hcju/isabelleps huggingface dataset:
option_eq_def: fixes eq :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and v1 :: "'b option"   and v2 :: "'c option" shows "option_eq eq v1 v2 \<equiv> case (v1, v2) of (None, None) \<Rightarrow> True | (None, Some a) \<Rightarrow> False | (Some x1, None) \<Rightarrow> False | (Some x1, Some x2) \<Rightarrow> eq x1 x2"
fps_of_poly_eq_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(fps_of_poly p = fps_of_poly q) = (p = q)"
arccos_lbound_abs: fixes y :: "real" assumes "\<bar>y\<bar> \<le> 1" shows "0 \<le> arccos y"
homE_2: fixes A :: "'b"   and B :: "'b"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "Cod f = B"
finsert_fsubset: fixes x :: "'c"   and A :: "'c fset"   and B :: "'c fset" shows "(finsert x A |\<subseteq>| B) = (x |\<in>| B \<and> A |\<subseteq>| B)"
that_2: fixes x :: "'a" assumes "x \<in> src.carrier" shows "open (U x)"
invar_of_list: fixes key :: "'b \<Rightarrow> 'a list"   and xs :: "'b list" shows "invar_trie (tries_of_list key xs)"
simps_2: fixes n :: "'b" shows "headconst (n)\<^sub>p = n"
l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_new_character_data_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
less: fixes k :: "nat" assumes "n < k" shows "n < 2 ^ k"
0: shows "((\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) has_integral i) {0..1}"
simps_1: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and a :: "'b"   and x :: "'a"   and xs :: "'a list" shows "applied_rule C a (x # xs) = (if a \<in> dom (C x) then \<lfloor>x\<rfloor> else applied_rule C a xs)"
succ_chain_always_r_valuation_id: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat"   and x :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "(\<forall>k. i \<le> k \<and> k \<le> j \<longrightarrow> x \<in> rvars (\<T> (l ! k))) \<longrightarrow> \<langle>\<V> (l ! i)\<rangle> x = \<langle>\<V> (l ! j)\<rangle> x"
0: shows "Outside' X a ,,, i \<subseteq> \<Union> {\<Union> (a `` (X \<union> {i}))}"
summands_equal: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a = b"   and "c = d" shows "a \<oplus>\<^bsub>V\<^esub> c = b \<oplus>\<^bsub>V\<^esub> d"
t: shows "supp_fun t \<subseteq> supp_fun s \<union> supp_fun t"
int_vertices_def: fixes p :: "('a \<times> 'a) list" shows "int_vertices p \<equiv> set (map fst p)"
asToHeap_graph_aux_def: shows "asToHeap_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>asToHeap_aux. x1 = ANil \<and> x2 = []) \<or> (\<exists>asToHeap_aux v e as. x1 = ACons v e as \<and> x2 = (v, e) # ((\<lambda>uu_. uu_) \<circ> asToHeap_aux) as \<and> p as (((\<lambda>uu_. uu_) \<circ> asToHeap_aux) as)))"
simps_4: fixes e1 :: "expr"   and e2 :: "expr" shows "\<lbrace>Addition e1 e2\<rbrace> = \<lbrace>e1\<rbrace> + \<lbrace>e2\<rbrace>"
size_annos_same: fixes C1 :: "'b acom"   and C2 :: "'c acom" assumes "strip C1 = strip C2" shows "length (annos C1) = length (annos C2)"
assms_3: shows "ipresIGSwapAll h' hA' MOD' MOD''"
dom_eq_empty_conv: fixes f :: "'g \<Rightarrow> 'h option" shows "(dom f = {}) = (f = Map.empty)"
2: shows "of_complex (1 / 2) \<in> unit_disc"
approx_subset_monad: fixes x :: "'a star"   and y :: "'a star" assumes "x \<approx> y" shows "{x, y} \<subseteq> monad x"
bg_impl_rel_eext_internal_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "bg_impl_rel_eext Rm Rv \<equiv> \<langle>Rm, \<langle>Rv\<rangle>fun_set_rel\<rangle>gen_bg_impl_rel_eext"
pre_ranks_state_obtain: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and r\<^sub>q :: "nat"   and r :: "'d \<Rightarrow> nat option"   and \<nu> :: "'c"   and q :: "'d" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "r\<^sub>q \<in> semi_mojmir_def.pre_ranks \<Sigma> \<delta> q\<^sub>0 r \<nu> q - {semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0}" obtains q' :: "'d" where "r q' = Some r\<^sub>q" and "q = \<delta> q' \<nu>"
effect_to_assignments_ii: fixes as :: "('a \<times> bool) list"   and op :: "'a strips_operator" assumes "as = effect_to_assignments op" obtains as\<^sub>1 :: "('a \<times> bool) list" and as\<^sub>2 :: "('a \<times> bool) list" where "as = as\<^sub>1 @ as\<^sub>2" and "as\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)" and "as\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)"
pareto_efficiency: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt" assumes "is_pref_profile R"   and "x \<in> pareto_losers R" shows "x \<notin> scf R"
induced_vector_indV: fixes H :: "'a set"   and G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and rrsmult :: "('b, 'a) aezfun \<Rightarrow> (('b, 'a) aezfun \<Rightarrow> 'c) \<Rightarrow> ('b, 'a) aezfun \<Rightarrow> 'c"   and v :: "'c" assumes "InducedFHModule H G smult V rrsmult"   and "v \<in> V" shows "InducedFHModule.induced_vector H G smult V v \<in> aezfun_scalar_mult.indspace smult G (aezfun_setspan H) V"
assms_1: shows "\<forall>PROB' as s. finite PROB \<and> P PROB' \<and> s \<in> valid_states PROB' \<and> as \<in> valid_plans PROB' \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB')"
new_configured_list_SecurityInvariant_complies: fixes m :: "('a, 'b) TopoS_packed"   and sinvar_spec :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and nP :: "'a \<Rightarrow> 'b"   and C :: "('a, 'b) TopoS_Params"   and Spec :: "'a SecurityInvariant_configured option"   and Impl :: "'a SecurityInvariant"   and description :: "char list" assumes "TopoS_modelLibrary m sinvar_spec"   and "nP = nm_node_props m C"   and "Spec = new_configured_SecurityInvariant (sinvar_spec, nm_default m, nm_receiver_violation m, nP)"   and "Impl = new_configured_list_SecurityInvariant m C description" shows "SecurityInvariant_complies_formal_def Impl (the Spec)"
delay_bisimulation_obs_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool" assumes "delay_bisimulation_obs trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2" shows "delay_bisimulation_obs trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
pHom_pj_csrp: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and n :: "nat"   and c' :: "nat \<Rightarrow> 'a set" assumes "Idomain S"   and "ideal S P"   and "PolynRg R' (S /\<^sub>r P) Y"   and "pol_coeff (S /\<^sub>r P) (n, c')" shows "erH R S X R' (S /\<^sub>r P) Y (pj S P) (polyn_expr R X n (n, cmp (csrp_fn S P) c')) = polyn_expr R' Y n (n, c')"
heap_upds_ok_to_heap: fixes \<Gamma> :: "(Vars.var \<times> 'b) list"   and x :: "Vars.var"   and S :: "stack_elem list"   and e :: "'b" assumes "heap_upds_ok (\<Gamma>, Upd x # S)" shows "heap_upds_ok ((x, e) # \<Gamma>, S)"
ch_dist: fixes x :: "'a"   and Y :: "'a set" shows "x \<sqinter> \<Squnion> Y = \<Squnion> ((\<sqinter>) x ` Y)"
simps_2: fixes e :: "aexp"   and sa :: "'a Abs_State.st" shows "aval'' e (Some sa) = aval' e sa"
is_extensional: fixes f :: "'d \<times> 'c" assumes "\<not> DopxC.arr f" shows "\<Phi> f = S.null"
induct: fixes P :: "tm \<Rightarrow> bool"   and tm :: "tm" assumes "\<And>x. P (CP x)"   and "\<And>x. P (tm.Bound x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (tm.Add x1a x2a)"   and "\<And>x1a x2a. P x2a \<Longrightarrow> P (tm.Mul x1a x2a)"   and "\<And>x. P x \<Longrightarrow> P (tm.Neg x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (tm.Sub x1a x2a)"   and "\<And>x1a x2a x3. P x3 \<Longrightarrow> P (CNP x1a x2a x3)" shows "P tm"
max_of_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and m :: "'a"   and n :: "'a" assumes "class.linorder less_eq less"   and "order.mono less_eq f" shows "max (f m) (f n) = f (ord.max less_eq m n)"
h1a: shows "(\<Sum>i = 0..<deg + 1. f i * SQ ^ (i div 2) * real (i mod 2) * sqrt SQ + f i * SQ ^ (i div 2) * (1 - real (i mod 2))) = (\<Sum>i = 0..<deg + 1. f i * sqrt SQ ^ i)"
not_Interp_imp_not_INTERP: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and D' :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "D' \<le> D"   and "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> ground_resolution_with_selection.Interp S N D \<Turnstile> D'" shows "\<not> ground_resolution_with_selection.INTERP S N \<Turnstile> D'"
DiffD1: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A - B" shows "c \<in> A"
that: fixes M :: "(nat \<times> nat) set" assumes "M \<in> [UU]\<^bsup>m\<^esup>"   and "\<And>u. u \<in> [M]\<^bsup>2\<^esup> \<Longrightarrow> form u k"   and "\<And>u. u \<in> [M]\<^bsup>2\<^esup> \<Longrightarrow> scheme u \<subseteq> N" shows "thesis"
elims: fixes x :: "'a Literal"   and y :: "'a Literal" assumes "complement x = y" obtains A :: "'a" where "x = Pos A__" and "y = Neg A__"   | A :: "'a" where "x = Neg A" and "y = Pos A"
distinct_set_subseqs: fixes xs :: "'a list" assumes "distinct xs" shows "distinct (map set (subseqs xs))"
comp_arr_inv: fixes f :: "'b"   and g :: "'b" assumes "\<phi>.B.inverse_arrows f g" shows "f \<cdot>\<^sub>B g = \<phi>.B.dom g"
fundamental_property_of_static_slicing: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and s :: "'state"   and S :: "'node set" assumes "n -as\<rightarrow>* n'"   and "preds (kinds as) s"   and "n' \<in> S" obtains as' :: "'edge list" where "preds (slice_kinds S as') s" and "\<forall>V\<in>Use n'. state_val (transfers (slice_kinds S as') s) V = state_val (transfers (kinds as) s) V" and "slice_edges S as = slice_edges S as'" and "n -as'\<rightarrow>* n'"
f7: shows "s'' \<otimes> s \<otimes> (s'' \<otimes> s') \<otimes> (r'' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r')) = s'' \<otimes> s \<otimes> (s'' \<otimes> s') \<otimes> (r'' \<otimes> (s' \<otimes> r)) \<oplus> s'' \<otimes> s \<otimes> (s'' \<otimes> s') \<otimes> (r'' \<otimes> (s \<otimes> r'))"
acc_lengths_plus: fixes m :: "nat"   and n :: "nat"   and as :: "'a list list" shows "acc_lengths (m + n) as = map ((+) m) (acc_lengths n as)"
fds_zeta_altdef: shows "fds_zeta = (\<chi>n. if n = 0 then 0::'b else (1::'b))"
lcm_D: fixes a :: "'a" shows "a r\<rightarrow> a = (1::'a)"
additiveD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "Transformers.additive t"   and "sound P"   and "sound Q" shows "t (\<lambda>s. P s + Q s) = (\<lambda>s. t P s + t Q s)"
exists_index_imp_Sup_llist: fixes i :: "nat"   and Xs :: "'b set llist"   and x :: "'b" assumes "enat i < llength Xs"   and "x \<in> lnth Xs i" shows "x \<in> Sup_llist Xs"
distinct_873: fixes x23 :: "v"   and x311 :: "t"   and x312 :: "cvtop"   and x313 :: "t"   and x314 :: "sx option" shows "C x23 \<noteq> Cvtop x311 x312 x313 x314"
Impl_nvalid_0: fixes P :: "assn"   and M :: "cname \<times> mname"   and Q :: "assn" shows "\<Turnstile>0: (P, Impl M, Q)"
limit_in_range_suffix: fixes x :: "nat \<Rightarrow> 'b"   and k :: "nat" shows "limit x \<subseteq> range (suffix k x)"
predecessors_reachable: fixes r :: "'a"   and v :: "'a"   and w :: "'a"   and t :: "'a"   and g :: "'a" assumes "r \<sqinter> top = r"   and "r \<sqinter> id r \<le> top"   and "id v = id r \<sqinter> top"   and "w \<sqinter> id w \<le> top \<and> w \<sqinter> top \<le> - top"   and "t \<le> w"   and "w \<le> id (id r \<sqinter> top) \<sqinter> (id r \<sqinter> top) \<sqinter> g"   and "id r \<sqinter> top \<le> id r \<sqinter> top" shows "w \<sqinter> v \<le> v"
set_Macaulay_list: fixes ps :: "('a \<Rightarrow>\<^sub>0 'd) list" shows "set (punit.Macaulay_list ps) = set (mat_to_polys (punit.Keys_to_list ps) (row_echelon (punit.Macaulay_mat ps))) - {0}"
closure_convex_Int_superset: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "interior S \<noteq> {}"   and "interior S \<subseteq> closure T" shows "closure (S \<inter> T) = closure S"
interior_of_subtopology_open: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" assumes "openin X U" shows "subtopology X U interior_of S = U \<inter> X interior_of S"
vertices_restrict: fixes G :: "('c, 'b) labeled_graph" shows "vertices (restrict G) = vertices G"
distributivity: fixes A1 :: "int poly fract list list"   and A2 :: "int poly fract list list"   and B1 :: "int poly fract list list"   and B2 :: "int poly fract list list" assumes "rat_poly.matrix_match A1 A2 B1 B2" shows "rat_poly.matrix_mult A1 A2 \<otimes> rat_poly.matrix_mult B1 B2 = rat_poly.matrix_mult (A1 \<otimes> B1) (A2 \<otimes> B2)"
span_add: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> ideal S"   and "y \<in> ideal S" shows "x + y \<in> ideal S"
gauss2complex_eq_0_iff: fixes x :: "gauss_int" shows "(gauss2complex x = 0) = (x = 0)"
WT_calleeD: fixes \<I> :: "('d, 'e) \<I>"   and callee :: "'d \<Rightarrow> ('e \<times> 'f) spmf"   and ret :: "'e"   and s :: "'f"   and out :: "'d" assumes "\<I> \<turnstile>c callee \<surd>"   and "(ret, s) \<in> set_spmf (callee out)"   and "out \<in> outs_\<I> \<I>" shows "ret \<in> responses_\<I> \<I> out"
srtc_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "relation.srtc r a b \<equiv> (\<lambda>a b. r a b \<or> r b a)\<^sup>*\<^sup>* a b"
tens_assoc: shows "\<forall>a xs ys. brickmat a \<otimes> (blockmat xs \<otimes> blockmat ys) = brickmat a \<otimes> blockmat xs \<otimes> blockmat ys"
decomp: shows "replicate n True @ replicate n False = u @ v @ w"
simps_4: fixes x2 :: "'b"   and x1 :: "'a" shows "Ok x2 \<noteq> Error x1"
canonical_proj_ext_ker: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R" shows "a_kernel R (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n])) (canonical_proj_ext I n) = \<Inter> (I ` {..n})"
graph_unionI: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" assumes "edges G\<^sub>1 \<subseteq> edges G\<^sub>2"   and "vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2" shows "graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum g A = sum h B"
axioms_2: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi> :: "'c \<times> 'c \<Rightarrow> 'd" assumes "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
simps_1: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and y :: "'v" shows "lift f (SOURCE, Inner y) = (if y = source \<Delta> then value_flow \<Delta> f else 0)"
new_tv_not_free_tv: fixes n :: "nat"   and A :: "'a" assumes "new_tv n A" shows "n \<notin> free_tv A"
arrI: fixes f1 :: "'a"   and f2 :: "'a" assumes "arr f1"   and "arr f2" shows "VxV.arr (f1, f2)"
set_simps_3: fixes x :: "'b"   and xs :: "['b]" shows "set (x : xs) = insert x (set xs)"
le_minus_one_simps_4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq one (uminus one)"
word_2p_lem: fixes n :: "nat"   and w :: "'b word" assumes "n < size w" shows "(w < 2 ^ n) = (uint w < 2 ^ n)"
hyps_3: shows "perm \<bullet> P \<Longrightarrow>\<^sub>\<tau> perm \<bullet> P''"
assms_1: fixes x :: "real" assumes "0 \<le> x" shows "0 \<le> f x"
in_set_replicate: fixes x :: "'g"   and n :: "nat"   and y :: "'g" shows "(x \<in> set (replicate n y)) = (x = y \<and> n \<noteq> 0)"
that: fixes \<chi>' :: "'a \<Rightarrow> complex" assumes "character G \<chi>'"   and "\<And>x. x \<in> H \<Longrightarrow> \<chi>' x = \<chi> x" shows "thesis"
bdd_belowI: fixes A :: "'a multiset set"   and m :: "'a multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># x" shows "subset_mset.bdd_below A"
mapping_of_pmfI': fixes m :: "('b, real) mapping"   and p :: "'b pmf" assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup_default 0 m x = pmf p x"   and "Mapping.keys m = set_pmf p" shows "mapping_of_pmf p = m"
Go_\<epsilon>o_terminal: fixes a :: "'c" assumes "\<exists>b e. terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F b a e" shows "terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go a) a (\<epsilon>o a)"
If_transC_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tst :: "'a"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com"   and s :: "'b"   and c' :: "('a, 'c) com"   and s' :: "'b" assumes "PL.transC tval aval (if tst then c1 else c2, s) (c', s')" obtains "tval tst s" and "c' = c1" and "s' = s"   | "\<not> tval tst s" and "c' = c2" and "s' = s"
tensor_preserves_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a"   and "partial_magma.ide (free_monoidal_category.comp C) b" shows "partial_magma.ide (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C a b)"
assms_6: shows "InvariantCnCharacterization (getCn state) (getC state) (getM state)"
Sigma_def: fixes A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" shows "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {(x, y)}"
alpha_tPred: fixes f1 :: "'d"   and \<phi>1 :: "'b"   and f2 :: "'d"   and \<phi>2 :: "'b" shows "(tPred f1 \<phi>1 =\<^sub>\<alpha> tPred f2 \<phi>2) = (f1 = f2 \<and> \<phi>1 = \<phi>2)"
apply_state_iso_def: shows "apply_state_iso \<equiv> \<lambda>(f, g). map_fun g (map_fun id (map_spmf (map_prod id f)))"
is_nth_power_mult_coprime_natD_1: fixes a :: "nat"   and b :: "nat"   and n :: "nat" assumes "coprime a b"   and "is_nth_power n (a * b)"   and "0 < a"   and "0 < b" shows "is_nth_power n a"
cltn2_three_point_line: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and l :: "proj2_line"   and C :: "cltn2"   and s :: "proj2" assumes "p \<noteq> q"   and "r \<noteq> p"   and "r \<noteq> q"   and "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident r l"   and "apply_cltn2 p C = p"   and "apply_cltn2 q C = q"   and "apply_cltn2 r C = r"   and "proj2_incident s l" shows "apply_cltn2 s C = s"
ring_finite_field_ops32: fixes pp :: "uint32" assumes "p = int_of_uint32 pp"   and "p \<le> 65535" shows "ring_ops (finite_field_ops32 pp) mod_ring_rel32"
intro: fixes f :: "'b list \<Rightarrow> 'c list" assumes "\<And>u v. f (u \<cdot> v) = f u \<cdot> f v" shows "morphism f"
simps_3: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n\<^sub>s :: "nat" assumes "\<not> Option.is_none (devBC t n nid n\<^sub>s)" shows "devExt t n nid n\<^sub>s 0 = bc (\<sigma>\<^bsub>the (devBC t n nid n\<^sub>s)\<^esub>t n\<^sub>s)"
nonempty: fixes G :: "'a set" assumes "Group G" shows "G \<noteq> {}"
of_rat_sum: fixes f :: "'b \<Rightarrow> rat"   and A :: "'b set" shows "of_rat (sum f A) = (\<Sum>a\<in>A. of_rat (f a))"
map_to_set_inverse: fixes m :: "'d \<Rightarrow> 'e option" shows "set_to_map (map_to_set m) = m"
share_all_until_volatile_write_Read_commute: fixes i :: "nat"   and ls :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d \<times> 'e"   and v :: "nat"   and \<D>' :: "'c"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, Read volatile a t # is, \<theta>, sb, \<D>, \<O>)" shows "share_all_until_volatile_write (ls[i := (p, is, \<theta>(t \<mapsto> v), sb @ [Read\<^sub>s\<^sub>b volatile a t v], \<D>', \<O>)]) S = share_all_until_volatile_write ls S"
assms_3: shows "i \<in>\<^sub>\<circ> I"
LSC_invs_1: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F \<^bold>\<and> G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F, G, \<Gamma> \<Rightarrow>\<^sub>n"
cod_in_codomains: fixes f :: "'a1" assumes "C1.codomains f \<noteq> {}" shows "C1.cod f \<in> C1.codomains f"
PU_decomposition: shows "{p. p permutes UNIV} = {p \<in> {p. p permutes UNIV}. evenperm p} \<union> {Fun.swap j k id \<circ> p |p. p \<in> {p \<in> {p. p permutes UNIV}. evenperm p}}"