hcju/isabelleps huggingface dataset:
refl_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. x \<^bsub>r\<^esub>\<preceq> x))"
app_nterm_def: fixes t :: "nterm"   and u :: "nterm" shows "app t u = t $\<^sub>n u"
vsplit_def: fixes c :: "V \<Rightarrow> V \<Rightarrow> 'a" shows "vsplit c \<equiv> \<lambda>p. c (vfst p) (vsnd p)"
bdd_aboveI2: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and M :: "'d" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "preorder.bdd_above less_eq (f ` A)"
minus_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "fset (xs - ys) = fset xs - fset ys"
assms: shows "\<forall>a\<in>supp x. p \<bullet> a = q \<bullet> a"
seq_char: fixes Obj :: "'b set"   and Null :: "'b"   and f :: "'b"   and g :: "'b" assumes "discrete_category Obj Null" shows "partial_magma.arr (discrete_category.comp Obj Null) (discrete_category.comp Obj Null f g) = (partial_magma.ide (discrete_category.comp Obj Null) f \<and> f = g)"
sorted_iff_nth_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> less_eq (xs ! i) (xs ! j))"
simps: fixes a :: "int \<Rightarrow> 'b"   and l :: "int"   and h :: "int" shows "lran a l h = (if l < h then a l # lran a (l + 1) h else [])"
idiff_infinity_right: fixes a :: "nat" shows "enat a - \<infinity> = 0"
bind_game_alt_def: fixes key_gen :: "('a \<times> 'b) spmf"   and verify :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and valid_msg :: "'c \<Rightarrow> bool"   and \<A> :: "'a \<Rightarrow> ('d \<times> 'c \<times> 'e \<times> 'c \<times> 'e) spmf" shows "abstract_commitment.bind_game key_gen verify valid_msg \<A> = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A> ck \<bind> (\<lambda>(c, m, d, m', d'). assert_spmf (m \<noteq> m' \<and> valid_msg m \<and> valid_msg m') \<bind> (\<lambda>_. let b = verify vk m c d; b' = verify vk m' c d' in assert_spmf (b \<and> b') \<bind> (\<lambda>_. return_spmf True)))) ELSE return_spmf False"
lcm_C: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a r\<rightarrow> b = (1::'a))"
abs_Tree\<^sub>p_eq: fixes p :: "perm"   and t :: "('a, 'b, 'c, 'd) Tree" shows "abs_Tree\<^sub>p (p \<bullet> t) = abs_Tree\<^sub>p t"
t1: shows "as' ! i \<otimes> (aa_1 \<otimes> aa_2) \<sim> a"
diff: shows "xs \<noteq> ys \<or> x \<noteq> y"
ord_isom_Pre11: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and a :: "'c"   and f :: "'c \<Rightarrow> 'e" assumes "Worder D"   and "Worder E"   and "a \<in> carrier D"   and "ord_isom D E f" shows "ExPre D a = ExPre E (f a)"
fiter_def: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> \<equiv> gfp (\<lambda>x::'a::refinement_lattice. nil \<sqinter> c ; x)"
II: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> \<alpha> m"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>\<alpha> m - it. k \<le> j" shows "P \<sigma>"
irreducible_monic_factor: fixes p :: "'a poly" assumes "0 < degree p" shows "\<exists>q r. irreducible q \<and> p = q * r \<and> monic q"
br'_initial_abs: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br'_\<alpha> ` br'_initial \<delta> = {br_initial \<delta>}"
pts_symbolic_c_preserves_constr_disj_vars: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"   and "fv\<^sub>e\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>e\<^sub>s\<^sub>t \<A> = {}" shows "fv\<^sub>e\<^sub>s\<^sub>t \<A>' \<inter> bvars\<^sub>e\<^sub>s\<^sub>t \<A>' = {}"
funpow_shift1: fixes bs :: "'a list"   and n :: "nat"   and p :: "poly" shows "\<lparr>(shift1 ^^ n) p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>Mul (Pw (Bound 0) n) p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
induct: fixes P :: "('c, 'd) strand_step \<Rightarrow> bool"   and a0 :: "('c, 'd) strand_step" assumes "\<And>t. P (send\<langle>t\<rangle>\<^sub>s\<^sub>t)"   and "\<And>t. P (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)"   and "\<And>uu_ t t'. P \<langle>uu_: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t"   and "\<And>X F. P \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t" shows "P a0"
finprod_cong: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'a" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier G) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finprod G f A = finprod G g B"
bdd_above_setle: fixes A :: "'a set" shows "bdd_above A = (\<exists>a. A *<= a)"
Rf_def: fixes N :: "'a literal multiset set" shows "sr.Rf N = {C. \<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD \<longrightarrow> I \<Turnstile> C) \<and> (\<forall>D. D \<in># DD \<longrightarrow> D < C)}"
atLeastLessThan_iff: fixes i :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
w: shows "weight (Fun g ts) \<le> sum_list (map weight (scf_list (scf (f, length (bef @ s # aft))) (bef @ s # aft)))"
log1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and op0 :: "lop"   and e2 :: "exp0"   and e' :: "exp0"   and bv :: "'b state \<times> (v, v) result" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> do_log op0 v1 e2 = Some (Exp e') \<and> evaluate ck env s2 e' bv" shows "evaluate ck env s1 (Log op0 e1 e2) bv"
hyps_3: shows "S closed_in \<Gamma> \<and> T closed_in \<Gamma>"
ch2ch_fst: fixes Y :: "nat \<Rightarrow> 'b \<times> 'c" assumes "chain Y" shows "chain (\<lambda>i. fst (Y i))"
order_trans: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and z :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<preceq> y"   and "y \<preceq> z" shows "x \<preceq> z"
ex_comm: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
simps_1: fixes dimen :: "nat" shows "matr_option dimen None = 1\<^sub>m dimen"
pred_inject_1: fixes P :: "'c \<Rightarrow> bool" shows "list_all P []"
os_prepend_impl: shows "imp_list_prepend os_list os_prepend"
ev_iff_sdrop: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" shows "ev P \<omega> = (\<exists>m. P (sdrop m \<omega>))"
wp_inv_def: fixes G :: "'b \<Rightarrow> bool"   and body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and I :: "'b \<Rightarrow> real" shows "wp_inv G body I = (\<forall>s. \<guillemotleft> G \<guillemotright> s * I s \<le> wp body I s)"
revis_consistency: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and ".\<not> \<phi> \<notin> Cn {}" shows "\<bottom> \<notin> K \<^bold>* \<phi>"
Sapp: shows "t = x41_ $\<^sub>s x42_"
bin_rest_numeral_simps_1: shows "0 div 2 = 0"
fmfilter_alt_defs_2: fixes A :: "'a set" shows "fmdrop_set A = fmfilter (\<lambda>a. a \<notin> A)"
execn_Normal_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> =n\<Rightarrow> t" obtains s' :: "('e, 'f) xstate" where "s \<in> b" and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> =n\<Rightarrow> t"   | "t = Normal s" and "s \<notin> b"
weakly_preferred_by_men_strictly_preferred_by_a_man: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set"   and \<mu>' :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw"   and "StableMarriage.match Pm Pw \<mu>"   and "StableMarriage.match Pm Pw \<mu>'" shows "StableMarriage.weakly_preferred_by_men Pm \<mu> \<mu>' = (\<not> StableMarriage.strictly_preferred_by_a_man Pm \<mu>' \<mu>)"
present_def: fixes h :: "heap"   and a :: "'a array" shows "Array.present h a = (addr_of_array a < lim h)"
the1_equality': fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<exists>\<^sub>\<le>\<^sub>1 x. P x"   and "P a" shows "(THE x. P x) = a"
mem: shows "x \<in> {True # w |w. w \<in> gener_bool_list n} \<union> {False # w |w. w \<in> gener_bool_list n}"
rel_simps_70: shows "\<not> (0::'a) < (0::'a)"
prime_factorization_unique': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<forall>p\<in>#M. normalization_semidom.prime zero times one normalize p"   and "\<forall>p\<in>#N. normalization_semidom.prime zero times one normalize p"   and "comm_monoid_mult.prod_mset times one {#i. i \<in># M#} = comm_monoid_mult.prod_mset times one {#i. i \<in># N#}" shows "M = N"
intro: assumes "\<And>x y. P x y = (PP x y \<or> x = y)"   and "\<And>x y. O x y = (\<exists>z. P z x \<and> P z y)"   and "\<And>x y z. \<lbrakk>PP x y; PP y z\<rbrakk> \<Longrightarrow> PP x z"   and "\<And>y x. PP y x \<Longrightarrow> \<exists>z. P z x \<and> \<not> O z y" shows "MM3"
compatible_r_def: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('a \<times> 'a) set" shows "compatible_r R1 R2 \<equiv> R2 O R1 \<subseteq> R2"
ltl_prop_implies_equiv: fixes \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "(\<phi> \<longrightarrow>\<^sub>P \<psi> \<and> \<psi> \<longrightarrow>\<^sub>P \<phi>) = \<phi> \<equiv>\<^sub>P \<psi>"
left_inverse: fixes x :: "'a" shows "inverse x \<odot> x = one"
rel_with_source_impl_SRel_weakly_preserves_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and SRel :: "('procS \<times> 'procS) set" assumes "rel_weakly_preserves_barbs Rel (STCalWB SWB TWB)"   and "\<forall>S1 S2. (S1, S2) \<in> SRel \<longrightarrow> (SourceTerm S1, SourceTerm S2) \<in> Rel" shows "rel_weakly_preserves_barbs SRel SWB"
dim_row_transfer_rule: fixes A :: "'a mat"   and A' :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_row A = CARD('c)"
HoHV_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "HoHV \<mu> \<equiv> if VVV.arr \<mu> then (fst \<mu> \<star> fst (snd \<mu>)) \<star> snd (snd \<mu>) else null"
size_eqvt_1: fixes p :: "perm"   and exp :: "exp" shows "size (p \<bullet> exp) = size exp"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(a = b \<oplus> \<ominus> c) = (b = a \<oplus> c)"
bisimilarity_implies_equivalence: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and Q :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition"   and "nominal_ts.bisimilar satisfies transition P Q" shows "indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q"
dining_cryptographers_def: shows "dining_cryptographers = ({None} \<union> Some ` {0..<n}) \<times> {xs. length xs = n}"
c: shows "\<phi> \<in> K \<div>\<^bsub>\<gamma>\<^esub> \<phi>"
wf_subst_elim_dom: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "\<forall>v\<in>subst_domain \<theta>. subst_elim \<theta> v"
append_queue_rep: fixes a :: "'a"   and q :: "'a queue" shows "linearize (append_queue a q) = linearize q @ [a]"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C E.FG.map E.FG.cmp I\<^sub>C.map I\<^sub>C.cmp \<epsilon>\<^sub>0 \<epsilon>\<^sub>1"
find_slot_slot_in_heap: fixes h :: "(_) heap"   and open_flag :: "bool"   and slotable :: "(_) node_ptr"   and slot :: "(_) element_ptr" assumes "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r Some slot" shows "slot |\<in>| element_ptr_kinds h"
game1_def: shows "game1 = (\<lambda>x y. coin_spmf \<bind> (\<lambda>b. exec_gpv hash_oracle'' (\<A>1 (\<^bold>g [^] x)) ({}, Map.empty) \<bind> (\<lambda>(((msg1, msg2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains msg1 msg2) \<bind> (\<lambda>_. hash.oracle s_h (\<^bold>g [^] (x * y)) \<bind> (\<lambda>(h, s_h'). let cipher = (\<^bold>g [^] y, h [\<oplus>] (if b then msg1 else msg2)) in exec_gpv hash_oracle'' (\<A>2 cipher \<sigma>) (s, s_h') \<bind> (\<lambda>(guess, s', s_h''). return_spmf (guess = b, \<^bold>g [^] (x * y) \<in> s')))) ELSE (coin_spmf \<bind> (\<lambda>b. return_spmf (b, \<^bold>g [^] (x * y) \<in> s))))))"
card_dim_span: fixes Wa :: "'a vec set"   and Va :: "'a vec set" assumes "Wa \<subseteq> carrier_vec n"   and "Va \<subseteq> local.span Wa"   and "lin_indpt Va"   and "card Va = dim_span Wa" shows "local.span Va = local.span Wa"
branch_existence': fixes \<rho> :: "'b run"   and \<S>\<^sub>1 :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list"   and k :: "nat" assumes "\<rho> \<in> \<lbrakk> \<S>\<^sub>1 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g" shows "\<exists>\<S>\<^sub>2. \<S>\<^sub>1 \<hookrightarrow>\<^bsup>k\<^esup> \<S>\<^sub>2 \<and> \<rho> \<in> \<lbrakk> \<S>\<^sub>2 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
add_poly_p_polynomial_of_mset: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and p' :: "(char list multiset \<times> int) multiset"   and q' :: "(char list multiset \<times> int) multiset"   and r' :: "(char list multiset \<times> int) multiset" assumes "add_poly_p (p, q, r) (p', q', r')" shows "polynomial_of_mset r + (polynomial_of_mset p + polynomial_of_mset q) = polynomial_of_mset r' + (polynomial_of_mset p' + polynomial_of_mset q')"
set_tree_ins: fixes k :: "'c"   and p :: "'d"   and t :: "('c \<times> 'd) tree" shows "set_tree (ins k p t) \<subseteq> {(k, p)} \<union> set_tree t"
iiws_imp_iipwn: fixes c :: "('loc, 't) configuration" assumes "inv_imps_work_sum c" shows "inv_imp_plus_work_nonneg c"
ex_spanning_tree: fixes g :: "'b ugraph" assumes "Undirected_Graph.connected g" shows "\<exists>t. is_spanning_tree g t"
poincare_line_0_real_is_x_axis: fixes x :: "complex_homo" assumes "x \<in> circline_set x_axis"   and "x \<noteq> 0\<^sub>h"   and "x \<noteq> \<infinity>\<^sub>h" shows "poincare_line 0\<^sub>h x = x_axis"
gt_wf: shows "wfP (\<lambda>x y. gt y x)"
agree_iff_subset: fixes G :: "('a, 'b) labeled_graph"   and X :: "('a, 'c) labeled_graph"   and f :: "('b \<times> 'c) set"   and g :: "('b \<times> 'c) set" assumes "graph_homomorphism G X f"   and "univalent g" shows "agree_on G f g = (f \<subseteq> g)"
BrnFT_Int_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "II \<in> UNpart1 cl dl P" shows "BrnFT cl dl \<inter> II = {}"
raw_term_inverse: fixes x :: "('a, 'b) strm" shows "simple_term (raw_term x) = x"
mult_cancel_less_iff: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "(\<alpha> * \<beta> < \<alpha> * \<gamma>) = (\<beta> < \<gamma> \<and> 0 < \<alpha>)"
alphaAll_sym: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(\<forall>X'. X #= X' \<longrightarrow> X' #= X) \<and> (\<forall>A'. A $= A' \<longrightarrow> A' $= A)"
less_le: fixes x :: "'a"   and y :: "'a" assumes "x \<in> U"   and "y \<in> U" shows "(x <\<^sub>o\<^sub>w y) = (x \<le>\<^sub>o\<^sub>w y \<and> x \<noteq> y)"
name_list_set_fresh: fixes xvec :: "name list"   and x :: "'d" shows "set xvec \<sharp>* x = xvec \<sharp>* x"
expand: fixes x :: "x"   and x' :: "x" assumes "a x = a x' \<and> b x = b x'" shows "x = x'"
par: assumes "(u, v) \<in> E"   and "(v, u) \<notin> E" shows "thesis"
a_normal: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "abelian_subgroup H G" shows "H \<lhd> add_monoid G"
set_less_eq_aux''_linear_order: shows "linear_order_on infinite_complement_partition {(A, B). A \<sqsubseteq>''' B}"
mult_sign_intros_3: fixes a :: "'a"   and b :: "'a" assumes "a \<le> (0::'a)"   and "(0::'a) \<le> b" shows "a \<cdot> b \<le> (0::'a)"
norm_pre_pure_iff: fixes P :: "assn"   and b :: "bool"   and f :: "'d Heap"   and Q :: "'d \<Rightarrow> assn" shows "<P * \<up> b> f <Q> = (b \<longrightarrow> <P> f <Q>)"
coprime_left_2_iff_odd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "algebraic_semidom.coprime times one (numeral.numeral one plus (num.Bit0 num.One)) a = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
type_weaken': fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile> t : T"   and "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "\<Delta> @ \<Gamma> \<turnstile> \<up> \<parallel>\<Delta>\<parallel> 0 t : \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (L l \<chi>)"
"2_5": shows "0 < zcount (c_imp c loc1) t"
eq_onp_to_Domainp: fixes P :: "'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "Quotient (eq_onp P) Abs Rep T" shows "Domainp T = P"
cases: fixes nodes :: "ref set"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and a :: "dag" assumes "Dagsp nodes low high a" obtains t :: "dag" and p :: "ref" where "a = t" and "set_of t \<subseteq> nodes" and "Dag p low high t" and "t \<noteq> Tip"
surjective: fixes r :: "('c, 'd, 'e) configuration_scheme" shows "r = \<lparr>c_records = c_records r, c_temp = c_temp r, c_msg = c_msg r, c_glob = c_glob r, \<dots> = more r\<rparr>"
append_Cons: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "(x \<Colon> xs) @ ys = x \<Colon> xs @ ys"
assms_1: fixes k :: "'a"   and v :: "'a" assumes "k \<in> K"   and "v \<in> E" shows "k \<otimes> v \<in> E"
face_of_refl: fixes S :: "'b set" assumes "convex S" shows "S face_of S"
vector_inf_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "vector x" shows "(x \<sqinter> y) * z = x \<sqinter> y * z"
poly_fun_eqI: fixes f :: "int \<Rightarrow> int"   and g :: "int \<Rightarrow> int" assumes "poly_fun f"   and "poly_fun g"   and "infinite {a. f a = g a}" shows "f = g"
init_Suc_snoc: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "f \<triangleright> Suc n = e_snoc (f \<triangleright> n) (the (f (Suc n)))"
assms: shows "t \<in> fmdom' (clearjunk0 xs)"
quotient_of_inject: fixes a :: "rat"   and b :: "rat" assumes "quotient_of a = quotient_of b" shows "a = b"
atLeastLessThan_iff: fixes i :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and l :: "'f"   and u :: "'f" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
lim_stream: shows "lim_stream \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)"
poly_exp_divides: fixes p :: "'a list"   and n :: "nat"   and q :: "'a list"   and m :: "nat" assumes "p %^ n divides q"   and "m \<le> n" shows "p %^ m divides q"
rat_cnv: fixes n :: "nat" shows "convergent (\<lambda>k. f (d k) (r n))"
eq2: shows "sort_oalist (x # xs) = update_by_pair (k, v) (sort_oalist xs)"
simps_2: fixes disc :: "'b \<Rightarrow> bool"   and a :: "'b" shows "has_disc disc (Match a) = disc a"
dual_iff: fixes dual :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.dual dual" shows "(dual x = y) = (x = dual y)"
simps_2: fixes r :: "real"   and x :: "octo" shows "Im1 (r *\<^sub>R x) = r * Im1 x"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> N"   and "b \<in> N" shows "a \<cdot> b \<in> N"
univalentD: fixes R :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "univalent R"   and "(x, y) \<in> R"   and "(x, z) \<in> R" shows "z = y"
simps_1: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
unit_disc_fix_moebius_inv: fixes M :: "moebius" assumes "unit_disc_fix M" shows "unit_disc_fix (moebius_inv M)"
snapshot_increases_message_length: fixes ev :: "('p, 's, 'm) event"   and p :: "'p"   and c :: "('p, 's, 'm) configuration"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat"   and q :: "'p"   and r :: "'p" assumes "ev = Snapshot p"   and "c \<turnstile> ev \<mapsto> c'"   and "channel i = Some (q, r)" shows "length (msgs c i) \<le> length (msgs c' i)"
res: shows "preprocess_opt X t1 ((x, p) # t2) = (t', tv)"
eq_iff: fixes a :: "'b list"   and b :: "'b list" shows "(a = b) = (Sublist.prefix b a \<and> Sublist.prefix a b)"
Sum_eq_pick_changed_elem: fixes M :: "'a set"   and m :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and \<Delta> :: "'b" assumes "finite M"   and "m \<in> M"   and "f m = g m + \<Delta>"   and "\<And>n. n \<noteq> m \<and> n \<in> M \<Longrightarrow> f n = g n" shows "sum f M = sum g M + \<Delta>"
length_sorted_list_of_set: fixes A :: "'a set" shows "length (sorted_list_of_set A) = card A"
leg1_simps_5: shows "D.dom (F g) = F g"
pred_def: fixes nat :: "nat" shows "nat.pred nat = (case nat of 0 \<Rightarrow> 0 | Suc x2 \<Rightarrow> x2)"
hiding_tickFree: fixes s :: "'b event list"   and A :: "'b set" shows "tickFree s = tickFree (trace_hide s (ev ` A))"
smult_closed: fixes r :: "'r"   and m :: "'m" assumes "r \<in> R"   and "m \<in> M" shows "r \<cdot> m \<in> M"
rtrancl_trans: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and z :: "'c" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
prj_chine_assoc_3: shows "\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0\<^sub>0 \<cdot> \<mu>\<nu>_\<pi>_\<rho>.chine_assoc = \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0"
bounds_lg_3: fixes c :: "'a"   and b :: "'a option"   and x :: "'a" assumes "c \<ge>\<^sub>u\<^sub>b b"   and "x \<le>\<^sub>u\<^sub>b b" shows "x \<le> c"
map_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<ll>' a = local.inv (\<ll> a)"
sterm_to_pterm_consts: fixes t :: "sterm" shows "consts (sterm_to_pterm t) = consts t"
fg: fixes x :: "'a" assumes "x \<in> T"   and "b \<bullet> x = 0" shows "f (g x) = x"