ShadowRootClass.type_wf"
freshInputTransition: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> u" shows "c \<sharp> P'"
vdisjnt_nin_right: fixes A :: "V"   and B :: "V"   and a :: "V" assumes "A \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> A" shows "a \<notin>\<^sub>\<circ> B"
factorial_euclidean_semiring_gcdI: shows "OFCLASS('a, euclidean_semiring_gcd_class)"
IH: assumes "(l, a) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)" shows "\<exists>b. (l, b) \<in> set A \<and> dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, b) = (l, a)"
pochhammer_of_nat_eq_0_lemma: fixes n :: "nat"   and k :: "nat" assumes "n < k" shows "pochhammer (- of_nat n) k = (0::'a)"
ax_nochange_lemma: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "'b"   and s :: "'c"   and w :: "'b" assumes "P Y s"   and "All ((=) w)" shows "P w s"
union_interior_of_subset: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X interior_of S \<union> X interior_of T \<subseteq> X interior_of (S \<union> T)"
split_seg: shows "segment G.face_cycle_succ a a = segment G.face_cycle_succ a b \<union> {b} \<union> segment G.face_cycle_succ b a"
adverse_def: fixes z :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat" shows "local.adverse z i j x \<equiv> Some (last (local.prefixes z i j x))"
finite_linorder_min_induct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "finite A"   and "P {}"   and "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. less b a; P A\<rbrakk> \<Longrightarrow> P (insert b A)" shows "P A"
sel_ren_list_inv: fixes \<rho>s :: "'s list"   and Cs :: "'a literal multiset list" assumes "length \<rho>s = length Cs"   and "is_renaming_list \<rho>s" shows "map S (Cs \<cdot>\<cdot>cl \<rho>s) = map S Cs \<cdot>\<cdot>cl \<rho>s"
sup_absorb1: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x" shows "sup x y = x"
sets_pre_sigmaD: fixes T :: "'a \<Rightarrow> 't"   and A :: "'a set"   and t :: "'t" assumes "stopping_time F T"   and "A \<in> sets (pre_sigma T)" shows "{\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)"
disj_subst: fixes P :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "P = R" shows "(P \<or> Q) = (R \<or> Q)"
FL_valid_Act_strong: fixes X :: "'a"   and P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act, 'effect) FL_Formula.formula" assumes "finite (supp X)" shows "P \<Turnstile> FL_Formula.Act f \<alpha> x = (\<exists>\<alpha>'' x' P'. FL_Formula.Act f \<alpha> x = FL_Formula.Act f \<alpha>'' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x' \<and> bn \<alpha>'' \<sharp>* X)"
abs_if: fixes abs :: "'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.abs_if abs uminus zero less" shows "abs a = (if less a zero then uminus a else a)"
1: shows "X \<in> obj\<^bsub>Op C\<^esub>"
sdrop_imp_alw: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi> :: "('state \<times> 'aprop set) stream"   and \<pi>' :: "('state \<times> 'aprop set) stream"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<And>i. (\<And>j. j \<le> i \<Longrightarrow> \<phi> [sdrop j \<pi>, sdrop j \<pi>']) \<Longrightarrow> \<psi> [sdrop i \<pi>, sdrop i \<pi>']" shows "imp (alw \<phi>) (alw \<psi>) [\<pi>, \<pi>']"
exp_lower_Taylor_quadratic: fixes x :: "real" assumes "0 \<le> x" shows "1 + x + x\<^sup>2 / 2 \<le> exp x"
normalize: fixes A :: "'a set" shows "normalize (Lcm\<^sub>f\<^sub>i\<^sub>n A) = Lcm\<^sub>f\<^sub>i\<^sub>n A"
prems: shows "character (G\<lparr>carrier := adjoin G H' a, carrier := H\<rparr>) \<chi>"
intros_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>e Q" shows "Q \<equiv>\<^sub>e P"
set_ConsD: fixes y :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "y \<in> list.set (x # xs)" shows "y = x \<or> y \<in> list.set xs"
rtrancl_tab_imp_rtrancl_path: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and x :: "'b"   and y :: "'b" assumes "rtrancl_tab r ys x y" obtains xs :: "'b list" where "rtrancl_path r x xs y"
space_empty_eq_bot: fixes a :: "'c measure" shows "(space a = {}) = (a = \<bottom>)"
redT_updLs_has_locks: fixes ls :: "('a, 'b) locks"   and t' :: "'b"   and las :: "'a lock_actions"   and t :: "'b"   and l :: "'a" assumes "lock_ok_las ls t' las"   and "t \<noteq> t'" shows "has_locks (redT_updLs ls t' las $ l) t = has_locks (ls $ l) t"
assms: shows "solve_depressed_cubic_Cardano_real e f = Some y"
DisjointPaths_axioms_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" shows "DisjointPaths_axioms G v0 v1 paths \<equiv> (\<forall>xs. xs \<in> paths \<longrightarrow> v0 \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> v1) \<and> (\<forall>xs ys v. xs \<in> paths \<longrightarrow> ys \<in> paths \<longrightarrow> xs \<noteq> ys \<longrightarrow> v \<in> set xs \<longrightarrow> v \<in> set ys \<longrightarrow> v = v0 \<or> v = v1)"
dbl_inc_simps_4: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (uminus one) = uminus one"
rem_UnPath_even: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "finite E"   and "finite V"   and "is_trail v ps v'"   and "even (degree v' G)" shows "num_of_odd_nodes (rem_unPath ps G) = num_of_odd_nodes G + (if even (degree v G) \<and> v \<noteq> v' then 2 else 0)"
mat_of_col_def: fixes v :: "'b vec" shows "mat_of_col v = (mat_of_row v)\<^sup>T"
HF: shows "(H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (d::'a::refinement_lattice) \<sqinter> (c::'a::refinement_lattice) ; c\<^sup>\<star> ; d \<sqinter> c ; x)"
class_add_relevant_entries_eq: fixes wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and P :: "jvm_prog"   and C :: "char list"   and i :: "instr"   and pc :: "nat"   and xt :: "ex_table"   and cdec :: "jvm_method class" assumes "wf_prog wf_md P"   and "\<not> is_class P C" shows "relevant_entries P i pc xt = relevant_entries (class_add P (C, cdec)) i pc xt"
conv_radius_norm: fixes f :: "nat \<Rightarrow> 'a" shows "conv_radius (\<lambda>x. norm (f x)) = conv_radius f"
compatible_sym: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.cpo below"   and "cpo.compatible below x y" shows "cpo.compatible below y x"
thm_2_3_6_indets: fixes fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list"   and b :: "nat" assumes "pp_pm.is_GB_cofactor_bound (set fs) b" shows "pp_pm.punit.is_Groebner_basis (set (pp_pm.punit.Macaulay_list (pp_pm.deg_shifts (\<Union> (indets ` set fs)) b fs)))"
Shift_def: fixes Kl :: "'b list set"   and k :: "'b" shows "Shift Kl k = {kl. k # kl \<in> Kl}"
int_eq: shows "fst (bezw x q) * int x + snd (bezw x q) * int q = 1"
j: shows "has_bochner_integral lebesgue (\<lambda>x. indicat_real {c..b} x *\<^sub>R f x) j"
distinct_73: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi1' :: "pi"   and pi2' :: "pi" shows "[name1\<noteq>name2]pi \<noteq> pi1' \<parallel> pi2'"
urel32_mod: fixes pp :: "uint32"   and x :: "uint32"   and x' :: "int"   and y :: "uint32"   and y' :: "int" assumes "p = int_of_uint32 pp"   and "p \<le> 65535"   and "urel32 x x'"   and "urel32 y y'" shows "urel32 (if y = 0 then x else 0) (if y' = 0 then x' else 0)"
B: fixes ys :: "'a list"   and zs :: "'a list" shows "xs @ ws \<cong> {ys @ ws, zs, P} \<longrightarrow> xs \<cong> {ys, zs, \<lambda>v vs. P v (vs @ ws)}"
add_mset_add_mset_same_iff: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(add_mset a A = add_mset a B) = (A = B)"
ksc_simps_2: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "short_cycles G k \<noteq> {}" shows "kill_short G k = kill_short (G -- choose_v G k) k"
ih3: fixes t :: "trm" assumes "e\<^sub>1'[x::=e\<^sub>2'] \<Down> t" shows "e' = t"
p_pp_comp: fixes x :: "'a"   and y :: "'a" shows "- (- - x \<sqinter> y) = - (x \<sqinter> y)"
union_eqclass_decreases_components: fixes F :: "'edge set"   and x :: "'a"   and y :: "'a"   and e :: "'edge" assumes "CC F x \<noteq> CC F y"   and "e \<notin> F"   and "x \<in> V"   and "y \<in> V"   and "F \<subseteq> E"   and "e \<in> E"   and "joins x y e" shows "Suc (card (CCs (insert e F))) = card (CCs F)"
rb_aux_alt_spp: fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_inv (fst args)" shows "app_args spp_of (rb_aux args) = rb_spp_aux (app_args spp_of args)"
monom_zero: fixes m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.monom m (0::'a) = 0"
similar_FnD: fixes f :: "Value \<rightarrow> Value"   and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue"   and x :: "Value"   and y :: "C \<rightarrow> CValue" assumes "Fn\<cdot>f \<triangleleft>\<triangleright> CFn\<cdot>g"   and "x \<triangleleft>\<triangleright> y\<cdot>C\<^sup>\<infinity>" shows "f\<cdot>x \<triangleleft>\<triangleright> g\<cdot>y\<cdot>C\<^sup>\<infinity>"