hcju/isabelleps huggingface dataset:
invar_rel_internal: fixes a :: "'i \<Rightarrow> 'h set"   and X :: "('g \<times> 'i) set"   and S :: "('f \<times> 'h set) set" shows "invar_rel a X S = {((x, s'), y). \<exists>s. (s', s) \<in> X \<and> (x, y) \<in> S \<and> y \<subseteq> a s}"
assms: shows "\<forall>s\<in>props p. i1 s = i2 s"
chain_def: fixes A :: "'e set set"   and C :: "'e set set" shows "subset.chain A C = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"
pseudo_mod_2: fixes g :: "'a poly"   and f :: "'a poly" assumes "g \<noteq> 0" shows "pseudo_mod f g = 0 \<or> degree (pseudo_mod f g) < degree g"
simps: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'b" shows "r\<^sup>+\<^sup>+ a1 a2 = ((\<exists>a b. a1 = a \<and> a2 = b \<and> r a b) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>+\<^sup>+ a b \<and> r b c))"
UP_pre_univ_prop_def: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "UP_pre_univ_prop R S h \<equiv> ring_hom_cring R S h \<and> UP_cring R"
hyps: fixes p :: "'p"   and sb' :: "'p memref list" assumes "history_consistent \<theta> (hd_prog p (sb\<^sub>1 @ sb')) (sb\<^sub>1 @ sb')"   and "last_prog p (sb\<^sub>1 @ sb') = p" shows "last_prog (hd_prog p (sb\<^sub>1 @ sb')) sb\<^sub>1 = hd_prog p sb'"
select_convs_7: fixes Obj :: "'e set"   and Mor :: "'f set"   and Dom :: "'f \<Rightarrow> 'e"   and Cod :: "'f \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'f"   and Comp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and more :: "'d" shows "Category.more \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = more"
fi: shows "f ` [H]\<^bsup>n\<^esup> \<subseteq> {i}"
apps_eq_Abs_conv: fixes s :: "dB"   and ss :: "dB list"   and r :: "dB" shows "(s \<degree>\<degree> ss = Abs r) = (s = Abs r \<and> ss = [])"
ax_no_hazard: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" assumes "G,A\<turnstile>{P \<and>. type_ok G t} t\<succ> {Q}" shows "G,A\<turnstile>{P} t\<succ> {Q}"
continuous_at_Inf_mono: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "mono f"   and "continuous (at_right (Inf S)) f"   and "S \<noteq> {}"   and "bdd_below S" shows "f (Inf S) = Inf (f ` S)"
S_Q_def: shows "S_Q = S_M S (Q_of_state (Liminf_state Sts))"
refl: fixes a :: "complex" shows "a \<le>\<^sub>\<complex> a"
vsv_vunion: fixes r :: "V"   and s :: "V" assumes "vsv r"   and "vsv s"   and "\<D>\<^sub>\<circ> r \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> s = []\<^sub>\<circ>" shows "vsv (r \<union>\<^sub>\<circ> s)"
multLessState_def: shows "multLessState \<equiv> {(stateA, stateB). getM stateA = getM stateB \<and> getConflictFlag stateA = getConflictFlag stateB \<and> (getC stateA, getC stateB) \<in> multLess (getM stateA)}"
val2FormEl: fixes literal :: "Literal"   and valuation :: "Literal list" shows "literal el valuation = [literal] el val2form valuation"
fixp_strong_induct_uc: fixes U :: "'f \<Rightarrow> 'd \<Rightarrow> 'e spmf"   and F :: "'f \<Rightarrow> 'f"   and C :: "('d \<Rightarrow> 'e spmf) \<Rightarrow> 'f"   and f :: "'f"   and P :: "('d \<Rightarrow> 'e spmf) \<Rightarrow> bool" assumes "\<And>x. spmf.mono_body (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (spmf.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "spmf.admissible P"   and "P (\<lambda>_. return_pmf None)"   and "\<And>f'. \<lbrakk>P (U f'); spmf.le_fun (U f') (U f)\<rbrakk> \<Longrightarrow> P (U (F f'))" shows "P (U f)"
simps_4: fixes \<phi> :: "'b ltl" shows "\<^bold>G (G \<phi>) = \<^bold>G \<phi> \<union> {G \<phi>}"
embedding_lemma_iii: fixes \<pi> :: "'a strips_operator list"   and \<Pi> :: "'a strips_problem" assumes "\<forall>op\<in>set \<pi>. op \<in> set (\<Pi>\<^sub>\<O>)" shows "\<forall>ops\<in>set (List_Supplement.embed \<pi>). \<forall>op\<in>set ops. op \<in> set (\<Pi>\<^sub>\<O>)"
fiter2: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_iteration.fiter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (finite_iteration.fiter seq nil c) = finite_iteration.fiter seq nil c"
linorder_cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "linorder less_eq less" obtains "less x y"   | "x = y"   | "less y x"
nsum_ideal_inc: fixes A :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "\<forall>i\<le>n. f i \<in> A" shows "\<Sigma>\<^sub>e R f n \<in> A"
sort_poly_spec_id': fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> unsorted_poly_rel_with0" shows "sort_poly_spec p \<le> \<Down> (sorted_repeat_poly_list_rel_with0 (rel2p (Id \<union> term_order_rel))) (RETURN p')"
constructive_security_sim_obsf: shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common \<A> adv"
exhaust: fixes y :: "val" obtains "y = Unit"   | "y = Null"   | x3 :: "bool" where "y = Bool x3"   | x4 :: "int" where "y = Intg x4"   | x5 :: "nat" where "y = Addr x5"
semiring_norm_137: shows "\<not> (0::'a) < (0::'a)"
unit_factor: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes> b \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<in> Units G"
simps: fixes f :: "'d"   and g :: "'e set"   and c :: "('e, 'f, 'd) com" shows "dest_Guard (Guard f g c) = (f, g, c)"
assms_4: shows "pathfinish p = a - complex_of_real e"
not_lfinite_lprefix_conv_eq: fixes xs :: "'f llist"   and ys :: "'f llist" assumes "\<not> lfinite xs" shows "lprefix xs ys = (xs = ys)"
isFDERIV_uptD: fixes n :: "nat"   and fas :: "floatarith list"   and vs :: "real list"   and i :: "nat"   and j :: "nat" assumes "isFDERIV n [0..<n] fas vs"   and "i < n"   and "j < n" shows "isDERIV i (fas ! j) vs"
senior_states_card_le: fixes p :: "'b"   and n :: "nat"   and q :: "'b" shows "(card (senior_states p n) < card (senior_states q n)) = (senior_states p n \<subset> senior_states q n)"
simps_1: fixes l :: "'b llist" shows "l \<down> 0 = LNil"
nonterminating_implies_div_free: fixes P :: "'b process" assumes "non_terminating P" shows "D P = {}"
exists_SeqAppendP: fixes g :: "name"   and f1 :: "tm"   and k1 :: "tm"   and f2 :: "tm"   and k2 :: "tm"   and H :: "fm set" assumes "atom g \<sharp> (f1, k1, f2, k2)" shows "H \<turnstile> SyntaxN.Ex g (SeqAppendP f1 k1 f2 k2 (Var g))"
c_top_welldefTr4: fixes N :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "x \<in> N \<bullet> (a \<cdot> b)" shows "x \<in> c_top G N (N \<bullet> a) (N \<bullet> b)"
Inter_iff: fixes A :: "'e"   and C :: "'e set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
pt_inject_on_ineq: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "inj_on ((\<bullet>) pi) X"
dbm_int_abstr: fixes g :: "('c, 'd) cconstraint"   and n :: "nat"   and v :: "'c \<Rightarrow> nat" assumes "\<forall>(x, m)\<in>collect_clock_pairs g. m \<in> \<int>" shows "dbm_int (abstr g (\<lambda>i j. \<infinity>) v) n"
bind_assoc: fixes m :: "('c, 'f) state"   and f :: "'f \<Rightarrow> ('c, 'e) state"   and g :: "'e \<Rightarrow> ('c, 'd) state" shows "m \<bind> f \<bind> g = m \<bind> (\<lambda>x. f x \<bind> g)"
mk_minsky_loop: fixes P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and v :: "nat"   and Q :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" assumes "mk_minsky P"   and "\<And>vs vs'. P vs vs' \<Longrightarrow> vs' v = vs v"   and "\<And>vs. vs v = 0 \<Longrightarrow> Q vs vs"   and "\<And>n vs vs' vs''. \<lbrakk>vs v = Suc n; P (\<lambda>x. if x = v then n else vs x) vs'; Q vs' vs''\<rbrakk> \<Longrightarrow> Q vs vs''" shows "mk_minsky Q"
mpt_translation: fixes c :: "'b" shows "mpt lborel (\<lambda>x. x + c)"
mult_not_zero: fixes a :: "'a"   and b :: "'a" assumes "a * b \<noteq> (0::'a)" shows "a \<noteq> (0::'a) \<and> b \<noteq> (0::'a)"
wlp_while_split: fixes M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and P :: "complex mat" assumes "well_com (While M S)"   and "is_quantum_predicate P" shows "wlp (While M S) P = adjoint (M 0) * P * M 0 + adjoint (M 1) * wlp S (wlp (While M S) P) * M 1"
length_tms_of: fixes aprog :: "abc_inst list" shows "length (tms_of aprog) = length aprog"
TYRELI: fixes R :: "('c \<times> 'd) set" shows "TYREL R"
mem_UnitsI: fixes u :: "'a" assumes "invertible u"   and "u \<in> M" shows "u \<in> Units"
apex_def: shows "\<epsilon>.cod.apex \<equiv> C.dom \<epsilon>.cod.leg0"
le_add_same_cancel2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b + a) = ({#} \<subseteq># b)"
of_nat_eq_iff: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
preal_downwards_closed: fixes A :: "rat set"   and y :: "rat"   and z :: "rat" assumes "Dedekind_Real.cut A"   and "y \<in> A"   and "0 < z"   and "z < y" shows "z \<in> A"
iMODb_card_if: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "card [ r, mod m, c ] = (if m = 0 then Suc 0 else Suc c)"
normal_form_disjunct_length_upper_bound_2: fixes M :: "'b ltln set"   and \<phi> :: "'b ltln"   and N :: "'b ltln set" assumes "M \<subseteq> subformulas\<^sub>\<mu> \<phi>"   and "N \<subseteq> subformulas\<^sub>\<nu> \<phi>" shows "normal_form_disjunct_with_flatten_pi_2_length \<phi> M N \<le> 2 ^ (size \<phi> + 1) + 2 * (size \<phi> + 2)\<^sup>2"
assms_3: shows "Broadcast (i2, Insert e2 n2) \<in> set (history j)"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat list"   and y :: "(nat \<times> nat) convnet" assumes "deep_model x xa xb = y" obtains Y :: "nat" and r :: "nat" and rs :: "nat list" where "x = Y" and "xa = r" and "xb = rs" and "y = Conv (Y, r) (deep_model' r rs)"
starset_def: fixes A :: "'a set" shows "*s* A = Iset (star_of A)"
oarrivemsg_simps_1: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and R :: "nat set"   and m :: "'b" shows "oarrivemsg P \<sigma> (R:*cast(m)) = True"
reduced_word_for_lists: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list" assumes "reduced_word_for A a as" shows "as \<in> lists A"
left_treeE: fixes G :: "('c, 'd) Graph_scheme"   and u :: "'c"   and s :: "'c"   and t :: "'c" assumes "Tree G"   and "u \<in> Tree.left_tree G s t" shows "u \<in> V\<^bsub>G\<^esub> \<and> s \<in> set (u \<leadsto>\<^bsub>G\<^esub> t)"
zip_def: shows "zip \<equiv> \<lambda>uu uua. rec_list (\<lambda>xs. []) (\<lambda>y ys ysa xs. case xs of [] \<Rightarrow> [] | z # zs \<Rightarrow> (z, y) # ysa zs) uua uu"
nsub_RIs_iconfs: fixes es :: "'c exp list"   and sh :: "sheap" assumes "\<not> sub_RIs es" shows "iconfs sh es"
quasiconvex_mono: fixes C :: "real"   and D :: "real"   and G :: "'b set" assumes "C \<le> D"   and "quasiconvex C G" shows "quasiconvex D G"
unique_connecting_path_properties_4: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "last (v \<leadsto>\<^bsub>G\<^esub> w) = w"
exhaust_sel: fixes prot_transaction :: "('e, 'f, 'g, 'h) prot_transaction" obtains "prot_transaction = Transaction (transaction_fresh prot_transaction) (transaction_receive prot_transaction) (transaction_selects prot_transaction) (transaction_checks prot_transaction) (transaction_updates prot_transaction) (transaction_send prot_transaction)"
bij_betw_trans: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and C :: "'c set" assumes "bij_betw f A B"   and "bij_betw g B C" shows "bij_betw (g \<circ> f) A C"
matrix_left_invertible_injective: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(\<exists>B. B ** A = mat (1::'a)) = inj ((*v) A)"
LAss1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and V :: "nat"   and a :: "'addr"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>V:=Throw a,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
P_no_deadend_v0: assumes "\<not> lnull (ltl P)" shows "\<not> deadend v0"
a12: shows "let dsc = sqrt (Re ((D - A)\<^sup>2 + 4 * (B * cnj B))) in a = (A - D + cor dsc) / (2 * C) \<or> a = (A - D - cor dsc) / (2 * C)"
heap_is_wellformed_children_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "child |\<in>| node_ptr_kinds h"
x2: shows "\<not> P (nP e1) e1 \<bottom> e2"
Collect_mem_eq: fixes A :: "'h set" shows "{x. x \<in> A} = A"
simps: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop6_exit x (l, r) = (\<exists>i j. i + j = x \<and> 0 < j \<and> (l, r) = (Oc \<up> i, Oc # Bk \<up> j @ Oc \<up> j))"
AxiomsSetMcL_axioms: shows "AxiomsSetMcL"
extg_range_prod: fixes i1 :: "('d \<times> 'e) set"   and i2 :: "'f set" shows "extg ` (i1 \<times> i2) = {(\<sigma>, s1, s2) |\<sigma> s1 s2. (\<sigma>, s1) \<in> i1 \<and> s2 \<in> i2}"
arr_iff_in_hom: fixes f :: "'a" shows "Left_a.arr f = Left_a.in_hom f (Left_a.dom f) (Left_a.cod f)"
Nmlize_red2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Nml t"   and "Nml u"   and "Src t = Trg u" shows "\<^bold>\<lfloor>t \<^bold>\<Down> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t \<^bold>\<star> u\<^bold>\<rfloor>"
g_square: fixes m :: "nat" shows "g (m\<^sup>2) = moebius_mu m"
onorm_scaleR_left_lemma: fixes r :: "'d \<Rightarrow> real"   and f :: "'e" assumes "bounded_linear r" shows "onorm (\<lambda>x. r x *\<^sub>R f) \<le> onorm r * norm f"
disjSD_2_def: fixes x :: "('a set set, 'b) Combinators"   and y :: "('a set set, 'c) Combinators" shows "disjSD_2 x y = (\<forall>a b c d. (a, b) \<in> sdnets x \<and> (c, d) \<in> sdnets y \<longrightarrow> twoNetsDistinct a b c d \<and> twoNetsDistinct a b d c)"
is_continuation_continue: fixes k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "is_continuation k u q ts"   and "charslength ts = 0"   and "t \<in> \<Z> k (Suc u)"   and "admissible (q @ ts @ [t])" shows "is_continuation k u q (ts @ [t])"
invol_dual: shows "\<partial> \<circ> \<partial> = id"
gen: fixes S :: "variable set" assumes "finite S"   and "S \<subseteq> - FVF \<phi>" shows "(\<omega>' \<in> fml_sem I \<phi>) = (stateinterpol \<omega> \<omega>' S \<in> fml_sem I \<phi>)"
left_diff_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ring plus zero minus uminus times" shows "times (minus a b) c = minus (times a c) (times b c)"
subsequence_prefix_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'a"   and j :: "nat" shows "prefix (j - i) (suffix i w) = w [i \<rightarrow> j]"
isViewOn_def: fixes \<V> :: "'a V_rec"   and E :: "'a set" shows "isViewOn \<V> E \<equiv> V_valid \<V> \<and> V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub> \<union> C\<^bsub>\<V>\<^esub> = E"
abs_neg_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (uminus one) = one"
us2: shows "unat (2 ^ s2 * word_of_nat nq) = 2 ^ s2 * nq"
preprocess_tableau_normalized: fixes cs :: "('i \<times> 'a ns_constraint) list"   and t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and trans_v :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and U :: "'i list" assumes "preprocess cs = (t, as, trans_v, U)" shows "\<triangle> t"
QSPS: fixes s :: "'a"   and s' :: "'a"   and a :: "'c" assumes "QS s s' a" shows "PS s s' a"
leg_simps_4: shows "C.dom \<mu>.dom.leg1 = \<mu>.dom.apex"
P2: shows "X \<noteq> Y \<and> A A' ReflectL X Y"
ex_incl_iff_subset: fixes a :: "'s"   and b :: "'s" assumes "ide a"   and "ide b" shows "(\<exists>f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> incl f) = (local.set a \<subseteq> local.set b)"
truncate_down_ge1: fixes x :: "real"   and p :: "nat" assumes "1 \<le> x" shows "1 \<le> truncate_down p x"
Rn: assumes "p = []" shows "hn_refine (hn_ctxt (list_assn P) p p' * F) f1' (hn_ctxt XX1 p p' * \<Gamma>1') R f1"
sub1_preserves_triangle: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub1 q k l (r, A) = (r', A')"   and "local.triangular_to l A"   and "l < dim_row A"   and "l < dim_col A"   and "l + k < dim_row A" shows "local.triangular_to l A'"
simple_rule_dtor_def: fixes r :: "'b simple_rule" shows "simple_rule_dtor r \<equiv> case r of SimpleRule m a \<Rightarrow> (m, a)"
sigma_Inter: fixes a :: "nat \<Rightarrow> 'b set"   and A :: "'b set set" assumes "\<And>i. a i \<in> sigma A" shows "\<Inter> (range a) \<in> sigma A"
Collect_mono_iff: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
strategy_attracts_does_not_leave: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set"   and v :: "'c"   and w :: "'c" assumes "ParityGame G"   and "ParityGame.strategy_attracts G p \<sigma> A W"   and "ParityGame.strategy G p \<sigma>"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<in> A - W"   and "w \<notin> A \<union> W" shows "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<and> \<sigma> v \<noteq> w"
limsup_INF_SUP: fixes f :: "nat \<Rightarrow> 'c" shows "limsup f = (INF n. Sup (f ` {n..}))"
next_plane0_incr_faceListAt: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "minGraphProps g" shows "|faceListAt g| \<le> |faceListAt g'| \<and> (\<forall>v<|faceListAt g|. |faceListAt g ! v| \<le> |faceListAt g' ! v|)"
\<omega>_limit_set_in_compact_existence: fixes K :: "'a set"   and x :: "'a"   and y :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K"   and "y \<in> rev.\<omega>_limit_set x" shows "rev.existence_ivl0 y = UNIV"
Gromov_hyperbolic_subsetI: fixes A :: "'b set"   and delta :: "real" assumes "\<And>x y z t. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; t \<in> A\<rbrakk> \<Longrightarrow> dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z) + 2 * delta" shows "Gromov_hyperbolic_subset delta A"
x: fixes \<Gamma> :: "(char list \<times> 'a rule list) list"   and rs :: "'a rule list" assumes "sanity_wf_ruleset \<Gamma>"   and "rs \<in> ran (map_of \<Gamma>)" shows "wf_chain (map_of \<Gamma>) rs"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and acc :: "'b list" shows "start_remove (x # xs) y acc = (if x = y then rev acc @ remove_cycles xs y [y] else start_remove xs y (x # acc))"
bdd_above_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq A = (\<exists>M. \<forall>x\<in>A. less_eq x M)"
compass_reflect_compass_5: shows "apply_cltn2 K2_centre meridian_reflect = K2_centre"
unity_root_add: fixes k :: "nat"   and m :: "int"   and n :: "int" shows "unity_root k (m + n) = unity_root k m * unity_root k n"
cblinfun_compose_1_left: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "1 o\<^sub>C\<^sub>L x = x"
App_eq_foldl_conv: fixes r :: "dB"   and s :: "dB"   and t :: "dB"   and ts :: "dB list" shows "(r \<degree> s = t \<degree>\<degree> ts) = (if ts = [] then r \<degree> s = t else \<exists>ss. ts = ss @ [s] \<and> r = t \<degree>\<degree> ss)"
zero_hom_0_axioms: shows "zero_hom_0 hom"
ir_While_triv: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and Q\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> \<not> bval b s \<and> \<not> bval b' s') SKIP SKIP Q\<^sub>2" shows "ir_hoare P (WHILE b DO c) (WHILE b' DO c') Q\<^sub>2"
rep_eq: shows "apply_bcontfun 0 = (\<lambda>_. 0::'d)"
add_mult_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and v :: "'b Matrix.vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A + B) *\<^sub>v v = A *\<^sub>v v + B *\<^sub>v v"
lfinite_chain_imp_rtranclp_lhd_llast: fixes xs :: "'b llist"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "lfinite xs"   and "chain R xs" shows "R\<^sup>*\<^sup>* (lhd xs) (llast xs)"
da_good_approx: fixes Env :: "env"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and T :: "tys"   and A :: "assigned" assumes "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "Env\<turnstile>t\<Colon>T"   and "Env\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "wf_prog (prg Env)" shows "(normal s1 \<longrightarrow> nrm A \<subseteq> dom (locals (snd s1))) \<and> (\<forall>l. fst s1 = Some (Jump (Break l)) \<and> normal s0 \<longrightarrow> brk A l \<subseteq> dom (locals (snd s1))) \<and> (fst s1 = Some (Jump Ret) \<and> normal s0 \<longrightarrow> Result \<in> dom (locals (snd s1)))"
hyps: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and ts' :: "'a \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "thread_oks ts tas = thread_oks ts' tas"
A_def: shows "A = (\<Union>n. {CHR ''a'' ^^^ n @ CHR ''b'' ^^^ n})"
ennreal_add_less_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a + b < top) = (a < top \<and> b < top)"
space_empty: fixes M :: "'b measure" assumes "space M = {}" shows "M = count_space {}"
WellDefinedExprInterp: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t" assumes "Sig iS \<triangleright> Vx : A \<turnstile> e : B" shows "\<exists>i. L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> i"
P_upper_bound: fixes x :: "real" assumes "0 < x" shows "P x \<le> 1 / (12 * x\<^sup>2)"
lc_eq_zero_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "(lc p = (0::'c)) = (p = 0)"
rtrancl_is_UN_relpow: fixes R :: "('b \<times> 'b) set" shows "R\<^sup>* = \<Union> (range ((^^) R))"
states_wellformed: shows "\<K>\<^sub>s \<noteq> []"