hcju/isabelleps huggingface dataset:
ExistsMsg: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and m :: "('a, 'c) message"   and cfg2 :: "('a, 'c, 'b) configuration" assumes "asynchronousSystem.steps trans sends cfg1 m cfg2" shows "m \<in># msgs cfg1"
that: fixes z :: "'a" assumes "(y, z) \<in> R"   and "(y', z) \<in> R" shows "thesis"
span_def: fixes S :: "'c set" shows "span S = {lincomb a A |a A. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier K}"
tp_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<cdot>\<^sub>r b = b \<cdot>\<^sub>r a"
vector_to_cblinfun_comp_one: fixes s :: "'b" shows "vector_to_cblinfun s o\<^sub>C\<^sub>L 1 = vector_to_cblinfun s"
a_l_repr_independence: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <+ H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "x <+ H = y <+ H"
acquired_reads_pending_write_mono_in: fixes x :: "nat"   and xs :: "'b memref list"   and A :: "nat set" assumes "x \<in> acquired_reads False xs A" shows "x \<in> acquired_reads True xs A"
the_apartment_isoD_3: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'" shows "fixespointwise (the_apartment_iso A A') (- \<Union> A)"
widen1: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<nabla> y"
Ch_unilateral_substitutes: shows "\<forall>h. Y.unilateral_substitutes_on UNIV (Ch h)"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_7: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>) \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<theta>) - set X"
resultant_impl_Lazard_def: shows "resultant_impl_Lazard = resultant_impl_main dichotomous_Lazard"
induct: fixes P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list \<Rightarrow> ('d \<Rightarrow> ('e, 'f) Term.term option) \<Rightarrow> bool"   and a0 :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) list"   and a1 :: "'d \<Rightarrow> ('e, 'f) Term.term option" assumes "\<And>\<sigma>. P [] \<sigma>"   and "\<And>x t P \<sigma>. (\<sigma> x = None \<or> \<sigma> x = Some t \<Longrightarrow> P P (\<sigma>(x \<mapsto> t))) \<Longrightarrow> P ((Var x, t) # P) \<sigma>"   and "\<And>f ss g ts P \<sigma>. (\<And>x2. decompose (Fun f ss) (Fun g ts) = Some x2 \<Longrightarrow> P (x2 @ P) \<sigma>) \<Longrightarrow> P ((Fun f ss, Fun g ts) # P) \<sigma>"   and "\<And>f ss x P \<sigma>. P ((Fun f ss, Var x) # P) \<sigma>" shows "P a0 a1"
subdg_Obj_vsubset: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and a :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
Nml_Src: fixes t :: "'a local.term" assumes "Nml t" shows "Nml (Src t)"
substitute''_NilE_2: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and xs :: "'b list" assumes "substitute'' f T [] xs" obtains "xs = []"
theorem3_part1: assumes "\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'b}" shows "All admits_triangular_reduction"
order_trans: fixes less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'i"   and y :: "'i"   and z :: "'i" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
Bex_def: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "Bex A P = (\<exists>x. x \<in> A \<and> P x)"
lex_ext_stri_imp_nstri: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and m :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "fst (lex_ext f m xs ys)" shows "snd (lex_ext f m xs ys)"
ptensor_mat_dim_col: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_col (st.ptensor_mat m1 m2) = st.d0"
add_le_same_cancel1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and b :: "'e"   and a :: "'e" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus b a) b = less_eq a zero"
select_convs_3: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'a" shows "more \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = more"
signs_smash_def: fixes s1 :: "'a list list"   and s2 :: "'a list list" shows "signs_smash s1 s2 = concat (map (\<lambda>l1. map ((@) l1) s2) s1)"
thetaParTT_BisT: shows "thetaParTT \<subseteq> BisT"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b" shows "(case LNil of LNil \<Rightarrow> f1 | x ## xa \<Rightarrow> f2 x xa) = f1"
span_diff: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> span S"   and "y \<in> span S" shows "x - y \<in> span S"
bounded_iff: fixes b :: "'c"   and c :: "'c"   and a :: "'c" shows "(sup b c \<le> a) = (b \<le> a \<and> c \<le> a)"
box_subset_cbox: fixes a :: "'a"   and b :: "'a" shows "box a b \<subseteq> cbox a b"
size_4: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "size \<langle>x21, x22, x23\<rangle> = size x21 + size x23 + Suc 0"
subdivFace'_holds_minGraphProps: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and ovl :: "nat option list" assumes "pre_subdivFace' g f v' v n ovl"   and "f \<in> \<F> g"   and "minGraphProps g" shows "minGraphProps (subdivFace' g f v n ovl)"
add_left_mono_trans: fixes x :: "'d multiset"   and a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "x \<subseteq># a + b"   and "b \<subseteq># c" shows "x \<subseteq># a + c"
has_integral_on_superset: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and S :: "'a set"   and T :: "'a set" assumes "(f has_integral i) S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'b)"   and "S \<subseteq> T" shows "(f has_integral i) T"
Sup_empty: fixes Inf :: "'h set \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h"   and inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and bot :: "'h"   and top :: "'h" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
preserves_reflects_arr: fixes f :: "'a" shows "hom'.arr (trg f \<star> f \<star> e) = hom.arr f"
closed_sum_zero_left: fixes A :: "'a set" shows "{0::'a} +\<^sub>M A = closure A"
simps_1: fixes uu :: "'b"   and u :: "'a \<Rightarrow> 'b" shows "[[]\<rightarrow>uu]u = u"
SUP_bot: fixes A :: "'e set" shows "Sup_prod ((\<lambda>x. bot_prod) ` A) = bot_prod"
nf_inres_SPEC: fixes \<Phi> :: "'c \<Rightarrow> bool"   and x :: "'c" shows "nf_inres (SPEC \<Phi>) x = \<Phi> x"
intro: fixes \<I> :: "('b, 'c) \<I>"   and I :: "'a \<Rightarrow> bool"   and \<I>' :: "('d, 'e) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a, 'd, 'e) gpv" assumes "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> {s. I s}"   and "\<And>x s. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>" shows "raw_converter_invariant \<I> \<I>' callee I"
coeffs_map_poly': fixes f :: "'b \<Rightarrow> 'c"   and p :: "'b poly" assumes "\<And>x. x \<noteq> (0::'b) \<Longrightarrow> f x \<noteq> (0::'c)" shows "coeffs (map_poly f p) = map f (coeffs p)"
closed_valid_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "valid_path g" shows "closed (path_image g)"
that: fixes xs :: "'a list" assumes "f = list_succ xs"   and "set xs = S"   and "distinct xs" shows "thesis"
pmf_pos: fixes p :: "'b pmf"   and x :: "'b" assumes "pmf p x \<noteq> 0" shows "0 < pmf p x"
decode_encode: fixes e :: "nat"   and n :: "nat" shows "local.decode e (local.encode e n) = n"
absorb_iff2: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (inf a b = b)"
exhaust: fixes y :: "'f \<times> 'g" obtains a :: "'f" and b :: "'g" where "y = (a, b)"
interior_of_def: fixes X :: "'b topology"   and S :: "'b set" shows "X interior_of S \<equiv> {x. \<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> S}"
intro: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "ContractsWithUnilateralSubstitutes Xd Xh Pd Ch"   and "ContractsWithIRC Xd Xh Pd Ch" shows "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch"
fst_transfer: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_prod A B) A fst fst"
True: shows "z \<in> \<real>\<^sub>\<le>\<^sub>0"
assms_2: fixes i :: "'a" assumes "i \<in> J" shows "S i \<in> sets (M i)"
charts_submanifold_def: fixes S :: "'a set" shows "W'.src.charts_submanifold S = restrict_chart S ` charts_submanifold W'"
cases: fixes r :: "'b sparc_state" obtains cpu_reg :: "CPU_register \<Rightarrow> word32" and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32" and sys_reg :: "sys_reg \<Rightarrow> word32" and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option" and mmu :: "MMU_state" and cache :: "cpu_cache" and dwrite :: "(int \<times> word32 \<times> CPU_register) list" and state_var :: "sparc_state_var" and traps :: "Trap set" and undef :: "bool" where "r = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef\<rparr>"
iMOD_iTILL_iMODb_conv: fixes r :: "nat"   and n :: "nat"   and m :: "nat" assumes "r \<le> n" shows "[ r, mod m ] \<inter> [\<dots>n] = [ r, mod m, (n - r) div m ]"
SeqSteps: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and cfg\<^sub>1 :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and cfg\<^sub>2 :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c\<^sub>1' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and c\<^sub>2 :: "('a, 'b, 'c) com" assumes "\<Gamma>\<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* cfg\<^sub>2"   and "cfg\<^sub>1 = (c\<^sub>1, s)"   and "cfg\<^sub>2 = (c\<^sub>1', s')" shows "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, s) \<rightarrow>\<^sup>* (Seq c\<^sub>1' c\<^sub>2, s')"
T_seq: fixes P :: "'a process"   and Q :: "'a process" shows "T (P `;` Q) = {t. \<exists>X. (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> T Q} \<union> {t1 @ t2 |t1 t2. t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> D Q} \<union> {t1 @ t2 |t1 t2. t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {t1 @ t2 |t1 t2. t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"
pos_bounded: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bounded_bilinear prod" shows "\<exists>K>0. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
GF_Inf_many: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<phi>) = (\<exists>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n \<phi>)"
mset_subseqs_size: fixes xs :: "'a list"   and n :: "nat" shows "mset ` {ys \<in> set (subseqs xs). length ys = n} = {ws. ws \<subseteq># mset xs \<and> size ws = n}"
wod_def: fixes n :: "'node"   and n\<^sub>1 :: "'node"   and n\<^sub>2 :: "'node" shows "n \<longrightarrow>\<^sub>w\<^sub>o\<^sub>d n\<^sub>1,n\<^sub>2 \<equiv> n\<^sub>1 \<noteq> n\<^sub>2 \<and> (\<exists>as. n -as\<rightarrow>* n\<^sub>1 \<and> n\<^sub>2 \<notin> set (sourcenodes as)) \<and> (\<exists>as. n -as\<rightarrow>* n\<^sub>2 \<and> n\<^sub>1 \<notin> set (sourcenodes as)) \<and> (\<exists>a. valid_edge a \<and> n = sourcenode a \<and> ((\<exists>as. targetnode a -as\<rightarrow>* n\<^sub>1 \<and> (\<forall>as'. targetnode a -as'\<rightarrow>* n\<^sub>2 \<longrightarrow> n\<^sub>1 \<in> set (sourcenodes as'))) \<or> (\<exists>as. targetnode a -as\<rightarrow>* n\<^sub>2 \<and> (\<forall>as'. targetnode a -as'\<rightarrow>* n\<^sub>1 \<longrightarrow> n\<^sub>2 \<in> set (sourcenodes as')))))"
smulextp_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and m :: "'b multiset"   and n :: "'b multiset" shows "smulextp f m n = ((m, n) \<in> s_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)})"
rep_eq: fixes x :: "(String.literal \<times> event_data list) prefix" shows "msteps_stateless x = msteps0_stateless (Rep_prefix x)"
big_step_t_part_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> int option"   and x2a :: "nat"   and x3a :: "char list \<Rightarrow> int option"   and P :: "com \<Rightarrow> (char list \<Rightarrow> int option) \<Rightarrow> nat \<Rightarrow> (char list \<Rightarrow> int option) \<Rightarrow> bool" assumes "(x1a, x1b) \<Rightarrow>\<^sub>A x2a \<Down> x3a"   and "\<And>s. P SKIP s (Suc 0) s"   and "\<And>a x ps v ps'. \<lbrakk>vars a \<union> {x} \<subseteq> dom ps; paval a ps = v; ps' = ps(x \<mapsto> v)\<rbrakk> \<Longrightarrow> P (x ::= a) ps (Suc 0) ps'"   and "\<And>c1 s1 x s2 c2 y s3 z. \<lbrakk>(c1, s1) \<Rightarrow>\<^sub>A x \<Down> s2; P c1 s1 x s2; (c2, s2) \<Rightarrow>\<^sub>A y \<Down> s3; P c2 s2 y s3; z = x + y\<rbrakk> \<Longrightarrow> P (c1;; c2) s1 z s3"   and "\<And>b ps ps' c1 x y c2. \<lbrakk>vars b \<subseteq> dom ps; dom ps' = dom ps; pbval b ps; (c1, ps) \<Rightarrow>\<^sub>A x \<Down> ps'; P c1 ps x ps'; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) ps y ps'"   and "\<And>b ps ps' c2 x y c1. \<lbrakk>vars b \<subseteq> dom ps; dom ps' = dom ps; \<not> pbval b ps; (c2, ps) \<Rightarrow>\<^sub>A x \<Down> ps'; P c2 ps x ps'; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2) ps y ps'"   and "\<And>b s c. \<lbrakk>vars b \<subseteq> dom s; \<not> pbval b s\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s (Suc 0) s"   and "\<And>b s1 c x s2 y s3 z. \<lbrakk>pbval b s1; vars b \<subseteq> dom s1; (c, s1) \<Rightarrow>\<^sub>A x \<Down> s2; P c s1 x s2; (WHILE b DO c, s2) \<Rightarrow>\<^sub>A y \<Down> s3; P (WHILE b DO c) s2 y s3; 1 + x + y = z\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s1 z s3" shows "P x1a x1b x2a x3a"
dom_in_domains: fixes f :: "'a \<times> 'a" assumes "VV.domains f \<noteq> {}" shows "VV.dom f \<in> VV.domains f"
simps_2: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" shows "input_i_stored_in_reg t' t i (Suc r) = (if is_generalisation_of t' t i (Suc r) then Some (i, Suc r) else input_i_stored_in_reg t' t i r)"
zero_prod_def: shows "0 = (0::'a, 0::'b)"
complex_norm_square: fixes z :: "complex" shows "complex_of_real ((cmod z)\<^sup>2) = z * Complex.cnj z"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c, $Call_indirect j] : ts _> ts'"
root: shows "a * r\<^sup>2 - b * r + c = 0"
lv_2: fixes n :: "nat" assumes "0 < n" shows "real n * lv \<le> v n"
continuous_map_on_empty: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "topspace X = {}" shows "continuous_map X Y f"
dedekind: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z \<le> x \<sqinter> (z \<sqinter> id y) \<sqinter> (y \<sqinter> (id x \<sqinter> z))"
filtration_measurable: fixes f :: "bool stream \<Rightarrow> 'a"   and t :: "nat"   and N :: "'a measure" assumes "f \<in> F t \<rightarrow>\<^sub>M N" shows "random_variable N f"
acyclicity_def: fixes E :: "(node fset \<times> command_gadget \<times> node fset) list" shows "acyclicity E \<equiv> acyclic (\<Union>e\<in>set E. fset (fst3 e) \<times> fset (thd3 e))"
exp_times_self_inj: shows "inj_on (\<lambda>x. x * exp x) {- 1..}"
assms_2: shows "\<gamma>2 C1_differentiable_on {0..1}"
vector_smult_component: fixes c :: "'b"   and y :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(c *s y) $h i = c * y $h i"
word_arith_nat_defs_6: fixes a :: "'a word"   and b :: "'a word" shows "a div b = word_of_nat (unat a div unat b)"
bdd_aboveI: fixes A :: "'d multiset set"   and M :: "'d multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<subseteq># M" shows "subset_mset.bdd_above A"
sat_eq_iff: fixes m :: "'b sat"   and n :: "'b sat" shows "(m = n) = (nat_of m = nat_of n)"
to_int_poly_i_def: fixes ops :: "'b arith_ops_record"   and f :: "'b list" shows "to_int_poly_i ops f = poly_of_list (map (arith_ops_record.to_int ops) f)"
disj: shows "disjoint_family (\<lambda>n. {l n<..r n})"
rel_T_neg_distr_imp: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "((('z, 'n, 'q, 't, 'w, 'aa) T \<times> (('ab, 'p, 's, 'v, 'y, 'aa) T \<Rightarrow> ('z \<times> 'ab, 'o, 'r, 'u, 'x, 'aa) T)) \<times> ((('z, 'n, 'q, 't, 'w, 'aa) T \<times> (('ab, 'p, 's, 'v, 'y, 'aa) T \<Rightarrow> ('z \<times> 'ab, 'o, 'r, 'u, 'x, 'aa) T)) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) T) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) T \<times> 'z \<times> ('z \<times> 'ab) \<times> 'ab \<times> 'aa) itself"   and tytok_T :: "('z \<times> 'ac \<times> 'ab \<times> 'aa) itself" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_T_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_T"
split_w: shows "w = \<alpha> @ snd r @ \<beta>"
Zfun_le: fixes g :: "'d \<Rightarrow> 'e"   and F :: "'d filter"   and f :: "'d \<Rightarrow> 'f" assumes "Zfun g F"   and "\<forall>x. norm (f x) \<le> norm (g x)" shows "Zfun f F"
3: shows "mat (row_length B1) (length B2) (B1 \<circ> B2)"
finiteI_independent: fixes B :: "'b set" assumes "vs1.independent B" shows "finite B"
tr_par_proj: fixes B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and n :: "'lbl" assumes "B \<in> set (tr\<^sub>p\<^sub>c A D)" shows "proj n B \<in> set (tr\<^sub>p\<^sub>c (proj n A) (proj n D))"
closure_is_csubspace: fixes A :: "'b set" assumes "csubspace A" shows "csubspace (closure A)"
retraction_map_product_projection: fixes i :: "'c"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd topology" assumes "i \<in> I" shows "retraction_map (product_topology X I) (X i) (\<lambda>x. x i) = (topspace (product_topology X I) = {} \<longrightarrow> topspace (X i) = {})"
Fract_less_zero_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(Fract a b < 0) = (a < 0)"
vars_scaleRat1: fixes c :: "rat"   and p :: "linear_poly" shows "vars (c *R p) \<subseteq> vars p"
new_conjoin: fixes c :: "'a"   and G :: "('a, 'b) form list" assumes "new c (conjoin G)" shows "news c G"
componentsI: fixes x :: "'b"   and U :: "'b set" assumes "x \<in> U" shows "connected_component_set U x \<in> components U"
check_poly_ge: fixes p :: "('b monom \<times> 'c) list"   and q :: "('b monom \<times> 'c) list" assumes "check_poly_ge p q" shows "p \<ge>p q"
Sum2: fixes Q :: "pi"   and V :: "residual"   and P :: "pi" assumes "Q \<longmapsto> V" shows "P \<oplus> Q \<longmapsto> V"
inorder2_leaves: fixes as :: "'a list" shows "inorder2 (leaves as) = as"
prems_1: fixes ys :: "'a list"   and zs :: "'a list" assumes "a # xs = ys @ x # zs" shows "thesis"
rat_to_normalized_int_poly_def: fixes p :: "rat poly" shows "rat_to_normalized_int_poly p \<equiv> if p = 0 then (1, 0) else case rat_to_int_poly p of (s, q) \<Rightarrow> (rat_of_int (content q) / rat_of_int s, primitive_part q)"
const_lookup_suc: fixes p :: "real"   and x :: "nat"   and i :: "nat" shows "isolate_variable_sparse (Const p) x (Suc i) = 0"
a_in_principal: fixes R :: "('e, 'f) Ring_scheme"   and a :: "'e" assumes "Ring R"   and "a \<in> carrier R" shows "a \<in> R \<diamondsuit>\<^sub>p a"
can_syncE: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>2" obtains taa :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" and x' :: "'x2" and m' :: "'m2" where "r2.r_syntax t x m taa x' m'" and "LT = collect_waits taa"
pre: fixes s :: "char list \<Rightarrow> int" shows "pre C (\<lambda>a. I a + 1) s \<le> I s + \<up> (bval b s)"
out_arcs_dv: shows "out_arcs (del_vert u) = out_arcs G"
rGamma_series_real: fixes x :: "real" shows "\<forall>\<^sub>F n in sequentially. rGamma_series x n = Re (rGamma_series (complex_of_real x) n)"
simps_1: fixes data_w32 :: "word32"   and ur :: "word5"   and state :: "'b sparc_state" shows "global_reg_mod data_w32 0 ur state = state"
Un_ac_1: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
size_if_complete: fixes t :: "'c tree" assumes "complete t" shows "size t = 2 ^ height t - 1"
construct_sa_def: fixes cnf :: "lit uprod set" shows "construct_sa cnf \<equiv> finite (vars_of_cnf cnf) \<and> (\<nexists>x. (Pos x, Neg x) \<in> (imp_graph cnf)\<^sup>* \<and> (Neg x, Pos x) \<in> (imp_graph cnf)\<^sup>*) \<and> is_2sat cnf"
assms_3: shows "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))"
write_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"
proper_map_polyfun: fixes S :: "'a set"   and K :: "'a set"   and c :: "nat \<Rightarrow> 'a"   and i :: "nat"   and n :: "nat" assumes "closed S"   and "compact K"   and "c i \<noteq> (0::'a)"   and "1 \<le> i"   and "i \<le> n" shows "compact (S \<inter> {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K})"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> orb" shows "\<exists>A. Says A B X \<in> set evs"
irreducible_multD: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.idom times one plus zero minus uminus"   and "comm_semiring_1.irreducible times one zero (times a b)" shows "dvd.dvd times a one \<and> comm_semiring_1.irreducible times one zero b \<or> dvd.dvd times b one \<and> comm_semiring_1.irreducible times one zero a"